# Unified theory of the nonlinear Schrödinger equation

David B. Reinhardt<EMAIL_ADDRESS><EMAIL_ADDRESS>German
Aerospace Center (DLR), Institute of Quantum Technologies, Wilhelm-Runge-
Straße 10, 89081 Ulm, Germany  Dean Lee  Facility for Rare Isotope Beams and
Department of Physics and Astronomy, Michigan State University, MI 48824, USA
Wolfgang P. Schleich  Institut für Quantenphysik and Center for Integrated
Quantum Science and Technology (IQST), Universität Ulm, D-89069 Ulm, Germany
Hagler Institute for Advanced Study at Texas A$\&$M University, Texas A$\&$M
AgriLife Research, Institute for Quantum Science and Engineering (IQSE), and
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843-4242, USA  Matthias Meister<EMAIL_ADDRESS><EMAIL_ADDRESS>German Aerospace Center (DLR), Institute of Quantum
Technologies, Wilhelm-Runge-Straße 10, 89081 Ulm, Germany

(July 14, 2023)

###### Abstract

The nonlinear Schrödinger equation (NLSE) is a rich and versatile model, which
in one spatial dimension has stationary solutions similar to those of the
linear Schrödinger equation as well as more exotic solutions such as solitary
waves and quantum droplets. We present a unified theory of the NLSE, showing
that all stationary solutions of the cubic-quintic NLSE can be classified
according to a single number called the cross-ratio. Any two solutions with
the same cross-ratio can be converted into one another using a conformal
transformation, and the same also holds true for traveling wave solutions. In
this way we demonstrate a conformal duality between solutions of cubic-quintic
NLSEs and lower-order NLSEs. The same analysis can be applied to the Newtonian
dynamics of classical particles with polynomial potentials. Our framework
provides a deeper understanding of the connections between the physics of the
NLSE and the mathematics of algebraic curves and conformal symmetry.

Introduction – The nonlinear Schrödinger equation (NLSE) is ubiquitous in
physics, where it plays a key role in plasma physics [1, 2, 3], hydrodynamics
[4, 5, 6], degenerate quantum gases [7, 8] and light propagation in nonlinear
fiber optics [9, 10, 11, 12]. Understanding the possible solutions of the NLSE
is therefore of great importance for a large variety of purposes whether they
are application-oriented or fundamental. In this Letter we point out a
conformal duality between different classes of solutions and even different
orders of the NLSE. This conformal mapping provides a unified picture of the
cubic- and the cubic-quintic NLSE and even establishes a direct link to the
linear Schrödinger equation. Moreover, our method allows us to systematically
classify the complete solution spaces of these equations.

The linear Schrödinger equation typically features oscillating and constant-
amplitude solutions which have their counterparts in the NLSE. However, there
also exist solutions which are uniquely nonlinear such as solitary waves [13,
14, 15, 16, 17, 18, 19] which are of versatile interest in physics [20, 21,
22, 23, 24]. Considering (multiple) higher-order self-modulating terms like in
the cubic-quintic NLSE drastically expands the solution space allowing for
instance for bright and dark soliton pairs [25] and solitons with power law
tail decay [26]. Although the different polynomial NLSEs have been studied in
great detail [27, 28, 29, 30, 31, 32, 33, 34, 35, 36] there exists so far no
unified theory linking their solution spaces.

In this work, we identify a large family of conformal dualities for the one-
dimensional time-independent cubic-quintic NLSE. These dualities allow us to
establish conformal maps between solutions of the cubic-quintic NLSE and even
to conformally reduce the cubic-quintic to the cubic NLSE and the linear
Schrödinger equation, highlighting that the lower-order equations essentially
are conformal limiting cases of the cubic-quintic NLSE. Conformal dualities
are of particular interest in physics [37, 38, 39] with famous instances being
the Kramers–Wannier duality in statistical mechanics [40] relating high-and
low temperature limits of the free energy in certain Ising models and the
Montonen–Olive duality [41] a generalization of the electro-magnetic symmetry
of Maxwell’s equations in quantum field theory.

The new and useful insights into the conformal duality of NLSEs presented in
this Letter can directly be transferred to Newtonian mechanics relating the
motion of classical particles in various harmonic and anharmonic potentials.
In fact, there is a remarkable similarity between the solutions of the NLSE
and Newtonian dynamics. Finally, our theory provides a fundamental
understanding of two-and three-body contact interactions being inherently
related in one-dimensional Bose-Einstein condensates [42, 43, 44, 45]
described by higher-order Gross-Pitaevskii equations [46, 47, 48, 14, 49, 17].

Nonlinear Schrödinger equation – We consider the dimensionless time-
independent cubic-quintic NLSE

$\displaystyle\left(-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+a_{3}\absolutevalue{\psi}^{2}+\frac{a_{4}}{2}\absolutevalue{\psi}^{4}\right)\psi=a_{2}\psi$
(1)

in one spatial dimension of coordinate $x$, where $\psi=\psi\left(x\right)$ is
the complex-valued wave function and $a_{2}$, $a_{3}$, and $a_{4}$ are
constants 111The indices $j$ of the coefficients $a_{j}$ are chosen such that
they correspond to the exponents of the different powers of $\sigma$ of the
polynomial P defined in Eq. (3). By omitting position-dependent potentials, we
focus on the homogeneous case with either box or periodic boundary conditions.

The amplitude-phase representation
$\psi\equiv\sqrt{\sigma}\exp\left(i\phi\right)$ casts Eq. (1) into the
differential equations [28, 30, 25]

$\displaystyle\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x}\right)^{2}=P\left(\sigma\right)$
(2)

for the density $\sigma=\sigma(x)$ with the quartic polynomial

$\displaystyle
P\left(\sigma\right)\equiv\frac{4}{3}a_{4}\,\sigma^{4}+4a_{3}\,\sigma^{3}-8a_{2}\,\sigma^{2}-16a_{1}\,\sigma-4a_{0}$
(3)

and

$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}$
$\displaystyle=\pm\frac{\sqrt{a_{0}}}{\sigma}$ (4)

for the phase $\phi=\phi(x)$. Here $a_{0},a_{1}$ are constants of integration
and the different signs in Eq. (4) refer to the two possible directions of the
flow induced by the phase gradient.

Obviously, the order of the polynomial $P=P(\sigma)$ directly depends on the
leading nonlinearity in Eq. (1) yielding a cubic or quartic polynomial in the
case of the cubic ($a_{4}=0$) or cubic-quintic NLSE, while the polynomial is
quadratic for the linear Schrödinger equation ($a_{3}=a_{4}=0$).

Classification of solutions – The stationary solutions of Eq. (1) are in
general determined [30] by the polynomial $P$, defined by Eq. (3), and its
discriminant $\Delta=a_{4}^{6}\prod_{j\neq k}(\sigma_{j}-\sigma_{k})$ given by
the roots $\sigma_{j}$ of $P$. Depending on the sign of $\Delta$, three
classes of solutions can be identified: (i) simple complex conjugated roots
($\Delta<0$), (ii) multiple roots ($\Delta=0$), or (iii) only simple roots
($\Delta>0$) of $P$.

In order to discuss the roots $\sigma_{j}$ of $P$ it is convenient to
introduce the tuple notation ($r_{4}$, $r_{3}$, $r_{2}$, $r_{1}$), where every
entry $r_{m}$ denotes the number of roots at order $m$. For instance
$(0,0,0,4)$ labels a polynomial with four simple real roots as displayed in
Fig. 1a, while a polynomial with two simple real roots and two simple complex-
conjugated roots as shown in Fig. 1d is labeled by
$(0,0,0,2+2_{{\mathbb{C}}})$.

Figure 1: Conformal mapping between two realizations of the cubic-quintic NLSE
with discriminant $\Delta>0$ (a–c) and $\Delta<0$ (d–f). (a,d) Polynomials
$P(\sigma)$ and $\tilde{P}(\tilde{\sigma})$, defined by Eq. (2), with four
simple roots $\sigma_{j}$ (a) or two simple and two complex roots
$\tilde{\sigma}_{j}$ (d), respectively. (b,e) Oscillating phase-space
trajectories corresponding to real (blue) and complex (red) solutions
determined by $P$ in (a,d). The two cases (a–c) and (d–f) are related by a
conformal transformation, Eq. (5), which maps the positions of the roots, the
polynomials, and the corresponding phase-space orbits into each other. (c,f)
Complex density plane of $P$ and $\tilde{P}$ shown in (a,d) illustrating the
positions of the roots $\sigma_{j}$ and $\tilde{\sigma}_{j}$ (black dots) as
well as the argument of the phase of the density $\sigma$ (color map). The
lines of constant real and imaginary part of the density $\sigma$ form a
square grid (c) which is mapped into a grid of circles (f) by the conformal
transformation due to changing the sign of $\Delta$. The roots $\sigma_{j}$
are thus mapped from a straight line (c) to a circle (f) with counter-
clockwise orientation starting from the first real root. Likewise, the
cloverleaf-shaped boundary $Q$ shown in (c) is the inverse image of the square
boundary shown in (f) while the center of the angle-shaped region in (f)
corresponds to the point at infinity in (c).

The explicit solutions of the stationary NLSE are obtained by direct
integration of Eqs. (2) and (4) in the region between two neighboring real
roots. Consequently, oscillatory solutions of Eq. (2) occur between two
neighboring simple real roots which define the minimum and maximum density of
the oscillation as displayed by the three closed phase-space orbits in Fig. 1b
which originate from the polynomial in Fig. 1a.

Complex conjugate roots with finite imaginary parts can therefore not be the
turning points of such solutions, but instead deform the resulting orbits
spanned between other real roots as illustrated in Fig. 1e. Due to the finite
order of $P$, there is thus only one oscillatory orbit possible for the
polynomial shown in Fig. 1d.

For polynomials with a multiple root (shown in Fig. 2b for the case (0,0,1,2),
solitonic and other more exotic solutions emerge 222See Supplemental Material
at the end of the document for details on the conformal transformation of the
NLSE and explicit expressions of the solutions discussed in context of the
conformal reduction shown in Fig. 2. The Supplemental Material cites Refs.
[52, 59, 58].. In fact, the multiple root acts as a bifurcation point in phase
space constituting a separatrix for the phase-space trajectories separating
the two other solution classes [25]. Moreover, there always exists a constant
amplitude solution at the density value of the multiple root.

Finally, the outer density regions which are restricted by only one real root
typically lead to unbounded solutions with poles. For instance the light
orange shaded region of the polynomial displayed in Fig. 2c yields such an
unbounded solution [51].

Consequently, the sign of the discriminant $\Delta$ and thus the nature of the
roots of $P$ not only determine the character and shape of the resulting
solutions, but also the total number of different solutions for a given set of
parameters. Indeed, according to Eq. (2) physically meaningful real solutions
require $P(\sigma)>0$ between the roots considered, in addition to any
restrictions set by the boundary conditions of the system under study, while
for $P(\sigma)<0$ complex density solutions emerge. Hence, this approach
enables a straightforward and systematic classification of all possible
stationary solutions of higher-order NLSEs.

Conformal duality – In the phase space $(\sigma,\sigma^{\prime})$ with
$\sigma^{\prime}\equiv\mathrm{d}\sigma/\mathrm{d}x$, the differential equation
Eq. (2) constitutes an elliptic curve. A key characteristic of elliptic curves
is the possibility to transform their underlying algebraic equation by
rational transformations [52]. Strikingly, in the case of the NLSE the Möbius
transformation can be adapted to the differential equation Eq. (2) leading to
the conformal map

$\displaystyle\sigma\left(x\right)=\frac{A\,\tilde{\sigma}\left(\tilde{x}\right)+B}{C\,\tilde{\sigma}\left(\tilde{x}\right)+D}$
(5)

of the densities $\sigma$ and $\tilde{\sigma}$ with the generally complex-
valued coefficients $A,B,C,D$. In contrast to the mapping of elliptic curves,
here the spatial coordinate $x$ also needs to be transformed according to the
affine transformation $x=x_{0}+\left(AD-BC\right)\tilde{x}$, where $x$ can
become complex-valued and $x_{0}$ is a constant.

The duality, Eq. (5), relates any two physical systems with the same real-
valued cross-ratio $k^{2}$ which is an invariant of the transformation
determined by the roots $\sigma_{j}$ of $P$ [51]. Note that the conformal
character of the Möbius transformation will preserve the angles in the complex
density plane by mapping every straight line of constant density into another
line or circle of constant density, and vice versa.

The gradient of the phase, determined by Eq. (4), enjoys a similar
transformation [51]

$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}=\pm\sqrt{a_{0}}\,\dfrac{D\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,C}{B\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,A}$
(6)

with the very same coefficients $A,B,C,D$ as in Eq. (5). As a result, the
combination of Eqs. (5) and (6) provides the complete conformal mapping of the
differential equations under study. Hence, different realizations of the
cubic-quintic NLSE are conformally related establishing a fundamental
connection between their solution spaces. In particular, these transformations
also apply to the solutions of density $\sigma$ and phase $\phi$ themselves
such that the complete complex wave function $\psi$ can be conformally mapped.

We emphasize that the conformal duality remains intact for traveling-wave
solutions of the NLSE such as solitary waves subjected to a velocity boost.
This effect is a direct consequence of the Galilean covariance of the NLSE
allowing to obtain arbitrary many additional solutions through the application
of Galilei-transformations [51].

Conformal mapping and reduction of the NLSE – Depending on the choice of the
transformation coefficients different scenarios can be realized. Indeed, the
conformal map, Eq. (5), directly relates different quartic polynomials with
each other and therefore their solution spaces. In this case the ratio $A/C$
must not match the value of any of the roots of the involved polynomials to
preserve their quartic order. Real-valued coefficients $A,B,C,D$ connect
polynomials within a given solution class, while complex coefficients enable
to change the solution class corresponding to a change of sign of $\Delta$.
Figure 1 shows an intriguing example of the latter case, where a
$(0,0,0,4)$-polynomial is mapped to one classified by
$(0,0,0,2+2_{{\mathbb{C}}})$. Despite the fact that the graphs of the
polynomials $P$ and $\tilde{P}$ (a,d) and their phase-space orbits (b,e)
appear quite distinct in Fig. 1 they are intimately connected as visualized by
the density maps (c,f). Here, the conformal character of the transformation
manifests itself by transforming the straight line connecting all four simple
roots in Fig. 1c to a circle which passes again through all (now partly
complex) roots in Fig. 1f. In the same way, the rectangular boundary of the
density plot in Fig. 1f is mapped into the cloverleaf-shaped boundary
displayed in Fig. 1c.

Figure 2: Conformal reduction from the cubic-quintic (b) to the cubic NLSE (c)
and the linear Schrödinger equation (a). Exemplary polynomials $P$ (green) and
$-P$ (orange), Eq. (2), with (a) two simple roots (black dots), (b) two simple
roots $\sigma_{3}$, $\sigma_{4}$ and one double root $\sigma_{1,2}$ (encircled
black dot), or (c) one simple and one double root. By moving the roots
$\sigma_{4}$ (or $\sigma_{1,2}$) to plus (minus) infinity the cubic-quintic
NLSE can be reduced to the cubic NLSE or the linear Schrödinger equation,
respectively. The green (soliton solutions) and orange (oscillating solutions)
fillings show which of the density regions (and corresponding solutions) are
mapped into each other featuring similar characteristics. The light shaded
green and orange areas in (a) and (c) illustrate unbounded solutions.

Moreover, as depicted in Fig. 2, it is possible to conformally reduce the
cubic-quintic NLSE to either the cubic NLSE or the linear Schrödinger equation
by mapping either an outer simple root or an outer double root to plus or
minus infinity, respectively. In these cases the ratio $A/C$ must match the
value of the roots to be moved. As a consequence, the overall degree of the
polynomial is reduced by one (simple root moved) or two (double root moved).
Analogously, the linear Schrödinger equation with an energy eigenvalue of zero
is obtained by removing a triple root.

By reducing the degree of the polynomial the solution space changes based on
Eq. (5) and as illustrated in Fig. 2: (i) one unbounded solution vanishes
because the root constituting its minimum or maximum density has been removed,
(ii) a bound solution becomes unbound since it is now only restricted by one
root, and (iii) the remaining solutions get transformed, but keep their main
characteristics as their roots retain their order.

The case shown in Fig. 2 highlights two prominent solitonic solutions, namely
the flat-top soliton [14] (green shaded area in b) and the elementary bright
soliton [53] (green shaded area in c) which are both governed by a hyperbolic
cosine in the denominator of their density profile. By the transformation from
the cubic-quintic to the cubic NLSE only the prefactor in front of the
hyperbolic cosine gets changed such that both solutions are quite similar
[51].

Likewise, the oscillatory solution in the cubic-quintic case (orange area in
Fig. 2b) governed by a cosine in the denominator as well changes its prefactor
when transformed. However, in this case the corresponding solution of the
cubic case (light orange area in Fig. 2c) becomes unbound due to the now
different prefactor and has thus completely changed its character by the
transformation.

Fascinatingly, the solutions of the green and orange regions in Fig. 2 are
also interconnected by a tranformation that maps a real position coordinate
$x$ to a purely imaginary position $\tilde{x}$ changing the functional
dependency from a hyperbolic sine (green) to a trigonometric sine (orange).
Effectively, this transformation thus flips the overall sign of the polynomial
from $P$ to $-P$. In this way all the solutions of the cubic and cubic-quintic
NLSE as well as the linear Schrödinger equation are fundamentally connected.

Connection to Newtonian mechanics – Besides the importance of the unified
theory of the NLSE, the conformal duality discussed in this Letter has also
strong implications for the dynamics of classical particles subjected to
anharmonic conservative potentials in Newtonian mechanics. Indeed, it is well-
known that the NLSE formally constitutes a classical Hamiltonian system for
the density $\sigma$ with the Hamiltonian function [54]

$\displaystyle\mathcal{H}(\sigma^{\prime},\sigma)\equiv\frac{1}{2}{\sigma^{\prime\,}}^{2}+U\left(\sigma\right)$
(7)

with the potential $U=U\left(\sigma\right)$. Here, the density $\sigma$ will
be analogous to the classical position, while the spatial coordinate $x$
corresponds to time in classical mechanics.

By constraining the energy of $\mathcal{H}$ to the value of $-4a_{0}$ and
considering the potential $U\left(\sigma\right)\equiv
2a_{0}-P\left(\sigma\right)/2$ one can recover [25] Eq. (2). Hence, in this
analogy the nonlinearites in Eq. (1) directly correspond to anharmonic
contributions in $U$ with the cubic or cubic-quintic NLSE giving rise to a
cubic or quartic potential, respectively, while the linear Schrödinger
equation yields a harmonic potential as usual.

The conformal map, Eq. (5), now allows us to transform the Hamiltonian, Eq.
(7), of a classical particle and consequently its underlying equations of
motions. Thus, we can map a double-well problem to another double-well
problem, or carry out the conformal reduction from a quartic to a cubic,
quadratic, linear or constant potential by mapping a simple, double, triple or
quadruple root of the potential to plus or minus infinity.

As a result, soliton solutions, as shown in Fig. 2, in our classical mechanics
analogy correspond to an oscillation with infinitely long period where the
particle in phase space approaches a bifurcation point similar to a
mathematical pendulum, where the angular coordinate approaches the unstable
fixed point at $\pi$ radians.

Analogously, unbounded solutions are the counterpart of scattering states of
the corresponding classical potentials. Hence, the ideas and concepts for
treating physics problems of classical particles in anharmonic potentials have
a direct correspondence to those required for the NLSE. Remarkably, both
systems enable the conformal mapping of their solutions within and between
different solution classes.

Conclusion – In summary we have provided a unified picture of the NLSE by
establishing a conformal duality between the solution spaces of the cubic-
quintic and cubic NLSE as well as the linear Schrödinger equation. This
connection gives rise to novel and elementary understanding of the physics of
nonlinear systems, in particular, when comparing the effect of nonlinearities
of different degrees.

Our results apply to stationary and travelling-wave solutions of the NLSE and
remain valid even under Galilean transformations. We therefore expect our
findings to have a wide variety of applications that include the dynamics of
solitons and their dual counterparts, mode structures in nonlinear fiber
optics, hydrodynamic wave-dynamics, and the interplay of two- and three-body
interactions in quasi-1D Bose-Einstein condensates as utilized for atomtronics
devices [55, 56].

In addition, the conformal duality can be employed for Newtonian mechanics
allowing us to classify and relate numerous different dynamical systems caused
by anharmonic conservative potentials. Finally, our algebraic-geometric
classification scheme can straightforwardly be extended to even higher order
NLSEs such as the cubic-quintic-septic NLSE [57] to search for new physics in
the form of exotic solutions that require strong nonlinearities of this kind.

Acknowledgments – We thank M.A. Efremov for fruitful discussions and helpful
suggestions. D.L. acknowledges financial support from the U.S. Department of
Energy (DE-SC0021152, DE-SC0013365, DE-SC0023658, SciDAC-5 NUCLEI
Collaboration). W.P.S. is grateful to Texas A$\&$M University for a Faculty
Fellowship at the Hagler Institute for Advanced Study at the Texas A$\&$M
University as well as to the Texas A$\&$M AgriLife Research.

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  * Note [1] The indices $j$ of the coefficients $a_{j}$ are chosen such that they correspond to the exponents of the different powers of $\sigma$ of the polynomial P defined in Eq. (3).
  * Note [2] See Supplemental Material at the end of the document for details on the conformal transformation of the NLSE and explicit expressions of the solutions discussed in context of the conformal reduction shown in Fig. 2. The Supplemental Material cites Refs. [52, 59, 58].
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Supplemental material for
Unified theory of the nonlinear Schrödinger equation
David B. Reinhardt1, Dean Lee2, Wolfgang P. Schleich3,4 and Matthias Meister1

1German Aerospace Center (DLR), Institute of Quantum Technologies,
Wilhelm-Runge-Straße 10, 89081 Ulm, Germany
2Facility for Rare Isotope Beams and Department of Physics and Astronomy,
Michigan State University, MI 48824, USA
3Institut für Quantenphysik and Center for Integrated Quantum Science and
Technology (IQST),
Universität Ulm, D-89069 Ulm, Germany
4Hagler Institute for Advanced Study at Texas A$\&$M University, Texas A$\&$M
AgriLife Research, Institute for Quantum Science and Engineering (IQSE), and
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
Texas 77843-4242, USA

(Dated: July 14, 2023)

##  I Conformal duality of the nonlinear Schrödinger equation

In this supplemental material we derive the transformation of the gradient of
the phase from the Möbius transformation of the density and further provide
more insights into the transformation of the differential equations with
quartic, cubic, and quadratic polynomial dependency in the density. In
addition, the cross-ratio of the transformation is discussed in more detail.
We conclude this supplement by providing explicit expressions for the
conformal reduction of typical oscillating solutions of the NLSE and of the
solitonic solutions shown in Fig. 2. In this context we also present the
involved analytical solutions of the NLSE.

### Basics of elliptic curves

A common way to define elliptic- and hyper-elliptic curves is to introduce the
relation

$\displaystyle Y^{2}=P(X)$ (8)

between the algebraic variables $X$ and $Y$ with the polynomial

$\displaystyle P(X)\equiv\sum_{n=0}^{N}\alpha_{n}X^{n}$ (9)

of order $N$ with coefficients $\alpha_{n}$. Note that for elliptic curves
$N=3,4$, while for hyper-elliptic curves $N>4$ [58].

A key feature of elliptic or hyper-elliptic curves is the possibility to
transform them into other elliptic or hyper-elliptic curves by rational
transformations [52]. For instance the bi-rational transformations

$\displaystyle X=\frac{A\tilde{X}+B}{C\tilde{X}+D}$ (10)

and

$\displaystyle Y=\frac{\tilde{Y}}{\left(CX+D\right)^{\frac{N}{2}}}$ (11)

connect both algebraic variables $X$ and $Y$ with the corresponding new
variables $\tilde{X}$ and $\tilde{Y}$ through the coefficients $A,B,C$ and
$D$.

###  I.1 Conformal transformation of the density and the phase gradient

In the case of the nonlinear Schrödinger equation (NLSE) discussed here we
identify the variables $X$ and $Y$ with $\sigma$ and
$\sigma^{\prime}\equiv\mathrm{d}\sigma/\mathrm{d}x$ such that the
transformation of the density is given by the relation

$\displaystyle\sigma\left(x\right)=\frac{A\,\tilde{\sigma}\left(\tilde{x}\right)+B}{C\,\tilde{\sigma}\left(\tilde{x}\right)+D}$
(12)

being of linear fractional type.

In contrast, the transformation of the derivative of the density
$\sigma^{\prime}$ is of nonlinear fractional type as is the mapping between
the algebraic variables $Y$ and $\tilde{Y}$, Eq. (11). Nevertheless, in case
of the NLSE the transformation of $\sigma^{\prime}$ is self-consistently
obtained by differentiation of Eq. (12)

$\displaystyle\frac{\mathrm{d}\sigma\left(x\right)}{\mathrm{d}x}$
$\displaystyle=\frac{\mathrm{d}\tilde{x}}{\mathrm{d}x}\frac{AD-
BC}{\left(C\,\tilde{\sigma}\left(\tilde{x}\right)+D\right)^{2}}\frac{\mathrm{d}\tilde{\sigma}\left(\tilde{x}\right)}{\mathrm{d}\tilde{x}}$
(13)
$\displaystyle=\frac{1}{\left(C\,\tilde{\sigma}\left(\tilde{x}\right)+D\right)^{2}}\frac{\mathrm{d}\tilde{\sigma}\left(\tilde{x}\right)}{\mathrm{d}\tilde{x}}\,,$
(14)

where we have used the linear coordinate transformation $x=x_{0}+\left(AD-
BC\right)\tilde{x}$ between $x$ and $\tilde{x}$ in the second step.

The origin of the Möbius transformation of the phase gradient

$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}$
$\displaystyle=\pm\frac{\sqrt{a_{0}}}{\sigma}$ (15)

is rooted in its reciprocal dependence on the density $\sigma$. Indeed, by
inserting the conformal map of the density, Eq. (12), into Eq. (15) and
dividing both the numerator and denominator by the density $\tilde{\sigma}$ we
immediately obtain the transformation

$\displaystyle\frac{\mathrm{d}\phi}{\mathrm{d}x}=\pm\sqrt{a_{0}}\,\dfrac{D\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,C}{B\,\frac{\mathrm{d}\tilde{\phi}}{\mathrm{d}\tilde{x}}\pm\sqrt{\tilde{a}_{0}}\,A}\,,$
(16)

of the phase gradient, where $\sqrt{\tilde{a}_{0}}$ is the flux in the
transformed system. Thus, the phase gradient also transforms according to a
Möbius transform with the same coefficients $A,B,C,D$ used for the density.

Consequently, the transformations Eq. (12) and Eq. (16) of the density and
gradient of the phase together enable a complete conformal mapping of the
complex field $\psi=\sqrt{\sigma}\,\mathrm{e}^{\mathrm{i}\phi}$ of the NLSE.

### Conformal transformation of the polynomial equation

The stationary solutions of the NLSE are governed by the differential equation
(2) which features a polynom $P(\sigma)$ of fourth order on the right-hand
side. In order to analyze this equation it is convenient to consider the
factored form

$\displaystyle\left(\frac{\mathrm{d}\sigma}{\mathrm{d}x}\right)^{2}$
$\displaystyle=a_{4}\,\left(\sigma-\sigma_{1}\right)\left(\sigma-\sigma_{2}\right)\left(\sigma-\sigma_{3}\
\right)\left(\sigma-\sigma_{4}\ \right)$ (17)

where $\sigma_{j}$ are the roots of $P$.

By applying the transformations Eq.(12) and (14) the transformed differential
equation is given by

$\displaystyle\left(\frac{\mathrm{d}\tilde{\sigma}}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=a_{4}\Big{(}\left(A-\sigma_{1}C\right)\sigma+B-\sigma_{1}D\Big{)}\cdot\Big{(}\left(A-\sigma_{2}C\right)\sigma+B-\sigma_{2}D\Big{)}\cdot\Big{(}\left(A-\sigma_{3}C\right)\sigma+B-\sigma_{3}D\Big{)}\cdot\Big{(}\left(A-\sigma_{4}C\right)\sigma+B-\sigma_{4}D\Big{)}$
$\displaystyle=\tilde{a}_{4}\,\Big{(}\tilde{\sigma}-\tilde{\sigma}_{1}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{2}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{3}\Big{)}\Big{(}\tilde{\sigma}-\tilde{\sigma}_{4}\Big{)}$
(18)

with

$\displaystyle\tilde{a_{4}}\equiv
a_{4}\,C^{4}\left(\frac{A}{C}-\sigma_{1}\right)\left(\frac{A}{C}-\sigma_{2}\right)\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\,,$
(19)

the new roots

$\displaystyle\tilde{\sigma}_{j}\equiv\frac{D\sigma_{j}-B}{-C\sigma_{j}+A}$
(20)

and $A/C\neq\sigma_{j}$. In general the conformal character of the
transformation maps straight lines to circles and vice versa such that the new
and original roots will always lie on a straight line or a circle. By choosing
a specific set of coefficients $A,B,C$ and $D$ Eq. (I) can be used to relate
two different solutions of the cubic-quintic NLSE as illustrated in Fig. 1.

Moreover, the quartic polynomial can conformally be reduced to a cubic
polynomial, describing the cubic NLSE, by mapping one of the outer roots to
plus or minus infinity as shown in Fig. 2. In this case $A/C$ needs to be
equal to the value of the root to be mapped. If for instance the first root
$\sigma_{1}$ is mapped to minus infinity ($A/C=\sigma_{1}$) the resulting
differential equation can be written as

$\displaystyle\left(\frac{\mathrm{d}\rho}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=\tilde{a}_{3}\left(\rho-\rho_{1}\right)\left(\rho-\rho_{2}\right)\left(\rho-\rho_{3}\right)$
(21)
$\displaystyle=\tilde{a}_{3}\rho^{3}+\tilde{a}_{2}\rho^{2}+\tilde{a}_{1}\rho+\tilde{a}_{0}$
(22)

with

$\displaystyle\tilde{a}_{3}\equiv
a_{4}C^{2}\left(\frac{A}{C}-\sigma_{2}\right)\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\left(BC-
AD\right)\,.$ (23)

Here we have introduced the density $\rho$ the of the cubic NLSE as to
distinguish between the various cases.

In the case that the quartic polynomial, Eq. (2), exhibits a vanishing
discriminant $\Delta$ multiple roots might emerge. If one of the outer roots
is a double root, the degree of the polynomial can be reduced by two, giving
rise to a simple parabola corresponding to the linear Schrödinger equation. By
choosing e.g. $A/C=\sigma_{1,2}$ we obtain a conic curve

$\displaystyle\left(\frac{\mathrm{d}\lambda}{\mathrm{d}\tilde{x}}\right)^{2}$
$\displaystyle=e_{2}\,\left(\lambda-\lambda_{1}\right)\left(\lambda-\lambda_{2}\right)$
(24) $\displaystyle=e_{2}\lambda^{2}+e_{1}\lambda+e_{0}$ (25)

describing the stationary states of the linear Schrödinger equation, where

$\displaystyle e_{2}\equiv
a_{4}\left(\frac{A}{C}-\sigma_{3}\right)\left(\frac{A}{C}-\sigma_{4}\right)\left(BC-
AD\right)^{2}$ (26)

and the density is labeled by $\lambda$.

###  I.2 Details of the cross-ratio

In the mathematics literature [59], the cross-ratio is sometimes also called
the anharmonic ratio of four co-linear or concyclic complex numbers
$z_{1},z_{2},z_{3}$ and $z_{4}$ and is typically defined as

$\displaystyle\left(z_{1},z_{2};z_{3},z_{4}\right)=\frac{z_{3}-z_{1}}{z_{3}-z_{2}}\frac{z_{4}-z_{2}}{z_{4}-z_{1}}\equiv\Lambda\,.$
(27)

Actually, this definition is not unique as the value will be unaltered for
pairwise interchange of all points . Furthermore, all other possibilities of
partitioning the four points can eventually be related to the cross-ratio
$\Lambda$ defined in Eq. (27).

For instance, the solutions of the “W”-shaped graph of the polynomial $P$
shown in Fig. 1a employ a cross-ratio or elliptic modulus given by the
relation

$\displaystyle k^{2}$
$\displaystyle\equiv\frac{\sigma_{4}-\sigma_{3}}{\sigma_{4}-\sigma_{2}}\frac{\sigma_{2}-\sigma_{1}}{\sigma_{3}-\sigma_{1}}=\frac{\Lambda-1}{\Lambda}\,.$
(28)

Similarly, the solutions of a “M”-shaped graph use the relation

$\displaystyle
k^{{\prime}^{2}}=\frac{\sigma_{3}-\sigma_{2}}{\sigma_{4}-\sigma_{2}}\frac{\sigma_{4}-\sigma_{1}}{\sigma_{3}-\sigma_{1}}=1-k^{2}=\frac{1}{\Lambda}$
(29)

often referred to as the complementary elliptic modulus [58]. In case of a
multiple root of the polynomial $P$, the cross-ratio employed in solutions
given in terms of Jacobi elliptic functions, will either correspond to the
trigonometric limit where $k^{2}=0$ or the hyperbolic limit with $k^{2}=1$.

### Galilean invariance of the conformal duality

####  I.2.1 Velocity boost for the time-dependent nonlinear Schrödinger
equation

Here, we discuss the Galilean covariance of the cubic-quintic NLSE by showing
that any Galilean transformation leaves the underlying equation of motion
invariant. We consider a solution $\psi\left(x,t\right)$ of the time-dependent
cubic-quintic NLSE

$\displaystyle i\frac{\partial}{\partial
t}\psi\left(x,t\right)=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+a_{3}\absolutevalue{\psi\left(x,t\right)}^{2}+a_{4}\absolutevalue{\psi\left(x,t\right)}^{4}\right)\psi\left(x,t\right)$
(30)

in the rest frame $\mathcal{F}$ described by the coordinates $x$ and $t$.
Applying a boost with the dimensionless velocity $u$ introduces the moving
frame $\mathcal{F}^{\prime}$ with coordinates $x^{\prime}$ and $t^{\prime}$
which are related to the rest frame via

$\displaystyle x$ $\displaystyle=x^{\prime}+ut^{\prime}$ (31) $\displaystyle
t$ $\displaystyle=t^{\prime}\,.$ (32)

To obtain the wave function $\psi^{\prime}\left(x^{\prime},t^{\prime}\right)$
in the boosted frame $\mathcal{F}^{\prime}$ we consider the transformation

$\displaystyle\psi\left(x,t\right)=\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$
(33)

of the wavefunction $\psi\left(x,t\right)$ in the rest frame with a phase
containing the boost velocity and the additional kinetic energy due to the
boost in the moving frame $\mathcal{F}^{\prime}$.

Next, we will show that the Galilean transformation of the wavefunction and
coordinates leaves the NLSE in Eq. (30) formally unchanged. For this purpose,
we consider the transformations of the derivatives in Eq. (30) which are given
by

$\displaystyle\frac{\partial}{\partial t}=\frac{\partial}{\partial
t^{\prime}}\cdot\frac{\partial t^{\prime}}{\partial
t}+\frac{\partial}{\partial x^{\prime}}\cdot\frac{\partial
x^{\prime}}{\partial t}=\frac{\partial}{\partial
t^{\prime}}-u\frac{\partial}{\partial x^{\prime}}$ (34)

and

$\displaystyle\frac{\partial}{\partial x}=\frac{\partial}{\partial
x^{\prime}}\frac{\partial x^{\prime}}{\partial x}+\frac{\partial}{\partial
t^{\prime}}\frac{\partial t^{\prime}}{\partial x}=\frac{\partial}{\partial
x^{\prime}}.$ (35)

By substituting these derivatives into Eq. (30) and also inserting the
transformation law for the wave function, Eq. (33), we obtain the differential
equation

$\displaystyle i$ $\displaystyle\left(\frac{\partial}{\partial
t^{\prime}}-u\frac{\partial}{\partial
x^{\prime}}\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$
(36)
$\displaystyle=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{4}\
\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$

describing the evolution of the boosted wave function
$\psi^{\prime}\left(x^{\prime},t^{\prime}\right)$ in the moving frame
$\mathcal{F}^{\prime}$.

By performing the partial derivatives on both sides of the equation and
factoring out the global phase factor
$\exp\left(iux^{\prime}+i\frac{u^{2}}{2}t^{\prime}\right)$ the differential
equation in the moving frame reduces to

$\displaystyle i\frac{\partial}{\partial
t^{\prime}}\psi^{\prime}\left(x^{\prime},t^{\prime}\right)=\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},t^{\prime}\right)}^{4}\right)\psi^{\prime}\left(x^{\prime},t^{\prime}\right)\,.$
(37)

The fact that Eqs. (30) and (37) have the same form proves the Galilean
invariance of the cubic-quintic NLSE.

####  I.2.2 Impact on stationary solutions

In order to derive the impact of the velocity boost on stationary solutions,
we start with the usual ansatz

$\displaystyle\psi\left(x,t\right)=\psi\left(x,0\right)\exp\left(-ia_{2}t\right)$
(38)

for the time-independent wave function $\psi(x,0)$ in the rest frame
$\mathcal{F}$ with eigenvalue $a_{2}$ determined by Eq. (1).

Applying the transformation, Eq. (33), to both the time-dependent and time-
independent wave functions in Eq. (38) yields the relation

$\displaystyle\psi^{\prime}\left(x^{\prime},t^{\prime}\right)=\psi^{\prime}\left(x^{\prime},0\right)\exp\left(-i\left(a_{2}+\frac{u^{2}}{2}\right)t^{\prime}\right)\,,$
(39)

where we have also used the relation $t=t^{\prime}$, Eq. (32).

Inserting this result into the time-dependent NLSE in the boosted frame, Eq.
(37) allows us to obtain the time-independent NLSE in the boosted frame

$\displaystyle\left(-\frac{1}{2}\frac{\partial^{2}}{\partial{x^{\prime}}^{2}}+a_{3}\absolutevalue{\psi^{\prime}\left(x^{\prime},0\right)}^{2}+a_{4}\absolutevalue{\psi^{\prime}\left(x^{\prime},0\right)}^{4}\right)\psi^{\prime}\left(x^{\prime},0\right)$
$\displaystyle=a_{2}^{\prime}\psi^{\prime}\left(x^{\prime},0\right)\,,$ (40)

where

$\displaystyle a_{2}^{\prime}\equiv a_{2}+\frac{u^{2}}{2}.$ (41)

Consequently, the boost to the moving frame adds a velocity-dependent term to
the eigenvalue $a_{2}^{\prime}$ of the stationary solution. This is just the
kinetic energy associated with the Galilean boost.

In the main text, we have shown that any two stationary solutions with the
same cross-ratio can be converted into each other using conformal
transformations. This fundamental result can now be extended to traveling wave
solutions. We simply take the dual stationary solutions and perform Galilean
boosts to produce the desired traveling waves.

### Explicit solutions for the conformal duality and reduction of the
nonlinear Schrödinger equation

####  I.2.3 Oscillating solutions

As a first application of the conformal duality of the NLSE we show how
typical oscillating solutions of the cubic and the cubic-quintic NLSE are
connected via the transformation, Eq. (12). For this purpose we consider the
real-valued solution

$\displaystyle\rho\left(x_{2}\right)$
$\displaystyle=\left(\rho_{2}-\rho_{1}\right)\operatorname{sn}^{2}\left(\kappa_{2}x_{2},k\right)+\rho_{1},$
(42)

which oscillates between the density values $\rho_{1}$ and $\rho_{2}$
corresponding to the roots of a cubic polynomial $P(\rho)$ with an “N”-shaped
graph. Here $\operatorname{sn}$ refers to the Jacobi elliptic sine function
with $\kappa_{2}=a_{3}\sqrt{\rho_{3}-\rho_{1}}/2$ and
$k^{2}=(\rho_{2}-\rho_{1})/(\rho_{3}-\rho_{1})$.

We now demonstrate the conformal duality between this solution and another
solution for a quartic polynomial $P(\sigma)$. By applying the solution of the
cubic NLSE, Eq. (42), to the transformation Eq. (12), and choosing
$A/C=\sigma_{1}<0$, to pull a root from minus infinity, we obtain the solution

$\displaystyle\sigma\left(x_{3}\right)$
$\displaystyle=\frac{\sigma_{2}\eta-\sigma_{1}\operatorname{sn}^{2}\left(\kappa_{3}\,x_{3},k\right)}{\eta-\operatorname{sn}^{2}\left(\kappa_{3}\,x_{3},k\right)},$
(43)

of the cubic-quintic NLSE. Here,
$\eta=\frac{\sigma_{3}-\sigma_{1}}{\sigma_{3}-\sigma_{2}}>1$ and
$\kappa_{3}=\kappa_{2}/(AD-BC)$ such that the solution given by Eq. (43)
oscillates between the densities $\sigma_{2}$ and $\sigma_{3}$ corresponding
to the roots of a quartic polynomial $P(\sigma)$ with a “W”-shaped graph.
Direct integration of the quartic polynomial of the cubic-quintic NLSE yields
exactly the same solution as given by Eq. (43) proving the validity of the
conformal transformation.

The mapping coefficients of this conformal reduction are given by

$\displaystyle A$
$\displaystyle\equiv\frac{\sigma_{1}}{\rho_{2}-\rho_{1}}\qquad$ $\displaystyle
B\equiv-\sigma_{2}\eta-\sigma_{1}\frac{\rho_{1}}{\rho_{2}-\rho_{1}}$ (44)
$\displaystyle C$ $\displaystyle\equiv\frac{1}{\rho_{2}-\rho_{1}}\qquad$
$\displaystyle D\equiv\eta-\frac{\rho_{1}}{\rho_{2}-\rho_{1}}.$ (45)

The coordinates are related via the linear map $x_{3}=\left(AD-
BC\right)x_{2}+\text{const.}$, where in this example the constant is set to
zero for simplicity.

####  I.2.4 Solitonic solutions

Here, we provide the explicit transformations of the conformal reduction shown
in Fig. 2. For the bright soliton solutions we naturally need a focusing or
attractive nonlinearity with $\tilde{a}_{3}<0$ while the flat-top soliton
likewise requires a focusing cubic nonlinearity $a_{3}<0$ and a defocusing
quintic nonlinearity $a_{4}>0$. The analytical solutions for all orange and
green shaded areas in Fig. 2 given by Eqs. (46) and (47), sorted according to
the appearance in the figure from left to right, are given by

$\displaystyle\lambda\left(x_{1}\right)$
$\displaystyle=\left(\lambda_{2}-\lambda_{1}\right)\sin^{2}\left(\kappa_{1}x_{1}\right)+\lambda_{1}\qquad$
$\displaystyle\sigma\left(x_{3}\right)=\sigma_{0}\eta\frac{1}{1+\sqrt{1-\eta}\cos\left(2\kappa_{3}x_{3}\right)}\qquad$
$\displaystyle\rho\left(x_{2}\right)=\rho_{0}\frac{1}{1+\cos\left(2\kappa_{2}x_{2}\right)}$
(46) $\displaystyle\tilde{\lambda}\left(\tilde{x}_{1}\right)$
$\displaystyle=\left(\lambda_{2}-\lambda_{1}\right)\sinh^{2}\left(\kappa_{1}\tilde{x}_{1}\right)+\lambda_{1}\qquad$
$\displaystyle\tilde{\sigma}\left(\tilde{x}_{3}\right)=\sigma_{0}\eta\frac{1}{1+\sqrt{1-\eta}\operatorname{cosh}\left(2\kappa_{3}\tilde{x}_{3}\right)}\qquad$
$\displaystyle\tilde{\rho}\left(\tilde{x}_{2}\right)=\rho_{0}\frac{1}{1+\cosh\left(2\kappa_{2}\tilde{x}_{2}\right)}\,,$
(47)

where $\kappa_{1}\equiv\sqrt{\absolutevalue{e_{2}}}$, $\rho_{0}\equiv
4\tilde{a}_{2}/\tilde{a}_{3}$,
$\kappa_{2}\equiv\sqrt{\absolutevalue{\tilde{a}_{2}}}$,
$\sigma_{0}\equiv-3a_{3}/2a_{4}$,
$\kappa_{3}\equiv\sqrt{\absolutevalue{a_{2}}}$, and
$\eta\equiv-8a_{2}a_{4}/3a_{3}^{2}<1$. Again $\lambda$ refers to the density
of the linear Schrödinger equation, $\rho$ corresponds to the density of the
cubic NLSE, and $\sigma$ gives the density of the cubic-quintic NLSE.

Now the transformation coefficients $A,B,C,D$, Eq. (12), of the conformal
reduction from the cubic-quintic to the cubic NLSE, connecting the solutions
$\sigma\left(x_{3}\right)$ and $\rho\left(x_{2}\right)$ or
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ and
$\tilde{\rho}\left(\tilde{x}_{2}\right)$, can be determined. For the
particular set of solutions given by Eqs. (46) and (47) the phase is
independent of position as they require a root at the origin such that the
coefficient $a_{0}=0$ in Eq. (2). As a consequence, the transformation
coefficient $B$ needs to be zero by construction in order to conformally
relate theses solutions. In addition, this transformation requires
$AD=\frac{\kappa_{2}}{\kappa_{3}}>0$ and
$A/C=\sigma_{0}\left(1+\sqrt{1-\eta}\right)=\sigma_{4}$. Ultimately, the
transformation coefficients are thus given by

$\displaystyle A$
$\displaystyle\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\rho_{0}\sigma_{0}\eta}{\sqrt{1-\eta}}}\qquad$
$\displaystyle B\equiv 0$ (48) $\displaystyle C$
$\displaystyle\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\rho_{0}}{\sigma_{0}}}\frac{1}{1+\sqrt{1-\eta}}\sqrt{\frac{\eta}{\sqrt{1-\eta}}}\qquad$
$\displaystyle
D\equiv\sqrt{\frac{\kappa_{2}}{\kappa_{3}}}\sqrt{\frac{\sqrt{1-\eta}}{\rho_{0}\sigma_{0}\eta}}.$
(49)

As the roots of the polynomial $P$ and $-P$ are identical the transformation
coefficients for the conformal reduction from $\sigma\left(x_{3}\right)$ to
$\rho\left(x_{2}\right)$ (orange shaded areas in Fig. 2) are formally
identical to those required for the reduction from
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ to
$\tilde{\rho}\left(\tilde{x}_{2}\right)$ (green shaded areas in Fig. 2).
Hence, the coefficients given by Eqs. (48) and (49) apply to both cases.

Finally, in order to connect the solutions of a polynomial $P$ to those of the
inverse polynomial $-P$ one can for instance employ the coefficients
$A=D=(1+i)/\sqrt{2},\,B=C=0$ which formally do not change the densities
according to Eq. (12), but instead yield the coordinate transformation
$x=\mathrm{i}\tilde{x}$. By transforming back to a real coordinate $\tilde{x}$
the imaginary unit is absorbed in the functional dependency of the density
inducing the change from a trigonometric function to a hyperbolic function. In
this way the conformal partnering of the trigonometric oscillating solution
${\sigma}\left({x}_{3}\right)$ and the the resting droplet solution
$\tilde{\sigma}\left(\tilde{x}_{3}\right)$ given by Eqs. (46) and (47) can be
realized.