Nonlinear Schr odinger equations with amplitude-dependent Wadati potentials Dmitry A. Zezyulin Department of Physics and Engineering ITMO University St. Petersburg 197101 Russia

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Nonlinear Schr¨odinger equations with amplitude-dependent Wadati potentials

Dmitry A. Zezyulin∗

Department of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia

(Dated: October 28, 2022)

Complex Wadati-type potentials of the form V(x) = −w2(x)+iwx(x), where w(x) is a real-valued

function, are known to possess a number of intriguing features, unusual for generic non-Hermitian

potentials. In the present work, we introduce a class of nonlinear Schr¨odinger-type problems which

generalize the Wadati potentials by assuming that the base function w(x) depends not only on the

transverse spatial coordinate but also on the amplitude of the ﬁeld. Several examples of prospective

physical relevance are discussed, including models with the nonlinear dispersion or with the deriva-

tive nonlinearity. The numerical study indicates that the generalized model inherits the remarkable

features of standard Wadati potentials, such as the existence of continuous soliton families, the

possibility of symmetry-breaking bifurcations when the model obeys the parity-time symmetry, the

existence of constant-amplitude waves, and the eigenvalue quartets in the linear-instability spec-

tra. Our results deepen the current understanding of the interplay between nonlinearity and non-

Hermiticity and expand the class of systems which enjoy the exceptional combination of properties

unusual for generic dissipative nonlinear models.

I. INTRODUCTION

The combination of nonlinearity and non-Hermiticity

can have a dramatic impact on the properties of a waveg-

uiding system. In particular, the presence of a non-

Hermitian complex potential (which in physical terms

takes into account the energy exchange with the environ-

ment) can heavily alter the structure of stationary local-

ized nonlinear modes or solitons propagating in the sys-

tem. While in conservative waveguides the solitons exist

as continuous families parameterized by an ‘internal’ pa-

rameter, such as the soliton frequency or amplitude, the

dissipative solitons most usually exist as isolated points

[1, 2] which, from the dynamical point of view, behave as

attractors (provided that the soliton is stable). A proto-

typical model that illustrates this dissimilarity between

conservative and dissipative systems is the generalized

nonlinear Schr¨odinger equation (GNLSE)

iΨt=−Ψxx +V(x)Ψ + W(x)F(|Ψ|2)Ψ,(1)

where complex-valued functions V(x) and W(x) can be

referred to as a linear and nonlinear potential, respec-

tively, and real-valued function F(·), such that F(0) = 0,

speciﬁes the nonlinearity. Equation (1) is a canonical

model describing evolution of nonlinear waves in various

physical settings. In particular, in optical applications, Ψ

corresponds to the dimensionless amplitude of the elec-

tric ﬁeld. Its dependence on coordinate tdescribes evo-

lution of the pulse along the propagation direction, and

xis the transverse coordinate. Eﬀective complex optical

potentials correspond to the presence of spatial regions

with gain and loss [3, 4], which, in particular, can be

implemented using coherent multilevel atoms driven by

external laser ﬁelds [5, 6].

∗email: d.zezyulin@gmail.com

When the imaginary parts of both potentials vanish,

the model becomes conservative, and its stationary non-

linear modes Ψ(x, t) = e−iµtψ(x) can be parameterized

by the continuous change of the real frequency µ. How-

ever, if at least one of the potentials is complex, then

the continuous families of solitons typically disappear.

At the same time, there exist at least two types of non-

Hermitian potentials that do support continuous fami-

lies of nonlinear modes (see a more detailed discussion in

[7]). Systems of the ﬁrst type obey the parity-time (PT )

symmetry [3, 8]: in this case real and imaginary parts of

functions V(x) and W(x) are even and odd functions of

x, respectively [9, 10]. The second type corresponds to

the so-called Wadati-type potentials [11] for which

V(x) = −w2(x) + iwx(x), W (x)≡1,(2)

where w(x) is a real-valued diﬀerentiable function which

is not required to have any particular symmetry (at the

same time, if w(x) is even, then the resulting Wadati po-

tential V(x) becomes PT symmetric, i.e., the two types

of potentials are partially overlapping). In what follows,

it will be convenient to say that the GNLSE (1) with

potentials (2) corresponds to linear Wadati potentials.

Apart from the existence of continuous families of non-

linear modes [12], linear Wadati potentials are known

to support several other remarkable features. When the

nonlinearity is absent, i.e., F≡0, linear Wadati poten-

tials can have all-real spectrum of eigenvalues [13] and

undergo distinctive phase transition from all-real to com-

plex spectrum through an exceptional point or a self-dual

spectral singularity [14, 15]. Returning to the nonlinear

setup, Wadati-type potentials support the existence of

constant-amplitude nonlinear waves [16], feature unusual

dynamical behavior near the phase-transition threshold

[17], and eigenvalue quartets in the linear-stability spec-

trum [18]. Additionally, when a Wadati potential is PT

symmetric, it allows for bifurcations of families of non-

PT -symmetric modes [19], which is impossible for PT -

symmetric potentials of general form. While the the-

arXiv:2210.15552v1 [nlin.PS] 27 Oct 2022

ory of Wadati potentials is far from being complete, on

the qualitative level some of their peculiar properties can

be explained by the existence of a ‘conserved’ (i.e., x-

independent) quantity which constrains the shape of sta-

tionary modes [20].

Unusual and not yet fully understood properties of Wa-

dati potentials encourage to look for their possible gener-

alizations. In the meantime, most of the presently avail-

able studies of Wadati potentials are basically limited by

GNLSE with spatially-uniform power-law and saturating

nonlinearities [21, 22]. In the present paper, we introduce

a more profound generalization of Wadati potentials by

considering the situation where the base function w(x)

depends not only on the spatial coordinate xbut also on

the amplitude of the ﬁeld Ψ. We argue that the extended

system preserves some of the properties of linear Wadati

potentials. Particular realizations of the introduced gen-

eralization lead to nonlinear systems of potential physical

relevance. Those include a GNLSE equation with the ad-

ditional higher-order nonlinearity that emerges due to the

stimulated response in optical ﬁbers and a GNLSE with

the derivative nonlinearity. Combining our analytical un-

derstandings with simple demonstrative computations,

we construct continuous families of nonlinear modes and

discuss their linearization spectra. Additionally, we show

that when the generalized system is PT symmetric, it un-

dergoes a symmetry-breaking bifurcation which results in

continuous families of non-PT -symmetric solitons.

The rest of the paper is organized as follows. Sec-

tion II outlines the derivation of the generalized model

and presents several particular realizations of the ex-

tended system. Section III contains a case study of a

model with the derivative nonlinearity. Section IV con-

cludes the paper.

II. CONSTRUCTION OF GENERALIZED

POTENTIALS

For stationary modes Ψ(x, t) = e−iµtψ(x), equation

(1) with linear Wadati potential (2) becomes

µψ =−ψxx + (−w2+iwx)ψ+F(|ψ|2)ψ. (3)

The latter stationary equation has been studied in the

previous literature [12, 18–24]. In particular, it is known

that if Eq. (3) is considered as a dynamical system with

xplaying the role of the evolution variable, then the re-

spective ‘dynamics’ is constrained by a conserved (i.e., x-

independent) quantity [20]. Our construction of the gen-

eralized model relies on a ‘gauge transformation’ which

converts the stationary equation (3) to another ordinary

diﬀerential equation, where the x-independent ‘integral

of motion’ has an especially simple form [25]. Indeed,

using the substitution ψ(x) = φ(x)eiRw(x)dx, where φ(x)

is a new stationary ﬁeld, from (3) we obtain

µφ =−φxx −2iw(x)φx+F(|φ|2)φ. (4)

Multiplying the latter equation by φ∗

xand adding it with

its complex conjugate, we obtain the integral of motion

µ|φ|2=−|φx|2+Z|φ|2

F(ξ)dξ + const,(5)

where ‘const’ is an arbitrary x-independent quantity.

Clearly, for localized modes with limx→±∞ φ(x) =

limx→±∞ φx(x) = 0 this constant must be zero. Using

the obtained integral of motion as a qualitative argu-

ment, continuous families of nonlinear localized modes

have been constructed in linear Wadati potentials [20].

In the meantime, since the integral (5) does not contain

function w(x) explicitly, Eq. (4) can be generalized nat-

urally by assuming that the base function w(x) depends

not only on the spatial coordinate xbut also on the am-

plitude of the ﬁeld. This idea suggests to replace Eq. (4)

with the following more general one:

µφ =−φxx −2iA(x, |φ|2)φx+F(|φ|2)φ, (6)

where A(·,·) is a real-valued function of two variables. It

is easy to check that the identity (5) remains valid for

the newly introduced Eq. (6). We further make the ‘in-

verse’ gauge transformation φ(x) = ψ(x)eiRz(x)dx, where

z(x) := B(x, |ψ|2) is another real-valued function, which

converts (6) into the following equation:

µψ =−ψxx +2AB +B2−i∂B(x, |ψ|2)

∂x −i∂B(x, |ψ|2)

∂|ψ|2(|ψ|2)xψ−2i(A+B)ψx+F(|ψ|2)ψ. (7)

The obtained stationary equation corresponds to the following temporal evolution problem:

iΨt=−Ψxx +2AB +B2−i∂B(x, |Ψ|2)

∂x −i∂B(x, |Ψ|2)

∂|Ψ|2(|Ψ|2)xΨ−2i(A+B)Ψx+F(|Ψ|2)Ψ.(8)

Equation (8) is the central point of the present paper.

It represents the generalization of the previously stud-

ied GNLSE with linear Wadati potentials, which can be

recovered from Eq. (8) by setting A=−B=w(x).

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NonlinearSchrodingerequationswithamplitude-dependentWadatipotentialsDmitryA.ZezyulinDepartmentofPhysicsandEngineering,ITMOUniversity,St.Petersburg197101,Russia(Dated:October28,2022)ComplexWadati-typepotentialsoftheformV(x)=w2(x)+iwx(x),wherew(x)isareal-valuedfunction,areknowntopossessanumberofintr...

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Nonlinear Schr odinger equations with amplitude-dependent Wadati potentials Dmitry A. Zezyulin Department of Physics and Engineering ITMO University St. Petersburg 197101 Russia

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