#  High-Frequency Instabilities of the Kawahara Equation: A Perturbative
Approach

Ryan Creedon1, Bernard Deconinck2, Olga Trichtchenko3

1 Dept. of Applied Mathematics, U. of Washington, Seattle, WA, 98105, USA
<EMAIL_ADDRESS>
2 Dept. of Applied Mathematics, U. of Washington, Seattle, WA, 98105, USA
<EMAIL_ADDRESS>
3 Dept. of Physics and Astronomy, U. of Western Ontario, London, ON, N6A 3K7,
CA<EMAIL_ADDRESS>

January 17, 2021

###### Abstract.

We analyze the spectral stability of small-amplitude, periodic, traveling-wave
solutions of the Kawahara equation. These solutions exhibit high-frequency
instabilities when subject to bounded perturbations on the whole real line. We
introduce a formal perturbation method to determine the asymptotic growth
rates of these instabilities, among other properties. Explicit numerical
computations are used to verify our asymptotic results.

###### Key words and phrases:

Kawahara equation, spectral stability, high-frequency instabilities,
perturbation methods, dispersive Hamiltonian systems, Stokes waves

##  1\. Introduction

We investigate small-amplitude, $L$-periodic, traveling-wave solutions of the
Kawahara equation

$\displaystyle u_{t}=\alpha u_{xxx}+\beta u_{5x}+\sigma(u^{2})_{x},$ (1.1)

where $\alpha$, $\beta$, and $\sigma$ are nonzero, real parameters [20].
Similar to Stokes waves of the Euler water wave problem [26, 29], these
solutions are obtained order by order as power series in a small parameter
that scales with the amplitude of the solutions; see [14] and Section 2 below
for more details. We refer to these solutions as the Stokes waves of the
Kawahara equation.

The Kawahara equation is dispersive with linear dispersion relation

$\displaystyle\omega(k)=\alpha k^{3}-\beta k^{5}.$ (1.2)

The equation is Hamiltonian,

$\displaystyle u_{t}=\partial_{x}\frac{\delta\mathcal{H}}{\delta u},$ (1.3)

with

$\displaystyle\mathcal{H}=\int_{0}^{L}\left(-\frac{\alpha}{2}u_{x}^{2}+\frac{\beta}{2}u_{xx}^{2}+\frac{\sigma}{3}u^{3}\right)dx.$
(1.4)

In an appropriate traveling frame, the Stokes waves of (1.1) are critical
points of the Hamiltonian, prompting an investigation of the flow generated by
(1.4) about the Stokes wave solutions.

Perturbing the Stokes waves by functions bounded in space and exponential in
time yields a spectral problem whose spectral elements characterize the
temporal growth rates of the perturbations; see Section 3 for more details. We
refer to this collection of spectral elements as the stability spectrum of the
Stokes waves.

A standard argument [18] shows that the stability spectrum is purely
continuous, but Floquet theory can decompose the spectrum into an uncountably
infinite collection of point spectra. Each point spectra is indexed by a real
number, called the Floquet exponent, that is contained within a compact
interval of the real line [15, 17].

For the Euler water wave problem, these point spectra depend analytically on
the amplitude of the Stokes waves [2, 3]. Based on numerical experiments [28],
similar results appear to hold for the Kawahara equation. The spectrum also
exhibits quadrafold symmetry due to the underlying Hamiltonian nature of (1.1)
[15],[23]. Therefore, for a Stokes wave with given amplitude to be spectrally
stable, all point spectra must be on the imaginary axis. Otherwise, there
exist perturbations to the Stokes waves that grow exponentially in time, and
the Stokes waves are spectrally unstable.

In contrast with the completely integrable KdV equation ($\beta=0$) [8, 24,
25], considerably less is known about the stability spectrum of Stokes waves
to (1.1). Haragus, Lombardi, and Scheel [14] prove that this spectrum lies on
the imaginary axis for small-amplitude Stokes waves in a particular scaling
regime. Such solutions are, therefore, spectrally stable. Work by
Trichtchenko, Deconinck, and Kollár [28] develops necessary criteria for the
stability spectrum of a broader class of small-amplitude Stokes waves to leave
the imaginary axis and provide numerical evidence for the high-frequency
instabilities that result.

High-frequency instabilities arise from pairwise collisions of nonzero,
imaginary elements of the stability spectrum. Upon colliding, these elements
may symmetrically bifurcate from the imaginary axis as the amplitude of the
Stokes wave grows, resulting in instability [11],[23]. An example of a high-
frequency instability for a small-amplitude Stokes wave of (1.1) is seen in
Figure 1. We refer to the locus of spectral elements off the imaginary axis
and bounded away from the origin as high-frequency isolas. The isolas of
Figure 1, as well as the rest of the stability spectrum, are obtained
numerically using the Floquet-Fourier-Hill (FFH) method; see [9] for a
detailed description of this method.

Figure 1. (Left) A stability spectrum of Stokes wave solutions of (1.1) with
$\alpha=1$, $\beta=0.7$, $\sigma=1$, and small-amplitude parameter
$\varepsilon=10^{-3}$, computed using the FFH method. A uniform grid of
$10^{3}$ Floquet exponents between $[-1/2,1/2]$ approximates purely imaginary
point spectra but misses the high-frequency isolas. A uniform grid of $4\times
10^{3}$ Floquet exponents in the interval described by (4.13), obtained in
Section 4, captures these isolas. (Right) Zoom-in of the high-frequency isola
boxed in the left plot (with fewer point spectra shown for ease of
visibility). The red curve is obtained in Section 5 and approximates the
isola.

High-frequency instabilities are not as well-studied as the modulational (or
Benjamin-Feir) instability that arises from collisions of spectral elements at
the origin of the complex spectral plane [5],[7]. Current understanding of
high-frequency instabilities is limited mostly to numerical experiments.
Exceptions include the works of Akers [4] and Trichtchenko, Deconinck, and
Kollár [28], which obtain asymptotic information about the high-frequency
isolas for the Euler problem in infinitely deep water and for the Kawahara
equation, respectively.

The purpose of our present work is to build on these results. In particular,
for sufficiently small-amplitude solutions, we seek the following:

  1. (i)

the asymptotic range of Floquet exponents that parameterize the high-frequency
isolas observed in numerical computations of the stability spectrum,

  2. (ii)

asymptotic estimates of the most unstable spectral elements of the high-
frequency isolas, and

  3. (iii)

expressions for curves asymptotic to these isolas, as seen in Figure 1.

To obtain these quantities, we develop a perturbation method inspired by [4]
that readily extends to higher-order calculations. Asymptotic results obtained
by this method are then compared with numerical results from the FFH method.

##  2\. Small-Amplitude Stokes Waves

We move to a frame traveling with velocity $c$ so that $x\rightarrow x-ct$.
Equation (1.1) becomes

$\displaystyle u_{t}=cu_{x}+\alpha u_{xxx}+\beta u_{5x}+\sigma(u^{2})_{x}.$
(2.1)

We seek $L$-periodic, steady-state solutions of (2.1). Equating time
derivatives to zero and integrating in $x$, we arrive at

$\displaystyle cu+\alpha u_{xx}+\beta u_{4x}+\sigma u^{2}=\mathcal{C},$ (2.2)

where $\mathcal{C}$ is a constant of integration. Using the Galilean symmetry
of (1.1), there exists a boost $\xi$ such that, with $c\rightarrow c+\xi$ and
$u\rightarrow u+\xi$, $\mathcal{C}$ can be omitted from (2.2):

$\displaystyle cu+\alpha u_{xx}+\beta u_{4x}+\sigma u^{2}=0.$ (2.3)

Scaling $x\rightarrow 2\pi x/L$ and $u\rightarrow 2\pi u/(\alpha L)$ allows us
to consider $2\pi$-periodic solutions of

$\displaystyle cu+u_{xx}+\beta u_{4x}+\sigma u^{2}=0,$ (2.4)

without loss of generality, provided $c$, $\beta$, and $\sigma$ are
appropriately redefined.

Let $u=u_{S}(x;\varepsilon)$ be a one-parameter family of $2\pi$-periodic
solutions of (2.4) with corresponding velocity $c=c(\varepsilon)$. The
existence of such a family is rigorously justified by Lyapunov-Schmidt
reduction; see [14]. In what follows, we define the parameter $\varepsilon$ as
twice the first Fourier coefficient of $u_{S}(x;\varepsilon)$:

$\displaystyle\varepsilon:=2\widehat{u_{S}(x;\varepsilon)}_{1}=\frac{1}{\pi}\int_{0}^{2\pi}u_{S}(x;\varepsilon)e^{ix}dx,$
(2.5)

where $~{}\widehat{\cdot}~{}$ is the Fourier transform on the interval
$(0,2\pi)$. Because the $\textrm{L}^{2}(0,2\pi)$ norm of
$u_{S}(x;\varepsilon)$ scales like $\varepsilon$ when $|\varepsilon|\ll 1$, we
call $\varepsilon$ the small-amplitude parameter.

From [14], expansions for $u_{S}(x;\varepsilon)$ and $c(\varepsilon)$ take the
form

$\displaystyle u_{S}(x;\varepsilon)$
$\displaystyle=\sum_{k=1}^{\infty}u_{k}(x)\varepsilon^{k},$ (2.6a)
$\displaystyle c(\varepsilon)$
$\displaystyle=\sum_{k=0}^{\infty}c_{2k}\varepsilon^{2k},$ (2.6b)

where $u_{k}(x)$ is analytic and $2\pi$-periodic for each $k$. Exploiting the
invariance of (2.4) under $x\rightarrow-x$ and $x\rightarrow x+\phi$, we
require $u_{k}(x)=u_{k}(-x)$ so that $u_{S}(x;\varepsilon)$ is even in $x$
without loss of generality. Substituting these expansions into (2.4) and
following a Poincaré-Lindstedt perturbation method [29], one finds corrections
to $u_{S}(x;\varepsilon)$ and $c(\varepsilon)$ order by order.

One difficulty occurs at leading order of the Poincaré-Lindstedt method.
Substituting expansions (2.6) into (2.4) and collecting terms of
$\mathcal{O}(\varepsilon)$, we find

$\displaystyle\left[c_{0}+\partial_{x}^{2}+\beta\partial_{x}^{4}\right]u_{1}(x)=0.$
(2.7)

From (2.5), $\widehat{u_{1}(x)}_{1}=1/2$. Taking the Fourier transform of
(2.7) and evaluating at the first mode, we find

$\displaystyle\left[c_{0}-1+\beta\right]\widehat{u_{1}(x)}_{1}=\frac{1}{2}(c_{0}-1+\beta)=0,$
(2.8)

which implies that

$\displaystyle c_{0}=1-\beta.$ (2.9)

By inspection,

$\displaystyle u_{1}(x)=\cos(x)$ (2.10)

is a solution to (2.7) that is analytic, $2\pi$-periodic, even in $x$, and
satisfies the normalization $\widehat{u_{1}(x)}_{1}=1/2$. If
$\beta=1/(1+N^{2})$ for any integer $N>1$, then

$\displaystyle u_{1}(x)=\cos(x)+C_{N}\cos(Nx),$ (2.11)

where $C_{N}$ is an arbitrary real constant, is an equally valid solution to
(2.7) with the requisite properties. In this case, the Stokes waves are said
to be resonant and exhibit Wilton ripples [30]. Expansions (2.6) must be
modified as a result; see [1, 16], for instance.

In this manuscript, we restrict to nonresonant Stokes waves:

$\displaystyle\beta\neq\frac{1}{1+N^{2}},$ (2.12)

for $N$ stated above, and (2.9) and (2.10) are the unique leading-order
behaviors of $c(\varepsilon)$ and $u_{S}(x;\varepsilon)$, respectively. The
remainder of the Poincaré-Lindstedt method follows as usual. We terminate the
method after third-order corrections, as this is sufficient for our
calculations that follow. We find

$\displaystyle u_{S}(x;\varepsilon)$ $\displaystyle=\varepsilon
u_{1}(x)+\varepsilon^{2}u_{2}(x)+\varepsilon^{3}u_{3}(x)+\mathcal{O}(\varepsilon^{4})$
(2.13a)
$\displaystyle=\varepsilon\cos(x)+\varepsilon^{2}\frac{\sigma}{2}\left(-\frac{1}{c_{0}}+\frac{2}{\Omega(2)}\cos(2x)\right)+\varepsilon^{3}\frac{3\sigma^{2}}{\Omega(2)\Omega(3)}\cos(3x)+\mathcal{O}(\varepsilon^{4}),$
$\displaystyle c(\varepsilon)$
$\displaystyle=c_{0}+c_{2}\varepsilon^{2}+\mathcal{O}(\varepsilon^{4})$
(2.13b)
$\displaystyle=1-\beta+\sigma^{2}\left(\frac{1}{c_{0}}-\frac{1}{\Omega(2)}\right)\varepsilon^{2}+\mathcal{O}(\varepsilon^{4}),$

where $\Omega(\cdot)$ is the linear dispersion relation of the Kawahara
equation (1.1) (with $\alpha=1$) in a frame traveling at velocity
$c(\varepsilon)$:

$\displaystyle\Omega(k)=-c_{0}k+k^{3}-\beta k^{5}.$ (2.14)

##  3\. Necessary Conditions for High-Frequency Instability

###  3.1.  The Stability Spectrum

We consider a perturbation to $u_{S}(x;\varepsilon)$ of the form

$\displaystyle u(x,t)=u_{S}(x;\varepsilon)+\rho v(x,t)+\mathcal{O}(\rho^{2}),$
(3.1)

where $|\rho|\ll 1$ is a small parameter independent of $\varepsilon$ and
$v(x,t)$ is a sufficiently smooth, bounded function of $x$ on the whole real
line for each $t\geq 0$. Substituting (1.1) (with $\alpha=1$) and collecting
terms of $\mathcal{O}(\rho)$, we find by formally separating variables

$\displaystyle v(x,t)=e^{\lambda t}W(x)+c.c.,$ (3.2)

where $c.c.$ denotes complex conjugation of what precedes and $W(x)$ satisfies
the spectral problem

$\displaystyle\lambda
W(x)=\mathcal{L}(u_{S}(x;\varepsilon),c(\varepsilon),\beta,\sigma)W(x),$ (3.3)

with

$\displaystyle\mathcal{L}(u_{S}(x;\varepsilon),c(\varepsilon),\beta,\sigma)=c(\varepsilon)\partial_{x}+\partial_{x}^{3}+\beta\partial_{x}^{5}+2\sigma
u_{S}(x;\varepsilon)\partial_{x}+2\sigma u_{S}^{\prime}(x;\varepsilon),$ (3.4)

where primes denote differentiation with respect to $x$. From Floquet theory
[15], all solutions of (3.3) that are bounded over $\mathbb{R}$ take the form

$\displaystyle W(x)=e^{i\mu x}w(x),$ (3.5)

where $\mu\in[-1/2,1/2]$ is the Floquet exponent and $w(x)$ is $2\pi$-periodic
in an appropriately chosen function space.

Remark. The conjugate of $W(x)$ is a solution of (3.3) with spectral parameter
$\overline{\lambda}$. Since the spectrum of $\mathcal{L}$ is invariant under
conjugation according to [15], one can restrict $\mu$ to the interval
$[0,1/2]$ without loss of generality.

Substituting (3.5) into (3.3), our spectral problem becomes a one-parameter
family of spectral problems:

$\displaystyle\lambda^{\mu}w(x)=\mathcal{L}^{\mu}(u_{S}(x;\varepsilon),c(\varepsilon),\beta,\sigma)w(x),$
(3.6)

where $\mathcal{L}^{\mu}$ is $\mathcal{L}$ with $\partial_{x}\rightarrow
i\mu+\partial_{x}$. In light of (3.6), we require
$w(x)\in\textrm{H}^{5}_{\textrm{per}}(0,2\pi)$ so that $\mathcal{L}^{\mu}$ is
a closed operator densely defined on the separable Hilbert space
$\textrm{L}_{\textrm{per}}^{2}(0,2\pi)$ for a given $\mu$. Then,
$\mathcal{L}^{\mu}$ has a discrete spectrum of eigenvalues $\lambda^{\mu}$ for
each $\mu$ and the union of $\lambda^{\mu}$ over all $\mu\in[0,1/2]$ yields
the purely continuous spectrum of $\mathcal{L}$, which is the stability
spectrum of Stokes waves. See [15] for more details.

As stated in the introduction, if there exists $\lambda^{\mu}$ with
$\textrm{Re}\left(\lambda^{\mu}\right)>0$, then there exists a perturbation to
the Stokes wave that grows exponentially in time, and we say that the Stokes
wave is spectrally unstable. Otherwise, the wave is spectrally stable. Since
(3.6) is obtained from a linearization of a Hamiltonian system (1.1), the
stability spectrum is invariant under conjugation and negation. As a result,
spectral stability implies that all eigenvalues of $\mathcal{L}^{\mu}$ are
purely imaginary.

###  3.2.  The Necessary Conditions for High-Frequency Instability

For fixed $\mu$, the operator $\mathcal{L}^{\mu}$ depends implicitly on the
small parameter $\varepsilon$ through its dependence on $u_{S}(x;\varepsilon)$
and $c(\varepsilon)$. If $\varepsilon=0$, $\mathcal{L}^{\mu}$ reduces to

$\displaystyle\mathcal{L}^{\mu}_{0}=c_{0}(i\mu+\partial_{x})+(i\mu+\partial_{x})^{3}+\beta(i\mu+\partial_{x})^{5},$
(3.7)

a constant-coefficient operator, and its eigenvalues $\lambda^{\mu}_{0}$ are
explicitly given by

$\displaystyle\lambda^{\mu}_{0,n}$ $\displaystyle=-i\Omega(\mu+n),$ (3.8)

where $n\in\mathbb{Z}$. For all $\mu$ and $n$, these eigenvalues are purely
imaginary, implying that the zero-amplitude solution of the Kawahara equation
is spectrally stable.

Importantly, not all eigenvalues given by (3.8) are simple. Using the theory
outlined in [21] and [28], one has

Theorem 1. For each $\Delta n\in\mathbb{N}$, there exists a unique Floquet
exponent $\mu_{0}\in[0,1/2]$ and unique integers $m$ and $n$ such that
$m-n=\Delta n$ and

$\displaystyle\lambda^{\mu_{0}}_{0,n}=\lambda^{\mu_{0}}_{0,m}\neq 0,$ (3.9)

provided that the parameter $\beta$ is nonresonant (2.12) and that $\beta$
satisfies the inequality111A similar statement holds for $\Delta n<0$. This
yields the complex conjugate eigenvalues that satisfy (3.9).:

$\displaystyle\textrm{max}\left(\frac{3}{5(\Delta n)^{2}},\frac{1}{1+(\Delta
n)^{2}}\right)<$ $\displaystyle~{}\beta<\textrm{min}\left(\frac{6}{5(\Delta
n)^{2}},\frac{1}{\left(\frac{\Delta n}{2}\right)^{2}+1}\right),\quad\Delta
n<3,$ (3.10a) $\displaystyle\frac{1}{1+(\Delta n)^{2}}<$
$\displaystyle~{}\beta<\frac{1}{1+\left(\frac{\Delta
n}{2}\right)^{2}},\quad\Delta n\geq 3.$ (3.10b)

The proof is found in the Appendix.

The eigenfunctions of these nonsimple eigenvalues take the form

$\displaystyle w_{0}(x)=\gamma_{0}e^{imx}+\gamma_{1}e^{inx},$ (3.11)

where $\gamma_{0},\gamma_{1}$ are arbitrary, complex constants. We assume the
eigenvalues that satisfy (3.9) are semi-simple with geometric and algebraic
multiplicity two. Then, these eigenvalues represent the collision of two
simple eigenvalues at $\varepsilon=0$, and (3.9) is referred to as the
collision condition.

Collision of eigenvalues away from the origin is a necessary condition for the
development of high-frequency instabilities. Inequality (3.10) guarantees that
there are a finite222For $\Delta n$ sufficiently large, $\beta$ fails to
satisfy inequality (3.10), and no high-frequency instabilities occur. number
of such collisions for a given $\beta$: this is in contrast to the water wave
problem, where a countably infinite number of collisions occur [11]. Each
collision site can be enumerated by $\Delta n$. The largest high-frequency
isola occurs from the $\Delta n=1$ collision, which we study in Section 4.1.

A second condition for high-frequency instabilities necessitates that the
Krein signatures [22] of the two collided eigenvalues have opposite signs
[23]. It is shown in [11, 21, 28] that this condition is equivalent to

$\displaystyle(\mu_{0}+n)(\mu_{0}+m)<0,$ (3.12)

where $\mu_{0}$, $m$, and $n$ are obtained from the collision condition (3.9).
For any $\beta$ that satisfies conditions (2.12) and (3.10) and any $\mu_{0}$,
$m$, and $n$ that satisfies the condition (3.9), (3.12) is automatically
satisfied; see [21] and [28] for the proof.

As $|\varepsilon|$ increases in magnitude, a neighborhood of spectral elements
around the collided eigenvalues of $\mathcal{L}^{\mu_{0}}_{0}$ (3.7) can leave
the imaginary axis, generating high-frequency instabilities. This is seen
explicitly in Figure 1 for the parameter choice $\beta=0.7$, where a $\Delta
n=1$ collision occurs at $\varepsilon=0$.

##  4\. Asymptotics of High-Frequency Instabilities

We obtain spectral data of $\mathcal{L}^{\mu}$ as a power series expansion in
$\varepsilon$ about the collided eigenvalues of $\mathcal{L}^{\mu_{0}}_{0}$.
First, we apply our method to the largest high-frequency instability
corresponding to $\Delta n=1$. Then, we consider $\Delta n\geq 2$.

###  4.1. High-Frequency Instabilities: $\Delta n=1$

Let $m$ and $n$ be the unique integers that satisfy the collision condition
(3.9) with $m-n=1$, and let $\mu_{0}$ be the corresponding unique Floquet
exponent in $[0,1/2]$. Then, the spectral data of $\mathcal{L}^{\mu_{0}}_{0}$
that gives rise to a $\Delta n=1$ high-frequency instability is

$\displaystyle\lambda_{0}:=\lambda^{\mu_{0}}_{0,n}$
$\displaystyle=-i\Omega(\mu_{0}+n)=\lambda^{\mu_{0}}_{0,m}=-i\Omega(\mu_{0}+m)\neq
0,$ (4.1a) $\displaystyle w_{0}(x)$
$\displaystyle:=\gamma_{0}e^{imx}+\gamma_{1}e^{inx}.$ (4.1b)

As $|\varepsilon|$ increases, we assume these data depend analytically on
$\varepsilon$:

$\displaystyle\lambda(\varepsilon)$
$\displaystyle=\lambda_{0}+\varepsilon\lambda_{1}+\mathcal{O}(\varepsilon^{2}),$
(4.2a) $\displaystyle w(x;\varepsilon)$ $\displaystyle=w_{0}(x)+\varepsilon
w_{1}(x)+\mathcal{O}(\varepsilon^{2}),$ (4.2b)

where $\lambda(\varepsilon)$ and $w(x;\varepsilon)$ solve the spectral problem
(3.6).

If $\lambda_{0}$ is a semi-simple and isolated eigenvalue of
$\mathcal{L}^{\mu_{0}}_{0}$, (4.2a) and (4.2b) may be justified using results
of analytic perturbation theory, provided $\mu_{0}$ is fixed [19]. Fixing the
Floquet exponent in this way, however, gives at most two elements on the high-
frequency isola (provided $|\varepsilon|$ is sufficiently small) and these
elements do not, in general, correspond to the spectral elements of largest
real part on the isola. For these reasons, we expand the Floquet exponent
about its resonant value as well:

$\displaystyle\mu=\mu(\varepsilon)=\mu_{0}+\varepsilon\mu_{1}+\mathcal{O}(\varepsilon^{2}).$
(4.3)

As we shall see, $\mu_{1}$ is constrained to an interval of values that
parameterizes an ellipse asymptotic to the high-frequency isola.

Like Akers [4], we impose the following normalization condition on our
eigenfunction $w(x;\varepsilon)$ for uniqueness:

$\displaystyle\widehat{w(x;\varepsilon)}_{n}=1.$ (4.4)

Substituting (4.2b) into this normalization condition, we find
$\widehat{w_{0}(x)}_{n}=1$ and $\widehat{w_{j}(x)}_{n}=0$ for
$j\in\mathbb{N}$, meaning $w_{0}(x)$ fully supports the $n^{\textrm{th}}$
Fourier mode of the eigenfunction $w(x;\varepsilon)$. As a consequence, (4.1)
becomes

$\displaystyle w_{0}(x)=e^{inx}+\gamma_{0}e^{imx}.$ (4.5)

Although $w_{0}(x)$ does not appear unique at this order, we find an
expression for $\gamma_{0}$ at the next order.

The $\mathcal{O}(\varepsilon)$ Problem

Substituting (4.2a), (4.2b), and (4.3) into (3.6) and collecting terms of
$\mathcal{O}(\varepsilon)$ yields

$\displaystyle(\mathcal{L}_{0}^{\mu_{0}}-\lambda_{0})w_{1}(x)=\lambda_{1}w_{0}(x)-\mathcal{L}_{1}w_{0}(x),$
(4.6)

where

$\displaystyle\mathcal{L}_{1}=ic_{0}\mu_{1}+3i\mu_{1}(i\mu_{0}+\partial_{x})^{2}+5i\beta\mu_{1}(i\mu_{0}+\partial_{x})^{4}+2\sigma
u_{1}(x)(i\mu_{0}+\partial_{x})+2\sigma u_{1}^{\prime}(x).$ (4.7)

Using (2.13) to replace $u_{1}(x)$, (4.5) to replace $w_{0}(x)$, and $m-n=1$,
(4.6) becomes

$\displaystyle(\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0})w_{1}(x)=$
$\displaystyle\left[\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+n)-i\sigma\gamma_{0}(\mu_{0}+n)\right]e^{inx}$
(4.8)
$\displaystyle+\left[\gamma_{0}\left(\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\right)-i\sigma(\mu_{0}+m)\right]e^{imx}$
$\displaystyle-i\sigma(\mu_{0}+n-1)e^{i(n-1)x}-i\sigma\gamma_{0}(\mu_{0}+m+1)e^{i(m+1)x},$

where $c_{g}(k)=\Omega^{\prime}(k)$ is the group velocity of $\Omega$.

If (4.8) can be solved for
$w_{1}(x)\in\textrm{H}^{5}_{\textrm{per}}\left(0,2\pi\right)$, the Fredholm
alternative necessitates that the inhomogeneous terms on the RHS of (4.8) must
be orthogonal333In the $\textrm{L}^{2}_{\textrm{per}}\left(0,2\pi\right)$
sense to the nullspace of $(\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0})^{\dagger}$,
the hermitian adjoint of $\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0}$. A quick
computation shows that $\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0}$ is skew-
Hermitian, and so its nullspace coincides with that of its Hermitian adjoint.
The nullspace of $\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0}$ is, by construction,

$\displaystyle\textrm{Null}(\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0})=\textrm{Span}\left(e^{inx},e^{imx}\right).$
(4.9)

Thus, the solvability conditions for (4.8) are

$\displaystyle\left<e^{inx},\left[\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+n)-i\sigma\gamma_{0}(\mu_{0}+n)\right]e^{inx}\right>$
$\displaystyle=0,$ (4.10a)
$\displaystyle\left<e^{imx},\left[\gamma_{0}\left(\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\right)-i\sigma(\mu_{0}+m)\right]e^{imx}\right>$
$\displaystyle=0,$ (4.10b)

where $\left<\cdot,\cdot\right>$ is the standard inner product on
$\textrm{L}_{\textrm{per}}^{2}(0,2\pi)$.

Remark. Both solvability conditions can be reinterpreted as removing secular
terms from (4.8). Moreover, solvability condition (4.10a) coincides with the
normalization condition $\widehat{w_{1}(x)}_{n}=0$.

The solvability conditions (4.10a) and (4.10b) yield a nonlinear system for
$\lambda_{1}$ and $\gamma_{0}$ with solution

$\displaystyle\lambda_{1}=$
$\displaystyle-i\mu_{1}\left(\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{2}\right)$
(4.11a)
$\displaystyle\quad\pm\sqrt{-\mu_{1}^{2}\left[\frac{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}{2}\right]^{2}-\sigma^{2}(\mu_{0}+m)(\mu_{0}+n)},$
$\displaystyle\gamma_{0}=$
$\displaystyle\frac{i\sigma(\mu_{0}+m)}{\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)}.$
(4.11b)

If $\mu_{1}\in(-M_{1},M_{1})$ with

$\displaystyle
M_{1}=\frac{2|\sigma|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)}}{\left|c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)\right|},$
(4.12)

it follows that $\lambda_{1}$ has nonzero real part, since
$(\mu_{0}+m)(\mu_{0}+n)<0$ by the choice of $\beta$. Therefore, to
$\mathcal{O}(\varepsilon)$, the $\Delta n=1$ high-frequency instability is
parameterized by

$\displaystyle\mu\in(\mu_{0}-\varepsilon M_{1},\mu_{0}+\varepsilon M_{1}).$
(4.13)

This interval is asymptotically close to the numerically observed interval of
Floquet exponents that parameterize the high-frequency isola for
$|\varepsilon|\ll 1$; see Figure 2.

Remark. The quantity $M_{1}$ is well-defined since $c_{g}(\mu_{0}+m)\neq
c_{g}(\mu_{0}+n)$. See the Appendix for the proof. The quantity $\gamma_{0}$
is also well-defined, as $\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)$ is guaranteed
to be a complex number with nonzero real part.

Equating $\mu_{1}=0$ maximizes the real part of $\lambda_{1}$ in (4.10a).
Thus, the Floquet exponent that corresponds to the most unstable spectral
element of the high-frequency isola has asymptotic expansion

$\displaystyle\mu_{*}=\mu_{0}+\mathcal{O}(\varepsilon^{2}).$ (4.14)

The corresponding real and imaginary components of this spectral element have
asymptotic expansions

$\displaystyle\lambda_{*,r}$
$\displaystyle=\varepsilon|\sigma|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)}+\mathcal{O}(\varepsilon^{2}),$
(4.15a) $\displaystyle\lambda_{*,i}$
$\displaystyle=-\Omega(\mu_{0}+n)+\mathcal{O}(\varepsilon^{2}),$ (4.15b)

respectively. The former of these expansions provides an estimate for the
growth rate of the $\Delta n=1$ high-frequency instabilities. Figure 2
compares these expansions with numerical results from FFH. Observe that, while
the expansion for the real part is accurate, the expansion for the imaginary
part requires a higher-order calculation; see Section 5.

Figure 2. (Left) Interval of Floquet exponents that parameterize the $\Delta
n=1$ high-frequency isola for parameters $\alpha=1$, $\beta=0.7$, and
$\sigma=1$ as a function of $\varepsilon$. Solid blue curves indicate the
asymptotic boundaries of this interval according to (4.13). Blue circles
indicate the numerical boundaries computed using FFH. The solid red curve
gives the Floquet exponent corresponding to the most unstable spectral element
of the isola according to (4.14). The red circles indicate the same but
computed numerically using FFH. (Right) The real (blue) and imaginary (red)
parts of the most unstable spectral element of the isola as a function of
$\varepsilon$. Solid curves illustrate asymptotic result (4.15). Circles
illustrate results of FFH.

If $\lambda$ is written as a sum of its real and imaginary components,
$\lambda_{r}$ and $\lambda_{i}$, respectively, then eliminating dependence on
$\mu_{1}$ between these quantities yields

$\displaystyle\frac{\lambda_{r}^{2}}{\varepsilon^{2}}+\frac{\left(\lambda_{i}+\Omega(\mu_{0}+n)\right)^{2}}{\varepsilon^{2}\left(\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right)^{2}}=-\sigma^{2}(\mu_{0}+m)(\mu_{0}+n)+\mathcal{O}(\varepsilon).$
(4.16)

Thus, the $\Delta n=1$ high-frequency isola is an ellipse to
$\mathcal{O}(\varepsilon)$ with center at the collision site of eigenvalues
$\lambda^{\mu_{0}}_{0,n}$ and $\lambda^{\mu_{0}}_{0,m}$ and with semi-major
and -minor axes

$\displaystyle a_{1}$
$\displaystyle=\varepsilon|\sigma|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)},$ (4.17a)
$\displaystyle b_{1}$
$\displaystyle=a_{1}\left|\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right|,$
(4.17b)

respectively.

Our asymptotic predictions agree well with numerics, particularly for the real
component of the isola. There is some discrepancy between asymptotic and
numerical results of the Floquet exponents and imaginary component of the
isola, even when $\varepsilon=10^{-3}$; see Figure 3. As noted before, this
discrepancy is resolved in Section 5.

Figure 3. (Left) $\Delta n=1$ high-frequency isola for $\alpha=1$,
$\beta=0.7$, $\sigma=1$, and $\varepsilon=10^{-3}$. The solid red curve is
ellipse (4.16). Blue circles are a subset of spectral elements from the
numerically computed isola using FFH. (Right) Floquet parameterization of the
real (blue) and imaginary (red) parts of the isola. Solid curves illustrate
asymptotic result (4.11a). Circles indicate results of FFH.

###  4.2. High-Frequency Instabilities: $\Delta n=2$

Suppose $m$, $n$, and $\mu_{0}$ satisfy the collision condition (3.9) for
$\Delta n=2$ and appropriately chosen $\beta$ parameter. Then, (4.1) gives a
semi-simple eigenpair of $\mathcal{L}^{\mu_{0}}_{0}$, and we assume (4.2a),
(4.2b), and (4.3) remain valid expansions for the eigenvalue, eigenfunction,
and Floquet exponents in the vicinity of this semi-simple eigenpair,
respectively. We obtain the coefficients of these expansions order by order in
much the same way as for the $\Delta n=1$ high-frequency instabilities.

The $\mathcal{O}(\varepsilon)$ Problem

Substituting expansions (4.2a), (4.2b), and (4.3) into the spectral problem
(3.6) and collecting $\mathcal{O}(\varepsilon)$ terms gives

$\displaystyle(\mathcal{L}^{\mu_{0}}_{0}-\lambda_{0})w_{1}(x)=$
$\displaystyle-i\sigma(\mu_{0}+n-1)e^{i(n-1)x}+\left[\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+n)\right]e^{inx}$
(4.18)
$\displaystyle-i\sigma(\mu_{0}+n+1)(1+\gamma_{0})e^{i(n+1)x}+\gamma_{0}\left[\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\right]e^{imx}$
$\displaystyle-i\sigma(\mu_{0}+m+1)e^{i(m+1)x},$

where we have used (2.13) to replace $u_{1}(x)$. Though equation (4.18) shares
similar features with (4.8), $m-n\neq 1$ in this case. Thus, (4.18) cannot be
simplified further.

The solvability conditions of (4.18) are

$\displaystyle\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+n)$ $\displaystyle=0,$ (4.19a)
$\displaystyle\gamma_{0}\left[\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\right]$
$\displaystyle=0.$ (4.19b)

Since $c_{g}(\mu_{0}+m)\neq c_{g}(\mu_{0}+n)$ by the corollary provided in the
Appendix and $\gamma_{0}\neq 0$444Otherwise, our unperturbed eigenfunction
$w_{0}(x)$ is not a superposition of two distinct Fourier modes., we must have

$\displaystyle\lambda_{1}=\mu_{1}=0.$ (4.20)

Solving (4.18) for $w_{1}(x)$ by the the method of undetermined coefficients,
one finds

$\displaystyle
w_{1}(x)=\tau_{1,n-1}e^{i(n-1)x}+\tau_{1,n+1}e^{i(n+1)x}+\tau_{1,m+1}e^{i(m+1)x}+\gamma_{1}e^{imx},$
(4.21)

where $\gamma_{1}$ is a constant to be determined at higher order,

$\displaystyle\tau_{1,n-1}$ $\displaystyle=Q_{n,n-1},$ (4.22a)
$\displaystyle\tau_{1,n+1}$ $\displaystyle=(1+\gamma_{0})Q_{n,n+1},$ (4.22b)
$\displaystyle\tau_{1,m+1}$ $\displaystyle=\gamma_{0}Q_{n,m+1},$ (4.22c)

and

$\displaystyle
Q_{N,M}=\frac{\sigma(\mu_{0}+M)}{\Omega(\mu_{0}+M)-\Omega(\mu_{0}+N)}.$ (4.23)

Note that $w_{1}(x)$ does not have an $n^{\textrm{th}}$ Fourier mode, which is
a consequence of the normalization (4.4).

The $\mathcal{O}(\varepsilon^{2})$ Problem

Substituting (4.2a), (4.2b), and (4.3) into (3.6) and collecting terms of
$\mathcal{O}(\varepsilon^{2})$ yields

$\displaystyle(\mathcal{L}_{0}^{\mu_{0}}-\lambda_{0})w_{1}(x)=\lambda_{2}w_{0}(x)-\mathcal{L}_{2}|_{\mu_{1}=0}w_{0}(x)-\mathcal{L}_{1}|_{\mu_{1}=0}w_{1}(x),$
(4.24)

where $\mathcal{L}_{1}|_{\mu_{1}=0}$ is as before (but evaluated at
$\mu_{1}=0$) and

$\displaystyle\mathcal{L}_{2}|_{\mu_{1}=0}=ic_{0}\mu_{2}+c_{2}(i\mu_{0}+\partial_{x})+3\mu_{2}i(i\mu_{0}+\partial_{x})^{2}+5\mu_{2}i(i\mu_{0}+\partial_{x})^{4}+2\sigma
u_{2}(x)(i\mu_{0}+\partial_{x})+2\sigma u_{2}^{\prime}(x).$ (4.25)

As in the previous order, we evaluate the RHS of (4.24) using (2.13) to
replace $u_{1}(x)$ and $u_{2}(x)$, (4.5) to replace $w_{0}(x)$, (4.21) to
replace $w_{1}(x)$, and $m-n=2$ to combine terms with exponential arguments
proportional to $m-1$ and $n+1$. After some work, one arrives at the
solvability conditions555To obtain (4.26a) and (4.26b), one also needs
evenness of $u_{2}(x)$ so that
$\widehat{u_{2}(x)}_{-2}=\widehat{u_{2}(x)}_{2}$.:

$\displaystyle\lambda_{2}+i\mathcal{C}_{\mu_{2},\mu_{0}}^{n}$
$\displaystyle=i\gamma_{0}\mathcal{S}_{2}(\mu_{0}+n),$ (4.26a)
$\displaystyle\gamma_{0}\left[\lambda_{2}+i\mathcal{C}_{\mu_{2},\mu_{0}}^{m}\right]$
$\displaystyle=i\mathcal{S}_{2}(\mu_{0}+m),$ (4.26b)

where

$\displaystyle\mathcal{C}_{\mu_{2},\mu_{0}}^{N}$
$\displaystyle=\mu_{2}c_{g}(\mu_{0}+N)-\mathcal{P}^{N}_{\mu_{0}},$ (4.27a)
$\displaystyle\mathcal{P}^{N}_{\mu_{0}}$
$\displaystyle=(\mu_{0}+N)\left[\sigma(Q_{n,N-1}+Q_{n,N+1}+2\widehat{u_{2}(x)}_{0})+c_{2}\right],$
(4.27b) $\displaystyle\mathcal{S}_{2}$
$\displaystyle=\sigma(Q_{n,n+1}+2\widehat{u_{2}(x)}_{2}).$ (4.27c)

Similar to the $\Delta n=1$ case, (4.26a) and (4.26b) are a nonlinear system
for $\lambda_{2}$ and $\gamma_{0}$. The solution of this system is

$\displaystyle\lambda_{2}=$
$\displaystyle~{}-i\left(\frac{\mathcal{C}_{\mu_{2},\mu_{0}}^{m}+\mathcal{C}_{\mu_{2},\mu_{0}}^{n}}{2}\right)\pm\sqrt{-\left[\frac{\mathcal{C}_{\mu_{2},\mu_{0}}^{m}-\mathcal{C}_{\mu_{2},\mu_{0}}^{n}}{2}\right]^{2}-\mathcal{S}_{2}^{2}(\mu_{0}+m)(\mu_{0}+n)},$
(4.28a) $\displaystyle\gamma_{0}=$
$\displaystyle~{}\frac{i\sigma(\mu_{0}+m)(Q_{n,n+1}+2\upsilon_{2,-2})}{\lambda_{2}+i\mathcal{C}_{\mu_{2},\mu_{0}}^{m}}.$
(4.28b)

Provided $\mathcal{S}_{2}\neq 0$, there exists an interval of
$\mu_{2}\in(M_{2,-},M_{2,+})$, where

$\displaystyle
M_{2,\pm}=\frac{\mathcal{P}^{m}_{\mu_{0}}-\mathcal{P}^{n}_{\mu_{0}}}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\pm
2\left|\frac{\mathcal{S}_{2}}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)},$
(4.29)

such that $\lambda_{2}$ has a nonzero real part. It is shown in the Appendix
that $\mathcal{S}_{2}\neq 0$ for all relevant values of $\beta$. Thus, the
interval of Floquet exponents that parameterizes the $\Delta n=2$ high-
frequency isola to $\mathcal{O}(\varepsilon^{2})$ is

$\displaystyle\mu\in\left(\mu_{0}+\varepsilon^{2}M_{2,-},\mu_{0}+\varepsilon^{2}M_{2,+}\right).$
(4.30)

Unlike when $\Delta n=1$ (4.13), the center of this interval changes at the
same rate as its width, and this width is an order of magnitude smaller than
for the $\Delta n=1$ instabilities. This explains why numerical detection of
$\Delta n=2$ instabilities presents a greater challenge than for $\Delta n=1$
instabilities; see Figure 4.

Figure 4. (Left) Interval of Floquet exponents that parameterize the $\Delta
n=2$ high-frequency isola for parameters $\alpha=1$, $\beta=0.25$, and
$\sigma=1$ as a function of $\varepsilon$. ($\beta=0.7$ only gives rise to a
$\Delta n=1$ isola: $\beta$ must be changed to satisfy (3.10) for a $\Delta
n=2$ isola to arise.) Solid blue curves indicate the asymptotic boundaries of
this interval according to (4.30). Blue circles indicate the numerical
boundaries computed using FFH. The solid red curve gives the Floquet exponent
corresponding to the most unstable spectral element of the isola according to
(4.31). The red circles indicate the same but computed numerically using FFH.
(Right) The real (blue) and imaginary (red) parts of the most unstable
spectral element of the isola as a function of $\varepsilon$. Solid curves
illustrate asymptotic result (4.32). Circles illustrate results of FFH.

From the results above, we obtain an asymptotic expansion for the Floquet
exponent of the most unstable spectral element of the $\Delta n=2$ high-
frequency isola:

$\displaystyle\mu_{*}=\mu_{0}+\frac{\mathcal{P}^{m}_{\mu_{0}}-\mathcal{P}^{n}_{\mu_{0}}}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\varepsilon^{2}+\mathcal{O}(\varepsilon^{3}).$
(4.31)

Asymptotic expansions for the real and imaginary component of this spectral
element are

$\displaystyle\lambda_{*,r}$
$\displaystyle=\varepsilon^{2}|\mathcal{S}_{2}|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)}+\mathcal{O}(\varepsilon^{3}),$
(4.32a) $\displaystyle\lambda_{*,i}$
$\displaystyle=-\Omega(\mu_{0}+n)-\left[\frac{\mathcal{P}_{\mu_{0}}^{m}c_{g}(\mu_{0}+n)-\mathcal{P}^{n}_{\mu_{0}}c_{g}(\mu_{0}+m)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right]\varepsilon^{2}+\mathcal{O}(\varepsilon^{3}).$
(4.32b)

These expansions are in excellent agreement with numerical computations from
FFH, as is seen in Figure 4. This is a consequence of resolving quadratic
corrections to the real and imaginary components of $\Delta n=2$ high-
frequency isolas simultaneously, unlike in the $\Delta n=1$ case.

Analogous to the derivation of (4.16), the ellipse given by

$\displaystyle\frac{\lambda_{r}^{2}}{\varepsilon^{4}}+$
$\displaystyle\frac{\left[\lambda_{i}+\Omega(\mu_{0}+n)+\varepsilon^{2}\left(\frac{\mathcal{P}_{\mu_{0}}^{m}c_{g}(\mu_{0}+n)-\mathcal{P}^{n}_{\mu_{0}}c_{g}(\mu_{0}+m)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right)\right]^{2}}{\varepsilon^{4}\left(\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right)^{2}}=-\mathcal{S}_{2}^{2}(\mu_{0}+m)(\mu_{0}+n)+O(\varepsilon).$
(4.33)

is asymptotic to the $\Delta n=2$ high-frequency isola. This ellipse has
center that drifts from the collision site at a rate comparable to its semi-
major and -minor axes,

$\displaystyle a_{2}$
$\displaystyle=\varepsilon^{2}|\mathcal{S}_{2}|\sqrt{-(\mu_{0}+m)(\mu_{0}+n)}$
(4.34a) $\displaystyle b_{2}$
$\displaystyle=a_{2}\left|\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right|,$
(4.34b)

respectively. This behavior contrasts with that seen in the $\Delta n=1$ case,
where the center drifts slower than the axes grow. Comparison with numerical
computations using FFH show that (4.33) is an excellent approximation for
$\Delta n=2$ high-frequency isolas; see Figure 5.

Figure 5. (Left) $\Delta n=2$ high-frequency isola for $\alpha=1$,
$\beta=0.25$, $\sigma=1$, and $\varepsilon=10^{-3}$. The solid red curve is
ellipse (4.33). Blue circles are a subset of spectral elements from the
numerically computed isola using FFH. Blue circles are a subset of spectral
elements from the numerically computed isola using FFH. (Right) Floquet
parameterization of the real (blue) and imaginary (red) parts of the isola.
Solid curves illustrate asymptotic result (4.28a). Circles indicate results of
FFH.

###  4.3. High-Frequency Instabilities: $\Delta n\geq 3$

The approach used to obtain leading-order behavior of the $\Delta n=1,2$ high-
frequency isolas generalizes to higher-order isolas. The method consists of
the following steps, each of which is readily implemented in a symbolic
programming language:

  1. (i)

Given $\Delta n\in\mathbb{N}$, determine the unique $\mu_{0}$, $m$, and $n$ to
satisfy collision condition (3.9), assuming $\beta$ satisfies (3.10).

  2. (ii)

Expand about the collided eigenvalues in a formal power series of
$\varepsilon$ and similarly expand their corresponding eigenfunctions and
Floquet exponents. To maintain uniqueness of the eigenfunctions, choose the
normalization (4.4).

  3. (iii)

Substitute these expansions into the spectral problem (3.6). Collecting like
powers of $\varepsilon$, construct a hierarchy of inhomogeneous linear
problems to solve.

  4. (iv)

Proceed order by order. At each order, impose solvability and normalization
conditions. Invert the linear operator against its range using the method of
undetermined coefficients. Use previous normalization and solvability
conditions as well as the collision condition to simplify problems if
necessary.

We conjecture that this method yields the first nonzero real part correction
to the $\Delta n^{\textrm{th}}$ high-frequency isola at
$\mathcal{O}(\varepsilon^{\Delta n})$. We have shown that this conjecture
holds for $\Delta n=1,2$. For $\Delta n=3$, one can show that the high-
frequency isola is asymptotic to the ellipse

$\displaystyle\frac{\lambda_{r}^{2}}{\varepsilon^{6}}+\frac{\left(\lambda_{i}+\Omega(\mu_{0}+n)+\varepsilon^{2}\left[\frac{\mathcal{P}^{m}_{\mu_{0}}c_{g}(\mu_{0}+n)-\mathcal{P}^{n}_{\mu_{0}}c_{g}(\mu_{0}+m)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right]\right)^{2}}{\varepsilon^{6}\left(\frac{c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n)}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}\right)^{2}}=-\mathcal{S}_{3}^{2}(\mu_{0}+m)(\mu_{0}+n)+O(\varepsilon),$
(4.35)

where

$\displaystyle\mathcal{S}_{3}=\sigma\left[Q_{n,n+1}Q_{n,n+2}+2\widehat{u_{2}(x)}_{2}(Q_{n,n+1}+Q_{n,n+2})+2\widehat{u_{3}(x)}_{3}\right].$
(4.36)

The semi-major and -minor axes of (4.35) scale as
$\mathcal{O}(\varepsilon^{3})$, as the conjecture predicts. If true for all
$\Delta n$, this conjecture explains why higher-order isolas are challenging
to detect both in numerical and perturbation computations of the stability
spectrum.

One notices that the center of (4.35) drifts similarly to that of the $\Delta
n=2$ high-frequency isolas (4.33). In fact, centers of higher-order isolas
(beyond $\Delta n=1$) all drift at a similar rate, as these isolas all satisfy
the same $\mathcal{O}(\varepsilon^{2})$ problem and, hence, yield corrections
at this order. Consequently, one can expect to incur corrections to the
imaginary component of the high-frequency isola before reaching
$\mathcal{O}(\varepsilon^{\Delta n})$, making it more difficult to prove our
conjecture about the first occurence of a nonzero real part correction.

##  5\. $\Delta n=1$ High-Frequency Instabilities at Higher-Order

As we saw in Section 4.1, the asymptotic formulas for the $\Delta n=1$ high-
frequency isola fail to capture its $\mathcal{O}(\varepsilon^{2})$ drift along
the imaginary axis. This is expected, as we only considered the
$\mathcal{O}(\varepsilon)$ problem. In this section, we go beyond the leading-
order behavoir of these instabilities. We expect similar calculations to arise
if one considered the $\mathcal{O}(\varepsilon^{p})$ problem for a generic
$\Delta n$ isola, where $p>\Delta n$.

###  5.1. The $\mathcal{O}(\varepsilon)$ Problem Revisited

Finishing our work from Section 4.1, we solve for $w_{1}(x)$ in (4.8). We find

$\displaystyle
w_{1}(x)=Q_{n,n-1}e^{i(n-1)x}+\gamma_{0}Q_{n,m+1}e^{i(m+1)x}+\gamma_{1}e^{imx},$
(5.1)

where $Q_{N,M}$ is defined as in (4.23) and $\gamma_{1}$ is an undetermined
constant at this order.

###  5.2. The $\mathcal{O}(\varepsilon^{2})$ Problem

At $\mathcal{O}(\varepsilon^{2})$, we have

$\displaystyle(L_{0}-\lambda_{0})w_{2}(x)=$
$\displaystyle~{}\lambda_{2}w_{0}(x)+\lambda_{1}w_{1}(x)-ic_{0}(\mu_{1}w_{1}(x)+\mu_{2}w_{0}(x))-c_{2}(i\mu_{0}+\partial_{x})w_{0}(x)$
(5.2)
$\displaystyle-3i(i\mu_{0}+\partial_{x})^{2}(\mu_{1}w_{1}(x)+\mu_{2}w_{0}(x))+3\mu_{1}^{2}(i\mu_{0}+\partial_{x})w_{0}(x)$
$\displaystyle-5\beta
i(i\mu_{0}+\partial_{x})^{4}(\mu_{1}w_{1}(x)+\mu_{2}w_{0}(x))+10\beta\mu_{1}^{2}(i\mu_{0}+\partial_{x})^{3}w_{0}(x)$
$\displaystyle-2\sigma(i\mu_{0}+\partial_{x})(u_{1}(x)w_{1}(x)+u_{2}(x)w_{0}(x))-2\sigma
i\mu_{1}u_{1}(x)w_{0}(x).$

After substituting $w_{0}(x)$, $w_{1}(x)$, $u_{1}(x)$, and $u_{2}(x)$ into
(5.2), solvability conditions of the second-order problem form the linear
system

$\displaystyle\displaystyle\begin{pmatrix}1&-i\sigma(\mu_{0}+n)\\\
\gamma_{0}&\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\end{pmatrix}\begin{pmatrix}\lambda_{2}\\\
\gamma_{1}\end{pmatrix}=i\begin{pmatrix}\sigma\gamma_{0}\mu_{1}-\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{n,-1}\\\
\sigma\mu_{1}-\gamma_{0}\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{m,1}\end{pmatrix},$
(5.3)

where

$\displaystyle\tilde{C}_{\mu_{2},\mu_{1},\mu_{0}}^{N,k}$
$\displaystyle=\mu_{2}c_{g}(\mu_{0}+N)-\tilde{\mathcal{P}}_{\mu_{0}}^{N,k}+\mu_{1}^{2}\mathcal{D}^{N}_{\mu_{0}},$
(5.4a) $\displaystyle\tilde{P}_{\mu_{0}}^{N,k}$
$\displaystyle=(\mu_{0}+N)\left[\sigma(Q_{n,N+k}+2\upsilon_{2,0})+c_{2}\right],$
(5.4b) $\displaystyle D_{\mu_{0}}^{N}$
$\displaystyle=3(\mu_{0}+N)-10\beta(\mu_{0}+N)^{3}.$ (5.4c)

For $\mu_{1}\in(-M_{1},M_{1})$ (4.12), one can show that

$\displaystyle\textrm{det}\begin{pmatrix}1&-i\sigma(\mu_{0}+n)\\\
\gamma_{0}&\lambda_{1}+i\mu_{1}c_{g}(\mu_{0}+m)\end{pmatrix}$
$\displaystyle=2\lambda_{1,r}.$ (5.5)

Since $\lambda_{1,r}\neq 0$ for this interval of $\mu_{1}$, it follows that
(5.3) is invertible. Using Cramer’s rule and (4.10a) and (4.10b), the
solvability conditions at $\mathcal{O}(\varepsilon)$, gives

$\displaystyle\lambda_{2}=-\frac{i}{2\lambda_{1,r}}\left(\mathcal{A}\lambda_{1}+i\mu_{1}\mathcal{B}\right),$
(5.6)

where

$\displaystyle\mathcal{A}$
$\displaystyle=\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{m,1}+\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{n,-1},$
(5.7a) $\displaystyle\mathcal{B}$
$\displaystyle=c_{g}(\mu_{0}+m)\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{n,-1}+c_{g}(\mu_{0}+n)\tilde{\mathcal{C}}_{\mu_{2},\mu_{1},\mu_{0}}^{m,1}-\sigma^{2}(2\mu_{0}+m+n).$
(5.7b)

###  5.3. Determination of $\mu_{2}$: The Regular Curve Condition

A quick calculation shows that $\lambda_{2}$ has two branches, $\lambda_{2,+}$
and $\lambda_{2,-}$, and, for any $\mu_{2}\in\mathbb{R}$,
$\lambda_{2,+}=-\overline{\lambda_{2,-}}$. Consequently, (5.6) results in a
spectrum that is symmetric about the imaginary axis regardless of $\mu_{2}$.
We want this spectrum to be a continuous, closed curve about the imaginary
axis. As we shall see, this additional constraint is enough to determine
$\mu_{2}$ uniquely. We call this additional constraint the regular curve
condition.

To motivate the regular curve condition, consider the real and imaginary parts
of (5.6):

$\displaystyle\lambda_{2,r}$
$\displaystyle=\frac{1}{2\lambda_{1,r}}\left(\mathcal{A}\lambda_{1,i}+\mu_{1}\mathcal{B}\right),$
(5.8a) $\displaystyle\lambda_{2,i}$ $\displaystyle=-\frac{\mathcal{A}}{2}.$
(5.8b)

As $|\mu_{1}|$ approaches $M_{1}$, $\lambda_{1,r}$ approaches zero. To avoid
unwanted blow-up of $\lambda_{2,r}$, we must impose

$\displaystyle\displaystyle\lim_{|\mu_{1}|\rightarrow
M_{1}}\left(\mathcal{A}\lambda_{1,i}+\mu_{1}\mathcal{B}\right)=0.$ (5.9)

Since $\mu_{1}$ appears in $\mathcal{A}$ only as $\mu_{1}^{2}$, we can rewrite
(5.9) with the help of (4.11a) as

$\displaystyle\lim_{\mu_{1}^{2}\rightarrow
M_{1}^{2}}\left(-\frac{\mathcal{A}}{2}(c_{g}(\mu_{0}+m)+c_{g}(\mu_{0}+n))+\mathcal{B}\right)=0.$
(5.10)

Equation (5.10) is the regular curve condition for second-order corrections to
the $\Delta n=1$ isola. Unpacking the definitions of $\mathcal{A}$ and
$\mathcal{B}$ above, the regular curve condition implies that

$\displaystyle\mu_{2}=\frac{\textrm{P}^{m,1}_{\mu_{0}}-\textrm{P}^{n,-1}_{\mu_{0}}}{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}-\frac{2\sigma^{2}(2\mu_{0}+m+n)}{(c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n))^{2}},$
(5.11)

where

$\displaystyle\textrm{P}^{N,k}_{\mu_{0}}=\tilde{\mathcal{P}}^{N,k}_{\mu_{0}}+\frac{4\sigma^{2}(\mu_{0}+m)(\mu_{0}+n)}{(c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n))^{2}}\mathcal{D}^{N}_{\mu_{0}}.$
(5.12)

Therefore, to $\mathcal{O}(\varepsilon^{2})$, the asymptotic interval of
Floquet exponents that parameterizes the $\Delta n=1$ high-frequency isola is

$\displaystyle\mu\in\left(\mu_{0}-\varepsilon
M_{1}+\varepsilon^{2}\mu_{2},\mu_{0}+\varepsilon
M_{1}+\varepsilon^{2}\mu_{2}\right).$ (5.13)

###  5.4. The Most Unstable Eigenvalue

To $\mathcal{O}(\varepsilon)$, the expression for the real part of the $\Delta
n=1$ high-frequency isola is

$\displaystyle\lambda^{(1)}_{r}:=\varepsilon\lambda_{1,r}=\pm\varepsilon\sqrt{-\mu_{1}^{2}\left[\frac{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}{2}\right]^{2}-\sigma^{2}(\mu_{0}+m)(\mu_{0}+n)}.$
(5.14)

The most unstable eigenvalue of (5.14) occurs when $\mu_{1}=\mu_{*,1}$, where
$\mu_{*,1}$ is a critical point of $\lambda^{(1)}_{r}$:

$\displaystyle\frac{\partial\lambda^{(1)}_{r}}{\partial\mu_{1}}\biggr{|}_{\mu_{*,1}}=0.$
(5.15)

Solving (5.15), one finds $\mu_{*,1}=0$, and we conclude that the Floquet
exponent that corresponds to the most unstable eigenvalue is
$\mu_{*}=\mu_{0}+O(\varepsilon^{2})$, as found in Section 4.1.

To $\mathcal{O}(\varepsilon^{2})$, the real part of our isola is

$\lambda^{(2)}_{r}:=\varepsilon\lambda_{1,r}+\varepsilon^{2}\lambda_{2,r},$
(5.16)

where $\lambda_{1,r}$ is given in (5.14) and $\lambda_{2,r}$ is given in
(5.8a). Without loss of generality, we choose the positive branch of
$\lambda_{1,r}$.

Taking inspiration from (5.15), we consider the critical points of (5.16):

$\frac{\partial\lambda_{r}^{(2)}}{\partial\mu_{1}}\biggr{|}_{\mu_{*,1}}=0.$
(5.17)

After some tedious calculations, (5.17) yields the following equation for
$\mu_{*,1}$:

$\displaystyle-\mu_{*,1}\lambda_{*,1,r}^{2}\left(\frac{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}{2}\right)^{2}+\frac{\varepsilon}{2}\biggr{[}\lambda_{*,1,r}^{2}\biggr{(}\lambda_{*,1,i}\frac{\partial\mathcal{A}}{\partial\mu_{1}}\biggr{|}_{\mu_{*,1}}+\mathcal{A}_{*}\frac{\partial\lambda_{1,i}}{\partial\mu_{1}}\biggr{|}_{\mu_{*,1}}$
(5.18)
$\displaystyle\phantom{[}\phantom{(}+\mu_{*,1}\frac{\partial\mathcal{B}}{\partial\mu_{1}}\biggr{|}_{\mu_{*,1}}+\mathcal{B}_{*}\biggr{)}+\mu_{*,1}\left(\mathcal{A}_{*}\lambda_{*,1,i}+\mu_{*,1}\mathcal{B}_{*}\right)\left(\frac{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}{2}\right)^{2}\biggr{]}=0,$

where it is understood that starred variables are evaluated at $\mu_{*,1}$.
Unpacking the definitions of $\mathcal{A}$, $\mathcal{B}$, $\lambda_{1,r}$,
and $\lambda_{1,i}$ reveals that (5.18) is a quartic equation for $\mu_{*,1}$
with the highest degree coefficient multiplied by the small parameter
$\varepsilon$. Rather than solving for $\mu_{*,1}$ directly, we obtain the
roots perturbatively.

An application of the method of dominant balance to (5.18) shows that all of
its roots have leading order behavior $\mathcal{O}(\varepsilon^{-1})$, except
for one. Because we anticipate that $\lim_{\varepsilon\rightarrow
0}\mu_{*,1}=0$ to match results at the previous order, it is this non-singular
root that we expect to yield the next order correction for $\mu_{*,1}$.
Therefore, we need not concern ourselves with singular perturbation methods
and, instead, make the following ansatz:

$\displaystyle\mu_{*,1}=0+\varepsilon\mu_{*,1,1}+O(\varepsilon^{2}).$ (5.19)

Plugging our ansatz into (5.18) and keeping terms of lowest power in
$\varepsilon$, we find the following linear equation to solve for
$\mu_{*,1,1}$:

$\displaystyle-\mu_{*,1,1}\left(\frac{c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)}{2}\right)^{2}+\frac{1}{2}\left(\mathcal{B}_{0}-\mathcal{A}_{0}\right)=0,$
(5.20)

where $\mathcal{A}_{0}$ and $\mathcal{B}_{0}$ are $\mathcal{A}$ and
$\mathcal{B}$ evaluated at $\mu_{1}=0$, respectively. Using the definition of
$\mathcal{A}$ and $\mathcal{B}$ together with the expression for $\mu_{2}$ in
(5.11) above, one finds that

$\displaystyle\mu_{*,1,1}=-4\sigma^{2}(\mu_{0}+m)(\mu_{0}+n)\left(\frac{\mathcal{D}^{m}_{\mu_{0}}-\mathcal{D}^{n}_{\mu_{0}}}{(c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n))^{3}}\right).$
(5.21)

It follows that the Floquet exponent corresponding to the most unstable
eigenvalue of the $\Delta n=1$ high-frequency isola is

$\displaystyle\mu_{*}=\mu_{0}+\varepsilon^{2}(\mu_{2}+\mu_{*,1,1})+O(\varepsilon^{3}).$
(5.22)

The most unstable eigenvalue is then

$\displaystyle\lambda_{*}=\lambda_{0}+\varepsilon\lambda_{1}|_{\mu_{1}=0}+\varepsilon^{2}\lambda_{2}|_{\mu_{1}=0,\mu_{2}=\mu_{2}+\mu_{*,1,1}}+O(\varepsilon^{3}).$
(5.23)

Figure 6 and Figure 7 show improvements to results in Figure 2 and Figure 3,
respectively, as a result of our higher-order calculations.

Figure 6. (Left) Interval of Floquet exponents that parameterize the $\Delta
n=1$ high-frequency isola for parameters $\alpha=1$, $\beta=0.7$, and
$\sigma=1$ as a function of $\varepsilon$. Solid blue curves indicate the
asymptotic boundaries of this interval according to (5.13), while the dotted
blue curves give the $\mathcal{O}(\varepsilon)$ result. Blue circles indicate
the numerical boundaries computed using FFH. The solid red curve gives the
Floquet exponent corresponding to the most unstable spectral element of the
isola according to (5.22), while the dotted red gives the
$\mathcal{O}(\varepsilon)$ result. The red circles indicate the same but
computed numerically using FFH. (Right) The real (blue) and imaginary (red)
parts of the most unstable spectral element of the isola as a function of
$\varepsilon$. Solid curves illustrate asymptotic result (5.23). Dotted curves
illustrate the asymptotic results only to $\mathcal{O}(\varepsilon)$. Circles
illustrate results of FFH. Figure 7. (Left) $\Delta n=1$ high-frequency isola
for $\alpha=1$, $\beta=0.7$, $\sigma=1$, and $\varepsilon=10^{-3}$. The solid
red curve is parameterized by (5.6). This curve is no longer an ellipse, but a
more complicated algebraic curve. For comparison, the dotted red curve is the
ellipse found at $\mathcal{O}(\varepsilon)$. Blue circles are a subset of
spectral elements from the numerically computed isola using FFH. (Right)
Floquet parameterization of the real (blue) and imaginary (red) parts of the
isola. Solid curves illustrate asymptotic result (5.6). Dotted curves
illustrate the asymptotic results only to $\mathcal{O}(\varepsilon)$. Circles
indicate results of FFH.

##  6\. Conclusions

In this work, we investigate the asymptotic behavior of high-frequency
instabilities of small-amplitude Stokes waves of the Kawahara equation. For
the largest of these instabilities ($\Delta n=1,2$), we introduce a
perturbation method to compute explicitly

  1. (i)

the asymptotic interval of Floquet exponents that parameterize the high-
frequency isola,

  2. (ii)

the leading-order behavior of its most unstable spectral elements, and

  3. (iii)

the leading-order curve asymptotic to the isola.

We outline the procedure to compute these quantities for higher-order isolas.
For the first time, we compare these asymptotic results with numerical results
and find excellent agreement between the two. We also obtain higher-order
asymptotic results for the $\Delta n=1$ high-frequency isolas by introducing
the regular curve condition.

The perturbation method used throughout our investigation holds only for
nonresonant Stokes waves (2.12). Resonant waves require a modified Stokes
expansion, and as a result of this modification, the leading-order behavior of
the high-frequency isolas will change. Some numerical work has been done to
investigate this effect [28], but no perturbation methods have been proposed.

##  7\. Appendix

Theorem 1. For each $\Delta n\in\mathbb{N}$, if $\beta$ satisfies (2.12) and
(3.10), there exists a unique $\mu_{0}\in[0,1/2]$ and unique $m\neq
n\in\mathbb{Z}$ such that the collision condition (3.9) is satisfied.

Proof. Define

$\displaystyle F(k;\Delta n)=\frac{\Omega(k+\Delta n)-\Omega(k)}{\Delta n}.$
(7.1)

Using the definition of the dispersion relation $\Omega$,

$\displaystyle F(k;\Delta n)=$ $\displaystyle~{}5\beta k^{4}+10\beta\Delta
nk^{3}+(10\beta(\Delta n)^{2}-3)k^{2}+(5\beta(\Delta n)^{3}-3\Delta n)k$ (7.2)
$\displaystyle+\beta(\Delta n)^{4}-(\Delta n)^{2}+1-\beta.$

A direct calculation shows that

$\displaystyle F(k;\Delta n)=F(-(k+\Delta n);\Delta n).$ (7.3)

Hence, the graph of $F$ is symmetric about $k=-\Delta n/2$. We prove the
desired result for the various cases of $\Delta n$.

Case 1. Suppose $\Delta n=1$. Then, $k_{1}=0$ and $k_{2}=-1$ are roots of $F$
by inspection. The remaining roots are

$\displaystyle k_{3,4}=\frac{-1\pm\sqrt{\frac{12}{5\beta}-3}}{2}.$ (7.4)

Because $\beta$ satisfies (3.10), one can show that

$\displaystyle 0<\frac{12}{5\beta}-3<1,$ (7.5)

so that $k_{3,4}\in(k_{2},k_{1})$. Because $F$ is symmetric about $k=-\Delta
n/2$, we have $k_{3}\in\left(-1/2,0\right)$ and
$k_{4}\in\left(-1,-1/2\right)$.

Each of these wavenumbers $k_{j}$ is mapped to a Floquet exponent
$\mu\in(-1/2,1/2]$ according to

$\displaystyle\mu(k)=k-[k],$ (7.6)

where $[\cdot]$ denotes the nearest integer function666Our convention is
$\left[p/2\right]=(p-1)/2$ for $p$ odd.. Both $k_{1}$ and $k_{2}$ map to
$\mu=0$. One checks that $\mu(k_{3})=-\mu(k_{4})\neq 0$ and
$|\mu(k_{3})|=|\mu(k_{4})|<1/2$, since $k_{4}=-(k_{3}+1)$ and $-1/2<k_{3}<0$.
Thus, the requisite $\mu_{0}\in(0,1/2)$ is $\mu(k_{j})$, where $j$ is either
$3$ or $4$ depending on which has the correct sign. Then, $n=[k_{j}]$ and
$m=n+\Delta n$. These are unique by the uniqueness of $k_{j}$.

Case 2. Suppose $\Delta n=2$. A calculation of $F(-1;2)$ and $F_{k}(-1;2)$
shows that $k_{1,2}=-1$ is a double root. The remaining roots are

$\displaystyle k_{3,4}=-1\pm\sqrt{\frac{3}{5\beta}-2}.$ (7.7)

Clearly, $\mu(k_{1,2})=0$. Since $k_{4}=-(k_{3}+2)$, we again have
$\mu(k_{3})=-\mu(k_{4})$. Also, from the formula for $k_{3}$ above, we have
that $-1<k_{3}<0$ by (3.10), so $\mu(k_{3})$ is nonzero. Thus, $\mu(k_{j})$ is
the requisite $\mu_{0}\in(0,1/2]$, where $j$ is either $3$ or $4$ depending on
which has the correct sign. Again, $n=[k_{j}]$ and $m=n+\Delta n$ are uniquely
defined.

Remark. Unlike in the first case, we cannot guarantee $\mu_{0}\neq 1/2$.
Indeed, this value can be achieved when $\beta=4/15$.

Case 3. Suppose $\Delta n\geq 3$. The discriminant of $F(k;\Delta n)$ with
respect to $k$ is

$\displaystyle\Delta_{k}[F]=5\beta\left[(\Delta
n)^{2}-4\right]\left[\beta((\Delta
n)^{2}+4)-4\right]\left[5\beta\left(\beta\left((\Delta
n)^{4}+4\right)-2\left((\Delta n)^{2}+2\right)\right)+9\right]^{2}.$ (7.8)

For $\beta$ satisfying inequality (3.10), we have $\Delta_{k}[F]<0$, implying
there are two distinct real roots of $F$. These roots must be nonpositive by
an application of Descartes’ Rule of Signs on $F$. Without loss of generality,
suppose $k_{2}<k_{1}$. Then, by the symmetry of $F$ about $k=-\Delta n/2$,
$k_{2}=-(k_{1}+\Delta n)$. It follows that $\mu(k_{1})=-\mu(k_{2})$. Thus,
$\mu(k_{j})$ is the requisite value of $\mu_{0}\in[0,\frac{1}{2}]$, where
$j=1$ or $2$ depending on which has the correct sign. The integers $n$ and $m$
are uniquely defined as before.

Remark. In [21], $\beta=1/((\Delta n/2)^{2}+1)$ is included in inequality
(3.10) when $\Delta n\geq 3$. For this $\beta$, $F(k;\Delta n)$ has a double
root at $k_{*}=-\Delta n/2$. However, $\Omega(k_{*})=0$, which corresponds to
an eigenvalue collision at the origin. Eigenvalue collisions at the origin do
not satisfy (3.9).

In each of these cases, we have found $k_{j}<0$ such that $F(k_{j};\Delta
n)=0$. Importantly, one must check that $\Omega(k_{j})\neq 0$ for such
$k_{j}$. Indeed, suppose $\Omega(k_{j})=0$. A direct calculation shows that
$k_{j}=\pm 1$, $0$, or $k_{j}^{2}=(1-\beta)/\beta$. Clearly $k_{j}=0$ or $1$
contradict that $k_{j}<0$. If $k_{j}=-1$, then $F(-1;\Delta n)=0$ implies
$\beta=1/[(\Delta n-1)^{2}+1]$, which contradicts (2.12) when $\Delta n\neq
2$. If $\Delta n=2$, $\beta=1/2$, which contradicts (3.10).

It remains to be seen if $k_{j}^{2}=(1-\beta)/\beta$ leads to contradiction.
Indeed, a straightforward (although tedious) calculation shows that, if
$k_{j}^{2}=(1-\beta)/\beta$, $F(k_{j};\Delta n)=0$ implies $\beta=0$,
$\beta=1/[1+(\Delta n/2)^{2}]$, $\beta=1/[1+(\Delta n-1)^{2}]$, or
$\beta=1/[1+(\Delta n+1)^{2}]$. All of these lead to contradictions of (2.12)
or (3.10). Therefore, we must have
$\Omega(k_{j})=\Omega(\mu_{0}+n)=\Omega(\mu_{0}+m)\neq 0$ in all cases, as
desired.

$\square$

In expressions for the isolas derived in Sections 4 and 5, factors of
$c_{g}(\mu_{0}+m)-c_{g}(\mu_{0}+n)$ appear in denominators. A consequence of
Theorem 1 is that this factor is never zero:

Corollary 1. Fix $\Delta n\in\mathbb{N}$ and choose $\beta$ to satisfy (2.12)
and (3.10). Consider $\mu_{0}\in[0,1/2]$ that solves
$\Omega(\mu_{0}+m)=\Omega(\mu_{0}+n)$ for unique integers $m,n$ such that
$m=n+\Delta n$. Suppose, in addition, that $\mu_{0}$ solves
$c_{g}(\mu_{0}+m)=c_{g}(\mu_{0}+n)$, where $c_{g}(k)=\Omega^{\prime}(k)$.
Then, $\Delta n=2$ and $\mu_{0}=0$.

Proof. If $\Omega(\mu_{0}+m)=\Omega(\mu_{0}+n)$ and
$c_{g}(\mu_{0}+m)=c_{g}(\mu_{0}+n)$, then $k_{0}=\mu_{0}+m$ is a double root
of $F(k;\Delta n)$. From the proof of the theorem above, the only double root
is $k_{0}=-1$ (i.e. $\mu_{0}=0$) when $\Delta n=2$.

$\square$

The corresponding eigenvalue collision for this $\Delta n$ and $\mu_{0}$
happens at the origin in the complex spectral plane and is not of interest to
us. Thus, $c_{g}(\mu_{0}+m)\neq c_{g}(\mu_{0}+n)$.

In Section 4.2, the quantity $\mathcal{S}_{2}$ (4.27c) must be nonzero in
order for $\Delta n=2$ high-frequency instabilities to exist at
$\mathcal{O}\left(\varepsilon^{2}\right)$. The following corollary shows
$\mathcal{S}_{2}\neq 0$ for $\beta$ satisfying inequality (3.10).

Corollary 2. For $\mathcal{S}_{2}$ defined in (4.27c) and $\beta$ satisfying
inequality (3.10) for $\Delta n=2$, $\mathcal{S}_{2}\neq 0$.

Proof. Since $\Delta n=2$, we have from (3.10) that $1/5<\beta<3/10$. In
addition, from Theorem 1 and Corollary 1, we know $k_{1,2}=1$ is a double root
of $F(k;\Delta n)$ for all $\beta$ in this interval, and the remaining roots
of $F$ are

$\displaystyle k_{3,4}=-1\pm\sqrt{\frac{3}{5\beta}-2}.$ (7.9)

These remaining roots correspond to the nonzero eigenvalue collisions that
give rise to the $\Delta n=2$ high-frequency instability.

The quantity $\mathcal{S}_{2}$ can be written in terms of $k_{3,4}$ as

$\displaystyle\mathcal{S}_{2}=\sigma^{2}\left[\frac{k_{3,4}+1}{\Omega(k_{3,4}+1)-\Omega(k_{3,4})}+\frac{1}{\Omega(2)}\right],$
(7.10)

Because $k_{3,4}$ are symmetric about $k=1$ (from the symmetry of $F$), the
value of $\mathcal{S}_{2}$ is independent of the choice of $k_{3,4}$. Using
the definition of the dispersion relation $\Omega$ (2.14), (7.10) simplifies
to

$\displaystyle\mathcal{S}_{2}=\frac{\sigma^{2}}{2(1-5\beta)},$ (7.11)

which is nonzero for $1/5<\beta<3/10$.

$\square$

## References

  * [1] B. Akers and W. Gao. Wilton ripples in weakly nonlinear model equations. _Commun. Math. Sci._ 10(3): 1015-1024, 2012.
  * [2] B. Akers and D. P. Nicholls. Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. _SIAM J. Appl. Math._ , 130(2): 81-107, 2012.
  * [3] B. Akers and D. P. Nicholls. The spectrum of finite depth water waves. _European Journal of Mechanics-B/Fluids_ , 46: 181-189, 2014.
  * [4] B. Akers. Modulational instabilities of periodic traveling waves in deep water. _Physica D_ , 300: 26-33, 2015
  * [5] T. B. Benjamin. Instability of periodic wave trains in nonlinear dispersive systems. _Proceedings, Royal Society of London, A,_ 299:59-79, 1967.
  * [6] T. B. Benjamin and J. E. Feir. The disintegration of wave trains on deep water. part i. theory. _Journal of Fluid Mechanics_ , 27:417-430, 1967.
  * [7] T. H. Bridges and A. Mielke. A proof of the Benjamin-Feir instability. _Archive for Rational Mechanics and Analysis_ , 133:145-198, 1995.
  * [8] B. Deconinck and T. Kapitula. The orbital stability of the cnoidal waves of the Korteweg-deVries equation. _Phys. Letters A_ , 374: 4018-4022, 2010.
  * [9] B. Deconinck and J. N. Kutz. Computing spectra of linear operators using the Floque-Fourier-Hill method. _Journal of Computational Physics_ , 219(1): 296-321, 2006.
  * [10] B. Deconinck and K. Oliveras. The instability of periodic surface gravity waves. _Journal of Fluid Mechanics_ , 675:141-167, 2011.
  * [11] B. Deconinck and O. Trichtchenko. High-frequency instabilities of small-amplitude of Hamiltonian PDE’s. _Discrete & Continuous Dynamical Systems-A_, 37(3):1323-1358, 2017.
  * [12] F. Dias and C. Kharif. Nonlinear gravity and gravity-capillary waves. _Annu. Rev. Fluid Mech._ , 31: 331-341, 1999.
  * [13] J. Hammack and D. Henderson. Resonant interations among surface water waves. _Annu. Rev. Fluid Mech._ , 25:55-97, 1993.
  * [14] M. Haragus, E. Lombardi, and A. Scheel. Spectral stability of wave trains in the Kawahara equation. _Journal of Mathematical Fluid Mechanics_ , 8(4):482-509, 2006.
  * [15] M. Haragus and T. Kapitula. On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, _Phys. D_ , 237: 2649-2671, 2008.
  * [16] S. E. Haupt and J. P. Boyd. Modeling nonlinear resonance: a modification to the Stokes’ perturbation expansion. _Wave Motion_ , 10(1):83-98, 1988.
  * [17] M. A. Johnson, K. Zumbrun, and J. C. Bronski].] On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions, _Phys. D_ , 239: 2067-2065, 2010.
  * [18] T. Kapitula and K. Promislow. _Spectral and Dynamical Stability of Nonlinear Waves._ Springer, New York, 2013.
  * [19] T. Kato. _Perturbation Theory for Linear Operators._ Springer-Verlag, Berlin, 1966.
  * [20] T. Kawahara. Oscillatory solitary waves in dispersive media. _J. Phys. Soc. Jpn._ , 33: 1015-1024, 1972.
  * [21] R. Kollár, B. Deconinck, and O. Trichtchenko. Direct characterization of spectral stability of small-amplitude periodic waves in scalar Hamiltonian problems via dispersion relation. _SIAM Journal on Mathematical Analysis_ , 51(4): 3145-3169, 2019.
  * [22] M. G. Krein. On the application of an algebraic proposition in the theory of matrices of monodromy. _Uspehi Matem. Nauk (N.S.)_ , 6(1(41)):171-177, 1951.
  * [23] R. S. MacKay and P. G. Saffman. Stability of water waves. _Proceedings of the Royal Society A_ , 406(1830):115-125, 1986.
  * [24] M. Nivala and B. Deconinck. Periodic finite-genus solutions of the KdV equation are orbitally stable. _Physica D_ , 239(13): 1147-1158, 2010.
  * [25] J. Pava and F. Natali. (Non)linear instability of periodic traveling waves: Klein-Gordon and KdV type equations. _Adv. Nonlinear Anal._ , 3(2): 95-123, 2014.
  * [26] G. G. Stokes. On the theory of oscillatory waves. _Trans. Camb. Phil. Soc._ , 8:441-455, 1847.
  * [27] O. Trichtchenko, B. Deconinck, and J. Wilkening. The instability of Wilton ripples. _Wave Motion_ , 66: 147-155, 2016.
  * [28] O. Trichtchenko, B. Deconinck, and R. Kollár. Stability of periodic traveling wave solutions to the Kawahara equation. _SIAM Journal on Applied Dynamical Systems_ , 17(4): 2761-2783, 2018.
  * [29] G. B. Whitham. Non-linear dispersion of water waves. _Journal of Fluid Mechanics_ , 27:399-412, 1967.
  * [30] J. R. Wilton. On ripples. _Phil. Mag. Ser._ 629(173): 688-700, 1915.


have been much more well-understood compared to pure YM theory. 212121The
infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills
theory has been considered in Gonzalez:2020tpi . So it would be interesting to
understand the fate of loop corrections in celestial amplitude. This is
important for further understanding the full-fledged quantum gravity via
celestial holography. Even at the tree level, in the discussion of soft and
collinear limits, we were mainly discussing the leading term. It would be nice
to work out the other sub-leading corrections.

Finally, a useful operation for celestial operator is shadow transformation
Pasterski:2017kqt . Therefore, it would be interesting to see how to perform
the shadow transformation for a super-operator. The shadow transformation
flips the spin of the operator and in particular, should be compatible with
supersymmetry. However, the supersymmetries considered here are 4D
supersymmetry in the bulk instead of the ordinary 2D supersymmetry on the
celestial sphere. So it seems to be not sensible to combine
$(z_{1},\bar{z}_{1},\eta_{1}^{A})$ and $(z_{2},\bar{z}_{2},\eta_{2}^{A})$ to
form a distance that could be used to construct the supershadow. Nevertheless,
it is worth studying whether one can assemble the shadow of each component
operator together in a supersymmetric way.

###### Acknowledgements.

We would like to thank Rodolfo Russo and Congkao Wen for discussions and
comments on the manuscript. We also thank the authors of AndiCelestial  for
sharing their draft before the publication, and Congkao Wen for informing us
the two parallel independent works on similar topics. This work was supported
by the Royal Society grant, “Relations, Transformations, and Emergence in
Quantum Field Theory”, and by the Science and Technology Facilities Council
(STFC) Consolidated Grant ST/T000686/1 “Amplitudes, strings & duality”.

##  Appendix A Soft and collinear limit in YM and SYM

###  A.1 Soft theorem in YM and SYM

The soft theorem for (color-ordered) YM gluon amplitude is

$\mathcal{A}_{n}(\cdots,a,s^{\pm},b,\cdots)\xrightarrow{p_{s}\to
0}{\mathsf{Soft}}^{\text{YM}}(a,s^{\pm},b)\mathcal{A}_{n}(\cdots,a,b,\cdots)~{},$
(225)

where the soft factor for positive helicity soft gluon at tree level is given
by He:2014bga

${\mathsf{Soft}}^{\text{YM}}(a,s^{+},b)=\frac{\langle ab\rangle}{\langle
as\rangle\langle sb\rangle}+\Bigg{(}\frac{1}{\langle
sb\rangle}\tilde{\lambda}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}_{b}^{\dot{\alpha}}}+\frac{1}{\langle
as\rangle}\tilde{\lambda}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}_{a}^{\dot{\alpha}}}\Bigg{)}~{}.$
(226)

The first term is the leading soft factor of order $\mathcal{O}(1/p_{s})$,
while the second term in parentheses is the subleading soft factor of order
$\mathcal{O}(1/p_{s}^{0})$. For negative helicity soft gluon, the soft factor
is given by the conjugate $\lambda_{i}\leftrightarrow\tilde{\lambda}_{i}$.

The discussion can be generalized to the superamplitude in $\mathcal{N}=4$ SYM
theory:

$\mathscr{A}_{n}(\cdots,a,s,b,\cdots)\xrightarrow{p_{s}\to
0}{\mathsf{Soft}}^{\text{SYM}}(a,s,b)\mathscr{A}_{n-1}(\cdots,a,b,\cdots)~{},$
(227)

where each particle is labeled by
$(\lambda_{i},\tilde{\lambda}_{i},\eta_{i})$. The physical soft limit
$p_{s}\to 0$ corresponds to setting
$(\lambda_{s},\tilde{\lambda}_{s},\eta_{s})\to(\epsilon\Lambda_{s},\epsilon\tilde{\Lambda}_{s},\eta_{s})$
and then taking $\epsilon\to 0$. However, this is a little complicated. In the
literature, one instead considers the holomorphic soft limit
$(\lambda_{s},\tilde{\lambda}_{s},\eta_{s})\to(\epsilon\Lambda_{s},\tilde{\lambda}_{s},\eta_{s})$
and anti-holomorphic soft limit
$(\lambda_{s},\tilde{\lambda}_{s},\eta_{s})\to(\lambda_{s},\epsilon\tilde{\Lambda}_{s},\eta_{s})$.

In the holomorphic soft limit
$(\lambda_{s},\tilde{\lambda}_{s},\eta_{s})\to(\epsilon\Lambda_{s},\tilde{\lambda}_{s},\eta_{s})$,
namely $\lambda_{s}\to 0$, the supersoft factor is given by He:2014bga

${\mathsf{Soft}}^{\text{SYM}}_{\text{hol}}(a,s,b)=\frac{\langle
ab\rangle}{\langle as\rangle\langle sb\rangle}+\Bigg{[}\frac{1}{\langle
sb\rangle}\Big{(}\tilde{\lambda}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}_{b}^{\dot{\alpha}}}+\eta_{s}^{A}\frac{\partial}{\partial\eta_{b}^{A}}\Big{)}+\frac{1}{\langle
as\rangle}\Big{(}\tilde{\lambda}_{s}^{\dot{\alpha}}\frac{\partial}{\partial\tilde{\lambda}_{a}^{\dot{\alpha}}}+\eta_{s}^{A}\frac{\partial}{\partial\eta_{a}^{A}}\Big{)}\Bigg{]}~{},$
(228)

while in the anti-holomorphic soft limit
$(\lambda_{s},\tilde{\lambda}_{s},\eta_{s})\to(\lambda_{s},\epsilon\tilde{\Lambda}_{s},\eta_{s})$,
namely $\tilde{\lambda}_{s}\to 0$, the supersoft factor is given by He:2014bga

${\mathsf{Soft}}^{\text{SYM}}_{\text{anti-hol}}(a,s,b)=\frac{[ab]}{\langle
as\rangle[sb]}\delta^{4}\Big{(}\eta_{s}+\frac{[as]}{[ab]}\eta_{b}+\frac{[sb]}{[ab]}\eta_{a}\Big{)}\Bigg{[}1+\Big{(}\frac{[sb]}{[ab]}\eta_{s}^{A}\frac{\partial}{\partial\eta_{b}^{A}}+\frac{[as]}{[ab]}\eta_{s}^{A}\frac{\partial}{\partial\eta_{a}^{A}}\Big{)}\Bigg{]}~{}.$
(229)

In these two limits above, we give both the leading and sub-leading soft
factors.

In the physical soft limit $p_{s}\to 0$ or equivalently
$\lambda_{s},\tilde{\lambda}_{s}\to 0$, the leading soft factor is given by
the sum of leading soft factors in holomorphic and anti-holomorphic soft
limit:

${\mathsf{Soft}}_{\text{leading}}^{\text{SYM}}(a,s,b)=\frac{\langle
ab\rangle}{\langle as\rangle\langle
sb\rangle}+\frac{[ab]}{[as][sb]}\delta^{4}(\eta_{s})~{}.$ (230)

The two terms obviously correspond to the soft gluons of positive and negative
helicity. Other components are absent here because their corresponding soft
factors are sub-dominant (for example, the leading soft factor for gluinos
scales as $1/\sqrt{p_{s}}$ Fotopoulos:2020bqj ).

###  A.2 Collinear limit in YM

When two particles point to the same direction, the amplitude also becomes
singular, knowns as the collinear singularity. More specifically, consider two
particles with momenta $p_{1}$ and $p_{2}$. The two particles can fuse into
one particle with momentum $P_{12}\equiv p_{1}+p_{2}$ and one can parametrize
the collinear momenta as $p_{1}={\sf z}P_{12},p_{2}=(1-{\sf z})P_{12}$. In the
collinear limit, the gluon amplitude in YM theory satisfies

$\mathcal{A}_{n}(1^{h_{1}},2^{h_{2}},\cdots)\xrightarrow{p_{1}/\\!/p_{2}}\sum_{h}\mathcal{A}_{n-1}(P_{12}^{h}\;,\cdots){\mathsf{Split}}_{-h}(1^{h_{1}},2^{h_{2}})~{},$
(231)

where the split factors ${\mathsf{Split}}_{-h}(1^{h_{1}},2^{h_{2}})$ are given
by Henn:2014yza ; Ferro:2020lgp

$\displaystyle{\mathsf{Split}}_{+}({\sf z};1^{+},2^{-})$ $\displaystyle=$
$\displaystyle\frac{(1-{\sf z})^{2}}{\sqrt{{\sf z}(1-{\sf
z})}}\frac{1}{\langle 12\rangle}~{},$ (232)
$\displaystyle{\mathsf{Split}}_{+}({\sf z};1^{-},2^{+})$ $\displaystyle=$
$\displaystyle\frac{{\sf z}^{2}}{\sqrt{{\sf z}(1-{\sf z})}}\frac{1}{\langle
12\rangle}~{},$ (233) $\displaystyle{\mathsf{Split}}_{+}({\sf z};1^{-},2^{-})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{{\sf z}(1-{\sf
z})}}\frac{1}{[12]}~{},$ (234) $\displaystyle{\mathsf{Split}}_{+}({\sf
z};1^{+},2^{+})$ $\displaystyle=$ $\displaystyle 0~{},$ (235)

and

${\mathsf{Split}}_{-}({\sf z};1^{-h_{1}},2^{-h_{2}})={\mathsf{Split}}_{+}({\sf
z};1^{h_{1}},2^{h_{2}})|_{\langle 12\rangle\leftrightarrow[12]}~{}.$ (236)

One can actually derive the soft theorem by taking the consecutive collinear
limits of the same type Ferro:2020lgp . The soft factor derived in this way is
in agreement with the leading term in (226) as well as its conjugate.

###  A.3 Collinear limit in SYM

We can also consider the collinear limit of SYM theory using the language of
superamplitude. Here we follow the discussion in Ferro:2020lgp . The SYM
amplitude can be written as

$\mathscr{A}_{n}=A_{n,2}+A_{n,4}+\cdots+A_{n,n-2}~{},$ (237)

which corresponds to MHV, NMHV,…, $\overline{\text{MHV}}$. In the collinear
limit, the SYM amplitude satisfies

$A_{n,k}(1,2,3,\cdots,n)\xrightarrow{P_{12}^{2}\to 0}\sum_{l=1}^{2}\int
d^{4}\eta_{P_{12}}\;{\mathsf{Split}}_{1-l}(1,2,P_{12})A_{n-1,k-l+1}(P_{12},3,\cdots,n)~{},$
(238)

where $l=1,2$ corresponds to helicity preserving and helicity decreasing
process. Summing over $k$ on both sides gives

$\mathscr{A}_{n}(1,2,3,\cdots,n)\xrightarrow{P_{12}^{2}\to
0}\sum_{l=1}^{2}\int
d^{4}\eta_{P_{12}}\;{\mathsf{Split}}_{1-l}(1,2,P_{12})\mathscr{A}_{n-1}(P_{12},3,\cdots,n)~{}.$
(239)

For helicity preserving case, the super-split factor is given by Ferro:2020lgp

${\mathsf{Split}}_{0}({\sf z};\eta_{1},\eta_{2},\eta_{3})=\frac{1}{\sqrt{{\sf
z}(1-{\sf z})}}\frac{1}{\langle
12\rangle}\prod_{A=1}^{4}\Big{(}\eta_{3}^{A}-\sqrt{\sf
z}\eta_{1}^{A}-\sqrt{1-{\sf z}}\eta_{2}^{A}\Big{)}~{}.$ (240)

While for helicity decreasing case, the super-split factor is given by
Ferro:2020lgp

${\mathsf{Split}}_{-1}({\sf z};\eta_{1},\eta_{2},\eta_{3})=\frac{1}{\sqrt{{\sf
z}(1-{\sf
z})}}\frac{1}{[12]}\prod_{A=1}^{4}\Big{(}\eta_{1}^{A}\eta_{2}^{A}-\sqrt{1-{\sf
z}}\eta_{1}^{A}\eta_{3}^{A}+\sqrt{{\sf z}}\eta_{2}^{A}\eta_{3}^{A}\Big{)}~{}.$
(241)

As in the pure Yang-Mills theory, one can also produce the soft limit by
taking consecutive collinear limits. Taking $p_{2}\to 0$ and thus ${\sf z}\to
1$, the split factors reduce to

$\displaystyle{\mathsf{Split}}_{0}({\sf z};\eta_{1},\eta_{2},\eta_{P_{12}})$
$\displaystyle\to$ $\displaystyle\frac{\langle 13\rangle}{\langle
12\rangle\langle
23\rangle}\prod_{A=1}^{4}(\eta_{P_{12}}^{A}-\eta_{1}^{A})~{},$ (242)
$\displaystyle{\mathsf{Split}}_{-1}({\sf z};\eta_{1},\eta_{2},\eta_{P_{12}})$
$\displaystyle\to$
$\displaystyle\frac{[13]}{[12][23]}\prod_{A=1}^{4}\eta_{2}^{A}(\eta_{P_{12}}^{A}-\eta_{1}^{A})~{}.$
(243)

Inserting into (238), one obtains

$\displaystyle A_{n,k}(1,2,3,\cdots,n)$ $\displaystyle\xrightarrow{p_{2}\to
0}$ $\displaystyle\frac{\langle 13\rangle}{\langle 12\rangle\langle
23\rangle}A_{n-1,k}(1,3,\cdots,n)~{},$ (244) $\displaystyle
A_{n,k}(1,2,3,\cdots,n)$ $\displaystyle\xrightarrow{p_{2}\to 0}$
$\displaystyle\delta^{4}(\eta_{2})\frac{[13]}{[12][23]}A_{n-1,k-1}(1,3,\cdots,n)~{}.$
(245)

Combining together, we have

$\mathscr{A}_{n}(1,2,3,\cdots,n)\xrightarrow{p_{2}\to 0}\Big{(}\frac{\langle
13\rangle}{\langle 12\rangle\langle
23\rangle}+\frac{[13]}{[12][23]}\delta^{4}(\eta_{2})\Big{)}\mathscr{A}_{n-1}(1,3,\cdots,n)~{}.$
(246)

This is in agreement with the leading supersoft factor in (230).

## References

  * (1) S. Pasterski, S.-H. Shao and A. Strominger, _Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere_ , _Phys. Rev. D_ 96 (2017) 065026 [1701.00049].
  * (2) J. de Boer and S. N. Solodukhin, _A Holographic reduction of Minkowski space-time_ , _Nucl. Phys. B_ 665 (2003) 545 [hep-th/0303006].
  * (3) C. Cheung, A. de la Fuente and R. Sundrum, _4D scattering amplitudes and asymptotic symmetries from 2D CFT_ , _JHEP_ 01 (2017) 112 [1609.00732].
  * (4) S. Pasterski and S.-H. Shao, _Conformal basis for flat space amplitudes_ , _Phys. Rev. D_ 96 (2017) 065022 [1705.01027].
  * (5) S. Pasterski, S.-H. Shao and A. Strominger, _Gluon Amplitudes as 2d Conformal Correlators_ , _Phys. Rev. D_ 96 (2017) 085006 [1706.03917].
  * (6) J. M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (7) G. ’t Hooft, _Dimensional reduction in quantum gravity_ , _Conf. Proc. C_ 930308 (1993) 284 [gr-qc/9310026].
  * (8) L. Susskind, _The World as a hologram_ , _J. Math. Phys._ 36 (1995) 6377 [hep-th/9409089].
  * (9) W. Fan, A. Fotopoulos and T. R. Taylor, _Soft Limits of Yang-Mills Amplitudes and Conformal Correlators_ , _JHEP_ 05 (2019) 121 [1903.01676].
  * (10) M. Pate, A.-M. Raclariu and A. Strominger, _Conformally Soft Theorem in Gauge Theory_ , _Phys. Rev. D_ 100 (2019) 085017 [1904.10831].
  * (11) D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, _Celestial Amplitudes: Conformal Partial Waves and Soft Limits_ , _JHEP_ 10 (2019) 018 [1904.10940].
  * (12) T. Adamo, L. Mason and A. Sharma, _Celestial amplitudes and conformal soft theorems_ , _Class. Quant. Grav._ 36 (2019) 205018 [1905.09224].
  * (13) M. Pate, A.-M. Raclariu, A. Strominger and E. Y. Yuan, _Celestial Operator Products of Gluons and Gravitons_ , 1910.07424.
  * (14) A. Guevara, E. Himwich, M. Pate and A. Strominger, _Holographic Symmetry Algebras for Gauge Theory and Gravity_ , 2103.03961.
  * (15) N. Arkani-Hamed, M. Pate, A.-M. Raclariu and A. Strominger, _Celestial Amplitudes from UV to IR_ , 2012.04208.
  * (16) D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, _2D Stress Tensor for 4D Gravity_ , _Phys. Rev. Lett._ 119 (2017) 121601 [1609.00282].
  * (17) E. Crawley, N. Miller, S. A. Narayanan and A. Strominger, _State-Operator Correspondence in Celestial Conformal Field Theory_ , 2105.00331.
  * (18) S. Pasterski, A. Puhm and E. Trevisani, _Celestial Diamonds: Conformal Multiplets in Celestial CFT_ , 2105.03516.
  * (19) S. Stieberger and T. R. Taylor, _Symmetries of Celestial Amplitudes_ , _Phys. Lett. B_ 793 (2019) 141 [1812.01080].
  * (20) A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, _Extended Super BMS Algebra of Celestial CFT_ , _JHEP_ 09 (2020) 198 [2007.03785].
  * (21) S. Pasterski and A. Puhm, _Shifting Spin on the Celestial Sphere_ , 2012.15694.
  * (22) A. Brandhuber, G. R. Brown, J. Gowdy, B. Spence and G. Travaglini, _Celestial Superamplitudes_ , to appear.
  * (23) P. Di Francesco, P. Mathieu and D. Senechal, _Conformal Field Theory_ , Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997, 10.1007/978-1-4612-2256-9.
  * (24) J.-F. Fortin, K. Intriligator and A. Stergiou, _Current OPEs in Superconformal Theories_ , _JHEP_ 09 (2011) 071 [1107.1721].
  * (25) J. M. Henn and J. C. Plefka, _Scattering Amplitudes in Gauge Theories_ , vol. 883. Springer, Berlin, 2014, 10.1007/978-3-642-54022-6.
  * (26) A. Schreiber, A. Volovich and M. Zlotnikov, _Tree-level gluon amplitudes on the celestial sphere_ , _Phys. Lett. B_ 781 (2018) 349 [1711.08435].
  * (27) A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , 1703.05448.
  * (28) A. Strominger, _On BMS Invariance of Gravitational Scattering_ , _JHEP_ 07 (2014) 152 [1312.2229].
  * (29) A. Strominger, _Asymptotic Symmetries of Yang-Mills Theory_ , _JHEP_ 07 (2014) 151 [1308.0589].
  * (30) S. Banerjee and S. Ghosh, _MHV Gluon Scattering Amplitudes from Celestial Current Algebras_ , 2011.00017.
  * (31) H. Elvang, Y.-t. Huang and C. Peng, _On-shell superamplitudes in N $<$4 SYM_, _JHEP_ 09 (2011) 031 [1102.4843].
  * (32) E. Casali and A. Puhm, _A Double Copy for Celestial Amplitudes_ , 2007.15027.
  * (33) J. M. Drummond, J. M. Henn and J. Plefka, _Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory_ , _JHEP_ 05 (2009) 046 [0902.2987].
  * (34) E. Witten, _Perturbative gauge theory as a string theory in twistor space_ , _Commun. Math. Phys._ 252 (2004) 189 [hep-th/0312171].
  * (35) H. A. González, A. Puhm and F. Rojas, _Loop corrections to celestial amplitudes_ , _Phys. Rev. D_ 102 (2020) 126027 [2009.07290].
  * (36) S. He, Y.-t. Huang and C. Wen, _Loop Corrections to Soft Theorems in Gauge Theories and Gravity_ , _JHEP_ 12 (2014) 115 [1405.1410].
  * (37) L. Ferro, T. Łukowski and R. Moerman, _From momentum amplituhedron boundaries toamplitude singularities and back_ , _JHEP_ 07 (2020) 201 [2003.13704].