# A Spectral Analysis of the Nonlinear Schrödinger Equation in the Co-
Exploding Frame

S. Jon Chapman  Mathematical Institute, University of Oxford, AWB, ROQ,
Woodstock Road, Oxford OX2 6GG  M. Kavousanakis  School of Chemical
Engineering, National Technical University of Athens, 15780, Athens, Greece
E.G. Charalampidis  Mathematics Department, California Polytechnic State
University, San Luis Obispo, CA 93407-0403, USA  I.G. Kevrekidis  Department
of Chemical and Biomolecular Engineering &
Department of Applied Mathematics and Statistics, Johns Hopkins University,
Baltimore, MD 21218, USA  P.G. Kevrekidis  Department of Mathematics and
Statistics, University of Massachusetts, Amherst MA 01003-4515, USA

###### Abstract

The nonlinear Schrödinger model is a prototypical dispersive wave equation
that features finite time blowup, either for supercritical exponents (for
fixed dimension) or for supercritical dimensions (for fixed nonlinearity
exponent). Upon identifying the self-similar solutions in the so-called “co-
exploding frame”, a dynamical systems analysis of their stability is natural,
yet is complicated by the mixed Hamiltonian-dissipative character of the
relevant frame. In the present work, we study the spectral picture of the
relevant linearized problem. We examine the point spectrum of 3 eigenvalue
pairs associated with translation, $U(1)$ and conformal invariances, as well
as the continuous spectrum. We find that two eigenvalues become positive, yet
are attributed to symmetries and are thus not associated with instabilities.
In addition to a vanishing eigenvalue, 3 more are found to be negative and
real, while the continuous spectrum is nearly vertical and on the left-half
(spectral) plane. Finally, the subtle effects of the boundaries are also
assessed and their role in the observed weak eigenvalue oscillations is
clarified.

##  I Introduction

One of the central dispersive nonlinear partial differential equations of
relevance to a wide range of physical systems is the nonlinear Schrödinger
(NLS) model [1, 2, 3, 4, 5]. Among the different research themes where the NLS
plays a central role, one can mention the study of the electric field of light
in nonlinear optical systems [6, 7], as well as in plasmas [8], the realm of
water waves and the evolution of their height, e.g., in deep water [9, 10], as
well as the condensate wavefunction for mean-field models of atomic Bose-
Einstein condensates (BECs) [11, 12, 5]. The prototypical variants of the
equation involve the self-focusing [4, 13] and the self-defocusing
nonlinearity [5], and the respective dynamics revolve around bright [14] and
dark [15] solitons.

In the case of self-focusing (self-attractive) nonlinearity, and for
sufficiently high dimension (for fixed nonlinearity) or for sufficiently
strong nonlinearity (for fixed dimension), a key feature of the NLS model is
the presence of collapse type phenomena, that have also been explored in
numerous books [4, 13, 16], as well as reviews [17, 18, 19]. Indeed, the topic
of finite time blow up of supercritical NLS solutions has been the objective
of continued study both in the mathematical and in the physical literature;
see, e.g., Refs. [20, 21, 22] and [23, 24] (and also references therein) for
only some recent examples. Importantly, the study of collapse is not only a
mathematical idealization but rather has become accessible to physical
experiments. In fact, on the one hand, there is the well-developed field of
nonlinear optics, where not only the well-known, two-dimensional collapsing
waveform of the Townes soliton has been observed [25] but also more elaborate
themes have been touched upon including the collapse of optical vortices [26],
the loss of phase information of collapsing filaments [27], and the
manipulation of the medium to avert optical collapse [28]. On the other hand,
a remarkable, very recent experimental development has been the emergence of 2
distinct works in the atomic physics realm of BECs, observing Townes solitons
in the $2d$ setting [29, 30]. Here, collapsing waveforms in higher dimensions
had been experimentally identified earlier [31, 32], and the ability to
manipulate the nonlinearity [33] and the initial conditions [34] has continued
to improve in recent times. In one of these recent works [29] the modulational
instability was manipulated to produce (in a less controllable, yet
experimentally observable) way such Townes waveforms. The authors of the
second work [30] leveraged a reduction of a minority component in a two-
component gas into a single-component one with effectively attractive
interactions to produce a collapsing Townes waveform.

In many of the above mathematical works that study the dynamics of collapse,
both in dispersive systems such as the NLS [4, 13], but also even in
dissipative systems such as reaction-diffusion ones [35], the emphasis is on
identifying the solution in a frame where it becomes steady, namely a self-
similar (or “co-exploding”) one [36, 37, 38, 39, 40]. A similar approach is
leveraged in dynamical systems and partial differential equations (PDEs) when
exploring traveling waves which are identified as steady solutions in a so-
called co-traveling frame. In such settings, a natural next step is to explore
the spectral stability of the solutions in such a frame [41, 42]. However, in
the realm of the self-similar solutions, far fewer studies appear to be
exploring the spectral properties of the wave in the co-exploding frame [43,
44, 45]. Indeed, in the context of NLS, the only earlier approach to
spectrally explore the collapse problem concerns the earlier work of some of
the authors [45]. In a recent work, we revisited this topic, attempting to
examine the self-focusing problem as a bifurcation one, identifying its
effective normal form [46]. In the present study, we complement this approach
by systematically examining the spectrum of the self-similarly collapsing
solitary wave.

Upon setting up the relevant linearization problem in the self-similar frame
(in Section II), our starting point will consist of observations of the
spectrum of the underlying Hamiltonian system before the bifurcation point (in
Section III). We will examine the relevant spectral picture when approaching
the limit point where collapsing solutions emerge, and also we will explore
the same picture for the dissipative system that results in the co-exploding
frame past the critical point. In Section IV, having observed the relevant
spectrum, we will then turn to a more refined analysis of the different
eigenvalues thereof, one-by-one. Finally, we will synthesize the picture and
its dynamical implications and offer some conclusions and future challenges in
Section V. The Appendices offer some additional insights, including about how
a symmetry of the original frame can turn into an unstable eigendirection in a
renormalized one, as well as about the role of the normal form obtained
previously in [46] in connection with the eigenvalues identified herein.

##  II Basic Mathematical Setup

Our model of interest will be the one-dimensional, general-nonlinearity-
exponent variant of NLS in the form

$\displaystyle\mathrm{i}\mathchoice{\frac{{\partial}\psi}{{\partial}z}}{{\partial}\psi/{\partial}z}{{\partial}\psi/{\partial}z}{{\partial}\psi/{\partial}z}+\mathchoice{\frac{\partial^{2}\psi}{{\partial}x^{2}}}{\partial^{2}\psi/{\partial}x^{2}}{\partial^{2}\psi/{\partial}x^{2}}{\partial^{2}\psi/{\partial}x^{2}}+|\psi|^{2\sigma}\psi=0.$
(1)

Notice that here we have used the typical optics notation, where $z$ is the
evolution variable, representing the propagation distance [4]. This model has
been studied extensively in [4, 13] and it is well-known that in
$d$-dimensions, the condition for its collapse is $\sigma d>2$. The model is
subcritical for $\sigma d<2$, and the special case of $\sigma d=2$ separates
the two regimes. We opt to consider the $d=1$ case for a number of practical
reasons, including (a) the availability of an analytical solution for all
values of $\sigma$, namely
$\psi=\mathrm{e}^{\mathrm{i}z}(1+\sigma)^{1/(2\sigma)}{\rm
sech}\left[(2\sigma)^{1/2}(x-x_{0})\right]$ and (b) the computational
convenience of the relevant spectral calculations. As we will see below, the
latter will be sensitively dependent on the domain size and its boundary
conditions, and associated considerations will be even more delicate (and
imposing a substantial additional computational overhead) in higher
dimensions. Nevertheless, we expect the main features and techniques proposed
herein to be directly reflected in such higher-dimensional settings, as will
be evident in what follows.

The Hamiltonian associated with Eq. (1) is given by

$\displaystyle
H=\int_{-\infty}^{\infty}\left(\left|\mathchoice{\frac{{\partial}\psi}{{\partial}x}}{{\partial}\psi/{\partial}x}{{\partial}\psi/{\partial}x}{{\partial}\psi/{\partial}x}\right|^{2}-\frac{1}{\sigma+1}|\psi|^{2\sigma+2}\right)\,\mathrm{d}x.$
(2)

The dynamical equations satisfy:

$\mathrm{i}\mathchoice{\frac{{\partial}\psi}{{\partial}z}}{{\partial}\psi/{\partial}z}{{\partial}\psi/{\partial}z}{{\partial}\psi/{\partial}z}=\frac{\delta
H}{\delta\psi^{*}},\qquad\mathrm{i}\mathchoice{\frac{{\partial}\psi^{*}}{{\partial}z}}{{\partial}\psi^{*}/{\partial}z}{{\partial}\psi^{*}/{\partial}z}{{\partial}\psi^{*}/{\partial}z}=-\frac{\delta
H}{\delta\psi}.$

We require that $H$ be finite.

In order to go to the co-exploding frame, we introduce the well-known [4, 13]
stretched variables, rescaling space by the length scale $L(z)$

$\displaystyle\xi=\frac{x}{L(z)},\quad\tau=\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{L^{2}(z^{\prime})},\quad\psi(x,z)=L^{-1/\sigma}u(\xi,\tau),$
(3)

to give

$\displaystyle\mathrm{i}\mathchoice{\frac{{\partial}u}{{\partial}\tau}}{{\partial}u/{\partial}\tau}{{\partial}u/{\partial}\tau}{{\partial}u/{\partial}\tau}+\mathchoice{\frac{\partial^{2}u}{{\partial}\xi^{2}}}{\partial^{2}u/{\partial}\xi^{2}}{\partial^{2}u/{\partial}\xi^{2}}{\partial^{2}u/{\partial}\xi^{2}}+|u|^{2\sigma}u-\mathrm{i}\xi
LL_{z}\mathchoice{\frac{{\partial}u}{{\partial}\xi}}{{\partial}u/{\partial}\xi}{{\partial}u/{\partial}\xi}{{\partial}u/{\partial}\xi}-\frac{\mathrm{i}LL_{z}}{\sigma}u=0,$
(4)

and the corresponding rescaling of the Hamiltonian:

$\displaystyle
H=L^{-2/\sigma-2}\int_{-\infty}^{\infty}\left(\left|\mathchoice{\frac{{\partial}u}{{\partial}\xi}}{{\partial}u/{\partial}\xi}{{\partial}u/{\partial}\xi}{{\partial}u/{\partial}\xi}\right|^{2}-\frac{1}{\sigma+1}|u|^{2\sigma+2}\right)\,\mathrm{d}x.$
(5)

We factor out the frequency of our solution without loss of generality and
assign the rate of width shrinkage/amplitude growth to be termed as $G$ by
setting

$\displaystyle u(\xi,\tau)=\Phi(\xi,\tau)\mathrm{e}^{\mathrm{i}\tau},\qquad
G=-LL_{z},$ (6)

which reduce Eq. (4) into

$\displaystyle\mathrm{i}\mathchoice{\frac{{\partial}\Phi}{{\partial}\tau}}{{\partial}\Phi/{\partial}\tau}{{\partial}\Phi/{\partial}\tau}{{\partial}\Phi/{\partial}\tau}+\mathchoice{\frac{\partial^{2}\Phi}{{\partial}\xi^{2}}}{\partial^{2}\Phi/{\partial}\xi^{2}}{\partial^{2}\Phi/{\partial}\xi^{2}}{\partial^{2}\Phi/{\partial}\xi^{2}}+|\Phi|^{2\sigma}\Phi-\Phi+\frac{\mathrm{i}G}{\sigma}\Phi+\mathrm{i}G\xi\mathchoice{\frac{{\partial}\Phi}{{\partial}\xi}}{{\partial}\Phi/{\partial}\xi}{{\partial}\Phi/{\partial}\xi}{{\partial}\Phi/{\partial}\xi}=0.$
(7)

It is particularly important for our considerations that will follow to
emphasize that the above system bears a rather unusual “mixed” character.
Along the manifold of $G=0$ (solitonic) solutions, the relevant model falls
back on the original one, retaining its Hamiltonian structure. Nevertheless,
for the genuinely self-similar solutions of $G\neq 0$, the system is no longer
conservative in nature. Hence, we are dealing with a mixed Hamiltonian-
dissipative system and the dissipativity for $G\neq 0$ should be mirrored in
the spectrum of the self-similar solutions. This is contrary to what is the
case for the four-fold symmetric spectrum of the $G=0$ solitons, for which if
$\lambda$ is an eigenvalue, so are $-\lambda$, $\lambda^{*}$ and
$-\lambda^{*}$.

We will find it convenient to also perform an additional transformation by
writing

$\displaystyle\Phi(\xi,\tau)=V(\xi,\tau)\mathrm{e}^{-\mathrm{i}G(\tau)\xi^{2}/4}$
(8)

to give

$\displaystyle\mathrm{i}\mathchoice{\frac{{\partial}V}{{\partial}\tau}}{{\partial}V/{\partial}\tau}{{\partial}V/{\partial}\tau}{{\partial}V/{\partial}\tau}+\frac{G^{\prime}\xi^{2}}{4}V+\mathchoice{\frac{\partial^{2}V}{{\partial}\xi^{2}}}{\partial^{2}V/{\partial}\xi^{2}}{\partial^{2}V/{\partial}\xi^{2}}{\partial^{2}V/{\partial}\xi^{2}}+|V|^{2\sigma}V-V-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}V+\frac{G^{2}\xi^{2}}{4}V=0,$
(9)

where
$G^{\prime}=\mathchoice{\frac{\mathrm{d}G}{\mathrm{d}\tau}}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}$,
since then (without loss of generality) the imaginary part of $V$ is
exponentially small in $G$ [46]. Notice that above we have suppressed the
dependence of $G$ on the parameter $\sigma$.

Our principal aim as indicated above is to consider the spectral stability of
the steady-state solutions (with $G\neq 0$) in the co-exploding frame. These
correspond to self-similar blowup solutions in the original frame. Such
steady-state solutions denoted as $\Phi_{s}$ satisfy

$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}\Phi_{s}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}+|\Phi_{s}|^{2\sigma}\Phi_{s}-\Phi_{s}+\frac{\mathrm{i}G}{\sigma}\Phi_{s}+\mathrm{i}G\xi\mathchoice{\frac{\mathrm{d}\Phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}=0,$
(10)

or equivalently

$\mathchoice{\frac{\mathrm{d}^{2}V_{s}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}+|V_{s}|^{2\sigma}V_{s}-V_{s}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}V_{s}+\frac{G^{2}\xi^{2}}{4}V_{s}=0.$
(11)

We now linearize Eq. (7) about the relevant steady-state solutions in the co-
exploding frame, $\Phi_{s}$, by setting:

$\displaystyle\phi(\xi,\tau)=\Phi_{s}(\xi)+\epsilon\left(X(\xi)\mathrm{e}^{\lambda\tau}+Y^{*}(\xi)\mathrm{e}^{\lambda^{*}\tau}\right),$
(12)

giving rise to the operator eigenvalue problem

$\displaystyle\mathrm{i}\lambda X$ $\displaystyle=$
$\displaystyle\left(-\mathchoice{\frac{\mathrm{d}^{2}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}-(\sigma+1)|\Phi_{s}|^{2\sigma}+1-\frac{\mathrm{i}G}{\sigma}-\mathrm{i}G\xi\mathchoice{\frac{\mathrm{d}}{\mathrm{d}\xi}}{\mathrm{d}/\mathrm{d}\xi}{\mathrm{d}/\mathrm{d}\xi}{\mathrm{d}/\mathrm{d}\xi}\right)X-\sigma|\Phi_{s}|^{2\sigma-2}\Phi_{s}^{2}Y,$
(13) $\displaystyle\mathrm{i}\lambda Y$ $\displaystyle=$
$\displaystyle\sigma|\Phi_{s}|^{2\sigma-2}(\Phi_{s}^{*})^{2}X+\left(\mathchoice{\frac{\mathrm{d}^{2}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}/\mathrm{d}\xi^{2}}+(\sigma+1)|\Phi_{s}|^{2\sigma}-1-\frac{\mathrm{i}G}{\sigma}-\mathrm{i}G\xi\mathchoice{\frac{\mathrm{d}}{\mathrm{d}\xi}}{\mathrm{d}/\mathrm{d}\xi}{\mathrm{d}/\mathrm{d}\xi}{\mathrm{d}/\mathrm{d}\xi}\right)Y.$
(14)

In a similar vein, for the stationary solution $V_{s}(\xi)$, we linearize in
$\epsilon$ by writing

$\displaystyle
V(\xi,\tau)=V_{s}(\xi)+\epsilon\left(f(\xi)\mathrm{e}^{\lambda\tau}+g^{*}(\xi)\mathrm{e}^{\lambda^{*}\tau}\right)$
(15)

which leads to

$\displaystyle\mathrm{i}\lambda
f+\mathchoice{\frac{\mathrm{d}^{2}f}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}g+(\sigma+1)|V_{s}|^{2\sigma}f-f-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f+\frac{G^{2}\xi^{2}}{4}f$
$\displaystyle=$ $\displaystyle 0,$ (16) $\displaystyle-\mathrm{i}\lambda
g+\mathchoice{\frac{\mathrm{d}^{2}g}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}f+(\sigma+1)|V_{s}|^{2\sigma}g-g+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g+\frac{G^{2}\xi^{2}}{4}g$
$\displaystyle=$ $\displaystyle 0.$ (17)

On the finite domain $[-K,K]$ we impose the boundary conditions

$\mathchoice{\frac{{\partial}\phi}{{\partial}\xi}}{{\partial}\phi/{\partial}\xi}{{\partial}\phi/{\partial}\xi}{{\partial}\phi/{\partial}\xi}=0\qquad\mbox{
at }\xi=\pm K,$

which corresponds to

$\mathchoice{\frac{{\partial}V}{{\partial}\xi}}{{\partial}V/{\partial}\xi}{{\partial}V/{\partial}\xi}{{\partial}V/{\partial}\xi}=\pm\frac{\mathrm{i}GKV}{2}\qquad\mbox{
at }\xi=\pm K.$

For the perturbation this gives, correspondingly,

$\mathchoice{\frac{{\partial}X}{{\partial}\xi}}{{\partial}X/{\partial}\xi}{{\partial}X/{\partial}\xi}{{\partial}X/{\partial}\xi}=\mathchoice{\frac{{\partial}Y}{{\partial}\xi}}{{\partial}Y/{\partial}\xi}{{\partial}Y/{\partial}\xi}{{\partial}Y/{\partial}\xi}=0,\qquad\mathchoice{\frac{{\partial}f}{{\partial}\xi}}{{\partial}f/{\partial}\xi}{{\partial}f/{\partial}\xi}{{\partial}f/{\partial}\xi}=\pm\frac{\mathrm{i}GKf}{2},\qquad\mathchoice{\frac{{\partial}g}{{\partial}\xi}}{{\partial}g/{\partial}\xi}{{\partial}g/{\partial}\xi}{{\partial}g/{\partial}\xi}=\mp\frac{\mathrm{i}GKg}{2}\qquad\mbox{
on }\xi=\pm K.$

Let us now try to explore, on the basis of the above principal setup, what we
should expect to see in the linearization around a collapsing waveform.

##  III Principal Numerical Results

The question of how the spectrum changes under the type of nontrivial scaling
transformation discussed above requires particular attention. This topic was
first addressed systematically, to the best of our knowledge, in a different
class of systems, in the pioneering work of [43, 44] who realized that such a
transformation that rescales space and time may have profound implications
within the renormalized frame as regards the interpretations of symmetries of
the original frame. To explain this subtle point, we provide arguably the
simplest possible example that we have been able to identify in Appendix A of
the present manuscript. There, and in the cleaner/simpler setting of an
autonomous ordinary differential equation, it can be seen that the symmetry of
time translation of the original system leads to an “apparent instability” in
the renormalized frame. This is because a shift in, e.g., the time of collapse
in the original frame, due to the exponential nature of the transformation
between the renormalized and the regular time, leads to an exponential
deviation in the renormalized frame and hence an apparent instability.

The key take-home message from this example is that symmetries of the original
frame may no longer correspond to ones such in the renormalized frame. The
even more dire consequence is that symmetries of the original frame may appear
as instabilities in the renormalized one. For example, differentiating Eq.
(10) with respect to $\xi$ gives

$\displaystyle\frac{\mathrm{d}^{3}\Phi_{s}}{\mathrm{d}\xi^{3}}+(\sigma+1)|\Phi_{s}|^{2\sigma}\mathchoice{\frac{\mathrm{d}\Phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}+\sigma|\Phi_{s}|^{2\sigma-2}\Phi_{s}^{2}\mathchoice{\frac{\mathrm{d}\Phi_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}^{*}/\mathrm{d}\xi}-\mathchoice{\frac{\mathrm{d}\Phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}+\frac{\mathrm{i}G}{\sigma}\mathchoice{\frac{\mathrm{d}\Phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}+\mathrm{i}G\xi\mathchoice{\frac{\mathrm{d}^{2}\Phi_{s}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}\Phi_{s}/\mathrm{d}\xi^{2}}+\mathrm{i}G\mathchoice{\frac{\mathrm{d}\Phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}{\mathrm{d}\Phi_{s}/\mathrm{d}\xi}=0,\qquad$
(18)

from which we observe that
$X=\mathchoice{\frac{\mathrm{d}\phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}$
and
$Y=\mathchoice{\frac{\mathrm{d}\phi_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}\phi_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}^{*}/\mathrm{d}\xi}$
satisfy Eqs. (13)-(14) if we choose $\lambda=G$. The eigenvector is associated
with the derivative, which is well-known to be the generator of translations.
However, instead of this vector being associated with a neutral direction, it
is now associated with an “apparently unstable” eigenmode (since $G>0$).
Nevertheless, that eigenmode is not a true instability in the original frame,
even though it appears as one in the renormalized frame. Rather, it only
involves spatial translation, i.e., a symmetry, and its suitable
reinterpretation in this renormalized frame.

Armed with this important piece of understanding, let us now scrutinize the
spectral picture in further detail. As is natural, we start with the
subcritical case of $\sigma d<2$. In the integrable limit of $d=\sigma=1$, it
is well-known [47] that the spectrum of the linearization of the NLS soliton
possesses two neutral directions, one associated with spatial translations,
and one associated with the phase or gauge (U$(1)$) invariance. We already saw
that the derivative
$\mathchoice{\frac{\mathrm{d}\phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}$
is connected to the translational eigenvector while the solution ($\phi_{s}$)
itself is associated with the corresponding phase eigenvector. In each case,
the generalized eigenvectors are known as well [47].

As we depart (parametrically in $\sigma$) from the integrable limit, an
eigenvalue pair bifurcates from the band edge of the continuous spectrum which
consists of the union of the intervals $\mathrm{i}[1,\infty)$ and
$-\mathrm{i}[1,\infty)$ and tends (along the imaginary axis) towards the
origin as $\sigma\rightarrow 2$. It is this eigenvalue pair that arrives at
the origin of the spectral plane, precisely at $\sigma=2$, instituting the
conformal invariance of the model, i.e., the invariance with respect to
rescaling that paves the way to collapse dynamics. The dependence of this
eigenvalue on the parameter $\sigma$ is shown in Fig. 1. The corresponding
eigenvector in this case is

$\displaystyle
X(\xi)=\mathrm{i}\phi_{s}+G\left(\frac{\phi_{s}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}\phi_{s}}{\mathrm{d}\xi}}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}{\mathrm{d}\phi_{s}/\mathrm{d}\xi}\right),$
(19)

and similarly its conjugate yields $Y(\xi)$. Past the critical point, the
relevant eigenvalue becomes real, giving rise to the dynamical instability of
the soliton and the emergence of the collapsing branch of solutions. The
spectra of the solitonic solution of $G=0$ for $\sigma$ below the critical one
(of $\sigma_{c}=2$ for $d=1$) and above the critical point are shown in Fig.
2.

\begin{overpic}[width=411.93767pt]{lambdasquare_variation.eps} \end{overpic}
Figure 1: Square of eigenvalue bifurcating from the band edge of the
continuous spectrum, tending towards the origin as $\sigma\rightarrow 2$, and
finally giving rise to real eigenvalues (one positive and one negative) past
the critical point, $\sigma=2$.
\begin{overpic}[width=411.93767pt]{spectra_from_1_9_to_2_1.eps} \end{overpic}
Figure 2: Spectra of the numerically obtained soliton solution of the rescaled
NLS equation with $K=20$ and $\sigma=1.9$, $\sigma=1.95$, $\sigma=2.05$ and
$\sigma=2.1$.

As the bifurcation of the self-similarly focusing branch of solutions occurs
[45, 46], the natural question is what becomes of the spectrum and what are
the corresponding dynamical implications of this spectral linearization
picture. Recall that at the critical point, the “parent branch” of solitary
waves has, in addition to the above mentioned continuous spectrum, 3
eigenvalue pairs at the origin. Hence, as this Hamiltonian system turns
dissipative for $G>0$, we have to determine the fate of the 6 eigenvalues
stemming from the origin, and the associated continuous spectrum band. Notice
that the 6 eigenvalues will no longer constitute pairs, except perhaps
approximately, as the dissipativity of $G\neq 0$ destroys the Hamiltonian
character and hence the eigenvalue pairing.

Before we systematically answer this question, it is relevant to remind the
reader of the established bifurcation of solutions with nontrivial $G\neq 0$
for $\sigma>2$ [4, 45, 46]. The bifurcation diagram of the relevant solutions
with a finite blowup rate $G$ is shown in Fig. 3, and a typical example of the
associated waveforms and the dynamics of approaching them within the realm of
the mixed Hamiltonian-dissipative system of Eq. (7) is shown in Fig. 4. The
latter suggests the attractivity of the structures and hence predisposes us
towards their (effective) spectral stability. Having obtained these solutions
with finite non-vanishing $G$ as stationary ones (see the details in [46]), we
are now ready to solve the corresponding spectral problem for the eigenvalues
$\lambda$ and eigenvectors $(X,Y)$. Some typical examples of the spectral
plane of the imaginary vs. the real part of the eigenvalues for specific
choices of $\sigma$ (and hence $G$, per Fig. 3) are shown in Fig. 5.

\begin{overpic}[width=411.93767pt]{bifurcation_diagram.eps} \end{overpic}
Figure 3: Variation of blowup rate $G$ as a function of $\sigma$, for domain
size $K=20$. The solitonic branch ($G=0$) remains stable up to $\sigma=2$
(black solid line) and becomes unstable for $\sigma>2$ (black dash-dotted
line). The stable collapsing branch ($G>0$) is illustrated with solid grey
line.

The answer to this central question of our manuscript for the spectrum at the
co-exploding frame is given in Fig. 6. There we can see that, in fact, only
one out of the 6 eigenvalues stays at the origin. Indeed, $X\propto\phi_{s}$
remains an eigenvector with vanishing eigenvalue, as the rescaled model
retains the original phase invariance. Nevertheless, as indicated above, the
generalized eigenvector is no longer there (due to dissipativity) and, thus,
the associated eigenvalue acquires a small negative value. In addition, there
are two pairs of eigenvalues that are only nearly symmetric. We find these to
be at $\lambda\approx\pm 2G$ and $\lambda\approx\pm G$. All of these point
spectrum eigenvalues are systematically captured in Fig. 6 to which we will
return shortly. Moreover, there are two more observations in place regarding
Fig. 5. One of the above 6 eigenvalues (and one of the ones shown in Fig. 6,
as well), the eigenvalue at $\lambda\approx-G$, is hard to detect. This is
because it almost coincides with a nearly vertical line of continuous spectrum
with real part $\lambda_{r}=-G$, i.e., the continuous spectrum is
approximately $\lambda=-G+\mathrm{i}s$ for arbitrary real $s$ (see also
Appendix B).

As we already discussed above, the pair at $\lambda\approx\pm G$ is associated
with spatial translation. Indeed, the derivative eigenvector, through the
exact calculation above, yields an eigenvalue of $\lambda=G$ in the infinite
domain. It can be discerned from Fig. 6, that this eigenvalue is no longer
exactly at $G$ on the finite domain but rather presents slight undulations in
its dependence. Indeed, one of our key aims in the detailed calculations that
follow will be to capture these finite-domain-induced undulatory corrections.
On the other hand, the eigenvalue at $-G$ is no longer exact even in the
infinite domain, due to the lack of symmetry, as induced by the dissipative
terms $\propto G$ in our linearized equation for $(X,Y)$ (or for $(f,g)$). In
a very similar tenor, the eigenvalue $\lambda=2G$ is also exact in the
infinite domain limit, as can be verified by direct calculation, upon
substituting the eigenvector of Eq. (19) in the linearization equations.
However, in this case too, the finite domain correction (to be also evaluated
below) induces an undulatory dependence on top of the $\lambda=2G$ leading
order. Furthermore, the eigenvalue $\lambda=-2G$ is also no longer exact even
for an infinite domain (due to dissipativity) and in addition, there is an
undulation (from the finite domain) on top of this eigenvalue as well. This
summary then accounts for all the point spectrum eigenvalues.

It is relevant to add here two important observations. The first one concerns
the dynamics of the collapsing solutions. On the one hand, we obtain that the
relevant waveforms have two unstable eigendirections in the co-exploding
frame. However, on the other hand, we have illustrated through our explicit
calculations above (see also the pertinent Appendix A) that such
eigendirections do not pertain to true instabilities, but rather to neutral
directions of the original frame (spatial translations and rescalings of the
original solution). Given the rescaling of space and time in the co-exploding
frame, both of these actions move solutions exponentially far from other
members of the family of such equivariant solutions, and thus appear as
instabilities in the co-exploding frame, yet this is not a true instability in
the original frame. Hence, in line with our above dynamical evolution results,
we expect such collapsing solutions to be dynamically robust (modulo
symmetries).

\begin{overpic}[width=411.93767pt]{rescaled_nls_evolution_power_v2.eps}
\end{overpic} Figure 4: Dynamics of $|\phi(\xi,\tau)|^{2}-|\phi(\xi,0)|^{2}$
in the co-exploding frame (rescaled NLS). The initial condition, $\phi(\xi,0)$
is the soliton solution for $\sigma=2.019$. Upon perturbing $\sigma$ to
$\sigma=2.02$, the co-exploding dynamics converges to a “steady-state”
solution. The inset on the bottom right illustrates the evolution of the
blowup rate, $G$, with the rescaled time, $\tau$.

The second observation is related to the results for the spectrum given in the
earlier work of [45]. There, only one of these positive eigenvalues was found
and moreover the continuous spectrum had a wider apparent extent around
$\lambda_{r}\approx-G$ (extending to values with more negative real part). The
former of these features was because the calculation of [45] was done in the
half-domain and hence, e.g., spatial translations were a priori excluded from
consideration. Furthermore, we believe that the observations of the continuous
spectrum had to do with the discretization used in the latter case. Our
refined numerics here suggest that the continuous spectrum progressively tends
to the vertical line with $\lambda_{r}=-G$ (asymptotically for large imaginary
part). Finally, we also note that the main features of the computed spectra
show only slight changes by increasing the size of the computational domain
(see Appendix C). Admittedly, in what follows we can only offer an asymptotic
prediction for the part of the spectral band with sufficiently large imaginary
part. For the part with small imaginary part, the situation is rather complex
and constitutes a technical challenge for potential future studies.
Nevertheless, we believe that we hereby offer a far more definitive
perspective of both the point and the continuous spectrum, than was previously
available.

\begin{overpic}[width=411.93767pt]{comparison_spectra_vertical.eps}
\end{overpic}\begin{overpic}[width=411.93767pt]{comparison_spectra_2_001_2_05.eps}
\end{overpic}

Figure 5: Spectra obtained from the numerical solution of the rescaled NLS
equation with $K=20$ and $\sigma$ values close to the critical value,
$\sigma=2$: $\sigma=2+10^{-9}$ (top left panel) and $\sigma=2+10^{-6}$ (top
right panel), as well as $\sigma=2.001$ (bottom left panel), $\sigma=2.05$
(bottom right panel).
\begin{overpic}[width=411.93767pt]{important_eigenvalues_new.eps}
\end{overpic} Figure 6: Variation of $\lambda\approx 2G$, $\lambda\approx G$,
$\lambda=0$, $\lambda\approx-G$, $\lambda\approx-2G$ and $\lambda$ close to 0
eigenvalues with $\sigma$ values as obtained from the numerical solution of
the rescaled NLS equation with $K=20$.

In the analytical calculations that follow (and their comparison with the
detailed numerical computations as regards the eigenvalue corrections), we
will consider each of these eigenvalues one by one. We will split their
dependence into a principal part (that we have effectively already discussed
above), and a correction that stems either from the finiteness of the
computational domain (in the case of $\lambda=G$ or $\lambda=2G$) or from both
the inexactness of the symmetry in the dissipative system and the finiteness
of the computational domain (in the case of the negative point spectrum
eigenvalues). We will develop a solvability based approach to calculate the
residual of each of these eigenvalues and will subsequently compare it to our
systematic eigenvalue computations. Finally, we will corroborate our
theoretical conclusion on the effective spectral stability (modulo the
symmetries) of the collapsing solutions via direct numerical simulations in
both the original and the co-exploding frame.

For the performance of numerical computations, we adopt a fourth-order central
finite difference scheme for the approximation of spatial derivatives. Space,
$\xi\in[-K,K]$ is uniformly discretized with step, $d\xi=0.01$. Time
integration (where needed) is performed utilizing MATLAB’s ode23t ODE solver.
Steady-state solutions are obtained through the iterative Newton-Raphson
algorithm. Finally, the eigenvalue computations were performed by utilizing
MATLAB’s eig solver and corroborated further by using the contour-integral
based FEAST eigenvalue solver [48] (and references therein). The spectral
stability analysis results we obtained through the use of both eigenvalue
solvers match precisely with each other.

##  IV Theoretical Analysis Approach

For our theoretical analysis, we work in terms of $V_{s}$, $f$ and $g$. We
first outline the general methodology, before we apply it to each eigenvalue
of the discrete spectrum in turn.

Suppose we have an asymptotic approximation to the eigenfunctions
$f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ and eigenvalue
$\lambda_{\mathrm{reg}}$ which is accurate to all orders in $G$ but misses
exponentially small terms. [Notice that in what follows, for mathematical
convenience, we will generally expand in powers of $G$, rather than the
parameter $\sigma$.] Let us write
$\lambda=\lambda_{\mathrm{reg}}+\lambda_{\mathrm{exp}}$. Then, Eqs. (16)-(17)
give:

$\displaystyle\mathrm{i}\lambda_{\mathrm{reg}}f+\mathchoice{\frac{\mathrm{d}^{2}f}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}g+(\sigma+1)|V_{s}|^{2\sigma}f-f-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f+\frac{G^{2}\xi^{2}}{4}f$
$\displaystyle=$ $\displaystyle-\mathrm{i}\lambda_{\mathrm{exp}}f,$
$\displaystyle-\mathrm{i}\lambda_{\mathrm{reg}}g+\mathchoice{\frac{\mathrm{d}^{2}g}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}f+(\sigma+1)|V_{s}|^{2\sigma}g-g+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g+\frac{G^{2}\xi^{2}}{4}g$
$\displaystyle=$ $\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}g.$

If we multiply by $f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ respectively, add
and integrate by parts, the left-hand side is

$\displaystyle\int_{-K}^{K}\left(\mathrm{i}\lambda_{\mathrm{reg}}f+\mathchoice{\frac{\mathrm{d}^{2}f}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}g+(\sigma+1)|V_{s}|^{2\sigma}f-f-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f+\frac{G^{2}\xi^{2}}{4}f\right)f_{\mathrm{reg}}\,\mathrm{d}\xi$
$\displaystyle+\int_{-K}^{K}\left(-\mathrm{i}\lambda_{\mathrm{reg}}g+\mathchoice{\frac{\mathrm{d}^{2}g}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}f+(\sigma+1)|V_{s}|^{2\sigma}g-g+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g+\frac{G^{2}\xi^{2}}{4}g\right)g_{\mathrm{reg}}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\int_{-K}^{K}\left(\mathrm{i}\lambda_{\mathrm{reg}}f_{\mathrm{reg}}+\mathchoice{\frac{\mathrm{d}^{2}f_{\mathrm{reg}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}+(\sigma+1)|V_{s}|^{2\sigma}f_{\mathrm{reg}}-f_{\mathrm{reg}}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f_{\mathrm{reg}}+\frac{G^{2}\xi^{2}}{4}f_{\mathrm{reg}}\right)f\,\mathrm{d}\xi$
$\displaystyle\int_{-K}^{K}\left(-\mathrm{i}\lambda_{\mathrm{reg}}g_{\mathrm{reg}}+\mathchoice{\frac{\mathrm{d}^{2}g_{\mathrm{reg}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}+(\sigma+1)|V_{s}|^{2\sigma}g_{\mathrm{reg}}-g_{\mathrm{reg}}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g_{\mathrm{reg}}+\frac{G^{2}\xi^{2}}{4}g_{\mathrm{reg}}\right)g\,\mathrm{d}\xi$
$\displaystyle\mbox{
}+\int_{-K}^{K}\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}gf_{\mathrm{reg}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}fg_{\mathrm{reg}}\,\mathrm{d}\xi+\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle=$
$\displaystyle\int_{-K}^{K}R_{\mathrm{f}}f+R_{\mathrm{g}}g+\sigma|V_{s}|^{2\sigma-2}(g_{\mathrm{reg}}f-f_{\mathrm{reg}}g)((V_{s}^{*})^{2}-V_{s}^{2})\,\mathrm{d}\xi$
$\displaystyle\mbox{
}+\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K},$

where

$\displaystyle R_{\mathrm{f}}$ $\displaystyle=$
$\displaystyle\mathrm{i}\lambda_{\mathrm{reg}}f_{\mathrm{reg}}+\mathchoice{\frac{\mathrm{d}^{2}f_{\mathrm{reg}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{reg}}/\mathrm{d}\xi^{2}}+(\sigma+1)|V_{s}|^{2\sigma}f_{\mathrm{reg}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}g_{\mathrm{reg}}-f_{\mathrm{reg}}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f_{\mathrm{reg}}+\frac{G^{2}\xi^{2}}{4}f_{\mathrm{reg}},$
$\displaystyle R_{\mathrm{g}}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\lambda_{\mathrm{reg}}g_{\mathrm{reg}}+\mathchoice{\frac{\mathrm{d}^{2}g_{\mathrm{reg}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{reg}}/\mathrm{d}\xi^{2}}+(\sigma+1)|V_{s}|^{2\sigma}g_{\mathrm{reg}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}f_{\mathrm{reg}}-g_{\mathrm{reg}}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g_{\mathrm{reg}}+\frac{G^{2}\xi^{2}}{4}g_{\mathrm{reg}},$

are the exponentially small residuals from the regular asymptotic expansion.
Since the imaginary part of $V_{s}$ is exponentially small, and
$f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ are exponentially close to $f$ and
$g$, the term
$(g_{\mathrm{reg}}f-f_{\mathrm{reg}}g)((V_{s}^{*})^{2}-V_{s}^{2})$ is doubly
exponentially small and can be neglected. Then, evaluating also the right-hand
side,

$\int_{-K}^{K}(R_{\mathrm{f}}f+R_{\mathrm{g}}g)\,\mathrm{d}\xi+\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}\sim\mathrm{i}\lambda_{\mathrm{exp}}\int_{-K}^{K}(g_{\mathrm{reg}}g-f_{\mathrm{reg}}f)\,\mathrm{d}\xi.$
(20)

Since $R_{\mathrm{f}}$, $R_{\mathrm{g}}$ and $\lambda_{\mathrm{exp}}$ are
already exponentially small, we can use $f_{\mathrm{reg}}$, $g_{\mathrm{reg}}$
in place of $f$ and $g$ except in the boundary terms, introducing only double-
exponentially-small errors. Thus, to exponential accuracy,

$\int_{-K}^{K}(R_{\mathrm{f}}f_{\mathrm{reg}}+R_{\mathrm{g}}g_{\mathrm{reg}})\,\mathrm{d}\xi+\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}\sim\mathrm{i}\lambda_{\mathrm{exp}}\int_{-K}^{K}(g_{\mathrm{reg}}^{2}-f_{\mathrm{reg}}^{2})\,\mathrm{d}\xi.$
(21)

This is the equation which determines the exponentially small correction to
the eigenvalue $\lambda_{\mathrm{exp}}$. To find the boundary terms we need to
examine the far field more carefully.

###  IV.1 Boundary condition on a finite domain

Consider first $f$. We write $f=f_{\mathrm{reg}}+f_{b}$, where $f_{b}$ is the
correction due to the fact that $f_{\mathrm{reg}}$ does not satisfy the
boundary conditions. Then, following the earlier work of [46], we have in the
far field with $\rho=G\xi$,

$f_{\mathrm{reg}}={A^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{\mathrm{i}\phi_{2}/G}+{B^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{-\mathrm{i}\phi_{2}/G},\qquad
f_{b}=A\mathrm{e}^{\mathrm{i}\phi_{2}/G}+B\mathrm{e}^{-\mathrm{i}\phi_{2}/G},$
(22)

where

$\phi_{2}^{\prime}=\sqrt{\frac{\rho^{2}}{4}-1},\qquad\mathrm{i}\lambda_{1}A+2\mathrm{i}\phi_{2}^{\prime}A^{\prime}+\mathrm{i}\phi_{2}^{\prime\prime}A=\mathrm{i}GA^{\prime\prime},\qquad\mathrm{i}\lambda_{1}B-2\mathrm{i}\phi_{2}^{\prime}B^{\prime}-\mathrm{i}\phi_{2}^{\prime\prime}B=\mathrm{i}GB^{\prime\prime},$
(23)

and $\lambda_{\mathrm{reg}}=\lambda_{1}G$. Note that
${A^{\mathrm{f}}_{\mathrm{reg}}}$ and ${B^{\mathrm{f}}_{\mathrm{reg}}}$ are
given, but $A$ and $B$ need to be determined. Expanding

$A=\sum_{n=0}^{\infty}A_{n}(\rho)(\mathrm{i}G)^{n},\qquad
B=\sum_{n=0}^{\infty}B_{n}(\rho)(-\mathrm{i}G)^{n},$ (24)

substituting into Eq. (23), and equating coefficients of powers of $G$ gives
at leading order

$\frac{A_{0}^{\prime}}{A_{0}}=-\frac{(\phi_{2}^{\prime\prime}+\lambda_{1})}{2\phi_{2}^{\prime}},\qquad\frac{B_{0}^{\prime}}{B_{0}}=-\frac{(\phi_{2}^{\prime\prime}-\lambda_{1})}{2\phi_{2}^{\prime}},$

so that

$A_{0}=\frac{a_{\mathrm{f}}}{(\rho^{2}-4)^{1/4}}\left(\frac{\rho-\sqrt{\rho^{2}-4}}{\rho+\sqrt{\rho^{2}-4}}\right)^{\lambda_{1}/2},\qquad
B_{0}=\frac{b_{\mathrm{f}}}{(\rho^{2}-4)^{1/4}}\left(\frac{\rho+\sqrt{\rho^{2}-4}}{\rho-\sqrt{\rho^{2}-4}}\right)^{\lambda_{1}/2},$

for some constants $a_{\mathrm{f}}$ and $b_{\mathrm{f}}$. At the next order

$\lambda_{1}A_{1}+2\phi_{2}^{\prime}A_{1}^{\prime}+\phi_{2}^{\prime\prime}A_{1}=A_{0}^{\prime\prime},\qquad-\lambda_{1}B_{1}+2\phi_{2}^{\prime}B_{1}^{\prime}+\phi_{2}^{\prime\prime}B_{1}=B_{0}^{\prime\prime}.$

Substituting for $\phi_{2}$, $A_{0}$ and $B_{0}$, and solving gives

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{a_{\mathrm{f}}}{(\rho^{2}-4)^{1/4}}\left(\frac{\rho-\sqrt{\rho^{2}-4}}{\rho+\sqrt{\rho^{2}-4}}\right)^{\lambda_{1}/2}\frac{(24(2\lambda_{1}^{2}-1)\rho+(1-12\lambda_{1}^{2})\rho^{3}-48\lambda_{1}\sqrt{\rho^{2}-4})}{48(\rho^{2}-4)^{3/2}},$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\frac{b_{\mathrm{f}}}{(\rho^{2}-4)^{1/4}}\left(\frac{\rho+\sqrt{\rho^{2}-4}}{\rho-\sqrt{\rho^{2}-4}}\right)^{\lambda_{1}/2}\frac{(24(2\lambda_{1}^{2}-1)\rho+(1-12\lambda_{1}^{2})\rho^{3}+48\lambda_{1}\sqrt{\rho^{2}-4})}{48(\rho^{2}-4)^{3/2}},$

where we fix the constants of integration by requiring that $A\leftrightarrow
B$ as we circle the branch point $\rho=2$. Continuing in this way, we find
that

$\displaystyle A$ $\displaystyle\sim$ $\displaystyle
a_{\mathrm{f}}\rho^{-1/2-\lambda_{1}}(1+\mathrm{i}G\mu_{1}-\mu_{2}G^{2}+\cdots),$
(25) $\displaystyle B$ $\displaystyle\sim$ $\displaystyle
b_{\mathrm{f}}\rho^{-1/2+\lambda_{1}}(1-\mathrm{i}G\mu_{1}+\mu_{2}G^{2}+\cdots),$
(26)

as $\rho\rightarrow\infty$, where

$\mu_{1}=\frac{(1-12\lambda_{1}^{2})}{48},\qquad\mu_{2}=\frac{\lambda_{1}(1-4\lambda_{1}^{2})}{48}.$

A similar asymptotic behaviour must hold for
${A^{\mathrm{f}}_{\mathrm{reg}}}$, ${B^{\mathrm{f}}_{\mathrm{reg}}}$, so that

$\displaystyle{A^{\mathrm{f}}_{\mathrm{reg}}}$ $\displaystyle\sim$
$\displaystyle{a^{\mathrm{f}}_{\mathrm{reg}}}\rho^{-1/2-\lambda_{1}}(1+\mathrm{i}G\mu_{1}-\mu_{2}G^{2}+\cdots),$
(27) $\displaystyle{B^{\mathrm{f}}_{\mathrm{reg}}}$ $\displaystyle\sim$
$\displaystyle{b^{\mathrm{f}}_{\mathrm{reg}}}\rho^{-1/2+\lambda_{1}}(1-\mathrm{i}G\mu_{1}+\mu_{2}G^{2}+\cdots)$
(28)

as $\rho\rightarrow\infty$. As we approach the turning point $\rho=2$,

$A\sim\frac{a_{\mathrm{f}}}{(4(\rho-2))^{1/4}},\quad
B\sim\frac{b_{\mathrm{f}}}{(4(\rho-2))^{1/4}}.$

Matching with the turning point region gives

$a_{\mathrm{f}}\mathrm{i}=b_{\mathrm{f}},$

which ensures that the extra contribution due to the reflection back from the
boundary is exponentially small in the near field.

The boundary condition gives

$\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\mathrm{i}\phi_{2}^{\prime}(A+{A^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{\mathrm{i}\phi_{2}/G}-\mathrm{i}\phi_{2}^{\prime}(B+{B^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{-\mathrm{i}\phi_{2}/G}+G(A^{\prime}+{A^{\mathrm{f}}_{\mathrm{reg}}}^{\prime})\mathrm{e}^{\mathrm{i}\phi_{2}/G}+G(B^{\prime}+{B^{\mathrm{f}}_{\mathrm{reg}}}^{\prime})\mathrm{e}^{-\mathrm{i}\phi_{2}/G}$\mbox{}\hfil\\\
=\frac{\mathrm{i}KG}{2}\left((A+{A^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{\mathrm{i}\phi_{2}/G}+(B+{B^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right).$

Equations (25)-(26) show that $A^{\prime}=O(A/K)$ for large $K$, so that the
term $AK$ dominates $A^{\prime}$ by a factor of $K^{2}$. Neglecting the third
and fourth terms on the left-hand side gives

$\left(\phi_{2}^{\prime}-\frac{KG}{2}\right)(A+{A^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{\mathrm{i}\phi_{2}/G}=\left(\phi_{2}^{\prime}+\frac{KG}{2}\right)(B+{B^{\mathrm{f}}_{\mathrm{reg}}})\mathrm{e}^{-\mathrm{i}\phi_{2}/G},$

so that

$\displaystyle\mathrm{e}^{2\mathrm{i}\phi_{2}/G}\frac{\sqrt{(KG)^{2}-4}-KG}{\sqrt{(KG)^{2}-4}+KG}$
$\displaystyle=$
$\displaystyle\frac{B+{B^{\mathrm{f}}_{\mathrm{reg}}}}{A+{A^{\mathrm{f}}_{\mathrm{reg}}}}.$

We now assume that $KG$ is large so that we can use the asymptotic behaviour
of Eqs. (25)-(28) to evaluate the right-hand side, giving

$\displaystyle\frac{\mathrm{i}a_{\mathrm{f}}+{b^{\mathrm{f}}_{\mathrm{reg}}}}{a_{\mathrm{f}}+{a^{\mathrm{f}}_{\mathrm{reg}}}}$
$\displaystyle\sim$
$\displaystyle-\frac{\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}}{(KG)^{2+2\lambda_{1}}}\left(\frac{1+\mathrm{i}G\mu_{1}-\mu_{2}G^{2}+\cdots}{1-\mathrm{i}G\mu_{1}+\mu_{2}G^{2}+\cdots}\right)=-S,$

say. Then

$a_{\mathrm{f}}\sim-\frac{{b^{\mathrm{f}}_{\mathrm{reg}}}+{a^{\mathrm{f}}_{\mathrm{reg}}}S}{\mathrm{i}+S},\qquad
b_{\mathrm{f}}\sim-\mathrm{i}\frac{{b^{\mathrm{f}}_{\mathrm{reg}}}+{a^{\mathrm{f}}_{\mathrm{reg}}}S}{\mathrm{i}+S}.$
(29)

As $KG\rightarrow\infty$ the behaviour of $S$ (and therefore $a_{\mathrm{f}}$
and $b_{\mathrm{f}}$) crucially depends on whether $\lambda_{1}$ is greater or
less than $-1$. For $\lambda_{1}>-1$, $S\rightarrow 0$ as
$KG\rightarrow\infty$ and

$a_{\mathrm{f}}\sim\mathrm{i}{b^{\mathrm{f}}_{\mathrm{reg}}},\qquad
b_{\mathrm{f}}\sim-{b^{\mathrm{f}}_{\mathrm{reg}}}.$ (30)

For $\lambda_{1}<-1$, $S\rightarrow\infty$ as $KG\rightarrow\infty$ and

$a_{\mathrm{f}}\sim-{a^{\mathrm{f}}_{\mathrm{reg}}},\qquad
b_{\mathrm{f}}\sim-\mathrm{i}{a^{\mathrm{f}}_{\mathrm{reg}}}.$ (31)

Now, for large $KG$, we can evaluate the boundary terms in Eq. (21) as

$\displaystyle\left.f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}\right|_{\rho=KG}=\left.f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f_{b}}{\mathrm{d}\xi}}{\mathrm{d}f_{b}/\mathrm{d}\xi}{\mathrm{d}f_{b}/\mathrm{d}\xi}{\mathrm{d}f_{b}/\mathrm{d}\xi}-f_{b}\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}\right|_{\rho=KG}$
$\displaystyle\sim$ $\displaystyle
G\left({A^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{\mathrm{i}\phi_{2}/G}+{B^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)\left(\frac{\mathrm{i}\phi_{2}^{\prime}}{G}\left(A\mathrm{e}^{\mathrm{i}\phi_{2}/G}-B\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)+A^{\prime}\mathrm{e}^{\mathrm{i}\phi_{2}/G}+B^{\prime}\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)$
$\displaystyle\mbox{
}-G\left(A\mathrm{e}^{\mathrm{i}\phi_{2}/G}+B\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)\left(\frac{\mathrm{i}\phi_{2}^{\prime}}{G}\left({A^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{\mathrm{i}\phi_{2}/G}-{B^{\mathrm{f}}_{\mathrm{reg}}}\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)+{A^{\mathrm{f}}_{\mathrm{reg}}}^{\prime}\mathrm{e}^{\mathrm{i}\phi_{2}/G}+{B^{\mathrm{f}}_{\mathrm{reg}}}^{\prime}\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\right)$
$\displaystyle\sim$
$\displaystyle-2\mathrm{i}\phi_{2}^{\prime}({A^{\mathrm{f}}_{\mathrm{reg}}}B-{B^{\mathrm{f}}_{\mathrm{reg}}}A)$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}({a^{\mathrm{f}}_{\mathrm{reg}}}(1+\mathrm{i}\mu_{1}G+\cdots)b_{\mathrm{f}}(1-\mathrm{i}\mu_{1}G+\cdots)-{b^{\mathrm{f}}_{\mathrm{reg}}}(1-\mathrm{i}\mu
G+\cdots)a_{\mathrm{f}}(1+\mathrm{i}\mu G+\cdots))$ $\displaystyle=$
$\displaystyle-\mathrm{i}({a^{\mathrm{f}}_{\mathrm{reg}}}b_{\mathrm{f}}-{b^{\mathrm{f}}_{\mathrm{reg}}}a_{\mathrm{f}})\left(1-(\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right).$

A similar calculation on $g$ shows that, when $\lambda_{1}$ is real,

$\displaystyle\left.g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right|_{\rho=KG}$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}({a^{\mathrm{g}}_{\mathrm{reg}}}b_{\mathrm{g}}-{b^{\mathrm{g}}_{\mathrm{reg}}}a_{\mathrm{g}})\left(1-(-\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right),$

where

$a_{\mathrm{g}}=\mathrm{i}\frac{{a^{\mathrm{g}}_{\mathrm{reg}}}+{b^{\mathrm{g}}_{\mathrm{reg}}}S^{*}}{-\mathrm{i}+S^{*}},\qquad
b_{\mathrm{g}}=-\frac{{a^{\mathrm{g}}_{\mathrm{reg}}}+{b^{\mathrm{g}}_{\mathrm{reg}}}S^{*}}{-\mathrm{i}+S^{*}},$

so that

$\displaystyle a_{\mathrm{g}}$
$\displaystyle\sim-{a^{\mathrm{g}}_{\mathrm{reg}}},$ $\displaystyle
b_{\mathrm{g}}$ $\displaystyle\sim-\mathrm{i}{a^{\mathrm{g}}_{\mathrm{reg}}},$
$\displaystyle\lambda_{1}$ $\displaystyle>-1,$ (32) $\displaystyle
a_{\mathrm{g}}$ $\displaystyle\sim\mathrm{i}{b^{\mathrm{g}}_{\mathrm{reg}}},$
$\displaystyle b_{\mathrm{g}}$
$\displaystyle\sim-{b^{\mathrm{g}}_{\mathrm{reg}}},$
$\displaystyle\lambda_{1}$ $\displaystyle<-1.$ (33)

A similar calculation of the boundary layer at $-K$ gives, finally,

$\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}\sim\\\
\begin{cases}\begin{split}&2\mathrm{i}{b^{\mathrm{f}}_{\mathrm{reg}}}({a^{\mathrm{f}}_{\mathrm{reg}}}+\mathrm{i}{b^{\mathrm{f}}_{\mathrm{reg}}})\left(1-(\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)\\\
&\mbox{
}-2\mathrm{i}{a^{\mathrm{g}}_{\mathrm{reg}}}(-\mathrm{i}{a^{\mathrm{g}}_{\mathrm{reg}}}+{b^{\mathrm{g}}_{\mathrm{reg}}})\left(1-(-\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)\end{split}&\mbox{
if }\lambda_{1}>-1,\\\\[14.22636pt]
\begin{split}&-2\mathrm{i}{a^{\mathrm{f}}_{\mathrm{reg}}}(-\mathrm{i}{a^{\mathrm{f}}_{\mathrm{reg}}}+{b^{\mathrm{f}}_{\mathrm{reg}}})\left(1-(\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)\\\
&\mbox{
}+2\mathrm{i}{b^{\mathrm{g}}_{\mathrm{reg}}}({a^{\mathrm{g}}_{\mathrm{reg}}}+\mathrm{i}{b^{\mathrm{g}}_{\mathrm{reg}}})\left(1-(-\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)\end{split}&\mbox{
if }\lambda_{1}<-1.\end{cases}$ (34)

We will see that
${a^{\mathrm{g}}_{\mathrm{reg}}}=\left({b^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*}$
and
${b^{\mathrm{g}}_{\mathrm{reg}}}=\left({a^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*}$
so that the right-hand side is real.

###  IV.2 Eigenvalues

We now apply the general methodology to each of the eigenvalues in turn. Since
the approximate eigenfunctions $f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ are
given in terms of the steady state solution $V_{s}$, to identify the
coefficients ${a^{\mathrm{f}}_{\mathrm{reg}}}$,
${a^{\mathrm{g}}_{\mathrm{reg}}}$, ${b^{\mathrm{f}}_{\mathrm{reg}}}$ and
${b^{\mathrm{g}}_{\mathrm{reg}}}$ that appear in the boundary terms it is
useful to recall the behaviour of $V_{s}$ in the far field, which was
determined in [46]. There we found that

$\displaystyle V_{s}$ $\displaystyle\sim$
$\displaystyle\alpha\mathrm{e}^{\mathrm{i}\phi_{2}/G}\sum_{n=0}^{\infty}A_{n}(\rho)(\mathrm{i}G)^{n}+\beta\mathrm{e}^{-\mathrm{i}\phi_{2}/G}\sum_{n=0}^{\infty}A_{n}(\rho)(-\mathrm{i}G)^{n},\qquad
A_{0}(\rho)=\frac{2^{1/2}a_{0}}{(\rho^{2}-4)^{1/2}},$ (35)

where

$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle 12^{1/4},$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\pi/2G}}{1-\mathrm{i}\nu\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}}=\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\pi/2G}+\frac{\mathrm{i}\nu\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\pi/2G}}{1-\mathrm{i}\nu\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\nu\alpha\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}=-\frac{\nu\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\pi/2G}}{1-\mathrm{i}\nu\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}},$
$\displaystyle\nu$ $\displaystyle\sim$
$\displaystyle\frac{KG-\sqrt{(KG)^{2}-4}}{KG+\sqrt{(KG)^{2}-4}}\sim\frac{1}{(KG)^{2}}.$

####  IV.2.1 The eigenvalue $\lambda_{\mathrm{reg}}=2G$

In terms of $f$ and $g$ the approximate eigenfunctions are

$f_{\mathrm{reg}}=\mathrm{i}V_{s}+G\left(\frac{V_{s}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}-\frac{\mathrm{i}G\xi^{2}V_{s}}{2}\right),\quad
g_{\mathrm{reg}}=-\mathrm{i}V_{s}^{*}+G\left(\frac{V_{s}^{*}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}+\frac{\mathrm{i}G\xi^{2}V_{s}^{*}}{2}\right).$
(36)

These satisfy the equations exactly so that $R_{\mathrm{f}}=R_{\mathrm{g}}=0$.
The perturbation to the eigenvalue arises solely because of the finiteness of
the domain, since $f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ do not satisfy the
boundary conditions. From the known expansion, Eq. (35) of the steady state
solution, we need to identify the amplitude coefficients
${a^{\mathrm{f}}_{\mathrm{reg}}}$, ${b^{\mathrm{f}}_{\mathrm{reg}}}$ in the
WKB expansion [cf. Eqs. (22)-(24)]. The easiest way to do this is to compare
the two representations of $f_{\mathrm{reg}}$ and $g_{\mathrm{reg}}$ as
$\rho\rightarrow\infty$. Comparing Eqs. (35)-(36) with Eqs. (27)-(28) as
$\rho\rightarrow\infty$ gives

$\displaystyle{a^{\mathrm{f}}_{\mathrm{reg}}}\left(1-\mathrm{i}\frac{47G}{48}\right)$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}\sqrt{2}a_{0}\alpha\left(1-\mathrm{i}\frac{95G}{48}+\cdots\right),$
$\displaystyle{b^{\mathrm{f}}_{\mathrm{reg}}}\left(1+\mathrm{i}\frac{47G}{48}\right)$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}\sqrt{2}a_{0}\beta\left(1-\mathrm{i}\frac{G}{48}+\cdots\right),$

since

$\mu_{1}=\frac{(1-12\lambda_{1}^{2})}{48}=-\frac{47}{48}.$

Thus,

${a^{\mathrm{f}}_{\mathrm{reg}}}\sim-\mathrm{i}\sqrt{2}a_{0}\alpha\left(1-\mathrm{i}G\right),\qquad{b^{\mathrm{f}}_{\mathrm{reg}}}\sim-\mathrm{i}\sqrt{2}a_{0}\beta\left(1-\mathrm{i}G\right),\qquad{a^{\mathrm{g}}_{\mathrm{reg}}}\sim\left({b^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*},\qquad{b^{\mathrm{g}}_{\mathrm{reg}}}\sim\left({a^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*}.$

Then

$\displaystyle\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle\sim$ $\displaystyle
2\mathrm{i}{b^{\mathrm{f}}_{\mathrm{reg}}}({a^{\mathrm{f}}_{\mathrm{reg}}}+\mathrm{i}{b^{\mathrm{f}}_{\mathrm{reg}}})-2\mathrm{i}{a^{\mathrm{g}}_{\mathrm{reg}}}(-\mathrm{i}{a^{\mathrm{g}}_{\mathrm{reg}}}+{b^{\mathrm{g}}_{\mathrm{reg}}})$
$\displaystyle\sim$
$\displaystyle-8a_{0}^{2}\nu\mathrm{e}^{-\pi/G}\mathrm{Re}\left((1-\mathrm{i}G)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right),$

as $\nu\rightarrow 0$. Evaluating the right-hand side of Eq. (21) gives

$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}^{2}-f_{\mathrm{reg}}^{2})\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}-f_{\mathrm{reg}})(g_{\mathrm{reg}}+f_{\mathrm{reg}})\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}\left(-2V_{s}+G^{2}\xi^{2}V_{s}\right)\left(\frac{V_{s}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}\right)\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}-2V_{s}^{2}\left(\frac{1}{\sigma}-\frac{1}{2}\right)+G^{2}\xi^{2}V_{s}^{2}\left(\frac{1}{\sigma}-\frac{3}{2}\right)\,\mathrm{d}\xi,$

since

$\int_{-\infty}^{\infty}\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}V_{s}\,\mathrm{d}\xi=-\frac{1}{2}\int_{-\infty}^{\infty}V_{s}^{2}\,\mathrm{d}\xi,\qquad\int_{-\infty}^{\infty}\xi^{3}\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}V_{s}\,\mathrm{d}\xi=-\frac{3}{2}\int_{-\infty}^{\infty}\xi^{2}V_{s}^{2}\,\mathrm{d}\xi.$

The dominant contribution to these integrals is from the near field [46].
Using the asymptotic expansion of $V_{s}$ in powers of $G$ [46] gives

$\int_{-\infty}^{\infty}V_{s}^{2}\,\mathrm{d}\xi=\frac{\sqrt{3}\pi}{2}+\frac{\sqrt{3}\,\pi^{3}G^{2}}{128}+O(G^{4}),\qquad\int_{-\infty}^{\infty}\xi^{2}V_{s}^{2}\,\mathrm{d}\xi=\frac{\sqrt{3}\pi^{3}}{32}+O(G^{2}),$
(37)

so that Eq. (21) becomes

$\displaystyle-8a_{0}^{2}\nu\mathrm{e}^{-\pi/G}\mathrm{Re}\left((1-\mathrm{i}G)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right)$
$\displaystyle=$
$\displaystyle-2\sqrt{3}G\pi\lambda_{\mathrm{exp}}\left(-\left(\frac{1}{\sigma}-\frac{1}{2}\right)+G^{2}\frac{\pi^{2}}{32}\left(\frac{1}{\sigma}-\frac{3}{2}\right)\right)$
$\displaystyle\sim$
$\displaystyle\frac{\sqrt{3}G^{3}\pi^{3}\lambda_{\mathrm{exp}}}{16},$

since $\sigma$ is exponentially close to 2. Thus, the correction to the
eigenvalue is

$\displaystyle\lambda_{\mathrm{exp}}$ $\displaystyle\sim$
$\displaystyle-\frac{256\nu\mathrm{e}^{-\pi/G}}{G^{3}\pi^{3}}\mathrm{Re}\left((1-\mathrm{i}G)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right).$
(38)

A comparison between Eq. (38) and the numerically calculated eigenvalue for
$K=20$ is shown in Fig. 7. This shows that our oscillatory correction
excellently captures the correction due to the finiteness of the domain around
the dominant $\lambda_{\mathrm{reg}}=2G$.

\begin{overpic}[width=411.93767pt]{comparison_lambda2G_asymptotics.eps}
\end{overpic} Figure 7: Comparison between numerical and the asymptotic
predictions, for $K=20$. The solid blue curve corresponds to Eq. (38), while
the purple dots to the numerical solution.

####  IV.2.2 The eigenvalue $\lambda_{\mathrm{reg}}=G$

In terms of $f$ and $g$, the approximate eigenfunctions are

$f_{\mathrm{reg}}=\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}-\frac{\mathrm{i}G\xi
V_{s}}{2},\qquad
g_{\mathrm{reg}}=\mathchoice{\frac{\mathrm{d}V_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}+\frac{\mathrm{i}G\xi
V_{s}^{*}}{2}.$ (39)

Again these satisfy the equations exactly, so that
$R_{\mathrm{f}}=R_{\mathrm{g}}=0$, and the perturbation to the eigenvalue
arises solely because of the finiteness of the domain.

Comparing Eqs. (39) with Eqs. (27)-(28) as $\rho\rightarrow\infty$ gives

${a^{\mathrm{f}}_{\mathrm{reg}}}\left(1-\mathrm{i}\frac{11G}{48}\right)\sim-\mathrm{i}\sqrt{2}a_{0}\alpha\left(1-\mathrm{i}\frac{23G}{48}+\cdots\right),\quad{b^{\mathrm{f}}_{\mathrm{reg}}}\left(1+\mathrm{i}\frac{11G}{48}\right)\sim-\mathrm{i}\sqrt{2}a_{0}\beta\left(1-\mathrm{i}\frac{G}{48}+\cdots\right),$

since

$\mu_{1}=\frac{(1-12\lambda_{1}^{2})}{48}=-\frac{11}{48}.$

Thus

${a^{\mathrm{f}}_{\mathrm{reg}}}\sim-\mathrm{i}\sqrt{2}a_{0}\alpha\left(1-\frac{\mathrm{i}G}{4}\right),\quad{b^{\mathrm{f}}_{\mathrm{reg}}}\sim-\mathrm{i}\sqrt{2}a_{0}\beta\left(1-\frac{\mathrm{i}G}{4}\right),\quad{a^{\mathrm{g}}_{\mathrm{reg}}}\sim{b^{\mathrm{f}}_{\mathrm{reg}}}^{*},\quad{b^{\mathrm{g}}_{\mathrm{reg}}}\sim{a^{\mathrm{f}}_{\mathrm{reg}}}^{*}.$

Then,

$\displaystyle\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle\sim$
$\displaystyle-8a_{0}^{2}\nu\mathrm{e}^{-\pi/G}\mathrm{Re}\left((1-\mathrm{i}G/4)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right).$

Evaluating the right-hand side of Eq. (21) gives

$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}^{2}-f_{\mathrm{reg}}^{2})\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}\xi
V_{s}\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}\,\mathrm{d}\xi=G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}V_{s}^{2}\,\mathrm{d}\xi.$

Using Eq. (37) gives

$\displaystyle-8a_{0}^{2}\nu\mathrm{e}^{-\pi/G}\mathrm{Re}\left((1-\mathrm{i}G/4)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right)$
$\displaystyle=$ $\displaystyle G\lambda_{\mathrm{exp}}\frac{\sqrt{3}\pi}{2},$

i.e.,

$\displaystyle\lambda_{\mathrm{exp}}$ $\displaystyle\sim$
$\displaystyle-\frac{32\nu\mathrm{e}^{-\pi/G}}{G\pi}\mathrm{Re}\left((1-\mathrm{i}G/4)^{2}\mathrm{e}^{2\mathrm{i}\phi_{2}(KG)/G}\right)$
(40)

In this case, a comparison between Eq. (40) and the numerically calculated
eigenvalue for $K=20$ is shown in Fig. 8. Once again, very good agreement is
observed with the numerical finite-domain-induced oscillations, even for
values of $G$ that are quite high (i.e., near $0.5$).

\begin{overpic}[width=411.93767pt]{comparison_lambdaG_asymptotics.eps}
\end{overpic} Figure 8: Comparison between numerical and the asymptotic
predictions, for $K=20$. The solid blue curve corresponds to Eq. (40), while
the purple dots pertain to numerical solution.

####  IV.2.3 The eigenvalue $\lambda_{\mathrm{reg}}=-2G$

In terms of $f$ and $g$ the approximate eigenfunctions are

$f_{\mathrm{reg}}=\mathrm{i}V_{s}-G\left(\frac{V_{s}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}+\frac{\mathrm{i}G\xi^{2}V_{s}}{2}\right),\quad
g_{\mathrm{reg}}=-\mathrm{i}V_{s}^{*}-G\left(\frac{V_{s}^{*}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}-\frac{\mathrm{i}G\xi^{2}V_{s}^{*}}{2}\right).$
(41)

This time, the approximate eigenfunctions do not satisfy the equation exactly,
but with an exponentially small residual. We find

$R_{\mathrm{f}}=-4\mathrm{i}G^{2}V_{s}\left(\frac{1}{2}-\frac{1}{\sigma}\right),\qquad
R_{\mathrm{g}}=4\mathrm{i}G^{2}V_{s}^{*}\left(\frac{1}{2}-\frac{1}{\sigma}\right),$

so that

$\displaystyle\int_{-\infty}^{\infty}R_{\mathrm{f}}f_{\mathrm{reg}}+R_{\mathrm{g}}g_{\mathrm{reg}}\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle
4G^{2}\left(\frac{1}{2}-\frac{1}{\sigma}\right)\int_{-\infty}^{\infty}\left(2V_{s}^{2}-G^{2}\xi^{2}V_{s}^{2}\right)\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle
4G^{2}\left(\frac{1}{2}-\frac{1}{\sigma}\right)\left(\sqrt{3}\pi+\frac{\sqrt{3}\pi^{3}G^{2}}{64}-\frac{\sqrt{3}\pi^{3}G^{2}}{32}+\cdots\right)$
$\displaystyle=$ $\displaystyle
4G^{2}\left(\frac{1}{2}-\frac{1}{\sigma}\right)\left(\sqrt{3}\pi-\frac{\sqrt{3}\pi^{3}G^{2}}{64}+\cdots\right).$

Unfortunately, for $\lambda_{\mathrm{reg}}=-2G$ we will find that we will need
to know more than the leading-order behaviour of
${a^{\mathrm{f}}_{\mathrm{reg}}}$ and ${b^{\mathrm{f}}_{\mathrm{reg}}}$ in
order to find the leading-order approximation to $\lambda_{\mathrm{exp}}$.
Comparing Eqs. (41) with Eqs. (27)-(28) at infinity, including higher-order
terms in both expansions (see [46]), gives

$\displaystyle{a^{\mathrm{f}}_{\mathrm{reg}}}\left(1+\mathrm{i}\mu_{1}G-\mu_{2}G^{2}-\mathrm{i}\mu_{3}G^{3}\right)$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}\sqrt{2}a_{0}\kappa\alpha\left(1+\frac{\mathrm{i}G}{48}+\frac{2021\mathrm{i}G^{3}}{1658880}+\cdots\right),$
$\displaystyle{b^{\mathrm{f}}_{\mathrm{reg}}}\left(1-\mathrm{i}\mu_{1}G+\mu_{2}G^{2}+\mathrm{i}\mu_{3}G^{3}\right)$
$\displaystyle\sim$
$\displaystyle-\mathrm{i}\beta\sqrt{2}a_{0}\kappa\left(1+\mathrm{i}\frac{95G}{48}-\frac{17G^{2}}{24}+\frac{23899\mathrm{i}G^{3}}{1658880}+\cdots\right),$

where

$\kappa\sim 1-\left(\frac{1+12\pi^{2}}{4608}\right)G^{2}+0.0152G^{4}+\cdots.$

Since

$\mu_{1}=\frac{(1-12\lambda_{1}^{2})}{48}=-\frac{47}{48},\quad\mu_{2}=\frac{\lambda_{1}(1-4\lambda_{1}^{2})}{48}=\frac{5}{8},\quad\mu_{3}=-\frac{450581}{1658880},$

we find

$\displaystyle{a^{\mathrm{f}}_{\mathrm{reg}}}$ $\displaystyle\sim$
$\displaystyle-\mathrm{i}\alpha\sqrt{2}a_{0}\kappa\left(1+\mathrm{i}G-\frac{17G^{2}}{48}+\frac{\mathrm{i}G^{3}}{128}+\cdots\right),$
$\displaystyle{b^{\mathrm{f}}_{\mathrm{reg}}}$ $\displaystyle\sim$
$\displaystyle-\mathrm{i}\beta\sqrt{2}a_{0}\kappa\left(1+\mathrm{i}G-\frac{17G^{2}}{48}+\frac{\mathrm{i}G^{3}}{128}+\cdots\right)$

as well as

${a^{\mathrm{g}}_{\mathrm{reg}}}=\left({b^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*},\qquad{b^{\mathrm{g}}_{\mathrm{reg}}}=\left({a^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*}.$

Then, as $KG\rightarrow\infty$,

$\displaystyle\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle\sim$
$\displaystyle-2\mathrm{i}{a^{\mathrm{f}}_{\mathrm{reg}}}(-\mathrm{i}{a^{\mathrm{f}}_{\mathrm{reg}}})\left(1-(\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)+2\mathrm{i}{b^{\mathrm{g}}_{\mathrm{reg}}}(\mathrm{i}{b^{\mathrm{g}}_{\mathrm{reg}}})\left(1-(-\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)$
$\displaystyle\sim$ $\displaystyle
8\sqrt{3}\mathrm{i}\mathrm{e}^{-\pi/G}\kappa^{2}\left(1+\mathrm{i}G-\frac{17G^{2}}{48}+\frac{\mathrm{i}G^{3}}{128}+\cdots\right)^{2}\left(1-(\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)$
$\displaystyle\mbox{
}-8\sqrt{3}\mathrm{i}\mathrm{e}^{-\pi/G}\kappa^{2}\left(1-\mathrm{i}G-\frac{17G^{2}}{48}-\frac{\mathrm{i}G^{3}}{128}+\cdots\right)^{2}\left(1-(-\mathrm{i}\mu_{1}G-\mu_{2}G^{2}+\cdots)^{2}\right)$
$\displaystyle\sim$
$\displaystyle-32\sqrt{3}\,\mathrm{e}^{-\pi/G}\kappa^{2}\left(G+\frac{G^{3}}{2304}+\cdots\right).$

Evaluating the right-hand side of Eq. (21) gives

$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}^{2}-f_{\mathrm{reg}}^{2})\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle
2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}\left(-2V_{s}+G^{2}\xi^{2}V_{s}\right)\left(\frac{V_{s}}{\sigma}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}\right)\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle
2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}-2V_{s}^{2}\left(\frac{1}{\sigma}-\frac{1}{2}\right)+G^{2}\xi^{2}V_{s}^{2}\left(\frac{1}{\sigma}-\frac{3}{2}\right)\,\mathrm{d}\xi,$

after integrating by parts. Using (37) we find that (21) becomes

$\displaystyle-32\sqrt{3}\,\mathrm{e}^{-\pi/G}\kappa^{2}G\left(1+\frac{G^{2}}{2304}+\cdots\right)\mbox{}+4G^{2}\left(\frac{1}{2}-\frac{1}{\sigma}\right)\left(\sqrt{3}\pi-\frac{\sqrt{3}\pi^{3}G^{2}}{64}\right)$
(42) $\displaystyle=$ $\displaystyle
2\sqrt{3}G\pi\lambda_{\mathrm{exp}}\left(-\left(\frac{1}{\sigma}-\frac{1}{2}\right)+G^{2}\frac{\pi^{2}}{32}\left(\frac{1}{\sigma}-\frac{3}{2}\right)\right)$
$\displaystyle\sim$
$\displaystyle-\frac{2\sqrt{3}G^{3}\pi^{3}\lambda_{\mathrm{exp}}}{32}$

Now, since (for $\nu\sim 0$) the relation between $\sigma$ and $G$ is (see
[46])

$G\left(\frac{1}{2}-\frac{1}{\sigma}\right)\left(\frac{\sqrt{3}\pi}{4}+\frac{\sqrt{3}\pi^{3}G^{2}}{256}+\cdots\right)=2\sqrt{3}\kappa^{2}\mathrm{e}^{-\pi/G},$

we find the leading terms on the left-hand side of (42) vanish. This is the
reason we needed to include the higher-order corrections; these give the
correction to the eigenvalue as

$\lambda_{\mathrm{exp}}\sim
G\,\left(1-\frac{2}{\sigma}\right)\left(1+\frac{1}{72\pi^{2}}\right)+\cdots.$
(43)

A comparison between (43) and the numerically calculated eigenvalue for $K=20$
is shown in Fig. 9. Note that although we derived (43) in the limit
$K\rightarrow\infty$, $\nu\rightarrow 0$, when we plot it in Fig. 9 we use the
finite-domain approximation to $\sigma$ as a function of $G$. We can see that
for this eigenvalue we do not purely observe the oscillatory effect induced by
the finite nature of the domain as in the two previous cases. Rather, the
relevant correction incorporates also the deviation from the exact scaling
symmetry (and hence from the symmetry of the eigenvalue pair at $\pm 2G$)
which provides the monotonic portion of the relevant correction.

\begin{overpic}[width=411.93767pt]{comparison_lambdamin2G_asymptotics.eps}
\end{overpic} Figure 9: Asymptotic prediction (43) (blue) compared to
numerical solution (purple) for the case of the eigenvalue with
$\lambda_{\mathrm{reg}}=-2G$.

####  IV.2.4 The eigenvalue $\lambda_{\mathrm{reg}}=-G$

In terms of $f$ and $g$, the approximate eigenfunctions are

$f_{\mathrm{reg}}=\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}+\frac{\mathrm{i}G\xi
V_{s}}{2},\qquad
g_{\mathrm{reg}}=\mathchoice{\frac{\mathrm{d}V_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}-\frac{\mathrm{i}G\xi
V_{s}^{*}}{2}.$ (44)

These satisfy the equations exactly, so that
$R_{\mathrm{f}}=R_{\mathrm{g}}=0$. However, they do not satisfy the correct
radiation condition at infinity. In the finite domain context, the
perturbation of the eigenvalue arises from the boundary terms in Eq. (21).

Comparing Eq. (44) with Eqs. (27)-(28) as $\rho\rightarrow\infty$ gives

${a^{\mathrm{f}}_{\mathrm{reg}}}\left(1-\mathrm{i}\frac{11G}{48}\right)\sim\mathrm{i}\sqrt{2}a_{0}\alpha\left(1+\mathrm{i}\frac{G}{48}+\cdots\right),\quad{b^{\mathrm{f}}_{\mathrm{reg}}}\left(1+\mathrm{i}\frac{11G}{48}\right)\sim\mathrm{i}\sqrt{2}a_{0}\beta\left(1+\mathrm{i}\frac{23G}{48}+\cdots\right),$

since

$\mu_{1}=\frac{(1-12\lambda_{1}^{2})}{48}=-\frac{11}{48}.$

Thus,

${a^{\mathrm{f}}_{\mathrm{reg}}}\sim\mathrm{i}\sqrt{2}a_{0}\alpha\left(1+\frac{\mathrm{i}G}{4}\right),\quad{b^{\mathrm{f}}_{\mathrm{reg}}}\sim\mathrm{i}\sqrt{2}a_{0}\beta\left(1+\frac{\mathrm{i}G}{4}\right),\quad{a^{\mathrm{g}}_{\mathrm{reg}}}\sim\left({b^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*},\quad{b^{\mathrm{g}}_{\mathrm{reg}}}\sim\left({a^{\mathrm{f}}_{\mathrm{reg}}}\right)^{*}.$

When $\lambda_{1}=-1$, $S\rightarrow\mathrm{e}^{2\mathrm{i}\phi_{2}/G}$ as
$\nu\rightarrow 0$, so we need to use the full expressions

$a_{\mathrm{f}}\sim-\frac{{b^{\mathrm{f}}_{\mathrm{reg}}}+{a^{\mathrm{f}}_{\mathrm{reg}}}S}{\mathrm{i}+S},\qquad
b_{\mathrm{f}}\sim-\mathrm{i}\frac{{b^{\mathrm{f}}_{\mathrm{reg}}}+{a^{\mathrm{f}}_{\mathrm{reg}}}S}{\mathrm{i}+S},$
(45)

for $a_{\mathrm{f}}$ and $b_{\mathrm{f}}$. Then

$\displaystyle\left[f_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\xi}}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}{\mathrm{d}f/\mathrm{d}\xi}-f\mathchoice{\frac{\mathrm{d}f_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}f_{\mathrm{reg}}/\mathrm{d}\xi}+g_{\mathrm{reg}}\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\xi}}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}{\mathrm{d}g/\mathrm{d}\xi}-g\mathchoice{\frac{\mathrm{d}g_{\mathrm{reg}}}{\mathrm{d}\xi}}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}{\mathrm{d}g_{\mathrm{reg}}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle\sim$
$\displaystyle\mathrm{Re}\left(-4\mathrm{i}({a^{\mathrm{f}}_{\mathrm{reg}}}b_{\mathrm{f}}-{b^{\mathrm{f}}_{\mathrm{reg}}}a_{\mathrm{f}})\left(1+\mu_{1}^{2}G^{2}+\cdots\right)\right)$
$\displaystyle\sim$
$\displaystyle\mathrm{Re}\left(-\frac{8\mathrm{i}a_{0}^{2}(1+\mathrm{i}G/4)^{2}}{\mathrm{i}+S}(\mathrm{i}\alpha-\beta)(S\alpha+\beta)\left(1+\mu_{1}^{2}G^{2}+\cdots\right)\right)$
$\displaystyle\sim$
$\displaystyle\mathrm{Re}\left(\frac{8\mathrm{i}a_{0}^{2}(1+\mathrm{i}G/4)^{2}}{\mathrm{i}+S}S\mathrm{e}^{-\pi/G}\left(1+\mu_{1}^{2}G^{2}+\cdots\right)\right)$
$\displaystyle\sim$
$\displaystyle\mathrm{Re}\left(\frac{16\mathrm{i}\sqrt{3}(1+\mathrm{i}G/4)^{2}}{1+\mathrm{i}\mathrm{e}^{-2\mathrm{i}\phi_{2}/G}}\mathrm{e}^{-\pi/G}\right)$

Since the integrals in Eq. (20) are dominated by the near field, where $V_{s}$
is real, using the near-field solution in Eq. (20) gives

$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}^{2}-f_{\mathrm{reg}}^{2})\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\mathrm{i}\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}(g_{\mathrm{reg}}-f_{\mathrm{reg}})(g_{\mathrm{reg}}+f_{\mathrm{reg}})\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle
2G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}\xi
V_{s}\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-G\lambda_{\mathrm{exp}}\int_{-\infty}^{\infty}V_{s}^{2}\,\mathrm{d}\xi\sim-G\lambda_{\mathrm{exp}}\frac{\sqrt{3}\pi}{2}.$

Thus the correction to the eigenvalue is

$\lambda_{\mathrm{exp}}\sim-\mathrm{Re}\left(\frac{32\mathrm{i}(1+\mathrm{i}G/4)^{2}}{\pi
G(1+\mathrm{i}\mathrm{e}^{-2\mathrm{i}\phi_{2}/G})}\mathrm{e}^{-\pi/G}\right).$
(46)

A comparison between Eq. (46) and the numerically calculated eigenvalue for
$K=20$ is shown in Fig. 10. A key feature to observe here is the presence of
vertical asymptotes in this exponentially small (in $1/G$) correction. These
represent the reason for the jumps observed in Fig. 10. Indeed, it is relevant
to note that a particularly careful observation of the orange line in Fig. 6
will reveal the outcome of these jumps to the particularly astute reader, as
can be discerned, e.g., near the outermost disconnect of the relevant
numerical line. Despite the fact that our theoretical approximation can no
longer be considered accurate when $\lambda_{\mathrm{exp}}$ becomes large, we
can still see that it very accurately captures our numerical results of Fig.
10.

We see from the numerical results that there is a very thin transition region
in the vicinity of each asymptote in which the eigenvalue perturbation
switches from large and positive to large and negative. We do not attempt to
capture this transition region, which requires a detailed calculation in the
vicinity of $S=-\mathrm{i}$.

\begin{overpic}[width=346.89731pt]{lambda_mG_asy.eps}
\put(-5.0,30.0){\rotatebox{90.0}{$\lambda+G$}} \put(90.0,-7.0){$G$}
\end{overpic} Figure 10: Asymptotic prediction [cf. Eq. (46)] (blue) compared
to numerical solution (purple), for the eigenvalues with
$\lambda_{\mathrm{reg}}=-G$ and for $K=20$.

####  IV.2.5 The eigenvalues near $\lambda=0$

In terms of $f$ and $g$ the approximate eigenfunctions are

$f_{\mathrm{reg}}=\mathrm{i}V_{s},\qquad
g_{\mathrm{reg}}=-\mathrm{i}V_{s}^{*}.$ (47)

In fact, these satisfy the equations and boundary conditions exactly, so that
$\lambda=0$ is an exact eigenvalue even for a finite domain. However, as we
have seen numerically, there is a second eigenvalue which is exponentially
close to zero, which we now approximate. This analysis does not fit into the
general framework of Section IV, but follows a similar methodology, which we
now outline.

We write

$\displaystyle f$ $\displaystyle=$
$\displaystyle\mathrm{i}V_{s}+{\lambda_{\mathrm{exp}}}_{1}f_{1}+f_{\mathrm{exp}},$
(48) $\displaystyle g$ $\displaystyle=$
$\displaystyle-\mathrm{i}V_{s}^{*}+{\lambda_{\mathrm{exp}}}_{1}g_{1}+g_{\mathrm{exp}},$
(49) $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle{\lambda_{\mathrm{exp}}}_{1}+{\lambda_{\mathrm{exp}}}_{2},$ (50)

where ${\lambda_{\mathrm{exp}}}_{2}\ll{\lambda_{\mathrm{exp}}}_{1}$ and

$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}f_{1}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}g_{1}+(\sigma+1)|V_{s}|^{2\sigma}f_{1}-f_{1}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f_{1}+\frac{G^{2}\xi^{2}}{4}f_{1}$
$\displaystyle=$ $\displaystyle V_{s}^{*},$ (51)
$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}g_{1}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s})^{2}f_{1}+(\sigma+1)|V_{s}|^{2\sigma}g_{1}-g_{1}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g_{1}+\frac{G^{2}\xi^{2}}{4}g_{1}$
$\displaystyle=$ $\displaystyle V_{s},$ (52)

with

$\mathchoice{\frac{\mathrm{d}f_{1}}{\mathrm{d}\xi}}{\mathrm{d}f_{1}/\mathrm{d}\xi}{\mathrm{d}f_{1}/\mathrm{d}\xi}{\mathrm{d}f_{1}/\mathrm{d}\xi}=\frac{\mathrm{i}G\xi
f_{1}}{2},\qquad\mathchoice{\frac{\mathrm{d}g_{1}}{\mathrm{d}\xi}}{\mathrm{d}g_{1}/\mathrm{d}\xi}{\mathrm{d}g_{1}/\mathrm{d}\xi}{\mathrm{d}g_{1}/\mathrm{d}\xi}=-\frac{\mathrm{i}G\xi
g_{1}}{2}\qquad\mbox{ at }\xi=K.$

Note that the linear operator here is slightly different from (but
exponentially close to) that of Eqs. (16)-(17), and is chosen so that the
solvability condition is exactly satisfied, so that we can be sure that
$f_{1}$, $g_{1}$ exist: multiplying Eq. (51) by $V_{s}$ and Eq. (52) by
$-V_{s}^{*}$ adding and integrating gives

$\displaystyle\int_{-K}^{K}\left(\mathchoice{\frac{\mathrm{d}^{2}f_{1}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}g_{1}+(\sigma+1)|V_{s}|^{2\sigma}f_{1}-f_{1}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f_{1}+\frac{G^{2}\xi^{2}}{4}f_{1}\right)V_{s}\,\mathrm{d}\xi$
$\displaystyle-\int_{-K}^{K}\left(\mathchoice{\frac{\mathrm{d}^{2}g_{1}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{1}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}f_{1}+(\sigma+1)|V_{s}|^{2\sigma}g_{1}-g_{1}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g_{1}+\frac{G^{2}\xi^{2}}{4}g_{1}\right)V_{s}^{*}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\int_{-K}^{K}\left(\mathchoice{\frac{\mathrm{d}^{2}V_{s}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}/\mathrm{d}\xi^{2}}+|V_{s}|^{2\sigma}V_{s}-V_{s}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}V_{s}+\frac{G^{2}\xi^{2}}{4}V_{s}\right)f_{1}\,\mathrm{d}\xi$
$\displaystyle-\int_{-K}^{K}\left(\mathchoice{\frac{\mathrm{d}^{2}V_{s}^{*}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}V_{s}^{*}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}^{*}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}V_{s}^{*}/\mathrm{d}\xi^{2}}+|V_{s}|^{2\sigma}V_{s}^{*}-V_{s}^{*}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}V_{s}^{*}+\frac{G^{2}\xi^{2}}{4}V_{s}^{*}\right)g_{1}\,\mathrm{d}\xi$
$\displaystyle+\left[V_{s}\mathchoice{\frac{\mathrm{d}f_{1}}{\mathrm{d}\xi}}{\mathrm{d}f_{1}/\mathrm{d}\xi}{\mathrm{d}f_{1}/\mathrm{d}\xi}{\mathrm{d}f_{1}/\mathrm{d}\xi}-f_{1}\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}-V_{s}^{*}\mathchoice{\frac{\mathrm{d}g_{1}}{\mathrm{d}\xi}}{\mathrm{d}g_{1}/\mathrm{d}\xi}{\mathrm{d}g_{1}/\mathrm{d}\xi}{\mathrm{d}g_{1}/\mathrm{d}\xi}+g_{1}\mathchoice{\frac{\mathrm{d}V_{s}^{*}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}{\mathrm{d}V_{s}^{*}/\mathrm{d}\xi}\right]^{K}_{-K}$
$\displaystyle=$ $\displaystyle\left[V_{s}\frac{\mathrm{i}G\xi
f_{1}}{2}-f_{1}\frac{\mathrm{i}G\xi V_{s}}{2}-V_{s}^{*}\frac{(-\mathrm{i}G\xi
g_{1})}{2}+g_{1}\frac{(-\mathrm{i}G\xi V_{s}^{*})}{2}\right]^{K}_{-K}=0.$

Now, substituting Eqs. (48)-(50) into Eqs. (16)-(17) gives

$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}f_{\mathrm{exp}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{\mathrm{exp}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{exp}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{\mathrm{exp}}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}V_{s}^{2}g_{\mathrm{exp}}+(\sigma+1)|V_{s}|^{2\sigma}f_{\mathrm{exp}}-f_{\mathrm{exp}}-\frac{\mathrm{i}(\sigma-2)G}{2\sigma}f_{\mathrm{exp}}+\frac{G^{2}\xi^{2}}{4}f_{\mathrm{exp}}$
$\displaystyle=$
$\displaystyle{\lambda_{\mathrm{exp}}}_{1}\sigma|V_{s}|^{2\sigma-2}((V_{s}^{*})^{2}-V_{s}^{2})g_{1}+{\lambda_{\mathrm{exp}}}_{1}(V_{s}-V_{s}^{*})-\mathrm{i}{\lambda_{\mathrm{exp}}}_{1}^{2}f_{1}+{\lambda_{\mathrm{exp}}}_{2}V_{s},$
$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}g_{\mathrm{exp}}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g_{\mathrm{exp}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{exp}}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g_{\mathrm{exp}}/\mathrm{d}\xi^{2}}+\sigma|V_{s}|^{2\sigma-2}(V_{s}^{*})^{2}f_{\mathrm{exp}}+(\sigma+1)|V_{s}|^{2\sigma}g_{\mathrm{exp}}-g_{\mathrm{exp}}+\frac{\mathrm{i}(\sigma-2)G}{2\sigma}g_{\mathrm{exp}}+\frac{G^{2}\xi^{2}}{4}g_{\mathrm{exp}}$
$\displaystyle=$
$\displaystyle{\lambda_{\mathrm{exp}}}_{1}\sigma|V_{s}|^{2\sigma-2}(V_{s}^{2}-(V_{s}^{*})^{2})f_{1}+{\lambda_{\mathrm{exp}}}_{1}(V_{s}^{*}-V_{s})+\mathrm{i}{\lambda_{\mathrm{exp}}}_{1}^{2}g_{1}+{\lambda_{\mathrm{exp}}}_{2}V_{s}^{*},$

where we have neglected triply-exponentially-small terms involving
${\lambda_{\mathrm{exp}}}_{2}{\lambda_{\mathrm{exp}}}_{1}$. Multiplying by
$\mathrm{i}V_{s}$, $-\mathrm{i}V_{s}^{*}$, adding and integrating the LHS is
triply exponentially small. After simplifying, and neglecting the triply-
exponentially-small term
$(V_{s}^{2}-(V_{s}^{*})^{2}){\lambda_{\mathrm{exp}}}_{2}$, the RHS gives

$\int_{-\infty}^{\infty}\mathrm{i}{\lambda_{\mathrm{exp}}}_{1}\sigma|V_{s}|^{2\sigma-2}((V_{s}^{*})^{2}-V_{s}^{2})(V_{s}g_{1}+V_{s}^{*}f_{1})\\\
+\mathrm{i}{\lambda_{\mathrm{exp}}}_{1}(V_{s}^{2}-(V_{s}^{*})^{2})+{\lambda_{\mathrm{exp}}}_{1}^{2}(V_{s}f_{1}+V_{s}^{*}g_{1})\,\mathrm{d}\xi\sim
0.$ (53)

This is the equation which will determine the eigenvalue
${\lambda_{\mathrm{exp}}}_{1}$; note that it is quadratic, and
${\lambda_{\mathrm{exp}}}_{1}=0$ is a solution as expected. In the outer
region $f_{1}$ and $g_{1}$ are exponentially small. Thus the integrals
involving $f_{1}$ and $g_{1}$ are dominated by the inner region. In the inner
region $f_{1}=g_{1}+$ exponentially small terms, and

$\mathchoice{\frac{\mathrm{d}^{2}f_{1}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{1}/\mathrm{d}\xi^{2}}+5V_{s}^{4}f_{1}-f_{1}+\frac{G^{2}\xi^{2}}{4}f_{1}=V_{s}.$

We find

$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{V_{s}}{2}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}\right)+G^{2}f_{12},$

where, up to exponentially small terms,

$\displaystyle\mathchoice{\frac{\mathrm{d}^{2}f_{12}}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f_{12}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{12}/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f_{12}/\mathrm{d}\xi^{2}}+5V_{s}^{4}f_{12}-f_{12}+\frac{G^{2}\xi^{2}}{4}f_{12}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\xi^{2}V_{s}.$

Unfortunately we need to find this correction term $f_{12}$ because the
leading-order term will integrate to zero. Expanding in powers of $G$, we find
$f_{12}\sim-2V_{1}$ where $V_{s}\sim V_{0}+G^{2}V_{1}+\cdots$ (see [46]), so
that

$\displaystyle\int_{-\infty}^{\infty}(V_{s}f_{1}+V_{s}^{*}g_{1})\,\mathrm{d}\xi$
$\displaystyle\sim$
$\displaystyle\int_{-\infty}^{\infty}V_{s}\left(\frac{V_{s}}{2}+\xi\mathchoice{\frac{\mathrm{d}V_{s}}{\mathrm{d}\xi}}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}{\mathrm{d}V_{s}/\mathrm{d}\xi}-4G^{2}V_{1}\right)\,\mathrm{d}\xi\sim-4G^{2}\int_{-\infty}^{\infty}V_{0}V_{1}\,\mathrm{d}\xi=-\frac{G^{2}\sqrt{3}\pi^{3}}{64}.$

The dominant contribution to the integral of $(V_{s}^{2}-(V_{s}^{*})^{2})$
comes from the outer region before the turning point, in which, with
$\xi=\rho/G$,

$V_{s}\sim\frac{2^{1/2}a_{0}}{(4-\rho^{2})^{1/4}}(\mathrm{e}^{\phi(\rho)/G}+\gamma\mathrm{e}^{-\phi(\rho)/G}),\qquad\phi=-\int_{0}^{\rho}\left(1-\frac{\bar{\rho}^{2}}{4}\right)^{1/2}\,\mathrm{d}\bar{\rho},\qquad\gamma=\frac{\mathrm{i}\mathrm{e}^{-\pi/G}}{2},$

(see [46]), so that

$\displaystyle\mathrm{i}\int_{\infty}^{\infty}(V_{s}^{2}-(V_{s}^{*})^{2})\,\mathrm{d}\xi$
$\displaystyle\sim$
$\displaystyle\frac{2\mathrm{i}}{G}\int_{0}^{2}(V_{s}+V_{s}^{*})(V_{s}-V_{s}^{*})\,\mathrm{d}\rho\sim\frac{2\mathrm{i}}{G}\int_{0}^{2}2\frac{2^{1/2}a_{0}\mathrm{e}^{\phi(\rho)/G}}{(4-\rho^{2})^{1/4}}2\frac{2^{1/2}a_{0}\gamma\mathrm{e}^{-\phi(\rho)/G}}{(4-\rho^{2})^{1/4}}\,\mathrm{d}\rho$
$\displaystyle\sim$
$\displaystyle-\frac{16\sqrt{3}\mathrm{e}^{-\pi/G}}{G}\int_{0}^{2}\frac{\mathrm{d}\rho}{(4-\rho^{2})^{1/2}}=-\frac{8\sqrt{3}\pi\,\mathrm{e}^{-\pi/G}}{G}.$

The final term in Eq. (53) is subdominant, so that, to leading order, Eq. (53)
gives

$\displaystyle{\lambda_{\mathrm{exp}}}_{1}$ $\displaystyle=$
$\displaystyle-\frac{8\sqrt{3}\,\pi\mathrm{e}^{-\pi/G}}{G}\frac{64}{G^{2}\sqrt{3}\pi^{3}}=-\frac{512\,\mathrm{e}^{-\pi/G}}{G^{3}\pi^{2}}.$
(54)

In appendix D we show that the asymptotic behaviour (54) can be determined
much more simply from the reduced system derived in [46], which describes the
slow evolution of $G$ in the vicinity of the bifurcation.

Figure 11 shows the asymptotic prediction Eq. (54) against a direct numerical
simulation. For this eigenvalue the convergence is slower as $G\rightarrow 0$
so that the leading-order approximation is not as close to the numerical
solution. This is because the higher-order corrections are significant when
estimating the integrals in Eq. (53). To demonstrate this we also show in Fig.
11 the approximation

${\lambda_{\mathrm{exp}}}_{1}\sim-\mathrm{i}\frac{\int_{-\infty}^{\infty}(V_{s}-(V_{s}^{*})^{2})\,\mathrm{d}\xi}{\int_{-\infty}^{\infty}(V_{s}f_{1}-V_{s}^{*}g_{1})\,\mathrm{d}\xi}$
(55)

with a numerical solution for $V_{s}$, $f_{1}$ and $g_{1}$, which converges
more quickly to the numerical value. It is clear that the latter expression of
Eq. (55) captures the dependence on $G$ more accurately than the leading-order
correction of the former.

\begin{overpic}[width=346.89731pt]{lambda_zero_asy.eps}
\put(-5.0,30.0){\rotatebox{90.0}{$\lambda$}} \put(90.0,-7.0){$G$}
\end{overpic} Figure 11: Asymptotic prediction [cf. Eq. (54)] (blue) compared
to numerical solution (purple), for the eigenvalue in the vicinity of
$\lambda_{\mathrm{reg}}=0$, for $K=20$. Also shown (green) is the
approximation (55).

###  IV.3 Continuous spectrum

In the far field with $\rho=G\xi$, neglecting the exponentially small terms
$|V_{s}|^{2\sigma}$ and $(\sigma-2)$, we have

$\mathrm{i}\lambda
f+G^{2}\mathchoice{\frac{\mathrm{d}^{2}f}{\mathrm{d}\rho^{2}}}{\mathrm{d}^{2}f/\mathrm{d}\rho^{2}}{\mathrm{d}^{2}f/\mathrm{d}\rho^{2}}{\mathrm{d}^{2}f/\mathrm{d}\rho^{2}}-f+\frac{\rho^{2}}{4}f=0,\qquad-\mathrm{i}\lambda
g+G^{2}\mathchoice{\frac{\mathrm{d}^{2}g}{\mathrm{d}\rho^{2}}}{\mathrm{d}^{2}g/\mathrm{d}\rho^{2}}{\mathrm{d}^{2}g/\mathrm{d}\rho^{2}}{\mathrm{d}^{2}g/\mathrm{d}\rho^{2}}-g+\frac{\rho^{2}}{4}g=0,$
(56)

along with the boundary conditions

$G\mathchoice{\frac{\mathrm{d}f}{\mathrm{d}\rho}}{\mathrm{d}f/\mathrm{d}\rho}{\mathrm{d}f/\mathrm{d}\rho}{\mathrm{d}f/\mathrm{d}\rho}=\frac{\mathrm{i}\rho
f}{2},\qquad
G\mathchoice{\frac{\mathrm{d}g}{\mathrm{d}\rho}}{\mathrm{d}g/\mathrm{d}\rho}{\mathrm{d}g/\mathrm{d}\rho}{\mathrm{d}g/\mathrm{d}\rho}=-\frac{\mathrm{i}\rho
g}{2}\qquad\mbox{ at }\rho=\pm KG.$

Let us start by imagining that these equations hold throughout the region
$[-KG,KG]$, before returning to investigate the impact of the inner region
near $\rho=0$.

Since the equations for $f$ and $g$ decouple (because we have ignored the
inner region), we can treat them separately, and each will give a set of
eigenvalues. In fact, we see that for any eigenfunction-eigenvalue pair
$(f,\lambda)$ the conjugates $(f^{*},\lambda^{*})$ satisfy the equations and
boundary conditions for $g$. We therefore start by focusing on the equation
for $f$.

Using the WKB expansion

$f=A_{f}\mathrm{e}^{\mathrm{i}\phi_{f}/G}+B_{f}\mathrm{e}^{-\mathrm{i}\phi_{f}/G},$
(57)

where

$A_{f}=\sum_{n=0}^{\infty}A_{fn}(\rho)(\mathrm{i}G)^{n},\qquad
B_{f}=\sum_{n=0}^{\infty}B_{fn}(\rho)(-\mathrm{i}G)^{n},$
$\mathrm{i}\lambda-(\phi_{f}^{\prime})^{2}-1+\frac{\rho^{2}}{4}=0,$
$2\mathrm{i}\phi_{f}^{\prime}A_{f0}^{\prime}+\mathrm{i}\phi_{f}^{\prime\prime}A_{f0}=0,\qquad-2\mathrm{i}\phi_{f}^{\prime}B_{f0}^{\prime}-\mathrm{i}\phi_{f}^{\prime\prime}B_{f0}=0,$

we find

$\phi_{f}=\int_{0}^{\rho}\left(\frac{\bar{\rho}^{2}}{4}-1+\mathrm{i}\lambda\right)^{1/2}\,\mathrm{d}\bar{\rho}=\frac{\rho}{4}\sqrt{\rho^{2}-4+4\mathrm{i}\lambda}+(-1+\mathrm{i}\lambda)\log\left(\frac{\rho+\sqrt{\rho^{2}-4+4\mathrm{i}\lambda}}{\sqrt{-4+4\mathrm{i}\lambda}}\right),$

with

$A_{f0}=a_{f}\left(\frac{{\rho}^{2}}{4}-1+\mathrm{i}\lambda\right)^{-1/4},\qquad
B_{f0}=b_{f}\left(\frac{{\rho}^{2}}{4}-1+\mathrm{i}\lambda\right)^{-1/4}.$

Note that this expansion differs from that performed previously in that we
have included $\lambda$ at leading order rather than assuming that
$\lambda=O(G)$.

There are two turning points, at

$\rho=\rho_{f\pm}=\pm 2(1-\mathrm{i}\lambda)^{1/2}.$

In order to define uniquely $\phi_{f}$ let us put branch cuts from these
turning points to $\pm\mathrm{i}\infty$ away from the real axis, as indicated
in Fig. 12 for an arbitrary but representative value of $\lambda$.

\begin{overpic}[width=173.44534pt]{Branches1.eps}
\put(-10.0,50.0){\rotatebox{90.0}{$\mathrm{Im}(\rho)$}}
\put(50.0,-7.0){$\mathrm{Re}(\rho)$} \put(54.0,57.0){$\rho_{f+}$}
\put(45.0,45.0){$\rho_{f-}$} \end{overpic} Figure 12: Branch points and branch
cuts in $\phi_{f}$ when $\lambda=-0.1-1.1\mathrm{i}$. The contour shading
corresponds to $\mathrm{Im}(\phi_{f})$.

\begin{overpic}[width=199.4681pt]{Stokes1.eps}
\put(58.0,48.0){{\small$\rho_{f+}$}} \put(30.0,38.0){{\small$\rho_{f-}$}}
\end{overpic} (a) $\lambda=-0.3-0.5\mathrm{i}$
\begin{overpic}[width=199.4681pt]{Stokes3.eps}
\put(48.0,55.0){{\small$\rho_{f+}$}} \put(40.0,32.0){{\small$\rho_{f-}$}}
\end{overpic} (b) $\lambda=-0.3-2\mathrm{i}$

Figure 13: Stokes lines (green) and anti-Stokes lines (red). The contour
shading corresponds to $\mathrm{Im}(\phi_{f})$. The path along the real axis
is indicated, as well as the two points at which Stokes lines are crossed.

To impose the boundary conditions on Eq. (57) we need to take account of the
change in the coefficients $a_{f}$ and $b_{f}$ due to Stokes phenomenon (see
e.g. [49, 50]). In Fig. 13 we illustrate the Stokes lines associated with each
of the turning points for various values of $\lambda$. Let us calculate the
change in the coefficients $a_{f}$ and $b_{f}$ as we cross Stokes lines when
moving from $\rho=-KG$ to $\rho=KG$. We suppose that $\mathrm{Re}(\lambda)<0$
so that $\rho_{+}$ is in the first quadrant. Although the topology of the
anti-Stokes lines changes as $\mathrm{Im}(\lambda)$ varies, as we move along
the real axis from minus infinity to infinity we always cross one Stokes line
from each turning point. Across these Stokes lines the dominant WKB
approximation will turn on a multiple of the subdominant WKB approximation.

The first Stokes line we cross, indicated by a “1” in Fig. 13, is that which
moves up from $\rho_{f-}$, on which
$\mathrm{Im}(\phi_{f}(\rho))>\mathrm{Im}(\phi_{f}(\rho_{f-}))$. Thus
$\mathrm{e}^{-\mathrm{i}\phi_{f}/G}$ is the exponentially dominant term, and
the coefficient $a_{f}$ changes by
$-\mathrm{i}b_{f}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G}$. The second
Stokes line we cross, indicated by a “2” in Fig. 13, is that which moves down
from $\rho_{f+}$, on which
$\mathrm{Im}(\phi_{f}(\rho))<\mathrm{Im}(\phi_{f}(\rho_{f+}))$. This on this
Stokes line $\mathrm{e}^{\mathrm{i}\phi_{f}/G}$ is the exponentially dominant
term, and the coefficient $b_{f}$ changes by
$\mathrm{i}a_{f}\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}$. Thus,
together, the change in the coefficient is

$\displaystyle(a_{f}^{-\infty},b_{f}^{-\infty})$ $\displaystyle\rightarrow$
$\displaystyle(a_{f}^{-\infty}-\mathrm{i}b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G},b_{f}^{-\infty})$
$\displaystyle\rightarrow$
$\displaystyle(a_{f}^{-\infty}-\mathrm{i}b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G},b_{f}^{-\infty}+\mathrm{i}(a_{f}^{-\infty}-\mathrm{i}b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G})\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}),$

so that

$\displaystyle a_{f}^{\infty}$ $\displaystyle=$ $\displaystyle
a_{f}^{-\infty}-\mathrm{i}b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G},$
$\displaystyle b_{f}^{\infty}$ $\displaystyle=$ $\displaystyle
b_{f}^{-\infty}+\mathrm{i}a_{f}^{-\infty}\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}+b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G}\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}.$

The boundary condition at $\rho=KG$ gives, at leading order,

$\displaystyle\mathrm{e}^{2\mathrm{i}\phi_{f}(KG)/G}\frac{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}-KG}{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}+KG}$
$\displaystyle=$
$\displaystyle\frac{B_{f0}(KG)}{A_{f0}(KG)}=\frac{b_{f}^{\infty}}{a_{f}^{\infty}}.$

The boundary condition at $\rho=-KG$ gives, at leading order,

$\displaystyle\mathrm{e}^{2\mathrm{i}\phi_{f}(-KG)/G}\frac{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}+KG}{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}-KG}$
$\displaystyle=$
$\displaystyle\frac{B_{f0}(-KG)}{A_{f0}(-KG)}=\frac{b_{f}^{-\infty}}{a_{f}^{-\infty}}.$

Noting that $\phi_{f}$ is odd, if we let

$\displaystyle
Q_{f}=\mathrm{e}^{2\mathrm{i}\phi_{f}(KG)/G}\frac{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}-KG}{\sqrt{(KG)^{2}-4+4\mathrm{i}\lambda}+KG},$
(58)

then we have the following homogeneous system of four equations in the four
unknowns $a_{f}^{\infty}$, $b_{f}^{\infty}$, $a_{f}^{-\infty}$,
$b_{f}^{-\infty}$,

$\displaystyle b_{f}^{\infty}$ $\displaystyle=$ $\displaystyle
Q_{f}a_{f}^{\infty},$ $\displaystyle a_{f}^{-\infty}$ $\displaystyle=$
$\displaystyle Q_{f}b_{f}^{-\infty},$ $\displaystyle a_{f}^{\infty}$
$\displaystyle=$ $\displaystyle
a_{f}^{-\infty}-\mathrm{i}b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G},$
$\displaystyle b_{f}^{\infty}$ $\displaystyle=$ $\displaystyle
b_{f}^{-\infty}+\mathrm{i}a_{f}^{-\infty}\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}+b_{f}^{-\infty}\mathrm{e}^{-2\mathrm{i}\phi_{f}(\rho_{f-})/G}\mathrm{e}^{2\mathrm{i}\phi_{f}(\rho_{f+})/G}.$

Noting that
$\phi_{f}(\rho_{f+})=-\phi_{f}(\rho_{f-})=-(\lambda+\mathrm{i})\pi/2$, the
condition for a non-trivial solution is

$(Q_{f}-\mathrm{i}\mathrm{e}^{(1-\mathrm{i}\lambda)\pi/G})^{2}=1.$ (59)

For finite $K$, Eq. (59) gives a discrete set of eigenvalues with the
separation between neighbouring eigenvalues becoming smaller as
$K\rightarrow\infty$, approximating the continuous spectrum. For large $K$,

$Q_{f}\sim\mathrm{e}^{\mathrm{i}K^{2}G/2}\mathrm{e}^{-\mathrm{i}(1+\mathrm{i}\lambda)/G}(-1+\mathrm{i}\lambda)^{1+(\mathrm{i}+\lambda)/G}(KG)^{-2-2(\mathrm{i}+\lambda)/G}.$

For $\mathrm{Im}(\lambda)<0$ we need $|Q_{f}|\sim 1$ as $G\rightarrow 0$ to
get a balance in Eq. (59), which requires $\mathrm{Re}(\lambda)\sim-G$. For
$\mathrm{Im}(\lambda)>0$ we need
$Q_{f}\sim\mathrm{i}\mathrm{e}^{(1-\mathrm{i}\lambda)\pi/G}$ as $G\rightarrow
0$ to get a balance in Eq. (59), which also requires
$\mathrm{Re}(\lambda)\sim-G$. Thus the continuous spectrum of the problem [cf.
Eq. (56)] lies close to (but not exactly on) the line
$\mathrm{Re}(\lambda)=-G$. The calculation for $g$ is similar and gives the
same equation as Eq. (59) with $\lambda\rightarrow\lambda^{*}$ as expected. In
Fig. 14 we compare the predictions of Eq. (59) (recalling the definition of
Eq. (58)) with the numerical evaluation of the eigenvalues of Eq. (56). A very
good agreement is found between the latter (identified as black dots) and the
former (identified via the intersection of the contours of the blue and red
curves associated with the real and the imaginary parts of Eq. (59).

\begin{overpic}[width=173.44534pt]{numoutG20.eps}
\put(-10.0,50.0){\rotatebox{90.0}{$\mathrm{Im}(\lambda)$}}
\put(50.0,-7.0){$\mathrm{Re}(\lambda)$} \end{overpic} Figure 14: Eigenvalues
for the problem of Eq. (56) for $f$ (black points), for $G=0.2$. The contours
show the asymptotic prediction of Eq. (59). The red curves correspond to
$\mathrm{Im}(Q_{f}-\mathrm{i}\mathrm{e}^{(1-\mathrm{i}\lambda)\pi/G})=0$,
while the blue curves correspond to
$\mathrm{Re}(Q_{f}-\mathrm{i}\mathrm{e}^{(1-\mathrm{i}\lambda)\pi/G})\in\\{-1,1\\}$.
The eigenvalues should lie at the intersections of these contours. The
approximation is very good, apart from near $\lambda=-\mathrm{i}$, at which
point the two turning points $\rho_{f\pm}=\pm 2(1-\mathrm{i}\lambda)^{1/2}$
coalesce.

We now consider how the picture above changes when we include the inner
region. Then, in addition to the Stokes lines already considered, there is a
change in the coefficients $a$ and $b$ as we pass from $\rho=0-$ to $\rho=0+$.
The connection formula comes from matching the solution in the inner region
with the far field expansions on each side. Note that $f$ and $g$ are coupled
in the inner region, so that we no longer have two separate eigenvalue
problems. Specifically, at leading order in the inner region $V_{s}$ is real
and

$\displaystyle\mathrm{i}\lambda
f+\mathchoice{\frac{\mathrm{d}^{2}f}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}f/\mathrm{d}\xi^{2}}+2V_{s}^{4}g+3V_{s}^{4}f-f$
$\displaystyle=$ $\displaystyle 0,$ (60) $\displaystyle-\mathrm{i}\lambda
g+\mathchoice{\frac{\mathrm{d}^{2}g}{\mathrm{d}\xi^{2}}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}{\mathrm{d}^{2}g/\mathrm{d}\xi^{2}}+2V_{s}^{4}f+3V_{s}^{4}g-g$
$\displaystyle=$ $\displaystyle 0,$ (61)

with

$\displaystyle f$ $\displaystyle\sim$ $\displaystyle
a_{f}^{0+}\frac{\mathrm{e}^{\mathrm{i}\phi_{f}(0)/G}}{(-1+\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{(1-\mathrm{i}\lambda)^{1/2}\xi}+b_{f}^{0+}\frac{\mathrm{e}^{\mathrm{i}\phi_{f}(0)/G}}{(-1+\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{-(1-\mathrm{i}\lambda)^{1/2}\xi}\mbox{
as }\xi\rightarrow\infty,$ $\displaystyle g$ $\displaystyle\sim$
$\displaystyle
a_{g}^{0+}\frac{\mathrm{e}^{\mathrm{i}\phi_{g}(0)/G}}{(-1-\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{-(1+\mathrm{i}\lambda)^{1/2}\xi}+b_{g}^{0+}\frac{\mathrm{e}^{\mathrm{i}\phi_{g}(0)/G}}{(-1-\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{(1+\mathrm{i}\lambda)^{1/2}\xi}\mbox{
as }\xi\rightarrow\infty,$ $\displaystyle f$ $\displaystyle\sim$
$\displaystyle
a_{f}^{0-}\frac{\mathrm{e}^{\mathrm{i}\phi_{f}(0)/G}}{(-1+\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{(1-\mathrm{i}\lambda)^{1/2}\xi}+b_{f}^{0-}\frac{\mathrm{e}^{\mathrm{i}\phi_{f}(0)/G}}{(-1+\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{-(1-\mathrm{i}\lambda)^{1/2}\xi}\mbox{
as }\xi\rightarrow-\infty,$ $\displaystyle g$ $\displaystyle\sim$
$\displaystyle
a_{g}^{0-}\frac{\mathrm{e}^{\mathrm{i}\phi_{g}(0)/G}}{(-1-\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{-(1+\mathrm{i}\lambda)^{1/2}\xi}+b_{g}^{0-}\frac{\mathrm{e}^{\mathrm{i}\phi_{g}(0)/G}}{(-1-\mathrm{i}\lambda)^{1/4}}\mathrm{e}^{(1+\mathrm{i}\lambda)^{1/2}\xi}\mbox{
as }\xi\rightarrow-\infty.$

The solution gives connection formulae between the incoming coefficients
$a_{f}^{0-}$, $b_{f}^{0-}$, $a_{g}^{0-}$ and $b_{g}^{0-}$ and the outgoing
coefficients $a_{f}^{0+}$, $b_{f}^{0+}$, $a_{g}^{0+}$ and $b_{g}^{0+}$.
Unfortunately it is not possible to determine these formulae analytically.

However, when $\lambda$ is large the first two terms in Eqs. (60) and (61)
dominate, and the solution is of WKB form even in the inner region. The phase
factor is trivial, with no turning points, so that there is no change in
coefficient. Thus for large $\lambda$ the eigenvalues should be well
approximated by Eq. (59).

On the other hand when $\lambda=0$ we find by solving Eqs. (60)-(61) as a
power series in $G$ that

$a_{f}^{0+}\sim a_{g}^{0-},\quad b_{f}^{0+}\sim-b_{g}^{0-},\quad
b_{g}^{0+}\sim b_{f}^{0-},\quad a_{g}^{0+}\sim-a_{f}^{0-},$

so that there must be some mixing of the coefficients in $f$ and $g$ for small
$\lambda$. Identifying the details of the relevant spectrum at small $\lambda$
remains a challenging question for future study.

##  V Conclusions & Future Challenges

In the present work, we have revisited the topic of stability of solutions
that are self-similarly blowing up. We followed up on the earlier work of [45]
with substantially improved numerical means and techniques, and also added a
systematic theoretical understanding, building also on important works in the
intermediate time interval (such as the key contributions of [43, 44]). This
has allowed us to obtain a systematic understanding of the 3 eigenvalue pairs
of the Hamiltonian system at the critical point of $\sigma d=2$ and its
continuous spectrum. We advocated the relevance of exploring the self-similar
solutions in the co-exploding frame, by analogy with the study of traveling
solutions in a co-traveling frame, as per the standard dynamical systems
perspective [41, 42]. We have also explained systematically why, despite the
presence of positive real eigenvalues, the relevant self-similar solution is
not genuinely unstable but only subject to effectively neutral
eigendirections. To corroborate the relevant results, we performed direct
numerical simulations in the renormalized frame, verifying (in line with
earlier computations) the attractivity of the relevant waveforms.

Naturally, this analysis raises a number of interesting questions for further
research. Understanding the dynamics (and the potential role of self-
similarity) slightly below the critical point $\sigma d=2$ is an example of
this type. Moreover, we have argued that the supercritical solutions
considered herein are effectively stable, upon explaining the origin of their
real eigendirections. Yet, it is well-known that there are other problems for
which multiple, higher-order collapsing solutions branches exist, some among
which are dynamically unstable: a notable example of this sort is, e.g., the
complex Ginzburg-Landau equation [51]. It is then of particular interest to
explore such waveforms via the type of spectral analysis proposed herein, and
corroborate in a systematic fashion their stability or instability, as well as
leverage such spectral information in an attempt to understand the
corresponding direct numerical simulations of the relevant system in the
renormalized frame. In a different vein, there are other important dispersive
PDE models that feature similar bifurcations towards the emergence of
collapsing solutions, such as the generalized KdV problem; for a recent
exposition of the collapsing solutions and asymptotics thereof, see, e.g.,
[40]. It is then of particular interest to adapt the methodology proposed
herein to the latter problem to explore the potential generality of the
eigenvalue phenomenology identified in the present work. Such topics are
presently under active investigation and relevant results will be reported in
future publications.

Acknowledgments. This material is based upon work supported by the US National
Science Foundation under Grants No. DMS-1809074 and PHY-2110030 (P.G.K.). PGK
and EGC are also grateful to Dionyssis Mantzavinos for numerous useful
discussions during the early stages of the present work.

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##  Appendix A An Instructive ODE Example

Consider, arguably, one of the simplest self-similar problems, namely the ODE:

$\displaystyle\dot{x}=x^{p},\quad x(t)\in\mathbb{R}.$ (62)

In order to leverage the self-similar frame to analyze this problem, we seek
to absorb the temporal dependence through a transformation to go to a frame
where the solution appears steady. In this (ODE) case, it is not a steady
spatial profile, as in the PDE example studied throughout this work, but
instead a “number.” We thus use $x(t)=A(\tau)\bar{x}$ with $\tau=\tau(t)$
corresponding to a rescaling of time (to be determined), and obtain the
steady-state problem $\bar{x}=\bar{x}^{p}$ leading to $\bar{x}=0$ or
$\bar{x}=1$. Then, in accordance to general self-similarity principles [52,
53], we select

$\displaystyle\frac{A_{\tau}}{A}=1\Rightarrow
A=A_{0}e^{\tau};\quad\tau_{t}=A^{p-1}\Rightarrow
A(t)=\left[\frac{1}{(p-1)(t^{\star}-t)}\right]^{\frac{1}{p-1}}=A_{0}\left[\frac{1}{(t^{\star}-t)}\right]^{\frac{1}{p-1}}.$
(63)

In Eq. (63), $t^{\star}$ denotes the blow-up time, i.e., $x(t)\to\infty$ as
$t\nearrow t^{\star}$. In the self-similar frame, we have indeed devised a
much more elaborate way to obtain a simple ODE result. The innate advantage of
the method, however, is that in this frame that “explodes” with the solution
it is possible to perform a stability analysis using:

$\displaystyle x(t)=A(\tau)\left[\bar{x}+\epsilon y(\tau)\right]\Rightarrow
y_{\tau}=(p-1)y,\quad\epsilon\ll 1,$ (64)

assuming that we keep the leading order (O$(\epsilon)$) terms in $y$. We thus
observe that the self-similar frame features a single eigenvalue (indeed,
since it is an ODE rather than a PDE) of $\lambda=p-1$. For collapsing
solutions with $p>1$, this is an eigenvalue associated with growth since
$\lambda>0$.

A natural question then is whether this is a true instability. The
perturbation $A(\tau)\epsilon y$ can be rewritten as $\epsilon
e^{p\tau}\sim\epsilon(t^{\star}-t)^{p/(1-p)}$. But then, considering a shift
in the collapse time $t^{\star}\rightarrow t^{\star}+\tilde{\epsilon}\delta t$
($\tilde{\epsilon}\ll 1$), and substituting it in the original solution, we
obtain:

$\displaystyle
x(t)\to\widetilde{x}(t)=\left[\frac{1}{p-1}\right]^{\frac{1}{p-1}}\left[\frac{1}{t^{\star}+\tilde{\epsilon}\delta
t-t}\right]^{\frac{1}{p-1}}=A_{0}\left[\frac{1}{t^{\star}-t}\right]^{\frac{1}{p-1}}-\left(\frac{\tilde{\epsilon}A_{0}\delta
t}{p-1}\right)(t^{\star}-t)^{\frac{p}{1-p}}.$ (65)

Namely, the positive eigenvalue does not correspond to a true instability but
rather is associated with the translational invariance of the ODE with respect
to the shifting of the collapse time.

##  Appendix B Continuous spectra obtained from the rescaled NLS equation

Fig. A1 depicts numerical spectra of the rescaled NLS equation obtained for
different $\sigma$ values. One can observe the alignment of eigenvalues on a
nearly vertical line of continuous spectrum, with real part, $\lambda_{r}=-G$.
As we move away from the critical value, $\sigma=2$, the distortions from the
vertical line become progressively larger.

\begin{overpic}[width=411.93767pt]{spectra_from_2_to_2_05_larger_scale.eps}
\end{overpic} Figure A1: Continuous spectra obtained from the numerical
solution of the rescaled NLS equation with $K=20$ and $\sigma$ values close to
the critical value, $\sigma=2$: $\sigma=2+10^{-9}$ (top left panel) and
$\sigma=2+10^{-6}$ (top right panel), as well as $\sigma=2.001$ (bottom left
panel), $\sigma=2.05$ (bottom right panel).

##  Appendix C Effect of size domain on the numerical computations

Fig. A2 presents a comparison of the computed spectra of the rescaled NLS
equation for size domain $K=20$ and $K=40$. Even by doubling the size of the
computational domain, the alignment of eigenvalues on a “vertical” line of the
continuous spectrum remains practically unchanged, and the real eigenvalues
deviations are also visually indistinguishable.

\begin{overpic}[width=411.93767pt]{comparison_K.eps} \end{overpic} Figure A2:
Comparison of spectra obtained from the numerical solution of the rescaled NLS
equation for size domain $K=20$ (left panel) and $K=40$ (right panel)

##  Appendix D Stability of the reduced system

In [46], asymptotic solutions of Eq. (9) were constructed which result in the
following reduced system for the growth rate $G$ and an associated variable
$\beta$, which can be thought of as the normal form associated with the
bifurcation:

$\displaystyle\mathchoice{\frac{\mathrm{d}G}{\mathrm{d}\tau}}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}$
$\displaystyle=$ $\displaystyle\beta-G^{2},$ (66) $\displaystyle
c_{0}\mathchoice{\frac{\mathrm{d}\beta}{\mathrm{d}\tau}}{\mathrm{d}\beta/\mathrm{d}\tau}{\mathrm{d}\beta/\mathrm{d}\tau}{\mathrm{d}\beta/\mathrm{d}\tau}$
$\displaystyle=$
$\displaystyle\frac{(\sigma-2)}{2\sigma}b_{0}G-A_{1}^{2}\operatorname{sign}(G)\mathrm{e}^{-\pi/|G|},$
(67)

where

$c_{0}=\frac{\sqrt{3}\,\pi^{3}}{512},\qquad
b_{0}=\frac{\sqrt{3}\,\pi}{4},\qquad A_{1}=12^{1/4}.$

###  D.1 Stability of the zero solution

Linearizing about the origin gives

$\displaystyle\mathchoice{\frac{\mathrm{d}G}{\mathrm{d}\tau}}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}$
$\displaystyle=$ $\displaystyle\beta,$
$\displaystyle\mathchoice{\frac{\mathrm{d}\beta}{\mathrm{d}\tau}}{\mathrm{d}\beta/\mathrm{d}\tau}{\mathrm{d}\beta/\mathrm{d}\tau}{\mathrm{d}\beta/\mathrm{d}\tau}$
$\displaystyle=$ $\displaystyle\frac{(\sigma-2)b_{0}}{2\sigma c_{0}}G.$

The characteristic equation is

$\lambda^{2}=\frac{(\sigma-2)b_{0}}{2\sigma c_{0}},$

showing a pair of eigenvalues moving from the imaginary axis to the real axis
as $\sigma$ passes through 2 as expected.

###  D.2 Stability of the non-zero solution

The steady solution ($G_{0},\beta_{0}$) given by

$\beta_{0}=G_{0}^{2},\qquad\frac{(\sigma-2)}{2\sigma}b_{0}G_{0}=A_{1}^{2}\mathrm{e}^{-\pi/G_{0}}.$

with $G_{0}>0$ corresponds to the blow-up solution in the original frame.
Perturbing about this solution by writing $G=G_{0}+x$, $\beta=G_{0}^{2}+y$ and
linearizing gives

$\displaystyle\mathchoice{\frac{\mathrm{d}x}{\mathrm{d}\tau}}{\mathrm{d}x/\mathrm{d}\tau}{\mathrm{d}x/\mathrm{d}\tau}{\mathrm{d}x/\mathrm{d}\tau}$
$\displaystyle=$ $\displaystyle y-2G_{0}x,$ $\displaystyle
c_{0}\mathchoice{\frac{\mathrm{d}y}{\mathrm{d}\tau}}{\mathrm{d}y/\mathrm{d}\tau}{\mathrm{d}y/\mathrm{d}\tau}{\mathrm{d}y/\mathrm{d}\tau}$
$\displaystyle=$
$\displaystyle\frac{(\sigma-2)}{2\sigma}b_{0}x-A_{1}^{2}\mathrm{e}^{-\pi/G_{0}}\frac{\pi}{G_{0}^{2}}x.$

The characteristic equation is

$\lambda(\lambda+2G_{0})-\frac{1}{c_{0}}\left(\frac{(\sigma-2)}{2\sigma}b_{0}-A_{1}^{2}\mathrm{e}^{-\pi/G_{0}}\frac{\pi}{G_{0}^{2}}\right)=0.$

One eigenvalue is exponentially close to zero, while one is exponentially
close to $-2G_{0}$. The exponentially small eigenvalue is approximately

$\displaystyle\lambda$ $\displaystyle\sim$
$\displaystyle\frac{1}{2c_{0}G_{0}}\left(\frac{(\sigma-2)}{2\sigma}b_{0}-A_{1}^{2}\mathrm{e}^{-\pi/G_{0}}\frac{\pi}{G_{0}^{2}}\right)\sim-
A_{1}^{2}\mathrm{e}^{-\pi/G_{0}}\frac{\pi}{2c_{0}G_{0}^{3}}\sim-\frac{\mathrm{e}^{-\pi/G_{0}}}{512G_{0}^{3}\pi^{2}},$

in agreement with the detailed calculation of Section IV.2.5. Note that this
is also the eigenvalue which remains when reducing Eqs. (66)-(67) to the slow
manifold giving

$\displaystyle
2c_{0}G\mathchoice{\frac{\mathrm{d}G}{\mathrm{d}\tau}}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}{\mathrm{d}G/\mathrm{d}\tau}$
$\displaystyle=$
$\displaystyle\frac{(\sigma-2)}{2\sigma}b_{0}G-A_{1}^{2}\operatorname{sign}(G)\mathrm{e}^{-\pi/|G|}.$
(68)

We note that the exponentially small correction to the eigenvalue
$\lambda\sim-2G$ does not agree with the detailed calculation in Section
IV.2.3. This is due to the fact that it corresponds to an exponentially small
correction of the fast timescale, and the system of Eqs. (66)-(67) is accurate
at leading-order only in that regime.