# Non-linear Schr$\ddot{o}$dinger equation with time-dependent balanced loss-
gain and space-time modulated non-linear interaction

Supriyo Ghosh<EMAIL_ADDRESS>Pijush K. Ghosh
<EMAIL_ADDRESS>Department of Physics, Siksha-Bhavana,
Visva-Bharati, Santiniketan, PIN 731 235, India.

###### Abstract

We consider a class of one dimensional vector Non-linear Schr$\ddot{o}$dinger
Equation(NLSE) in an external complex potential with Balanced Loss-Gain(BLG)
and Linear Coupling(LC) among the components of the Schr$\ddot{o}$dinger
field. The solvability of the generic system is investigated for various
combinations of time modulated LC and BLG terms, space-time dependent strength
of the nonlinear interaction and complex potential. We use a non-unitary
transformation followed by a reformulation of the differential equation in a
new coordinate system to map the NLSE to solvable equations. Several
physically motivated examples of exactly solvable systems are presented for
various combinations of LC and BLG, external complex potential and nonlinear
interaction. Exact localized nonlinear modes with spatially constant phase may
be obtained for any real potential for which the corresponding linear
Schr$\ddot{o}$dinger equation is solvable. A method based on supersymmetric
quantum mechanics is devised to construct exact localized nonlinear modes for
a class of complex potentials. The real superpotential corresponding to any
exactly solved linear Schr$\ddot{o}$dinger equation may be used to find a
complex-potential for which exact localized nonlinear modes for the NLSE can
be obtained. The solutions with singular phases are obtained for a few complex
potentials.

##  I Introduction

The Non-linear Schr$\ddot{o}$dinger equation(NLSE) appears in the context of
mathematical modelling of many physical phenomena in different domains of
science, like optics in nonlinear media Kivshar ; Serkin ; Serkin1 , Bose-
Einstein Condensates(BEC)Dalfovo ; Pitaevskii1 ; Kevrekidis ; Kengne , plasma
physicsDodd , gravity wavesTrulsen , Bio-molecular dynamicsDavydov  etc. NLSE
exhibits solitonZabusky ; Newell ; Zakharov3 ; Infeld  solutions and is an
integrable system with a very rich mathematical structureSulem ; Bourgain ;
Ablowitz ; Lamb . The quantized NLSE describes a hardcore Bose-gas with delta-
function interaction for which various physical properties including
correlations functions can be computed analyticallykorepin . Several
generalisations of the NLSE have been considered to model different emerging
phenomena in physics over the last few decades. For example, the NLSE with an
external confining potential is known as Gross-Pitaevskii equationPitaevskii1
and is relevant in the context of BEC. The investigations on collapse and
growth of quasi-condensates in NLSE with harmonic confinement offers a rich
variety of mathematical structurepkg2 ; pkg3 ; Wadati . Localized, unbounded
and periodic potentials have also been investigated with interesting results.
The NLSE with space Abdullaev ; Sakaguchi ; Carpentier ; Beitia ; Verde ;
Theocharis ; Porter  and time Saito ; Kevrekidis1 ; Brazhnyi ; Berge ; Towers
; Kevrekidis2 ; Abdullaev1 ; Garcia ; Abdullaev2 ; Montesinos ; Itin ; Konotop
modulated nonlinear strengths is also of immense interest in the same context.

The generalizations of NLSE based on underlying ${\cal{PT}}$-symmetryBenderl
may be broadly classified into three different classes: (i) NLSE with
${\cal{PT}}$-symmetric complex confining potential, (ii) the non-local NLSE,
and (iii) NLSE with LC and BLG terms. Solvable models of NLSE with
${\cal{PT}}$-symmetric complex localized potentialKonotop1 ; Shi ; Ortiz1 ;
Ortiz2 ; Cruz ; Midya ; Midya1 ; Musslimani  have been considered which admits
soliton solutions. The investigations on ${\cal{PT}}$-symmetry breaking as a
function of the system parameters reveal a host of interesting issues
including the behaviour of solitons at the exceptional pointsKonotop1 . The
non-local NLSEAblowitz1  is integrable and admits bright as well as dark
soliton for the same range of the nonlinear strength. The non-local vector
NLSE is also integrable and amenable to exact solutions and real-valued
physical quantitiesds-pkg . Further, it has lead to a new branch of integrable
models where non-local generalizations of previously known integrable models
have been consideredpkg1 ; Ablowitz-2 ; Avadh .

The central focus of this article is on NLSE with LC and BLG terms for which,
in addition to real-valued linear and nonlinear couplings, the components of
NLSE are subjected to loss or gain such that the power or a modified form of
it named as pseudo-power is conservedpiju . Soliton solution was found in the
discrete non-linear Schr$\ddot{o}$dinger equation(DNLS) with gain and
lossSuchkov . The NLSE with balanced loss and gain has been investigated over
the last one decade or so and is known to admit brightBludov1 ; Driben1  and
dark solitons Bludov , breathersBarashenkov ; Driben2 , rogue wavesKharif ,
exceptional pointsDriben2 ; Konotop1 , etc. Further, the power oscillation is
observed in some specific models which may have interesting technological
applicationspkg . The NLSE with time-modulated loss-gain terms have also been
investigated and shown its efficiency in the stabilization of solitonsDriben2
and soliton switchingAbdullaev3 ; Abdullaev4 ; Konotop1 . The NLSE with LC and
BLG has also been investigated from the viewpoint of exact solvability and
exact analytical solutions of some models are constructed under specific
reduction of the original equationDriben1 ; Alexeeva . Recently, a method was
prescribed to construct exactly solvable models of NLSE with LC and BLG by
using a non-unitary transformation which removes the LC and BLG terms
completely at the cost of modifying the strength of the nonlinear termpkg .
Further, the LC and BLG terms are removed completely without modifying the
nonlinear term at all for some specific models for which the non-unitary
transformation can be identified as a pseudo-unitary transformation. The
method has been used to construct exactly solvable models exhibiting power-
oscillation.

In this article, we consider a class of NLSE with time dependent LC and BLG as
well as space-time modulated nonlinear term in an external complex potential.
Both time dependentAbdullaev4  and constantDriben1 ; Driben2  BLG with
particular types of NLSE and vanishing confining potential have been studied
earlierKonotop1 . However, most of those studies are based on approximation
and/or numerical methods . The sole purpose of this article is to investigate
the exact solvability of a generic system for various combinations of the
time-modulated LC and BLG terms, space-time modulated nonlinear strengths and
external potentials.

We follow a two-step approach. The LC and the BLG terms are removed completely
in the first step via a non-unitary transformation by generalizing the
technique prescribed in Ref. pkg . This results in modifying the time-
dependence of the nonlinear term and in general, a real-valued non-linear term
becomes complex after the non-unitary transformation. We find that the LC and
BLG terms are removed completely provided they have the same time-modulation.
The method of separation of variables requires that the modified non-linear
term is real. This is ensured by fixing the space-time modulated strengths of
the nonlinear term. In the second step, the method of similarity
transformation is used to investigate the solvability of the resulting
equations. The method involves defining the differential equation in a new co-
ordinate system accompanied by the multiplication of a scale-factor to the
amplitude. This results in a system of equations with solvable limits for
appropriate choice of co-ordinates and scale-factor.

We construct several exactly solvable models to exemplify the general method.
The scheme for obtaining solvable limits for real and complex external
potentials are different. Exact localized nonlinear modes with spatially
constant phase may be obtained for a real potential for which the
corresponding linear Schr$\ddot{o}$dinger equation is solvable. Exact
analytical solutions with space-time dependent phase are also obtained for a
few real potential. The complex potential offers more flexibility in
comparison with real potential in constructing solvable models. We devise a
general method based on supersymmetric quantum mechanicskhare  to find a large
class of complex potentials admitting localized nonlinear modes. In general,
phases of the nonlinear modes are space-time dependent. Phase singularity is
present in the nonlinear mode for a few systems with complex potentials.

A few model independent features of the exact solutions are the following. The
exact solutions are insensitive to the detail form of the space-modulated
nonlinear strength. It is not obvious whether this is specific to the ansatz
chosen for obtaining exact solutions or a manifestation of some underlying
symmetry. Further, the power-oscillation in time is absent for time-
independent non-linear strength. There are NLSE with constant LC, BLG and
nonlinear strength for which power-oscillation has been observedpkg . The
class of NLSE considered in this article has nonlinear interaction which is
different from the model considered in Ref. pkg . Thus, the power oscillation
in time seems to be dependent on the nonlinear interaction.

The plan of the article is the following. The model is introduced in Sec. II
along with Lagrangian and Hamiltonian formulations of the system. In Sec. III,
the two-step approach is implemented to map the original equation into a
solvable model. Explicit examples are constructed for systems without and with
external real potentials in Sec. IV and V, respectively. Results pertaining to
complex confining potential are presented in Sec. VI. Finally, the findings
are summarized in Sec. VII with discussions and outlook. In Appendix-I,
solutions of a nonlinear equation arising in the discussions of solvable
models are described.

##  II The Model

We consider one dimensional 2-component NLSE as

$\displaystyle i\psi_{1t}$ $\displaystyle=$
$\displaystyle-\psi_{1xx}+k^{*}(t)\psi_{2}+i\gamma(t)\psi_{1}+V(x)\psi_{1}+\left(g_{11}(x,t)|\psi_{1}|^{2}+g_{12}(x,t)|\psi_{2}|^{2}\right)\psi_{1}$
$\displaystyle i\psi_{2t}$ $\displaystyle=$
$\displaystyle-\psi_{2xx}+k(t)\psi_{1}-i\gamma(t)\psi_{2}+V(x)\psi_{2}+\left(g_{21}(x,t)|\psi_{1}|^{2}+g_{22}(x,t)|\psi_{2}|^{2}\right)\psi_{2}$
(1)

where $k(t)$ is the strength of the linear coupling among the two fields and
$k^{*}$ denotes the complex conjugate of $k$. The time-modulated strength of
the balanced loss gain terms is $\gamma(t)$. The space-time modulated
strengths of the non-linear interaction are denoted as $g_{ij}(x,t)$. In the
terminology of optics, $g_{11},g_{22}$ are self phase modulation terms, while
$g_{21},g_{12}$ correspond to cross-phase modulation. The external potential
$V(x)$ is relevant for the mean-field description of Bose-Einstein
condensates. Several ${\cal{PT}}$-symmetric complex potentials $V(x)$ exhibit
many interesting phenomenon. We discuss both real and complex potentials
separately in this article and choose $V(x)=s(x)+i\tilde{s}(x)$, where
functions $s(x),\tilde{s}(x)$ are real. There are several solvable limits of
Eq.(1), starting from the simplest case of $g_{ij}=0,k=\gamma=0$ which models
the optical wave propagation under paraxial approximationGanainy ; Markis .
The integrable canonical form of Manakaov-Zakharov-Schulman(MZS) systemkk ; zs
is obtained for constant coefficients $g_{ij}$ with
$g_{11}=g_{21},g_{12}=g_{22}$ and vanishing linear coupling, loss-gain terms
and external potential. Exact solutions of MZS system with non-vanishing
constant loss-gain terms and time-dependent $g_{ij}$ with specific time-
modulation have been constructed via non-unitary transformationspkg . The
solutions for constant $g_{ij}$ appear as a special case. The Gross-Pitaevskii
equation, which provides a mean-field description of BEC, is obtained for
vanishing loss-gain terms and non-vanishing $V(x)$. Several exact solutions
for different choices of $V(x)$ and space-modulated nonlinear strengths have
been constructedBeitia . It appears that solvable limits of Eq. (1) in its
generic form, particularly with time-dependent $k,\gamma$, space-time
dependent $g_{ij}$ and with or without the external potential $V(x)$, have not
been investigated so far. The method of non-unitary transformationpkg  will be
employed to investigate solvability of Eq. (1) in its generic form.

It is convenient to rewrite Eq.(1) in a compact form in terms of the Pauli
matrices $\sigma_{1},\sigma_{2},\sigma_{3}$, $2\times 2$ identity matrix
$\sigma_{0}$ and two-component complex vector
$\Psi(x,t)=(\psi_{1},\psi_{2})^{T}$, where the superscript T denotes
transpose. We define projection operators
$P_{\pm}=\frac{1}{2}\left(\sigma_{0}\pm\sigma_{3}\right)$ with the properties
$P_{+}\Psi=(\psi_{1},0)^{T},P_{-}\Psi=(0,\psi_{2})^{T}$ and the matrices
$\sigma_{\pm}:=\frac{1}{2}\left(\sigma_{1}\pm i\sigma_{2}\right)$. The
matrices $P_{\pm},\sigma_{\pm}$ form a basis for $2\times 2$ matrices. We
introduce a non-hermitian scalar gauge potential $A(t)$ and an operator
$D_{0}$,

$\displaystyle
A(t)=k\sigma_{-}+k^{*}\sigma_{+}+i\gamma\sigma_{3},D_{0}:=\sigma_{0}\frac{\partial}{\partial
t}+iA(t).$ (2)

The term $i\gamma(t)\sigma_{3}\Psi$ corresponds to time-dependent BLG, while
$k(t)\sigma_{-}+k^{*}(t)\sigma_{+}$ denote time-dependent LC. The time-
dependence of $A(t)$ can be controlled by choosing $k(t)$ and $\gamma(t)$
appropriately. The operator $D_{0}$ resembles the temporal component of
covariant derivative with non-hermitian scalar gauge-potential $A$. It may be
noted that we have not included any term of the form $f(t)\sigma_{0}$ to $A$,
since it can always be trivially gauged away through a $U(1)$ transformation
of the form $\Psi\rightarrow e^{-i\int^{t}f(t^{\prime})dt^{\prime}}\Psi$. The
$U(1)$ phase-factor $e^{-i\int^{t}f(t^{\prime})dt^{\prime}}$ does not affect
the dynamical variables like power, width of the wave-packet and its speed.
Thus, the form of $A$ is quite general.

Eq. (1) can be rewritten as,

$\displaystyle iD_{0}\Psi$ $\displaystyle=$
$\displaystyle-\Psi_{xx}+V(x)\Psi+\left[\left(g_{11}(x,t)\Psi^{\dagger}P_{+}\Psi+g_{12}(x,t)\Psi^{\dagger}P_{-}\Psi\right)P_{+}\right.$
(3) $\displaystyle+$
$\displaystyle\left.\left(g_{21}(x,t)\Psi^{\dagger}P_{+}\Psi+g_{22}(x,t)\Psi^{\dagger}P_{-}\Psi\right)P_{-}\right]\Psi$

which is convenient for further analysis using non-unitary transformation.
Defining the power $P=\Psi^{\dagger}\Psi$ and $Q=\int dxP$, it can be checked
easily that $Q$ is not a constant of motion for $\gamma\neq 0$. The power-
oscillation is a hallmark of some systems with balanced loss-gain and it is a
manifestation of fact that $Q$ is not a constant of motion. In fact, the
expression of $P$ in Ref. pkg  contains a time-dependent periodic-function
which becomes unity in the limit of vanishing loss-gain terms multiplied by a
space-dependent function. The quantity $\tilde{Q}=\int dx\tilde{P}$ with
$\tilde{P}=\Psi^{\dagger}\eta\Psi$ is a constant of motion for $\gamma\neq
0,g_{11}=g_{21},g_{12}=g_{22}$, provided the matrix $A$ is $\eta$-pseudo-
hermitian, i.e. $A^{\dagger}=\eta A\eta^{-1}$. The positive-definite matrix
$\eta$ for $A$ in Eq. (2) is given in Ref. pkg ; piju . The system admits a
Lagrangian and Hamiltonian formulation for $g_{12}=g_{21}$:

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left[\Psi^{\dagger}D_{0}\Psi-\left(D_{0}\Psi\right)^{\dagger}\Psi\right]+\frac{1}{2}\Psi^{\dagger}\left(A^{\dagger}-A\right)\Psi-{\Psi_{x}}^{\dagger}\Psi_{x}-V(x)\Psi^{\dagger}\Psi-\frac{g_{11}}{2}\left(\Psi^{\dagger}P_{+}\Psi\right)^{2}$
$\displaystyle-$
$\displaystyle\frac{g_{22}}{2}\left(\Psi^{\dagger}P_{-}\Psi\right)^{2}-g_{12}\left(\Psi^{\dagger}P_{+}\Psi\right)\left(\Psi^{\dagger}P_{-}\Psi\right),$
$\displaystyle{\cal{H}}$ $\displaystyle=$
$\displaystyle\Psi_{x}^{\dagger}\psi_{x}+V(x)\Psi^{\dagger}\Psi+\Psi^{\dagger}A\Psi+\frac{g_{11}}{2}\left(\Psi^{\dagger}P_{+}\Psi\right)^{2}+\frac{g_{22}}{2}\left(\Psi^{\dagger}P_{-}\Psi\right)^{2}+g_{12}\left(\Psi^{\dagger}P_{+}\Psi\right)\left(\Psi^{\dagger}P_{-}\Psi\right).$
(4)

The Hamiltonian density ${\cal{H}}$ is complex-valued for $\gamma\neq 0$ and
the corresponding quantum ${\cal{H}}$ is non-hermitian with appropriate
quantization condition. For $g_{11}=g_{22}=g_{12}$, the Lagrangian density is
invariant under global phase transformation $\Psi\rightarrow
e^{i\theta\eta}\Psi,\theta\in\mathbb{R}$ and the corresponding conserved
charge is $\tilde{Q}$.

##  III Transformation to solvable equation

The purpose of this section is to generalize the technique outlined in Ref.
pkg  to remove the time-modulated BLG and LC by using a non-unitary
transformation. The transformation modifies the non-linear term and suitable
choices of the space-time modulated coefficients lead to solvable models. We
consider a transformation relating $\Psi$ to a two-component complex scalar
field $\Phi$ as follows,

$\displaystyle\Psi(x,t)=U(t)\Phi(x,t)$ (5)

It will appear later that the operator $U(t)$ in general is non-unitary. The
BLG and LC terms are contained only in $D_{0}\Psi$ and it can be checked that
$D_{0}\Psi=U\partial_{t}\Phi$ provided $U$ satisfies the equation,

$\displaystyle\frac{dU}{dt}=-iAU(t)$ (6)

A similar equation appears in the study of scattering theory in the
interaction picture and the operator analogous to $U$ in that context is known
as Dyson operator, which describes the time-evolution of a state
$|\Psi(t_{0})\rangle$ to $|\Psi(t)\rangle$, i.e.
$|\Psi(t)\rangle=U(t,t_{0})|\Psi(t_{0})\rangle$. The operator $U$ can not be
identified as Dyson operator in the present case since it maps $\Phi(x,t)$ to
$\Psi(x,t)$ at the same time $t$ and the physical context is also different.
However, the solution of $U$ in Eq. (6) may be obtained as an infinite series
along the line of derivation of Dyson series. We are interested in a closed
form expression of $U(t)$ from the viewpoint of exact solvability. This leads
to the solution of $U(t)$ for specific choice of $A$ as,

$\displaystyle U(t)=e^{-i\int^{t}A(t^{\prime})dt^{\prime}},\ \
\left[A,\int^{t}A(t^{\prime})dt^{\prime}\right]=0$ (7)

Note that $U(t)$ is non-unitary, since $A(t)$ is non-hermitian. Unitary
transformations have been used in physics in different contexts, particularly
in the context of field theory, for past several decades. To the best of our
knowledge, the use of non-unitary transformation to construct exact solution
in a systematic way has not been considered earlier. Within this background,
it may be noted that a unitary transformation is a change of basis in the
field-space by keeping the norm fixed, while the norm is not preserved under a
non-unitary transformation. This is a major difference —systems related by
unitary transformation are gauge equivalent, while the same can not be claimed
for systems related by non-unitary transformation. This is manifested in the
result that the power of the standard Manakov system is dif- ferent from the
Manakov system with balanced loss-gain, although they are connected via a non-
unitary/pseudo-unitary transformationpkg . The same is true for the system
considered in this article, since $\Psi^{\dagger}\Psi\neq\Phi^{\dagger}\Phi$.
Similarly, one can show that the time-dependence of other observables like
width of the wave-packet and its speed of growth are different for systems
connected via non-unitary/pseudo-unitary trans- formation. However, for
systems connected by unitary transformation, observables like power, width of
the wave-packet and its growth are identical. The unitary trans- formation
only mixes different components of the field leading to different expression
for the solutions.

The second condition of (7) implies that $k(t)$ and $\gamma(t)$ should have
the same time dependence. We choose
$k(t)=\mu_{0}(t)\beta,\gamma(t)=\mu_{0}(t)\Gamma$ in terms of an arbitrary
real function $\mu_{0}(t)$ and $\beta\in\mathbb{C},\Gamma\in\mathbb{R}$
leading to the following expression of $A(t)$:

$\displaystyle A(t)=\mu_{0}(t)A_{0},\
A_{0}:=\beta\sigma_{-}+\beta^{*}\sigma_{+}+i\Gamma\sigma_{3}.$ (8)

We introduce the following quantities:

$\displaystyle\mu(t)=\int^{t}\mu_{0}(t^{\prime})dt^{\prime},\
\epsilon_{0}=\sqrt{{|\beta|}^{2}-\Gamma^{2}},\
\epsilon(t)=\mu(t)\epsilon_{0}.$ (9)

The qualitative behaviour of the operator $U(t)$ primarily depends on
$\epsilon_{0}$ being positive, zero and purely imaginary. We denote the
corresponding $U(t)$ as $U_{+}(t),U_{0}(t)$ and $U_{I}(t)$, respectively, with
the following expressions:

$\displaystyle U_{+}(t)$ $\displaystyle=$
$\displaystyle\sigma_{0}\cos(\epsilon)-\frac{i{A}_{0}}{\epsilon_{0}}\sin(\epsilon),\epsilon_{0}>0$
$\displaystyle U_{0}(t)$ $\displaystyle=$
$\displaystyle\sigma_{0}-iA_{0}\mu(t)$ $\displaystyle U_{I}(t)$
$\displaystyle=$
$\displaystyle\sigma_{0}\cosh({|\epsilon_{0}|}\mu(t))-i\frac{A_{0}}{|\epsilon_{0}|}\sinh({|\epsilon_{0}|}\mu(t))$
(10)

The effect of the time-dependent scale-factor $\mu_{0}(t)$ in $A$ is to change
the time-modulation of $U(t)$. For a time-independent $\mu_{0}(t)$, i.e.
$\epsilon(t)=\epsilon_{0}t$, the matrix $U_{+}(t)$ becomes periodic in time.
However, the matrices $U_{0}(t)$ and $U_{I}(t)$ becomes unbounded and the
solutions $\Psi(x,t)$ becomes unbounded even for a bounded $\Phi(x,t)$. An
interesting point to note is that bounded sloutions may be obtained for all
three cases, namely, $U_{+},U_{0},U_{I}$, by suitably choosing the
$\mu_{0}(t)$. For example, $\mu(t)$ reduces to the Error function, i.e.
$\mu(t)=erf(t)$, for the choice

$\displaystyle\mu_{0}(t)=\frac{2}{\sqrt{\pi}}e^{-t^{2}}$ (11)

and $U(t)$ corresponding to all the three cases discussed above are bounded.
Thus, appropriate time-modulation may be used to stabilize a system whose
unboundedness comes solely from $U(t)$.

The removal of the BLG and LC terms through the non-unitary transformation
modifies the nonlinear term and imparts additional time-dependence on it. In
particular, substituting the expression of $\Psi(x,t)$ in Eq. (5) into Eq.
(3), yields the following

$\displaystyle i\Phi_{t}$ $\displaystyle=$
$\displaystyle-\Phi_{xx}+V(x)\Phi+\left[K(g_{11},g_{21})\
\Phi^{\dagger}F_{+}\Phi+K(g_{12},g_{22})\ \Phi^{\dagger}F_{-}\Phi\right]\Phi,$
(12)

where
$K(\xi_{1},\xi_{2})=\xi_{1}\left(U^{-1}P_{+}U\right)+\xi_{2}\left(U^{-1}P_{-}U\right)$
and $F_{\pm}=U^{\dagger}P_{\pm}U$. With the introduction of the functions
$T_{\pm}$,

$\displaystyle
T_{\pm}=\frac{\Gamma\mu(t)}{\epsilon^{2}}\sin^{2}(\epsilon)\pm\frac{\sin(2\epsilon)}{2\epsilon},$
(13)

the explicit expression of $F_{\pm}$ and $K$ may be obtained as follows:

$\displaystyle F_{\pm}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}+\Gamma\mu(t)T_{\pm}\right)\sigma_{0}-iT_{\pm}\mu(t)\left(\beta^{*}\sigma_{+}-\beta\sigma_{-}\right)+\left(\frac{\Gamma\mu(t)}{2\epsilon}\sin(2\epsilon)\pm\frac{1}{2}\cos(2\epsilon)\right)\sigma_{3},$
$\displaystyle K(\xi_{1},\xi_{2})$ $\displaystyle=$
$\displaystyle\frac{\xi_{1}+\xi_{2}}{2}\sigma_{0}+i\left(\xi_{1}-\xi_{2}\right)\mu(t)\left(\beta^{*}\sigma_{+}T_{-}+\beta\sigma_{-}T_{+}\right)+\left(\xi_{1}-\xi_{2}\right)\left(\frac{1}{2}-\frac{\mu^{2}(t){|\beta|}^{2}}{\epsilon^{2}}\sin^{2}(\epsilon)\right)\sigma_{3}$
(14)

We have presented the above expressions for the generic allowed values of
$\epsilon_{0}$ and appropriate limits may be taken depending on the physical
context of the problem. It may be noted that $T_{\pm}$ is always real valued
independent of whether $\epsilon_{0}$ is positive, zero or purely imaginary.
Further, the operator $F_{\pm}$ is necessarily hermitian, while
$K(\xi_{1},\xi_{2})$ is hermitian either for (i) $\xi_{1}=\xi_{2}$ or (ii)
$\xi_{1}\neq\xi_{2},T_{+}=-T_{-}$. The condition $T_{+}=-T_{-}$ is achieved
for $\Gamma\mu(t)=0$. The choice $\mu(t)=0\ \forall\ t$ is excluded, since LC
as well as BLG terms also vanish in this limit. Thus, for
$\xi_{1}\neq\xi_{2}$, $K(\xi_{1},\xi_{2})$ is hermitian for $\Gamma=0$. i.e.
the limit of vanishing loss-gain terms. In general, the nonlinear term in Eq.
(12) is non-hermitian —the non-unitary transformation involving $U(t)$ removes
the loss-gain terms at the cost of introducing non-hermiticity in the non-
linear term. The nonlinear term in Eq. (12) becomes hermitian either for (i)
$g_{11}=g_{21},g_{22}=g_{12},\Gamma\neq 0$ or (ii)$\Gamma=0$, since both
$K(g_{11},g_{21})$ and $K(g_{11},g_{21})$ are hermitian in these limits. The
second condition corresponds to vanishing loss-gain term for which $U(t)$ is
unitary and the result is expected. The nonlinear term for the first condition
takes the form of MZS systemkk ; zs
$\left[\Psi^{\dagger}\left(g_{11}P_{+}+g_{22}P_{-}\right)\Psi\right]\Psi$,
which remains real-valued after the non-unitary transformation. This
particular form of nonlinear interaction with $V(x)=0,\mu_{0}(t)=1$ and space-
time independent $g_{11},g_{22}$ has been investigated earlierpkg .

A pertinent question at this juncture is whether or not the solution $\Psi$ of
Eq. (3) corresponding to a stable solution $\Phi$ of Eq. (12) is also stable.
The answer is that the transformation (5) that connects the original system
described by Eq. (3) to the Eq. (12) does not alter the stability property of
$\Phi$ for a bounded $U(t)$. In order to see this, we consider
$\Phi(x,t)=\tilde{\Phi}(x,t)+\chi(x,t)$, where $\tilde{\Phi}(x,t)$ is an exact
solution of Eq. (12) and $\chi(x,t)$ is small perturbation. Plugging the
expression of $\Phi(x,t)$ in Eq. (12), we obtain the following equation in the
leading order of the perturbation,

$\displaystyle i\chi_{t}$ $\displaystyle=$
$\displaystyle-\chi_{xx}+V(x)\chi+\left[K(g_{11},g_{21})\left\\{\tilde{\Phi}^{\dagger}F_{+}\chi+\chi^{\dagger}F_{+}\tilde{\Phi}\right\\}+K(g_{12},g_{22})\left\\{\tilde{\Phi}^{\dagger}F_{-}\chi\right.\right.$
(15) $\displaystyle+$
$\displaystyle\left.\left.\chi^{\dagger}F_{-}\tilde{\Phi}\right\\}\right]\tilde{\Phi}+\left[K(g_{11},g_{21})\tilde{\Phi}^{\dagger}F_{+}\tilde{\Phi}+K(g_{12},g_{22})\tilde{\Phi}^{\dagger}F_{-}\tilde{\Phi}\right]\chi+{\cal{O}}\left(\chi^{2}\right)$

The Eq. (15) determines the stability of the exact solution $\tilde{\Phi}$ and
the solution is stable if $\chi$ is a bound state. The exact solution of Eq,
(3) corrresponding to $\tilde{\Phi}$ is $\tilde{\Psi}=U\tilde{\Phi}$. We
consider $\Psi=\tilde{\Psi}+U\eta(x,t)$, where $U(t)$ is taken to be bounded
in time and $\eta(x,t)$ is an arbitrary function. The perturbation to the
exact solution $\tilde{\Psi}$ is $U\eta(x,t)$ and arbitrariness of the small
fluctuations is contained in $\eta(x,t)$. We find after substituting the
expression of $\Psi(x,t)$ in Eq. (3) and keeping only the leading order terms
that $\eta$ satisfies the Eq. (15). Thus, the stability of the original Eq.
(3) and the transformed Eq. (12) is governed by the same Eq. (15) and the
transformation (5) can not change the stability property for a bounded $U$ in
time and under identical initial conditions.

The investigations on the complete integrability of the system described by
Eq. (12) is a highly nontrivial problem in presence of $V(x)$, space-time
modulated nonlinear-strengths and loss-gain terms. We are interested in this
article in finding solvable limits of Eq. (1) so that the exact solutions can
be used in plethora of physical systems in which it appears. The reduction of
Eq. (1) to Eq. (12) with a closed form expression for $U(t)$ has been
performed on general ground. In order to proceed further for analyzing two-
component vector non-linear Schr$\ddot{o}$dinger equation of the form given in
Eq. (12), we use the method of separation of variables by choosing one of the
standard ansatzes which is consistent with the group-theory based
analysisBeitia  of Eq. (12) for time-independent and real-valued nonlinear
strength. We consider the solution of Eq. (12) as,

$\Phi(x,t)=W\ R(x)\ e^{i\big{(}\theta(x)-Et\big{)}},$ (16)

where $W=(W_{1}e^{i\theta_{1}},W_{2}e^{i\theta_{2}})^{T}$ is a two-component
constant, complex vector, $R(x)$ and $\theta(x)$ are real functions of their
arguments and $E$ is a constant. Inserting the above expression into Eq. (12)
we find,

$\displaystyle R_{xx}$ $\displaystyle=$
$\displaystyle-i\left(R\theta_{xx}+2R_{x}\theta_{x}-R\tilde{s}(x)\right)-\left(E-s(x)\right)R+R\theta^{2}_{x}$
(17) $\displaystyle+$
$\displaystyle\frac{1}{W^{\dagger}W}\left[W^{\dagger}K(g_{11},g_{21})W\
W^{\dagger}F_{+}W+W^{\dagger}K(g_{12},g_{22})W\
W^{\dagger}F_{-}W\right]R^{3}(x).$

It is to be noted that in general the potential $V(x)$ in Eq.(12) is complex
and has the form $V(x)=s(x)+\tilde{s}(x)$. The nonlinear term is complex and
time-dependent. The method of separation of variables with the ansatz as above
fails unless the coefficient of the nonlinear term is real and time-
independent, which can be achieved with the judicious choice of the space-time
modulated coefficients $g_{ij}(x,t)$. The imaginary part of the coefficient
vanishes for the condition,

$\displaystyle\frac{g_{11}-g_{21}}{g_{12}-g_{22}}=-\frac{W^{\dagger}F_{-}W}{W^{\dagger}F_{+}W}$
(18)

while the real part of the coefficient of the nonlinear term is time-
independent and equals to $f(x)=\frac{1}{2}\left(f_{1}(x)+f_{2}(x)\right)$
provided,

$\displaystyle g_{11}+g_{21}=\frac{f_{1}(x)}{W^{\dagger}F_{+}W},\ \
g_{12}+g_{22}=\frac{f_{2}(x)}{W^{\dagger}F_{-}W}$ (19)

where $f_{1}(x)$ and $f_{2}(x)$ are arbitrary functions. The space-time
modulated strength $g_{ij}$ of the nonlinear interaction may be obtained by
solving Eqs. (18) and (19) which constitute an undetermined system. We solve
the equations by keeping $g_{22}(x,t)$ arbitrary:

$\displaystyle g_{11}(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{W^{\dagger}F_{+}W}\left[f_{1}(x)-f(x)+g_{22}(x,t)W^{\dagger}F_{-}W\right],$
$\displaystyle g_{21}(x.t)$ $\displaystyle=$
$\displaystyle\frac{1}{W^{\dagger}F_{+}W}\left[f(x)-g_{22}(x,t)W^{\dagger}F_{-}W\right],$
$\displaystyle g_{12}(x.t)$ $\displaystyle=$
$\displaystyle\frac{1}{W^{\dagger}F_{-}W}\left[f_{2}(x)-g_{22}(x,t)W^{\dagger}F_{-}W\right]$
(20)

It should be emphasized that each choice of
$g_{22}(x,t),f_{1}(x),f_{2}(x),\mu(t)$ leads to a different classes of the
space-time modulation $g_{ij}$ of the nonlinear term. However, the nonlinear
equation (17) depends only on $f(x)$ and does not keep track of the specific
forms of $g_{22}(x,t),f_{1}(x),f_{2}(x)$, and hence of $g_{ij}$, for fixed
$f(x)$. This implies that the spatial dependence of the solution of Eqn. (1)
in terms of $R(x)$ and $\theta(x)$ is same for a large class of models
characterized by different $g_{ij}$. This result is very important, since
solvability of Eq. (17) leads to solvability of a very large class of NLSE
with LC and BLG terms characterized by various forms of $g_{ij}$.

We choose a symmetric form of the $g_{ij}$ for presenting our results:

$\displaystyle
g_{11}=\frac{f_{1}(x)+G(x,t)}{2W^{\dagger}F_{+}W},g_{21}=\frac{f_{1}(x)-G(x,t)}{2W^{\dagger}F_{+}W},$
$\displaystyle
g_{22}=\frac{f_{2}(x)+G(x,t)}{2W^{\dagger}F_{-}W},g_{12}=\frac{f_{2}(x)-G(x,t)}{2W^{\dagger}F_{-}W},$
(21)

where $G(x,t)$ is an arbitrary function. The introduction of $G(x,t)$ is to
keep track of the arbitrariness present in the solution of the undetermined
system of Eqs. (18) and (19) so that the generality is not lost. The
expressions for $g_{11},g_{21}$ and $g_{22}$ may be obtained by substituting
$g_{22}$ from Eq. (21) in Eq. (20). The expressions for $W^{\dagger}F_{\pm}W$
are,

$\displaystyle W^{\dagger}F_{\pm}W$ $\displaystyle=$ $\displaystyle
b_{0}\left(\frac{1}{2}+T_{\pm}(t)\mu(t)D\right)+b_{3}\left[\frac{\Gamma\mu(t)}{2\epsilon}\sin(2\epsilon)\pm\frac{1}{2}\cos(2\epsilon)\right]$
(22)

where $\beta={|\beta|}e^{i\theta_{3}}$,
$b_{j}=W^{\dagger}\sigma_{j}W,j=0,1,2,3$ and the constant $D$ is defined as,

$\displaystyle
D\equiv\Gamma+\frac{2}{b_{0}}W_{1}W_{2}{|\beta|}\sin\left(\theta_{2}-\theta_{1}-\theta_{3}\right).$
(23)

The time-dependence of the space-time modulated coefficients is related to the
choice of $\mu(t)$ and $G(x,t)$. There are several interesting possibilities,
including time-independent $W^{\dagger}F_{\pm}W=\frac{b_{0}}{2}$ for
$b_{3}=D=0$. The choice $W_{1}=W_{2}\equiv W$ leads to $b_{3}=0$ and $D=0$
gives the relation
$\Gamma=-{|\beta|}\sin\left(\theta_{2}-\theta_{1}-\theta_{3}\right)$. The
condition $\Gamma^{2}<{|\beta|}^{2}$ for time-periodic $U(t)$ is ensured
provided $\theta_{2}-\theta_{1}-\theta_{3}\neq(2n+1)\frac{\pi}{2},\
n\in\mathbb{Z}$. Thus, $g_{ij}$’s are also independent of time in this limit
provided $G(x,t)$ is taken as time independent. The space-dependence may be
tailored by choosing appropriate functions $f_{1}(x),f_{2}(x),G(x,t)\equiv
G(x)$. The choice $G(x,t)=0$ corresponds to $g_{11}=g_{21},g_{22}=g_{12}$
which has been noted earlier as the limit for hermitian $K$.

The imaginary part of the nonlinear interaction of Eq. (17) vanishes for
$g_{ij}$’s chosen as in Eq.(21). The imaginary and real parts of Eq. (17) can
be separated as,

$\displaystyle\Big{(}R^{2}\theta_{x}\Big{)}_{x}=\tilde{s}(x)R^{2},$ (24)
$\displaystyle R_{xx}+\big{(}E-s(x)\big{)}R-R{\theta_{x}}^{2}=f(x)R^{3}.$ (25)

Note that the above equations are independent of $G(x,t)$, although the space-
time modulated coefficients $g_{ij}$ explicitly depend on it. This is a
surprising result that $G(x,t)$ has no effect at all on the solutions $\Psi$.
It seems that the specific ansatz for $\Phi$ in Eq. (16) leads to this result.
The system defined by Eq. (1) with $g_{ij}$ given by Eq. (21) may admit
solutions which depend on the choice of $G(x,t)$. However, the chosen ansatz
is not suitable for any such exploration —different analytic and/or numerical
methods may have to be employed for the purpose which is beyond the scope of
this article. The solutions of Eqs. (24,25) do not depend on $f_{1}(x)$ and
$f_{2}(x)$ either, but on their average $f(x)$. This is again possibly related
to the chosen ansatz and has to be confirmed independently through other
means. The task is to solve Eqs. (24,25) for given $s(x),\tilde{s}(x)$ and
$f(x)$ which characterize the forms of the potential and space-modulation of
the nonlinear strengths, respectively. The method involves defining a new co-
ordinate $\zeta(x)$ and expressing $R(x)$ as the product of a scale-factor
$\rho(x)$ and $\zeta$-dependent function $u(\zeta)$,

$\displaystyle R(x)=\rho(x)\ u\big{(}\zeta(x)\big{)},\
\zeta(x)=\int^{x}\frac{ds}{\rho^{2}(s)}.$ (26)

The treatment for obtaining exact solutions for real and complex potentials
are different and discussed separately:

Real Potential: The imaginary part vanishes, i.e. $\tilde{s}(x)=0$ and
$\theta(x)$ is determined from Eq.(24) as,

$\displaystyle\theta(x)=\int\frac{C}{R^{2}}dx,$ (27)

where $C$ is an integration constant. The decoupled equation for $R(x)$ is
obtained by substituting $\theta(x)$ into Eq.(25):

$\displaystyle R_{xx}+\big{(}E-s(x)\big{)}R-\frac{C^{2}}{R^{3}}=f(x)R^{3}.$
(28)

Eq. (28) reduces to the famous Ermakov-Pinney equationep  for $f(x)=0$, while
$C=0=s(x),f(x)=1$ leads to the standard NLSE. The equation (28) is solvable in
both the limits. There is a very interesting reduction of Eq. (28) for
$C=0=f(x)$ for which it reduces to the linear Schr$\ddot{o}$dinger equation.
The vanishing constant $C$ implies that the phase $\theta$ is constant. The
condition $f(x)=0$ can be implemented by taking $f_{2}(x)=-f_{1}(x)$, where
$f_{1},f_{2}$ may be considered as constants or space-dependent. Eq. (28)
reduces to the standard linear Schr$\ddot{o}$dinger equation with real
potential $s(x)$ for $C=0=f(x)$. We have the important result that the system
admits exact localized nonlinear modes for any $s(x)$ for which the
corresponding linear Schr$\ddot{o}$dinger equation is solvable. We do not
present any example in this regard. For the general case, the substitution of
Eq.(26) into Eq.(28) results in the following sets of equations:

$\displaystyle u_{\zeta\zeta}+mu-\frac{C^{2}}{u^{3}}-2\sigma u^{3}=0,$ (29)
$\displaystyle\rho_{xx}+\big{(}E-s(x)\big{)}\rho=\frac{m}{\rho^{3}},\ \
f(x)=\frac{2\sigma}{\rho^{6}},$ (30)

where $m$ and $\sigma$ are real constants. The $\theta$ dependence is
contained solely in Eq. (29). The solutions of Eq. (30) for a given $s(x)$ is
used to fix $f(x)$ and $\xi(x)$. The equation for $u$ can be solved
independently and substitutions of $\xi(x)$ along with $\rho(x)$ determines
$R(x)$. Exact solutions of Eq. (29) is discussed in Appendix-I.

Note that Eq. (30) reduces to the linear Schrödinger equation with $\rho$
playing the role of the eigenfunction for $m=0$. The complete spectra of the
linear equation are known for a large number of potential $s(x)$. Each
eigenstate $\rho$ for a given $s(x)$ and $E$, determines
$f(x)=\frac{2\sigma}{\rho^{6}}$ trivially, which corresponds to a unique Eq.
(3) via the $f(x)$ dependence of $g_{ij}$. It should be noted that different
$\rho$ corresponding to different $E$ for a given $s(x)$ does not correspond
to linearly independent solutions of Eq. (3) for fixed $g_{ij}$, rather it
defines different NLSE. Thus, the method can be used to find exact solution of
a large class of NLSE given by Eq. (3). If closed from expressions for the
integrations appearing in Eqs. (26) and (27) are available, a complete
analytic solution for $\Psi$ may be obtained. This only shows that Eq. (3)
with a wide class of nonlinear strengths are amenable to exact solutions by
using the method proposed herein. We shall present only a physically relevant
prototype for a given $V(x)$ and $f(x)(g_{ij})$ to exemplify the general
method in Sec. V & VI.

Complex Potential: The function $\theta(x)$ can not be expressed as the
integral of $R(x)$ alone due to non-vanishing imaginary part of the potential
i.e. $\tilde{s}(x)\neq 0$. Substitution of Eq.(26) into Eq.(25) results in the
following sets of equations

$\displaystyle u_{\zeta\zeta}+mu-2\sigma u^{3}=0$ (31)
$\displaystyle\frac{\rho_{xx}}{\rho}+E-{\theta_{x}}^{2}-\frac{m}{\rho^{4}}=s(x),\
\ f(x)=\frac{2\sigma}{\rho^{6}},$ (32)

which are completely different from Eqs. (29,30) for the case of real
potential. The effect of the imaginary part is contained in Eq. (32) via
$\theta_{x}$ and taking the limit of vanishing $\tilde{s}(x)$ does neither
reproduce Eq. (30) nor a decoupled equation for $\rho$ is obtained. The scheme
for constructing the solvable system is the following —we fix $g_{ij}$ by
choosing constant $f(x)$, say $f(x)=2\sigma$ for simplicity, which determines
$\rho$, and a relation between $\theta_{x}$ and $s(x)$,

$\displaystyle\theta_{x}^{2}=E-m-s(x).$ (33)

The complex potential is chosen such that the Eqs.(24, 25) and (33) are
consistent. This prescription is applicable for specific choices of $g_{ij}$
and complex potentials, nevertheless, it exhausts a large class of exactly
solvable models.

The solution of Eq. (1) has the form,

$\displaystyle\Psi(x,t)=U(t)\ W\ \rho(x)\ u\big{(}\zeta(x)\big{)}\
e^{i(\theta(x)-Et)},$ (34)

where the expressions for $\rho(x),u(\zeta)$ is determined from Eqs. (29,30)
for the real potential and from Eqs. (31,32) for the complex potential. The
power $P(x,t)$ is factorised in terms of time-dependent and space-dependent
parts as,

$\displaystyle P(x,t)=P_{1}(t)R^{2}(x),P_{1}(t)\equiv
W^{\dagger}U^{\dagger}(t)U(t)W.$ (35)

The time-dependent part $P_{1}(t)$ is solely determined in terms of the non-
unitary matrix $U$ and has the expression:

$\displaystyle P_{1}(t)=b_{0}\left[1+\frac{2\Gamma
D}{\epsilon_{0}^{2}}\sin^{2}(\epsilon)+\left(\frac{b_{3}\Gamma}{b_{0}\epsilon_{0}}\right)\sin(2\epsilon)\right]$
(36)

The expression of $P_{1}(t)$ for constant $k$ and $\gamma$ for which
$\mu(t)=t,\epsilon=\epsilon_{0}t$ has been obtained earlierpkg . The effect of
allowing $k$ and $\gamma$ to have identical time dependence specified by the
common scale factor $\mu_{0}(t)$ is solely contained in the argument of the
sine function. The power-oscillation vanishes for $b_{3}=0,D=0$ which is also
the limit for time-independent $W^{\dagger}F_{\pm}W=\frac{b_{0}}{2}$.

##  IV Systems without external potential $V(x)$

We have described the general method for obtaining analytic solutions of Eq.
(1) in section III. In this section, we present some specific examples for
$V(x)=0$ by considering time-independent and time-dependent LC and BLG terms
separately. The case of non-vanishing $V(x)$ will be considered in the next
section.

###  IV.1 Time-independent LC and BLG terms

We consider time-independent LC and BLG terms for which $\mu_{0}$,
$k=\mu_{0}\beta$ and $\gamma=\mu_{0}\Gamma$ are constants. The expression of
the non-unitary matrix $U(t)$ may be obtained by adjusting Eqs. (9) and
Eq.(10) as,

$\displaystyle
U(t)=\sigma_{0}\cos(\epsilon_{0}t)-\frac{iA_{0}}{\epsilon_{0}}\sin(\epsilon_{0}t)$
(37)

where we have chosen $\mu_{0}=1$ without loss of any generality. The time-
dependent function $\epsilon(t)$ appearing in the expression of $P_{1}$ in Eq.
(36) takes the form $\epsilon(t)=\epsilon_{0}t$. The space-time modulations of
the nonlinear strength are discussed by considering (i) constant, (ii) purely
time-dependent, (ii) purely space dependent and (iv) space-dependent $g_{ij}$
separately.

####  IV.1.1  Constant $g_{ij}$

The functions $g_{ij}$ in Eq. (21) are space-time independent provided
$f_{1},f_{2},G$ and $W^{\dagger}F_{\pm}W$ are chosen to be constants. It may
be recalled that $W^{\dagger}F_{\pm}W=\frac{b_{0}}{2}$ is constant for
$W_{1}=W_{2}$ and $D=0$, i.e.
$\Gamma=-{|\beta|}\sin\left(\theta_{2}-\theta_{1}-\theta_{3}\right)$. The
expressions of $g_{ij}$ for these choices are obtained as,

$\displaystyle g_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{b_{0}}(f_{1}+G)\ ;\ g_{21}=\frac{1}{b_{0}}(f_{1}-G)$
$\displaystyle g_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{b_{0}}(f_{2}+G)\ ;\ g_{12}=\frac{1}{b_{0}}(f_{2}-G)$
(38)

where the constants $f_{1},f_{2},G$ may be chosen independently for describing
different physical situations. For example, the choice $f_{1}=f_{2}$ leads to
$g_{11}=g_{22}\equiv g,g_{12}=g_{21}\equiv\tilde{g}$. The system admits a
Lagrangian-Hamiltonian formulation for $g_{12}=g_{21}$ and the relevant
expressions are given in Eq. (4). In the terminology of optics, $g$ and
$\tilde{g}$ may be identified as self-phase modulation and cross-phase
modulation, respectively. The self-phase and cross-phase modulation terms may
be made vanishing by choosing $f_{1}=f_{2}=G$ and $f_{1}=f_{2}=-G$,
respectively. The nonlinear interaction becomes $SU(2)$ invariant for the
choice $f_{1}=f_{2},G=0$ for which $g=\tilde{g}$. This particular system has
been studied earlier in the context of optics and to the best of our
knowledge, no exact solution has been found for generic values of $g_{ij}$. We
present below exact analytical solutions for arbitrary $g_{ij}$ for the first
time, which automatically includes the specific values of $g_{ij}$ discussed
above.

With the choice of $f=\frac{1}{2}(f_{1}+f_{2})\equiv 2\sigma$ and $V(x)=0$,
Eq.(28) reduces to,

$\displaystyle R_{xx}+ER-\frac{C^{2}}{R^{3}}-2\sigma R^{3}=0,$ (39)

which is exactly solvable. Hence, no further transformation as in Eq.(26) is
required. The exact solutions of Eq. (39) is discussed in Appendix-I. We
denote the solutions of Eq. (39) for $C=0,E<0,\sigma<0$ as $R_{0}(x)$ with its
analytical expression given by,

$\displaystyle
R_{0}(x)=\sqrt{\frac{|E|}{|\sigma|}}\operatorname{sech}\left(\sqrt{|E|}x\right)$
(40)

The solution describes a soliton and has been obtained earlierDriben2 . We
present new solutions by taking $C\neq 0$ and $\sigma\geq 0$. For $\sigma=0$,
i.e. $f_{1}=-f_{2}$, Eq. (39) is the famous Ermakov-Pinney equation and admits
periodic solution $R_{P}(x)$ for $E>0$,

$\displaystyle R_{P}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}(\sqrt{E}x)+\frac{C^{2}}{E}\cos^{2}(\sqrt{E}x)},\
E>0.$ (41)

while diverges for $E<0$. The solutions of Eq. (39) for $\sigma>0$ are
expressed in terms of Weierstrass Elliptic functions and choosing $\sigma=1$
for simplicity, one of the solutions reads,

$\displaystyle R(x)=\sqrt{Q_{1}+(Q_{2}-Q_{1})sn^{2}(\lambda x,r)},$ (42)

where $\lambda=\sqrt{Q_{3}-Q_{1}}$, $r^{2}=\frac{Q_{2}-Q_{1}}{Q_{3}-Q_{1}}$
and $Q_{1},Q_{2},Q_{3}$ are constants depending on the arbitrary real
constants $E$, $C$, $\sigma$. Thus, the constants $Q_{i}$’s may be chosen
conveniently by fixing the values of $E,C,\sigma$. The solution is chosen to
be bounded as $0<Q_{1}\leq R^{2}\leq Q_{2}<Q_{3}$. In the limit
$Q_{3}\rightarrow Q_{2}$, the solution reduces to

$\displaystyle R(x)=\sqrt{Q_{1}+(Q_{2}-Q_{1})\tanh^{2}(\lambda x)}.$ (43)

Following the discussions in Appendix-I, other solutions may be written down
easily. The power $P$ of the system for constant $g_{ij}$ is time independent,
i.e. $P=b_{0}R^{2}$, where $R$ is given by Eq. (42) or in the limit
$Q_{3}\rightarrow Q_{2}$ by Eq. (43). The power-oscillation in time has been
observed for models described in Ref. pkg  where the nonlinear potential is of
the form $(\Psi^{\dagger}M\Psi)^{2}$ where the scalar gauge potential $A$ is
pseudo-hermitian with respect to the constant matrix $M$, i.e.
$A^{\dagger}=MAM^{-1}$. The nonlinear potential defined by $g_{ij}$’s in Eq.
(38) can not be cast in the form $(\Psi^{\dagger}M\Psi)^{2}$ for some $M$ and
the absence of power oscillation for the present case is not in contradiction
with the results of Ref. pkg  .

####  IV.1.2 Purely time-dependent $g_{ij}$

We consider Eq.(1) with purely time-dependent nonlinear strengths by choosing
$g_{ij}$’s to be dependent on time alone. It may be recalled that time
dependent non-linear strength arises in optics for transverse beam propagation
in layered optical mediaBerge ; Towers . Further, NLSE with periodic time
variation of nonlinear strength in an external magnetic field has been used to
produce matter wave breathers in quasi one-dimensional Bose-Einstein
condensateKevrekidis2 . As evident from Eq. (21), purely time dependent
$g_{ij}$’s may be obtained by taking $f_{1},f_{2}$ as constants and allowing
$G$ to be a function of time only. For simplicity, we consider
$\theta_{2}-\theta_{1}-\theta_{3}=n\pi$ and $W_{1}=W_{2}$, for which the
expressions of $g_{ij}$ are,

$\displaystyle g_{11}$ $\displaystyle=$
$\displaystyle\frac{f_{1}+G(t)}{b_{0}(1+2\Gamma q_{+})},\ \
g_{21}=\frac{f_{1}-G(t)}{b_{0}(1+2\Gamma q_{+})},$ $\displaystyle g_{22}$
$\displaystyle=$ $\displaystyle\frac{f_{2}+G(t)}{b_{0}(1+2\Gamma q_{-})},\ \
g_{12}=\frac{f_{2}-G(t)}{b_{0}(1+2\Gamma q_{-})},$ (44)

where $q_{\pm}$ are given as,

$\displaystyle
q_{\pm}=\frac{\Gamma}{\epsilon^{2}_{0}}\sin^{2}(\epsilon_{0}t)\pm\frac{\sin(2\epsilon_{0}t)}{2\epsilon_{0}}.$
(45)

The functions $q_{\pm}$ introduce periodic time-modulation of the nonlinear
strengths. Eq.(28) reduces to Eq.(39) for this case also, since $f_{1},f_{2}$
are constants and $V(x)=0$. Consequently, $R(x)$ have the expressions given by
Eqs. (40), (42) and (43) for the specified choices of the constants. Unlike
the case of constant $g_{ij}$, the power is time-dependent,

$\displaystyle
P=b_{0}\left[1+\frac{2\Gamma^{2}}{\epsilon_{0}^{2}}\sin^{2}(\epsilon_{0}t)\right]R^{2}(x)$
(46)

where any one of the expressions of $R$ given in Eqs. (40), (42) and (43) may
be used within their ranges of validity.

####  IV.1.3 Purely space-dependent $g_{ij}$

It is evident from Eq. (21) that purely space-dependent $g_{ij}$’s are
obtained for constant $W^{\dagger}F_{\pm}W$ and choosing purely space
dependent $G(x,t)\equiv G(x)$. As discussed in Sec. III,
$W^{\dagger}F_{\pm}W=\frac{b_{0}}{2}$ is constant in the limit $W_{1}=W_{2}$
and $D=0$. The space modulation is determined by the space-dependent functions
$f_{1}(x),f_{2}(x)$ and $G(x)$. Unlike the previous cases, solution of Eq.
(28) for arbitrary $f(x)$ necessitates the transformation given by Eq. (26)
resulting in Eqs. (29,30). The solutions of Eq.(30) for $V(x)=0$ are,

$\displaystyle\rho^{2}(x)$ $\displaystyle=$ $\displaystyle
ae^{2\sqrt{|E|}x}+\frac{m}{4|E|a}e^{-2\sqrt{|E|}x},\ \ E<0$
$\displaystyle\rho^{2}(x)$ $\displaystyle=$ $\displaystyle
a\sin^{2}(\sqrt{E}x)+\frac{m}{aE}\cos^{2}(\sqrt{E}x),\ \ E>0$ (47)

where the constant $a>0$. The solutions for $E<0$ diverges for large $x$ and
will not be considered further in this article. We consider $E>0$ and make the
following choices for a simplified expression of $\rho$,
$E=\frac{\omega^{2}}{4},a=(1-\alpha),m=\frac{\omega^{2}(1-\alpha^{2})}{4}$,
where $\omega$ and $\alpha$ are real constants. The expressions of
$\rho^{2}(x)$ and hence, $f(x)$ have the following simplified expressions,

$\displaystyle\rho^{2}(x)=1+\alpha\cos(\omega x),\ \
f(x)=\frac{2\sigma}{[1+\alpha\cos(\omega x)]^{3}}.$ (48)

The functions $f_{1}(x)$ and $f_{2}(x)$ may be chosen independently subjected
to the constraint $f(x)=\frac{1}{2}(f_{1}(x)+f_{2}(x))$. We choose
$f_{1}(x)=f_{2}(x)=2\sigma[1+\alpha\cos(\omega x)]^{-3}$ for which space-
dependent $g_{ij}$ have the expressions,

$\displaystyle
g_{11}=g_{22}=\frac{G(x)}{b_{0}}+\frac{2\sigma}{b_{0}[1+\alpha\cos(\omega
x)]^{3}},$ $\displaystyle
g_{12}=g_{21}=-\frac{G(x)}{b_{0}}+\frac{2\sigma}{b_{0}[1+\alpha\cos(\omega
x)]^{3}}$ (49)

For $G=0$, all $g_{ij}$’s become identical. The expression of $\zeta$ is
obtained by using the second equation of Eq. (26) and the first equation of
(48),

$\displaystyle\zeta=\frac{2}{\omega\sqrt{1-\alpha^{2}}}\
\arctan\left[\sqrt{\frac{1-\alpha}{1+\alpha}}\ \tan(\frac{\omega
x}{2})\right],$ (50)

where ${|\alpha|}<1$. It remains to find $u(\zeta)$ in order to completely
specify the solution. Note that the equations (39) and (29) satisfied by
$R(x)$ and $u(\zeta)$, respectively, are identical with the identification of
$E\leftrightarrow m,x\leftrightarrow\zeta,R(x)\leftrightarrow u(\zeta)$. The
solutions of Eq. (39) are given in Eqs. (40) and (42) which may be used to
write down the solutions of $u(\zeta)$ with the identification stated above.
The power $P=b_{0}\rho^{2}(x)u(\zeta)^{2}$ is independent of time.

####  IV.1.4 Space-time dependent $g_{ij}$

The results for space-time dependent $g_{ij}$’s have been discussed in Sec.
III in detail except for the solution of Eq. (28) for $R(x)$ or equivalently
of Eq. (29,30) for $u(\zeta)$ and $\rho(x)$. It may be noted that Eq. (29) is
the same for purely space dependent $g_{ij}$ as well as space-time dependent
$g_{ij}$. Thus, the expressions of $\\{\rho(x),f(x)\\}$ and $\zeta(x)$ are
given by Eqs. (48) and (50), respectively, within the specified ranges of
validity. The solutions for $u(\zeta)$ are given by Eqs. (40) and (42) with
the replacement of $E\rightarrow
m,x\rightarrow\zeta,\\{R_{0}(x),R_{P}(x),R(x)\\}\rightarrow\\{u_{0}(\zeta),u_{P}(\zeta,u(x)\\}$.
The exact solution $\Psi(x,t)$ for space-time dependent $g_{ij}$ is different
from purely space-dependent $g_{ij}$ due to the time-dependence. In
particular, the power is time-independent for purely space-dependent $g_{ij}$,
while it is time-dependent for space-time dependent $g_{ij}$ and given by Eqs.
(35) and (36) with $\epsilon=\epsilon_{0}t$ and $\mu=1$.

###  IV.2 Time-dependent $k,\gamma$

The equations determining $R(x)$ and $\theta(x)$ are the same as in the case
of constant $k,\gamma$. However, the non-unitary matrix $U(t)$, the strengths
$g_{ij}$, and hence, expression of power will change depending on specific
forms of $k(t)$ and $\gamma(t)$. We have already discussed solutions for
$R(x)$ and $\theta(x)$ depending on constant, purely time-dependent, purely
space-dependent and space-time dependent $g_{ij}$ which are valid for time-
dependent $k(t),\gamma(t)$ independent of specific time-dependence. In order
to avoid repetition of the same results, we present results related to
$U(t),g_{ij}(t),P(t)$ for specific time-dependence of $k(t)$ and $\gamma(t)$.
We choose periodic modulation of the LC and BLG terms by choosing
$\mu_{0}(t)=\cos(\omega_{0}t),\omega_{0}\in\mathbb{R}$ so that,

$\displaystyle\mu(t)=\frac{1}{\omega_{0}}\sin(\omega_{0}t),\
\epsilon=\frac{\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t),\
T_{\pm}(t)=\frac{\omega_{0}p_{\pm}}{\sin(\omega_{0}t)},$ $\displaystyle
p_{\pm}\equiv\frac{\Gamma}{\epsilon^{2}_{0}}\sin^{2}\left(\frac{\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)\pm\frac{1}{2\epsilon_{0}}\sin\left(\frac{2\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right).$

The limit $\omega_{0}\rightarrow 0$ corresponds to constant LC and BLG terms.
The expressions for $U,W^{\dagger}F_{\pm}W,g_{ij},P$ may be obtained from Eqs.
(10), (22), (21) and (36), respectively by using the above expressions for
$\mu(t),\epsilon(t),T_{\pm}(t)$.

The expressions of $W^{\dagger}F_{\pm}W$,

$\displaystyle W^{\dagger}F_{\pm}W$ $\displaystyle=$
$\displaystyle\frac{b_{0}}{2}\left(1+2Dp_{\pm}\right)+\frac{b_{3}}{2}\left[\frac{\Gamma}{\epsilon_{0}}\sin\left(\frac{2\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)\pm\cos\left(\frac{2\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)\right],$
(51)

when substituted in Eq. (21) gives the space-time dependent $g_{ij}$ for
periodic time-modulation of the LC and BLG terms. Different types of space-
time modulations of the nonlinear strengths $g_{ij}$ may be considered as
follows:

  * •

Constant $g_{ij}$: $D=0,b_{3}=0,(G,f_{1},f_{2})=\textrm{constant}$

  * •

Purely time-dependent $g_{ij}$: Either $D$ or $b_{3}$ or both $D$ and $b_{3}$
are non-vanishing; $f_{1}$ and $f_{2}$ are constants and $G(x,t)\equiv G(t)$

  * •

Purely space-dependent $g_{ij}$: $D=0=b_{3}$, $G(x,t)\equiv G(x)$

The condition $D=0$ may be imposed by using Eq. (23) which gives a specific
relation among the parameters, while $b_{3}=0$ for the choice $W_{1}=W_{2}$.
The functions $f_{1}(x),f_{2}(x),G(x,t)$ may be chosen as per the requirement,
since they are arbitrary. The non-unitary operator $U$ and $P$ have the
expressions:

$\displaystyle U$ $\displaystyle=$
$\displaystyle\sigma_{0}\cos\left(\frac{\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)-\frac{iA_{0}}{\epsilon_{0}}\sin\left(\frac{\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)$
$\displaystyle P(x,t)$ $\displaystyle=$ $\displaystyle
b_{0}\left[1+\frac{2\Gamma
D}{\epsilon_{0}^{2}}\sin^{2}\left(\frac{\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)+\frac{b_{3}\Gamma}{b_{0}\epsilon_{0}}\sin\left(\frac{2\epsilon_{0}}{\omega_{0}}\sin(\omega_{0}t)\right)\right]R^{2}(x),$
(52)

appropriate expressions for $R(x)$ are to be substituted for a given $g_{ij}$.

The general explicit solution of Eq. (1) for $V(x)=0$ and space-time dependent
$g_{ij}$ is

$\displaystyle\psi_{1}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{1}e^{i\theta_{1}}\cos(\epsilon)-\left\\{i|\beta|W_{2}e^{i\left(\theta_{2}-\theta_{3}\right)}-\Gamma
W_{1}e^{i\theta_{1}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]\sqrt{1+\alpha\cos(\omega
x)}$ $\displaystyle\times$
$\displaystyle\sqrt{Q_{1}+(Q_{2}-Q_{1})sn^{2}(\lambda\zeta,r)}\
e^{i\left(\theta(x)-Et\right)}$ $\displaystyle\psi_{2}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{2}e^{i\theta_{2}}\cos(\epsilon)-\left\\{i|\beta|W_{1}e^{i\left(\theta_{1}+\theta_{3}\right)}+\Gamma
W_{2}e^{i\theta_{2}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]\sqrt{1+\alpha\cos(\omega
x)}$ (53) $\displaystyle\times$
$\displaystyle\sqrt{Q_{1}+(Q_{2}-Q_{1})sn^{2}(\lambda\zeta,r)}\
e^{i\left(\theta(x)-Et\right)}$

The expression of $\zeta$ is given in Eq. (50) and $\theta(x)$ can be obtained
from Eq. (27). The power $P=\Psi^{\dagger}\Psi$ is plotted in FIG. 1 for
various time-modulations. The case of vanishing BLG and LC is considered Fig.
1(a) and no variation of the power with time is seen. The power-oscillation is
seen in Fig. 1(b) for constatnt $\mu_{0}$ and non-vanishing $\beta,\Gamma$
satisfying ${|\beta|}>\Gamma$. The power-oscillation in time is a
manifestation of the balanced loss-gain in the system. The solution $\Psi$
becomes unstable for non-vanishing $\beta,\Gamma$ satisfying
$|\beta|\leq\Gamma$ and constant $\mu_{0}$. The respective plot is given in
Fig 1.(c). We can manage the instability by choosing $\mu_{0}(t)$ properly. In
Fig. 1(d), we have considered $|\beta|=\Gamma$ and a periodic modulation
function $\mu_{0}(t)=\cos(t)$ for which $P(x,t)$ shows periodic behaviour. We
have plotted power for $|\beta|=\Gamma$ and
$\mu_{0}(t)=\frac{2}{\sqrt{\pi}}e^{-t^{2}}$ in Fig. 1(e) which is also bounded
in time. The instability in the region $|\beta|<\Gamma$ can again be
controlled by periodic modulation. In particular, the plot in Fig. 1(f) shows
power oscillation for $\mu_{0}(t)=\cos t$ in the region $|\beta|<\Gamma$. The
figures are shown upto time $t\approx 15$, but the stated features have been
checked upto time $t\approx 200$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1:  (Color online) Plot of Power i.e $\Psi^{\dagger}\Psi$ for $V(x)=0$
and the expression of $\Psi$ is shown in Eq. (53). Parameters:
$W_{1}=W_{2}=1,\theta_{1}=\frac{\pi}{6},\theta_{2}=\frac{2\pi}{3},\theta_{3}=\frac{\pi}{3},Q_{1}=0.1,Q_{2}=0.29,Q_{3}=0.3,\sigma=1,\alpha=0.15,\omega=2$.
In Fig(a): $\Gamma=|\beta|=0$, Fig(b): $\Gamma=0.5,|\beta|=0.7,\mu_{0}=1$,
Fig(c): $\Gamma=|\beta|=0.5,\mu_{0}=1$, Fig(d):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=cos(t)$, Fig(e):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=\frac{2}{\sqrt{\pi}}e^{-t^{2}}$, Fig(f):
$\Gamma=0.5,|\beta|=0.3,\mu_{0}(t)=cos(t)$

##  V Systems with Real $V(x)$

In this section, results for non-vanishing real potential $V(x)$ will be
presented for purely space-dependent and space-time dependent $g_{ij}$. The
method outlined in Sec. III is not suitable for constant or purely time-
dependent $g_{ij}$ for which $f_{1},f_{2}$ are necessarily constants.
Consequently, it follows from Eq. (30) that $f(x),\rho$ and $s(x)$ are also
constants for real $V(x)$. Further, a direct solution of Eq. (28) for constant
$f(x)$ and spatially varying $V(x)$ is also unknown. Within this background,
we discuss a few examples of real $V(x)$ for space-time dependent $g_{ij}$ and
the limit of purely space-dependent $g_{ij}$ may be considered by choosing
$D=0=b_{3}$ and $G(x,t)\equiv G(x)$. The sole effect of adding a non-vanishing
real potential $V(x)$ to a system with $V(x)=0$ is modified expressions of
$\rho(x)$ and hence, of $\zeta(x)$ and $f(x)$. The equation determining
$u(\zeta)$ remains unaffected by the choice of $V(x)$, although $u(\zeta(x))$
is dependent on the choice of $V(x)$ via $\zeta(x)$. The knowledge of $f(x)$
specifies the space-dependent part of $g_{ij}$111$G(x,t)$ appearing in the
expressions of $g_{ij}$ does not affect any results and may be chosen to be
zero, while space-dependent part of the power $P$ is modified due to the
expression of $R(x)$ which depends on $\rho(x)$ and $u(\zeta(x))$. We present
the expressions of $\rho(x),f(x),\zeta(x),u(\zeta)$ for a given $V(x)$ and do
not repeat the results and discussions which have been presented in earlier
sections. The space-modulated nonlinear strengths are determined in terms of
$f_{1}(x)$ and $f_{2}(x)$. One may choose $f_{1}=f_{2}=f$ for simplicity or
choose $f_{1}$ and $f_{2}$ such that $f(x)=\frac{1}{2}(f_{1}(x)+f_{2}(x))$ is
satisfied. There is a freedom in tailoring the nonlinear strengths $g_{ij}$
for a given $f(x)$ which may be exploited to construct physically interesting
systems.

###  V.1 Reflection-less Potential

We choose the reflection-less potential,

$\displaystyle V(x)=N^{2}-N(N-1)sech^{2}(x),\ N\in\mathbb{Z}^{>}.$ (54)

The solutions for $\rho(x)$ and $f(x)$ for $E=m=0$ may be obtained from
Eq.(30) as,

$\displaystyle\rho(x)=\cosh^{N}(x),\ \ f(x)=2\sigma sech^{6N}(x).$ (55)

The potential $V(x)$ has been considered earlier in the context of single
component localBeitia1  as well as non-localpkg1  NLSE. The nonlinear
strengths $g_{ij}$’s may be tailored for the above $f(x)$ satisfying
$\frac{1}{2}(f_{1}(x)+f_{2}(x))=f(x)$. Following the discussions in
Appendix-I, the sum of the roots $Q_{1},Q_{2},Q_{3}$ is zero for $m=0$,
implying that all three roots can not either be positive or negative. We
choose $Q_{1}<0,Q_{2}>0,Q_{3}>0$. The solution of Eq.(29) for $\sigma=-1$ is
obtained as,

$\displaystyle u(\zeta)=\sqrt{Q_{3}-(Q_{3}-Q_{2})sn^{2}(\lambda\zeta,r)}$ (56)

where $\lambda=\sqrt{Q_{3}-Q_{1}}$ and
$r^{2}=\frac{Q_{3}-Q_{2}}{Q_{3}-Q_{1}}$. The expression of $\zeta$ is obtained
from Eq. (26),

$\displaystyle\zeta=\int\operatorname{sech}^{2N}(x)dx,$ (57)

which can be evaluated for fixed $N$ by using the formula, $\int
sech^{n}(x)=\frac{sech^{n-2}(x)tanh(x)}{n-1}+\frac{n-2}{n-1}\int
sech^{n-2}(x)$ repeatedly.

###  V.2 Quadratic Potential

We consider the quadratic potential $V(x)=E^{2}x^{2}$ and $E>0$. For $m=0$,
the solution of $\rho(x)$ and $f(x)$ is obtained from Eq.(29) as,

$\displaystyle\rho(x)=e^{-\frac{1}{2}Ex^{2}},\ \ f(x)=2\ \sigma\ e^{3Ex^{2}}.$
(58)

An unpleasant feature is that the nonlinear strengths $g_{ij}$ grow in space
for this specific $f(x)$. The freedom in choosing $f_{1}(x),f_{2}(x)$
satisfying the constraint $f(x)=\frac{1}{2}(f_{1}(x)+f_{2}(x))$ and arbitrary
function $G(x,t)$ is of no help to construct localized $g_{ij}$’s due to their
specific dependence on $f_{1},f_{2},G$. The solutions for $u(\zeta)$ is same
as (56) with the expression of $\zeta$ obtained from Eq.(26) as,

$\displaystyle\zeta=\frac{1}{2}\sqrt{\frac{\pi}{E}}\ erfi(\sqrt{E}x),$ (59)

where $erfi$ is imaginary error function. The function $\Psi(x,t)$ is finite
in all regions of space, since $R(x)\rightarrow 0$ as $|x|\rightarrow\infty$.
On other hand, $\Psi(x,t)$ diverges for $E<0$.

The general explicit solution of Eq. (1) for Quadratic Potential and space-
time dependent $g_{ij}$ is

$\displaystyle\psi_{1}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{1}e^{i\theta_{1}}\cos(\epsilon)-\left\\{i|\beta|W_{2}e^{i\left(\theta_{2}-\theta_{3}\right)}-\Gamma
W_{1}e^{i\theta_{1}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]\
e^{-\frac{1}{2}Ex^{2}}$ $\displaystyle\times$
$\displaystyle\sqrt{Q_{3}-(Q_{3}-Q_{2})sn^{2}(\lambda\zeta,r)}\
e^{i\left(\theta(x)-Et\right)}$ $\displaystyle\psi_{2}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{2}e^{i\theta_{2}}\cos(\epsilon)-\left\\{i|\beta|W_{1}e^{i\left(\theta_{1}+\theta_{3}\right)}+\Gamma
W_{2}e^{i\theta_{2}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]\
e^{-\frac{1}{2}Ex^{2}}$ (60) $\displaystyle\times$
$\displaystyle\sqrt{Q_{3}-(Q_{3}-Q_{2})sn^{2}(\lambda\zeta,r)}\
e^{i\left(\theta(x)-Et\right)}$

The expression of $\theta(x)$ is obtained from Eq. (27). The qualitative
behaviour of the plots for identical $\mu_{0}(t)$ and condition on
$\beta,\Gamma$ are the same with that of Fig. 1. There is no power oscillation
for vanishing $\beta,\Gamma$ as seen in Fig. 2(a). The power-oscillation is
seen in Fig. 2(b) for constant $\mu_{0}$ and ${|\beta|}>\Gamma$. The solution
grows without any upper bound for constant $\mu_{0}$ and ${|\beta|}=\Gamma$ as
is evident from Fig. 2.(c). The management of instabilities for
${|\beta|}\leq\Gamma$ by choosing suitable $\mu_{0}(t)$ is shown in Figs.
2(d), 2(e) and 2(f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2:  (Color online) Plot of Power i.e $\Psi^{\dagger}\Psi$ for
$V(x)=E^{2}x^{2}$ and the expression of the components of $\Psi$ is given in
Eq. (60). Parameters:
$W_{1}=W_{2}=1,\theta_{1}=\frac{\pi}{6},\theta_{2}=\frac{2\pi}{3},\theta_{3}=\frac{\pi}{3},Q_{1}=-0.3,Q_{2}=0.1,Q_{3}=0.2,\sigma=-1,E=0.12$.
In Fig(a): $\Gamma=|\beta|=0$, Fig(b): $\Gamma=0.5,|\beta|=0.7,\mu_{0}=1$,
Fig(c): $\Gamma=|\beta|=0.5,\mu_{0}=1$, Fig(d):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=cos(t)$, Fig(e):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=\frac{2}{\sqrt{\pi}}e^{-t^{2}}$, Fig(f):
$\Gamma=0.5,|\beta|=0.3,\mu_{0}(t)=cos(t)$

##  VI Systems with Complex $V(x)$

The complex potentials appear in the context of optics through refractive
index. The imaginary part of the refractive index is introduced to describe
the attenuation of propagating electromagnetic waves in the given material.
The ${\cal{PT}}$-symmetric complex potentials have some interesting properties
and have been studied extensively in the context of NLSEKonotop1 . We present
a few examples of complex confining potential for NLSE with time-dependent LC
and BLG, and space-time modulated nonlinear strengths.

The nonlinear strengths $g_{ij}$ may be chosen to be constant, purely time or
space dependent or space-time modulated. One notable aspect is that the
constant and purely time-dependent $g_{ij}$ may be considered for a complex
confining potential, which is not allowed for a real $V(x)$. The function
$\theta(x)$ is determined by Eq. (33) for a given $s(x)$ for constant $f(x)$
and $\rho$. Such a freedom to consider a space-dependent $s(x)$ for constant
$f$ is absent in Eq. (30) for the case of real potential. There is a freedom
in tailoring the nonlinear strengths $g_{ij}$ for a given $f(x)$ by choosing
$f_{1}(x)$ and $f_{2}(x)$ appropriately, which may be exploited to construct
physically interesting systems.

Scarf II Potential: The ${\cal{PT}}$-symmetric Scarf II complex potential has
the form,

$\displaystyle V(x)=-\frac{V^{2}}{9}sech^{2}(x)-iVsech(x)\tanh(x)$ (61)

The analytic and numerical solutions of one component NLSE with $V(x)$ has
been discussed in Ref. Shi ; Musslimani . We consider constant $f(x)=2\sigma$
for which $f_{1}$ and $f_{2}$ may be chosen as constants or space-dependent
subject to the condition $f(x)=\frac{1}{2}(f_{1}(x)+f_{2}(x))$. We choose
$E=m=-1$ for simplicity and the phase $\theta(x)$ is determined from Eq. (33)
as $\theta(x)=\frac{V}{3}\arctan(\sinh(x))$. We have to choose $\sigma<0$ so
that Eqs. (24,25) and (32) are consistent leading to the expression
$R(x)=\frac{1}{\sqrt{|\sigma|}}\operatorname{sech}(x)$. The solution of
Eq.(12) is obtained as,

$\displaystyle\Phi=W{{|\sigma|}}^{-\frac{1}{2}}\operatorname{sech}(x)e^{i\big{(}{\frac{V}{3}}\arctan(\sinh(x))+t\big{)}},$
(62)

which describes a soliton. This completes the extension of the result to the
NLSE with time-modulated LC and BLG terms, and space-time modulated nonlinear
strengths for the Scarf II potential. The expression of the components of
$\Psi$ is given below,

$\displaystyle\psi_{1}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{1}e^{i\theta_{1}}\cos(\epsilon)-\left\\{i|\beta|W_{2}e^{i\left(\theta_{2}-\theta_{3}\right)}-\Gamma
W_{1}e^{i\theta_{1}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]{{|\sigma|}}^{-\frac{1}{2}}\operatorname{sech}(x)\
e^{i\left({\frac{V}{3}}\arctan(\sinh(x))+t\right)}$
$\displaystyle\psi_{2}(x,t)$ $\displaystyle=$
$\displaystyle\left[W_{2}e^{i\theta_{2}}\cos(\epsilon)-\left\\{i|\beta|W_{1}e^{i\left(\theta_{1}+\theta_{3}\right)}+\Gamma
W_{2}e^{i\theta_{2}}\right\\}\frac{\sin(\epsilon)}{\epsilon_{0}}\right]{{|\sigma|}}^{-\frac{1}{2}}\operatorname{sech}(x)\
e^{i\left({\frac{V}{3}}\arctan(\sinh(x))+t\right)}$ (63)

The qualitative behaviour of the plots for identical $\mu_{0}(t)$ and
condition on $\beta,\Gamma$ are the same with that of Figs. 1 and 2. There is
no power oscillation for vanishing $\beta,\Gamma$ as seen in Fig. 3(a). The
power-oscillation is seen in Fig. 3(b) for constant $\mu_{0}$ and
${|\beta|}>\Gamma$. The solution grows without any upper bound for constant
$\mu_{0}$ and ${|\beta|}=\Gamma$ as is evident from Fig. 3.(c). The management
of instabilities for ${|\beta|}\leq\Gamma$ by choosing suitable $\mu_{0}(t)$
is shown in Figs. 3(d), 3(e) and 3(f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3:  (Color online) Plot of Power i.e $\Psi^{\dagger}\Psi$ for Scarf-II
potential and the expression of the components of $\Psi$ in Eq. (63).
Parameters:
$W_{1}=W_{2}=1,\theta_{1}=\frac{\pi}{6},\theta_{2}=\frac{2\pi}{3},\theta_{3}=\frac{\pi}{3},\sigma=-1$.
In Fig(a): $\Gamma=|\beta|=0$, Fig(b): $\Gamma=0.5,|\beta|=0.7,\mu_{0}=1$,
Fig(c): $\Gamma=|\beta|=0.5,\mu_{0}=1$, Fig(d):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=cos(t)$, Fig(e):
$\Gamma=|\beta|=0.5,\mu_{0}(t)=\frac{2}{\sqrt{\pi}}e^{-t^{2}}$, Fig(f):
$\Gamma=0.5,|\beta|=0.3,\mu_{0}(t)=cos(t)$

Periodic Potential : The ${\cal{PT}}$-symmetric complex potential,

$\displaystyle V(x)=-\cos^{2}(x)+i\sin(x)$ (64)

has been considered in Ref. Musslimani  in the context of one component NLSE.
Following the procedure described above for $E=m=1$ and $\sigma>0$, the
solution is determined as,

$\displaystyle\Phi=W\frac{1}{\sqrt{\sigma}}\sec(x)e^{i\big{(}\sin(x)-t\big{)}}.$
(65)

We get exact solutions for NLSE with time-modulated LC and BLG terms, and
space-time modulated nonlinear strengths for the above periodic potential.

###  VI.1 Supersymmetry-inspired solution for $f(x)=0$

The techniques of supersymmetric quantum mechanicskhare  may be used for
$f(x)=0$ to construct a large number of complex potentials for which the NLSE
with LC and BLG terms are exactly solvable. The choice $f(x)$=0 can be
implemented by fixing $f_{2}(x)=-f_{1}(x)$ and $f_{1},f_{2}$ may be chosen to
be either constants or space-dependent. There is a freedom in choosing a large
class of space-time modulated $g_{ij}$’s for a given complex $V(x)$. The exact
solutions of the NLSE are insensitive to the specific forms of $f_{2}=-f_{1}$
and $G(x,t)$.

The choice $f(x)=0$ reduces Eq.(25) to the linear Schr$\ddot{o}$dinger
equation with an effective potential $V_{eff}$:

$\displaystyle R_{xx}+(E-V_{eff})R=0,\ \ V_{eff}\equiv s(x)+\theta_{x}^{2}.$
(66)

We may fix $\theta(x)$ in terms of $\tilde{s}(x)$ such that Eqs. (24) and (25)
are consistent. We define a function $h(x)$ which is related to $s(x)$ and
$\theta(x)$ as follows,

$\displaystyle s(x)=-h_{x},\ \ \theta(x)$ $\displaystyle=$
$\displaystyle\int^{x}h(x^{\prime})dx^{\prime},$ (67)

for which $V_{eff}$ has the form of one of the supersymmetric partner
potentials,

$\displaystyle V_{eff}=h^{2}-h_{x}.$ (68)

The function $h(x)$ is known as superpotential in the context of
supersymmetric quantum mechanics. The function $R(x)$ plays the role of eigen-
function and $E$ as the energy eigen-value. The zero-energy eigen-function
$R_{0}(x)$ has the form,

$\displaystyle R_{0}(x)$ $\displaystyle=$ $\displaystyle N_{0}\
e^{-\theta(x)},$ (69)

where $N_{0}$ is a normalization constant. The expression of $R(x)$ for $E\neq
0$ can also be obtained for shape-invariant potentialskhare . We restrict our
discussions for $E=0$ in this article and $R_{0}(x)$ can be determined for any
$h(x)$ independent of whether it corresponds to shape-invariant potential or
not. The factorization of the effective potential $V_{eff}$ as in Eq. (68)
ensures exact solution for $R_{0}(x)$. The consistency of Eqs. (24) and (25)
fixes the imaginary part and the complex potential has the expression,

$\displaystyle V(x)=-h_{x}(x)+i\left[h_{x}(x)-2h^{2}(x)\right].$ (70)

The superpotential corresponding to all the known examples of solvable
supersymmetric quantum system may be used to construct complex potential
$V(x)$ and the corresponding solution for $R(x)$. The solution $R(x)$
corresponding to these potentials describe localized nonlinear modes. The
expression for $\Phi(x,t)$ is determined as,

$\displaystyle\Phi(x,t)=N_{0}We^{-\theta(x)+i\theta(x)}.$ (71)

The important point to note is that corresponding to each exactly solved
quantum mechanical problem by using supersymmetry, the corresponding
superpotential may be used to find a complex potential for which exact
localized nonlinear modes are obtained.

We present a few examples to complement the general discussions by including
expressions for $h(x),V(x)$ and $\Phi(x)$. Note that $\Phi(x,t)$ is time-
independent, since we have chosen $E=0$ for the presentation of results:
Polynomial potential :

$\displaystyle h(x)=\omega_{0}+\omega_{1}x+\omega_{2}x^{3},\
(\omega_{0},\omega_{1},\omega_{2})\in\ \mathbb{R}^{+},$ $\displaystyle
V(x)=-(1-i)\left(\omega_{1}+3\omega_{2}x^{2}\right)-2i\left(\omega_{0}+\omega_{1}x+\omega_{2}x^{3}\right)^{2},$
$\displaystyle\Phi(x)=N_{0}We^{-(1-i)\left(\omega_{0}x+\frac{\omega_{1}}{2}x^{2}+\frac{\omega_{2}}{4}x^{4}\right)}.$

The real part of the potential describes a harmonic oscillator, while the
imaginary part is a sextic potential. The function $\Phi(x)$ is localized in
space even for $\omega_{2}=0$ for which the real part of the potential is
constant and the imaginary part is given by an quadratic potential.

Exponential Potential :

$\displaystyle h(x)=A-Be^{-ax},\ (a,A,B)\in\mathbb{R}^{+},$ $\displaystyle
V(x)=-Bae^{-ax}+i[2A^{2}-B(a+4A)e^{-ax}+2B^{2}e^{-2ax}],$
$\displaystyle\Phi(x)=N_{0}W\ exp[-\frac{B}{a}e^{-ax}-Ax]\
e^{i\big{(}\frac{B}{a}e^{-ax}+Ax\big{)}}.$

The real part of the $V(x)$ describes a potential well of depth $Ba$ and
$\Phi(x)$ is localized in space.

Systems with singular phase: The phase singularity in two and higher
dimensional optical systems has interpretation in terms of vortices, wavefront
dislocation, etc.berry1 . We present two examples exhibiting phase singularity
in one dimension. The first example corresponds to the superpotential for a
system describing quantum particle in a box of length $L$:

$\displaystyle h(x)=-\frac{\pi}{L}cot(\frac{\pi x}{L}),$ $\displaystyle
V(x)=-\frac{\pi^{2}}{L^{2}}cosec^{2}(\frac{\pi
x}{L})+i[\frac{\pi^{2}}{L^{2}}\big{(}1-cot^{2}(\frac{\pi x}{L})\big{)}],$
$\displaystyle\Phi(x)=W\ \sqrt{\frac{2}{L}}\ \sin(\frac{\pi x}{L})\ e^{-i\
ln(\sin(\frac{\pi x}{L}))}.$ (72)

The nonlinear mode is localized within the box and vanishes at the boundary.
However, the phase is singular at the boundaries of the box. The second
example corresponds to the superpotential of Rosen-Morse potential:

$\displaystyle h(x)=N\tanh(x),\ N\in\mathbb{R}^{+},$ $\displaystyle
V(x)=-Nsech^{2}(x)-i[2N^{2}-N(2N+1)sech^{2}(x)],$
$\displaystyle\Phi(x)=Wsech^{N}(x)exp[-i\big{(}Nln(sech(x))\big{)}].$

Both the the real and imaginary parts of $V(x)$ define potential-wells of
finite depth. The potential is well-defined on the whole line. The nonlinear
mode ‘$\Phi(x)$ is localized in space. The phase of $\Phi(x)$ is singular for
${|x|}\rightarrow\infty$.

##  VII Discussion and Conclusion

We have investigated exact solvability of a class of vector NLSE with time-
modulated LC and BLG terms, and space-time modulated cubic nonlinear terms in
presence of an external complex potential. We have taken a two-step approach
to find exact solutions. The BLG and LC terms are removed completely through a
non-unitary transformation at the cost of modifying the time-modulation of the
nonlinear strength. In general, the real-valued nonlinear interaction becomes
complex after the non-unitary transformation. Further, the method of non-
unitary transformation is applicable only if LC and BLG terms have identical
time-modulation. The separation of time and space co-ordinates as well as real
and imaginary parts of the equation requires the nonlinear strengths to be
time-independent and real-valued. This is achieved by fixing the nonlinear
strengths appropriately such that the method of separation works. The time-
dependence of the nonlinear strengths is determined in terms of the time-
modulation of LC and BLG terms along with an arbitrary function $G(x,t)$,
while the space-dependence may be chosen in terms of two arbitrary functions
$f_{1}(x),f_{2}(x)$ and $G(x,t)$.

In the second step, the resulting equations are analyzed by using the method
of similarity transformation which involves writing the differential equation
in a new co-ordinate system and multiplying the amplitude by a space-dependent
scale-factor. The treatment for analyzing solvability for real and complex
potentials are different. The space-time dependence of the power $P$ is
factorised in terms of the product of space-dependent and time-dependent
functions. The time-dependence of $P$ solely depends on the form LC and BLG
couplings and independent of the specific form of external potential or the
strengths of the space-time modulated cubic nonlinearity. However, the space-
dependence of $P$ is determined in terms of the choice of the external
potential as well as space-modulation of the nonlinear strengths.

We have constructed several examples of exactly solvable models for constant,
purely time dependent, purely space dependent and space-time dependent
nonlinear-strengths for vanishing external potential. This has been done for
constant as well as time-modulated LC and BLG terms. On the other hand, the
method employed in this article allows to construct exact solutions for the
non-vanishing real potential for purely space dependent or space-time
modulated nonlinear strengths. Exact solutions for constant or purely time
dependent nonlinear strengths can not be constructed by using the method. The
complex external potential allows more flexibility in constructing exactly
solvable models. In particular, exactly solvable models for constant, purely
time dependent, purely space dependent and space-time dependent nonlinear-
strengths, and time-modulated LC and BLG terms are constructed. One
interesting result is that exact localized nonlinear modes with spatially
constant phase may be obtained for any real $V(x)$ for which the corresponding
linear Schr$\ddot{o}$dinger equation is solvable. Further, for the case of
complex potential, we have developed a method based on supersymmetric quantum
mechanics to construct several exactly solvable models. In fact, corresponding
to each exactly solved quantum mechanical problem by using supersymmetry, the
corresponding superpotential may be used to find a complex potential for which
exact localized nonlinear modes are obtained. We find a few complex potentials
for which exact nonlinear modes exhibit singular phases.

A few notable features independent of specific models are the following:

  * •

The exact solutions do not depend at all on the choice of the function
$G(x,t)$. The reasons may be attributed to the particular ansatz chosen for
the separation of variables, and real and imaginary parts of the differential
equation. It is possible that new solutions depending on $G(x,t)$ may be found
for the system. Such solution, if exists, have to be found numerically or by
different analytical methods.

  * •

The exact solutions do not depend on the specific choices of $f_{1}(x)$ and
$f_{2}(x)$, but, on their average $f(x)$. This may again be attributed to the
specific ansatz chosen for finding exact solutions. However, analytical and/or
numerical methods may be employed to verify whether or not the solutions are
insensitive to the choice of specific $f_{1}(x)$ and $f_{2}(x)$.

  * •

The power-oscillation in time is absent for time-independent strength of
nonlinear terms. This holds for constant as well as time-modulated LC and BLG
terms. The result is to be contrasted with the system considered in Ref. pkg ,
where power-oscillation is seen for constant LC, BLG and nonlinear strength.
There is no contradiction, since the nonlinear term in Eq. (1) for $g_{ij}$
given by Eq. (38) can not be cast in the form $(\psi^{\dagger}M\psi)^{2}$ such
that $A$ is pseudo hermitian with respect to $M$. The power oscillation is
observed in ref. pkg  for constant LC, BLG and nonlinear term whenever $A$ is
$M$ pseudo hermitian. Thus, the power-oscillation for constant LC and BLG
terms in pkg  may be attributed to the specific form of the nonlinear
interaction.

  * •

The case $f(x)=0$ for which exact localized nonlinear modes are obtained for
real as well as complex potentials is one of the salient features of the class
of NLSE considered in this article. The method based on supersymmetric quantum
mechanics to construct localized nonlinear modes for complex potential needs
to be explored further for its applicability to other types of NLSE with
external potential.

The system defined by Eq. (1) is directly relevant in the context of optics.
The NLSE has been studied through approximate and/or numerical methods
previously for constant nonlinear strengths and constant or specific time-
modulated LC and BLG terms. We have given a generic framework in which a class
of exact solutions may be found for various combinations of space-time
modulated nonlinear strengths, time-modulated LC and BLG terms and external
potential. Specific realizations of some of these models in realistic physical
scenario may enrich the current understanding on the subject.

###### Acknowledgements.

The work of PKG is supported by a grant (SERB Ref. No. MTR/2018/001036) from
the Science & Engineering Research Board(SERB), Department of Science and
Technology, Govt.of India, under the MATRICS scheme. SG acknowledges the
support of DST INSPIRE fellowship of Govt.of India(Inspire Code No. IF190276).

##  VIII Appendix-I: Solutions of a nonlinear equation

We present the solutions of the following equation:

$\displaystyle u_{\zeta\zeta}+\omega^{2}(\zeta)u-\frac{C^{2}}{u^{3}}-2\sigma
u^{3}=0,$ (73)

where $\sigma$ and $C$ are real constants. We discuss two special cases before
embarking on the general solutions:
(I) ${\bf\sigma=0}$: Eq. (73) describes Ermakov-Pinney equation. The general
solution is given by,

$\displaystyle u(\zeta)=\sqrt{v_{1}^{2}+\frac{C^{2}}{W^{2}}v_{2}^{2}},\
W(\zeta)=v_{1}v_{2,\zeta}-v_{2}v_{1,\zeta},$ (74)

where $v_{1}$, $v_{2}$ are two independent solutions of the equation
$v_{\zeta\zeta}+\omega^{2}(\zeta)v=0$ satisfying the initial conditions at
$\zeta=\zeta_{0}$:
$v_{1}(\zeta_{0})=u(\zeta_{0}),v_{1,\zeta}(\zeta_{0})=u_{\zeta}(\zeta_{0}),v_{2}(\zeta_{0})\neq
0,v_{2,\zeta}(\zeta_{0})=0$. We have used the convention
$v_{i,\zeta}=\frac{dv_{i}}{d\zeta},i=1,2$. For constant $\omega$, the choice
$v_{1}=\sin(\omega\zeta),v_{2}=\cos(\omega\zeta)$ leads to the periodic
solution,

$\displaystyle u(\zeta)$ $\displaystyle=$
$\displaystyle\sqrt{\sin^{2}(\omega\zeta)+\frac{C^{2}}{w^{2}}\cos^{2}(\omega\zeta)}.$
(75)

For a periodic $\omega(\zeta)$ with period $T$, we obtain Hill’s equation
which has been studied extensively in the literaturehill . The solutions for a
generic periodic $\omega(\zeta)$ with the condition
$v_{1}(0)=0,v_{1,\zeta}=1,v_{2}(0)=1$ are stable for
${|v_{1}(T)+v_{2,\zeta}(T)|}<2$hill .
(II) ${\bf C=0}$: In Eq.(73) we replace $\omega^{2}(\zeta)$ with constant $m$
which can take both positive and negative values. This describes a NLSE and
it’s solution is well known,

$\displaystyle u(\zeta)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{m}{\sigma}}\ \sec\big{(}\sqrt{m}\zeta\big{)},\
(m,\sigma)>0$ (76) $\displaystyle=$ $\displaystyle\sqrt{\frac{|m|}{|\sigma|}}\
sech\big{(}\sqrt{|m|}\zeta\big{)},\ (m,\sigma)<0.$

III. ${\bf C\neq 0,\sigma\neq 0,\omega^{2}\equiv m=}\textrm{\bf Constant}$:
The transformation $u(\zeta)=\sqrt{Q(\zeta)}$ along with a change in the
variable from $\zeta$ to $y=\sqrt{|\sigma|}\zeta$ followed by integration
transforms Eq. (73) in the following form:

$\displaystyle Q_{y}^{2}=\operatorname{sgn}(\sigma)\ 4\
(Q-Q_{1})(Q-Q_{2})(Q-Q_{3})$ (77)

where $Q_{1},Q_{2},Q_{3}$ are the roots of the cubic equation
$Q^{3}-\frac{m}{\sigma}Q^{2}+\frac{C_{1}}{\sigma}Q-\frac{C^{2}}{\sigma}=0$,
$C_{1}$ is an integration constant and $\operatorname{sgn}(\sigma)$ is the
signum function. The expressions for the roots in terms of
$\omega,\sigma,C,C_{1}$ are too lengthy and will not be presented here.
However, for given values of $Q_{i}$’s in a particular solution, corresponding
values of $\omega,\sigma,C,C_{1}$ may be obtained by using the properties of
the roots: $Q_{1}Q_{2}Q_{3}=\frac{C^{2}}{\sigma},\ \
Q_{1}Q_{2}+Q_{2}Q_{3}+Q_{1}Q_{3}=\frac{C_{1}}{\sigma},Q_{1}+Q_{2}+Q_{3}=\frac{m}{\sigma}$.
The real solutions of Eq. (77) satisfy $(Q-Q_{1})(Q-Q_{2})(Q-Q_{3})\geq 0$ for
$\sigma>0$, while $(Q-Q_{1})(Q-Q_{2})(Q-Q_{3})\geq 0$ for $\sigma<0$. The
finite and stable solutions of Eq.(77) are presented below based on
boundedness of the solution $Q$ in terms $Q_{i}$. A particular ordering among
thee roots are considered for presentation of the results following the
discussions in Ref. Beitia :
(a) ${\bf\sigma>0,0<Q_{1}\leq Q\leq Q_{2}<Q_{3}}$ : The solution of the
Eq.(77) reads,

$\displaystyle Q=Q_{1}+(Q_{2}-Q_{1})sn^{2}[\lambda y,r],$ (78)

where $\lambda=\sqrt{Q_{3}-Q_{1}}$ and
$r^{2}={\frac{Q_{2}-Q_{1}}{Q_{3}-Q_{1}}}$. The solution can be expressed in
terms of hyperbolic function in the limit of $Q_{3}\rightarrow Q_{2}$:

$\displaystyle Q=Q_{1}+(Q_{2}-Q_{1})\tanh^{2}[\lambda y]$ (79)

(b) ${\bf\sigma>0,0<Q_{1}<Q_{2}<Q_{3}\leq Q}$: The solution of Eq.(77) reads,

$\displaystyle Q=Q_{1}+\frac{Q_{3}-Q_{1}}{sn^{2}(\lambda y,r)},$ (80)

where $\lambda^{2}=Q_{3}-Q_{1}$ and $r^{2}=\frac{Q_{2}-Q_{1}}{Q_{3}-Q_{1}}$.
The solution reduces to an elementary singular periodic function for
$Q_{1}=Q_{2}$, while it gives a singular soliton if $Q_{3}=Q_{2}$.

(c) ${\bf\sigma<0,Q_{1}<0<Q_{2}\leq Q\leq Q_{3}}$: In this case, we get real
finite stable solution. The solution for $\sigma<0$ has the form:

$\displaystyle Q=Q_{3}-(Q_{3}-Q_{2})sn^{2}(\lambda y,r)$ (81)

where $\lambda^{2}=Q_{3}-Q_{1}$ and $r^{2}=\frac{Q_{3}-Q_{2}}{Q_{3}-Q_{1}}$.

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