# On explicit soliton solutions and blow-up for Coupled variable coefficient
nonlinear Schrödinger equations

José M. Escorcia  Escuela de Ciencias Aplicadas e Ingeniería, Universidad
EAFIT, Carrera 49 No. 7 Sur-50, Medellín 050022, Colombia.
<EMAIL_ADDRESS>and  Erwin Suazo  School of Mathematical and
Statistical Sciences, University of Texas Rio Grande Valley, 1201 W.
University Drive, Edinburg, Texas, 78539-2999<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

This work is concerned with the study of explicit solutions for a generalized
coupled nonlinear Schrödinger equations (NLS) system with variable
coefficients. Indeed, we show, employing similarity transformations, the
existence of Rogue wave and dark-bright soliton like-solutions for such a
generalized NLS system, provided the coefficients satisfy a Riccati system. As
a result of the multiparameter solution of the Riccati system, the nonlinear
dynamics of the solution can be controlled. Finite-time singular solutions in
the $L^{\infty}$ norm for the generalized coupled NLS system are presented
explicitly. Finally, an n-dimensional transformation between a variable
coefficient NLS coupled system and a constant coupled system coefficient is
presented. Soliton and Rogue wave solutions for this high-dimensional system
are presented as well. A Mathematica file has been prepared as supplementary
material, verifying the Riccati systems used in the construction of the
solutions

Keywords: Coupled Nonlinear Schr$\ddot{\mbox{o}}$dinger Equations, Soliton
solution, Rogue wave solution, Blowup solution, Similarity transformations and
Riccati systems.

###### 1991 Mathematics Subject Classification:

Primary 81Q05, 35C05. Secondary 42A38

##  1\. Introduction

The coupled nonlinear Schr$\ddot{\mbox{o}}$dinger equation models several real
phenomena such as the interaction of Bloch-wave packets in a periodic system
[37], the evolution of two surface wave packets in deep water, and in
wavelength-division-multiplexed systems [28]. The starting point of many
studies of these kinds of models was the well-known Manakov system introduced
in 1974 [25] (see Equations (2.1)-(2.2)). The Manakov system is an integrable
coupled NLS characterized by equal nonlinear interactions within and between
components [46]. A great effort has been made to figure out the dynamics of
the soliton-like solutions for coupled NLS. Some of these structures are dark-
bright (DB), bright-bright (BB), and dark-dark (DD) solitons. These states
have been observed in several experiments in optics [11, 29] and in the
context of Bose-Einstein condensates [7]. Another important class of solutions
is Rogue waves. These types of solutions are strong wavelets that can appear
in the ocean from nowhere and disappear without a trace. In fact, it is this
particular property that makes them candidates for the study of predicting
catastrophic phenomena such as tsunamis, thunderstorms, and earthquakes [4].

In order to describe more realistically some phenomena dynamics, the space-
time variation of the coefficients must be taken into account in the majority
of cases. In this sense, new nonlinear Schr$\ddot{\mbox{o}}$dinger-type
coupled systems with variable coefficients have been introduced [14, 15, 17,
19, 24, 26, 30, 31, 34, 45]. The difficulty of obtaining non-constant
coefficients in the construction of explicit solutions can be overcome by
making use of different techniques such as the Hirota bilinear method [10,
47], similarity transformations [17, 26, 45], Darboux transformations [8, 14]
and Bäcklund transformation [9], to mention a few.

During recent year and due to many applications, there has been a growing
interest in the study of solitons and Rogue wave type solutions for coupled
variable coefficient NLS equations [15, 24, 30, 31, 34]. At present, the field
interested in these kinds of solution and moved by the necessity of data
transfer are optical fiber technology and soliton transmission. More
precisely, these special solutions can overcome some difficulties related to
physical properties of the fiber and the interaction of it and the transmitted
soliton. Although some important advances have been made in understanding the
dynamics of soliton solutions for generalized coupled NLS systems from the
mathematical and physical point of view, it remains a difficult problem. In
this context and under the revolution of hardware equipment and software
technology, recent studies of these fascinating solutions have been carried
out by using a deep learning approach as well [33, 35].

In this work, the objective will be to study the Schr$\ddot{\mbox{o}}$dinger-
type coupled system with quadratic Hamiltonian (VCNLS hereafter):

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\psi_{xx}+\left(b(t)x^{2}-id(t)-xf(t)\right)\psi+i\left(g(t)-c(t)x\right)\psi_{x}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\psi,$
(1.1) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\varphi_{xx}+\left(b(t)x^{2}-id(t)-xf(t)\right)\varphi+i\left(g(t)-c(t)x\right)\varphi_{x}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\varphi.$
(1.2)

Here $\psi(x,t)$ and $\varphi(x,t)$ represent complex wavefunctions,
$a(t),b(t),c(t),d(t),f(t),g(t),h(t)$ are given time-dependent functions and
$s>0.$ The coefficient $a(t)$ is the dispersion management parameter, $b(t)$
determines the time-dependence of the quadratic potential, $d(t)$ is the
gain/loss term, $f(t)x$ standardizes an arbitrary linear potential and $h(t)$
is the nonlinear management. The system (1.1)-(1.2) appears to describe the
propagation of solitons in birefringence fibers, see for example [17, 45], and
also, in the framework of solitary waves in multi-component Bose-Einstein
condensates [18, 43]. We point out that the proposed system (1.1)-(1.2)
extends many of the models studied in the literature.

The uncoupled case ($\varphi\equiv 0$), has been the source of many studies in
the last years [3, 5, 12, 16, 39, 40]. By employing the similarity
transformation and imposing the coefficients satisfy a Riccati system (see
Equations (3.5)-(3.10) and (4.5)-(4.10)), soliton-like solutions (bright,
dark, and Peregrine), singular solutions, and the fundamental solution
($h(t)\equiv 0$) were constructed. Also, the Riccati system allowed the
construction and study of explicit solutions for diffusion-type equations [41]
and reaction–diffusion equations [32].

The paper’s aim is to establish a relationship between dispersion, damping,
and nonlinearity terms by the Riccati system in order to obtain solutions such
as Dark-Bright and Rogue wave soliton solutions for the coupled VCNLS
(1.1)-(1.2). The multiparameters solution of the Riccati system allows
controlling the soliton dynamics; this is crucial in practice, because one
needs to control the soliton transmission of information since standard vector
solitons for NLS systems don’t have controllable parameters. This relation in
the coefficients provides the construction of solutions with blowup in finite
time in the $L^{\infty}$ norm. Finally, an $n-$dimensional generalization of
the VCNLS is studied and the existence of soliton solutions is shown as well.

The paper is organized as follows: In Section 2 we introduce the classical
coupled Manakov system and its Rogue wave and soliton type solutions. The
explicit construction of Rogue wave and soliton type-solutions for the
generalized coupled NLS system (1.1)-(1.2) is presented in Section 3. This
objective is obtained by using Theorem 1 and solutions of the Manakov system
given in the previous section. In Section 4, we construct explicit
$L^{\infty}$ singular solutions for the generalized coupled NLS system, as can
be seen in Theorem 2. Section 5 is devoted to the extension of the
transformation established in Theorem 1. Such a new transformation allows us
to connect an n-dimensional coupled NLS system with a n-dimensional Manakov
system, see Theorem 3. Also, soliton and Rogue wave solutions are constructed
for this model. A Mathematica file has been prepared as supplementary
material, verifying the Riccati systems used in the construction of the
solutions in all the previous sections. Conclusions and some final remarks are
given in Section 6. Section 7 corresponds to the appendix, in which the
solution of the Riccati system is presented.

##  2\. Classical solutions of coupled NLS

The coupled NLS system, or Manakov system

$\displaystyle
i\chi_{\tau}+\chi_{\xi\xi}+2(\left|\phi\right|^{2}+\left|\chi\right|^{2})\chi$
$\displaystyle=$ $\displaystyle 0,$ (2.1) $\displaystyle
i\phi_{\tau}+\phi_{\xi\xi}+2(\left|\phi\right|^{2}+\left|\chi\right|^{2})\phi$
$\displaystyle=$ $\displaystyle 0,$ (2.2)

is a nonlinear wave system that governs a wide variety of physical phenomena
ranging from the dynamics of two-component Bose-Einstein condensate [7, 20],
nonlinear optics [11, 29], and the dynamics of deep-sea waves under the
interaction of two-layer stratification [44]. This system (2.1)-(2.2) was
shown by Manakov to be integrable [25, 46]. It admits the soliton-like
structures such as dark-bright (DB) solutions. In fact, (2.1)-(2.2) possesses
the DB solitons, see for example [7, 11, 45]:

$\chi(\xi,\tau)=C\tanh\left[C(\xi-2e\tau)+E\right]\exp\left[i(e\xi+(2C^{2}-e^{2})\tau)\right],$
(2.3)

$\phi(\xi,\tau)=-4C^{2}(a_{3}+ib_{3})\operatorname{sech}\left[C(\xi-2e\tau)+E\right]\exp\left[i\left(e\xi+(3C^{2}-e^{2})\tau\right)\right],$
(2.4)

where $E=\frac{1}{2}\ln\left(\frac{a_{3}^{2}+b_{3}^{2}}{2C^{2}}\right)$ and
$C,e,a_{3},b_{3}$ are arbitrary constants. These kinds of solutions were first
experimentally observed in photorefractive crystals [11]. Other interesting
solutions with fascinating properties are the Rogue waves. Indeed, the coupled
NLS system (2.1)-(2.2) admits the exact Rogue wave solution of type I [6, 45]:

$\chi(\xi,\tau)=A\exp(i\theta_{1})\left(-1-i\sqrt{3}+\frac{-6AB\sqrt{3}-36\tau
A^{2}\sqrt{3}-3+i(36A^{2}\tau+6AB+5\sqrt{3})}{12A^{2}B^{2}+8AB\sqrt{3}+144\tau^{2}A^{4}+5}\right),$
(2.5)

$\phi(\xi,\tau)=A\exp(i\theta_{2})\left(-1+i\sqrt{3}+\frac{-6AB\sqrt{3}+36\tau
A^{2}\sqrt{3}-3+i(36A^{2}\tau-6AB-5\sqrt{3})}{12A^{2}B^{2}+8AB\sqrt{3}+144\tau^{2}A^{4}+5}\right),$
(2.6)

where $B=\xi+6q\tau,$
$\theta_{i}=d_{i}\xi+(2c_{1}^{2}+2c_{2}^{2}-d_{i}^{2})\tau$ with $i=1,2.$ The
parameters $A=d_{2}+3q,$ $d_{1}=d_{2}-2A,$ $c_{i}=\pm 2A,$ with $i=1,2$ and
$d_{2},q$ are arbitrary constants. Similarly, the Rogue wave solution of type
II [6, 45] is given by:

$\chi(\xi,\tau)=A\left(-1-i\sqrt{3}+\frac{G_{1}+iH_{1}}{D}\right)\exp(i\theta_{1}),$
(2.7)

$\phi(\xi,\tau)=A\left(-1+i\sqrt{3}+\frac{G_{2}+iH_{2}}{D}\right)\exp(i\theta_{2}),$
(2.8)

where

$\displaystyle D$ $\displaystyle=$ $\displaystyle
1+4\sqrt{3}AB+24A^{2}B^{2}+16\sqrt{3}A^{3}B^{3}+12A^{4}B^{4}+48A^{4}(9+8\sqrt{3}AB+6A^{2}B^{2})\tau^{2}+1728A^{8}\tau^{4},$
$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle-3\left(-1+6A^{2}B^{2}+4\sqrt{3}A^{3}B^{3}+4A^{2}(\sqrt{3}+12AB+6\sqrt{3}A^{2}B^{2})\tau+24A^{4}(3+2\sqrt{3}AB)\tau^{2}+288\sqrt{3}A^{6}\tau^{3}\right),$
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
3\left(1-6A^{2}B^{2}-4\sqrt{3}A^{3}B^{3}+4A^{2}(\sqrt{3}+12AB+6\sqrt{3}A^{2}B^{2})\tau-24A^{4}(3+2\sqrt{3}AB)\tau^{2}+288\sqrt{3}A^{6}\tau^{3}\right),$
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\sqrt{3}+12AB+18\sqrt{3}A^{2}B^{2}+12A^{3}B^{3}+12A^{2}(9+8\sqrt{3}AB+6A^{2}B^{2})\tau+24A^{4}(13\sqrt{3}+6AB)\tau^{2}+864A^{6}\tau^{3},$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle-\sqrt{3}-12AB-18\sqrt{3}A^{2}B^{2}-12A^{3}B^{3}+12A^{2}(9+8\sqrt{3}AB+6A^{2}B^{2})\tau-24A^{4}(13\sqrt{3}+6AB)\tau^{2}+864A^{6}\tau^{3}.$

In the subsequent section, we will use these solutions to construct explicit
solutions for the generalized coupled NLS system.

##  3\. Generalized Coupled NLS System

The aim of this section is to establish the existence of soliton-type and
Rogue wave solutions (by the explicit construction of these solutions) of the
generalized coupled NLS system (1.1)-(1.2). More precisely, by using
similarity transformations and by imposing the coefficients to satisfy the
Riccati system, classical solutions of the standard Manakov system (2.1)-(2.2)
are extended to this general framework. Indeed, the main result of this
section is:

######  Theorem 1.

Let $l_{0}=\pm 1.$ Then, the following NLS coupled system (or VCNLS)

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\psi_{xx}+(b\left(t\right)x^{2}-id(t)-xf(t))\psi+i(g(t)-c\left(t\right)x)\psi_{x}+h(t)(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\psi,$
(3.1) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\varphi_{xx}+(b\left(t\right)x^{2}-id(t)-xf(t))\varphi+i(g(t)-c\left(t\right)x)\varphi_{x}+h(t)(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\varphi,$
(3.2)

admits dark-bright (DB) solitons and Rogue wave type solutions, where $h(t)$
satisfies the integrability condition (3.14).

###### Proof.

We look for solutions for (3.1)-(3.2) of the form

$\psi(x,t)=\frac{1}{\sqrt{\mu(t)}}e^{i(\alpha(t)x^{2}+\delta(t)x+\kappa(t))}\chi(\xi,\tau),\qquad\xi=\beta(t)x+\varepsilon(t),\qquad\tau=\gamma(t),$
(3.3)

and

$\varphi(x,t)=\frac{1}{\sqrt{\mu(t)}}e^{i(\alpha(t)x^{2}+\delta(t)x+\kappa(t))}\phi(\xi,\tau).\qquad$
(3.4)

After substituting (3.3)-(3.4) in (3.1)-(3.2) and taking the real and
imaginary parts, we obtain the Riccati system [12, 16, 39, 40, 41]:

$\dfrac{d\alpha}{dt}+b(t)+2c(t)\alpha+4a(t)\alpha^{2}=0,$ (3.5)
$\dfrac{d\beta}{dt}+(c(t)+4a(t)\alpha(t))\beta=0,$ (3.6)
$\dfrac{d\gamma}{dt}+a(t)\beta^{2}(t)l_{0}=0,$ (3.7)
$\dfrac{d\delta}{dt}+(c(t)+4a(t)\alpha(t))\delta=f(t)+2\alpha(t)g(t),$ (3.8)
$\dfrac{d\varepsilon}{dt}=(g(t)-2a(t)\delta(t))\beta(t),$ (3.9)
$\dfrac{d\kappa}{dt}=g(t)\delta(t)-a(t)\delta^{2}(t).$ (3.10)

Considering the standard substitution

$\alpha=\left(\dfrac{1}{4a(t)}\dfrac{\mu^{\prime}(t)}{\mu(t)}-\dfrac{d(t)}{2a(t)}\right),$
(3.11)

it follows that the Riccati equation (3.5) becomes

$\mu^{\prime\prime}-\eta(t)\mu^{\prime}+4\sigma(t)\mu=0,$ (3.12)

with

$\eta(t)=\frac{a^{\prime}}{a}-2c+4d,\hskip 28.45274pt\sigma(t)=ab-
cd+d^{2}+\frac{d}{2}\left(\frac{a^{\prime}}{a}-\frac{d^{\prime}}{d}\right).$
(3.13)

We will refer to (3.12) as the characteristic equation of the Riccati system.
Further, if we choose by integrability conditions for the uncoupled case

$h(t)=\lambda a(t)\beta^{2}(t)\mu(t)$ (3.14)

with $\lambda\in\mathbb{R}$, the functions $\chi$ and $\phi$ will satisfy the
NLS system in the new variables $\tau$ and $\xi$:

$\displaystyle
i\chi_{\tau}-l_{0}\chi_{\xi\xi}+l_{0}\lambda(\left|\phi\right|^{2}+\left|\chi\right|^{2})\chi$
$\displaystyle=$ $\displaystyle 0,$ (3.15) $\displaystyle
i\phi_{\tau}-l_{0}\phi_{\xi\xi}+l_{0}\lambda(\left|\phi\right|^{2}+\left|\chi\right|^{2})\phi$
$\displaystyle=$ $\displaystyle 0,$ (3.16)

where $\alpha(t),$ $\beta(t),$ $\gamma(t),$ $\delta(t),$ $\epsilon(t)$ and
$\kappa(t)$ are given by the Equations (7.1)-(7). Therefore, the existence of
these kinds of solutions (DB, and Rogue wave solutions) are established by
letting $\chi$ and $\phi$ be the solutions discussed in the previous section,
see Equations (2.3)-(2.8). ∎

###  3.1. Rogue wave solutions

The existence of Rogue wave solutions can be guaranteed by the extension of
the solutions (2.5)-(2.6) and (2.7)-(2.8) by the previous Theorem 1. We start
with the Type I Rogue Wave.

####  3.1.1. Case $a=d=1,\ b=t,\ c=f=g=0,\
h=\frac{-2e^{2t}}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)}$

In this case, the coupled nonlinear Schr$\ddot{\mbox{o}}$dinger equations have
the form

$i\psi_{t}=-\psi_{xx}+tx^{2}\psi-i\psi-\frac{2e^{2t}}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)}\left(|\varphi|^{2}+|\psi|^{2}\right)\psi,$
(3.17)
$i\varphi_{t}=-\varphi_{xx}+tx^{2}\varphi-i\varphi-\frac{2e^{2t}}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)}\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi.$
(3.18)

Then, according to Theorem 1, we get

$\displaystyle\alpha(t)=\frac{1}{4t}-\frac{1+t^{3}{}_{0}F_{1}\left(2/3,-4t^{3}/9\right){}_{0}F_{1}\left(7/3,-4t^{3}/9\right)}{4t{}_{0}F_{1}\left(2/3,-4t^{3}/9\right){}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},\quad\quad\quad\beta(t)=\dfrac{1}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)},$
$\displaystyle\gamma(t)=\frac{t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)},\quad\quad\quad\delta(t)=\dfrac{1}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)},\quad\quad\varepsilon(t)=-\dfrac{2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)},$
$\displaystyle\kappa(t)=-\frac{t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)},\quad\quad\mu(t)=e^{2t}{}_{0}F_{1}\left(2/3,-4t^{3}/9\right),\quad\quad\quad\quad$

where ${}_{0}F_{1}\left(a,z\right)$ is the confluent hypergeometric function
given by the series
${}_{0}F_{1}\left(a,z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{(a)_{k}k!}$ with
$(a)_{k}=\Gamma(a+k)/\Gamma(a)$ and $\Gamma(z)$ is the gamma function. Here,
we have solved the Riccati system

(a) Profile of $|\psi|^{2}$.

(b) Profile of $|\varphi|^{2}$.

(c) Difference $|\psi-\varphi|$.

Figure 1. Solutions for the system (3.17)-(3.18) for the parameters $d_{2}=1,$
$q=0$.

with the initial conditions $\alpha(0)=0$, $\beta(0)=1$, $\gamma(0)=0,$
$\delta(0)=1$, $\varepsilon(0)=0,$ $\kappa(0)=0$ and $\mu(0)=1.$
Therefore, we can construct a solution in the form:

$\psi(x,t)=\dfrac{e^{-t}}{\sqrt{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.19)
$\varphi(x,t)=\dfrac{e^{-t}}{\sqrt{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau),$
(3.20)

where $\chi$ and $\phi$ are given by Equations (2.5)-(2.6). The profiles of
these solutions for the parameters $d_{2}=1$ and $q=0$ are shown in Figure
1(a)-(b). Figure 1(c) shows the difference between the two solutions to show
that they are different.

####  3.1.2. Case $a=-\frac{\cos t}{2},\ b=c=f=g=0,\ d=-1,\ h=e^{-2t}\cos t$

The corresponding system is given by

$i\psi_{t}=\frac{\cos t}{2}\psi_{xx}+i\psi+e^{-2t}\cos
t\left(|\varphi|^{2}+|\psi|^{2}\right)\psi,$ (3.21) $i\varphi_{t}=\frac{\cos
t}{2}\varphi_{xx}+i\varphi+e^{-2t}\cos
t\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi.$ (3.22)

(a) Profile of $|\psi|^{2}$.

(b) Profile of $|\varphi|^{2}$.

(c) Difference $|\psi-\varphi|$.

Figure 2. Solutions for the system (3.21)-(3.22) with parameters $d_{2}=1,$
$q=-1$.

Then,

$\displaystyle\alpha(t)=0,\quad\quad\quad\quad\beta(t)=1,\quad\quad\quad\gamma(t)=-\frac{\sin
t}{2},\quad\quad\quad\delta(t)=1,$ $\displaystyle\varepsilon(t)=\sin
t,\quad\quad\quad\quad\quad\quad\kappa(t)=\frac{\sin
t}{2},\quad\quad\quad\quad\quad\quad\quad\mu(t)=e^{-2t}.\quad\quad\quad\quad$

The initial conditions used to solve the Riccati system are the same as those
given in the previous case. In fact, except for the section on bending
dynamics, we shall use these initial conditions for the rest of the study.
Solutions for the system (3.21)-(3.22) are

$\psi(x,t)=e^{\arccos(\cos
t)}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.23) $\varphi(x,t)=e^{\arccos(\cos
t)}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau).$
(3.24)

Again $\chi$ and $\phi$ are given by the Equation (2.5)-(2.6). Figure 2 shows
the time evolution of $|\psi|^{2}$ and $|\varphi|^{2}$ for the corresponding
solutions. The difference between the two solutions are shown in the Figure
2(c).

The next two cases correspond to the Type II Rogue Wave solutions.

####  3.1.3. Case $a=-\operatorname{sech}t/2,\ b=-4a,\ c=4a,\ d=2a,\ f=g=0,\
h=\frac{-2\operatorname{sech}t}{-2\cosh\left(\sqrt{8}\operatorname{gd}t\right)+\sqrt{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right)}$

Assume the system

$i\psi_{t}=\operatorname{sech}t\left(\dfrac{\psi_{xx}}{2}+(i+2x^{2})\psi+2ix\psi_{x}-\frac{2\left(|\varphi|^{2}+|\psi|^{2}\right)\psi}{-2\cosh\left(\sqrt{8}\operatorname{gd}t\right)+\sqrt{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right)}\right),$
(3.25)
$i\varphi_{t}=\operatorname{sech}t\left(\dfrac{\varphi_{xx}}{2}+(i+2x^{2})\varphi+2ix\varphi_{x}-\frac{2\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi}{-2\cosh\left(\sqrt{8}\operatorname{gd}t\right)+\sqrt{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right)}\right),$
(3.26)

where $\operatorname{gd}t$ is the Gudermannian function given by
$\operatorname{gd}t=2\arctan\left[\tanh\left(\tfrac{1}{2}t\right)\right]$. The
expressions for the functions $\alpha(t)$, $\beta(t)$, $\gamma(t)$,
$\delta(t)$, $\varepsilon(t)$, $\kappa(t)$ and $\mu(t)$ are

$\displaystyle\alpha(t)=\dfrac{1}{1-\sqrt{2}\coth\left(\sqrt{8}\operatorname{gd}t\right)},\quad\quad\quad\beta(t)=\dfrac{\sqrt{2}}{\sqrt{2}\cosh\left(\sqrt{8}\operatorname{gd}t\right)-\sinh\left(\sqrt{8}\operatorname{gd}t\right)},$
$\displaystyle\gamma(t)=\dfrac{1}{4-4\sqrt{2}\coth\left(\sqrt{8}\operatorname{gd}t\right)},\quad\quad\quad\quad\quad\delta(t)=\dfrac{\sqrt{2}}{\sqrt{2}\cosh\left(\sqrt{8}\operatorname{gd}t\right)-\sinh\left(\sqrt{8}\operatorname{gd}t\right)},$
$\displaystyle\
\varepsilon(t)=-\dfrac{1}{2-\sqrt{8}\coth\left(\sqrt{8}\operatorname{gd}t\right)},\quad\quad\quad\quad\kappa(t)=-\dfrac{1}{4-4\sqrt{2}\coth\left(\sqrt{8}\operatorname{gd}t\right)},$
$\displaystyle\mu(t)=\cosh\left(\sqrt{8}\operatorname{gd}t\right)-\frac{\sqrt{2}}{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right).\quad\quad\quad\quad\quad\quad$

(a) Profile of $|\psi|^{2}$.

(b) Profile of $|\varphi|^{2}$.

(c) Difference $|\psi-\varphi|$.

Figure 3. Solutions for the system (3.25)-(3.26) for the parameters
$d_{2}=-0.5,$ $q=0$.

Now, we can give a form for the solutions, that is,

$\psi(x,t)=\dfrac{1}{\sqrt{\cosh\left(\sqrt{8}\operatorname{gd}t\right)-\frac{\sqrt{2}}{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.27)
$\varphi(x,t)=\dfrac{1}{\sqrt{\cosh\left(\sqrt{8}\operatorname{gd}t\right)-\frac{\sqrt{2}}{2}\sinh\left(\sqrt{8}\operatorname{gd}t\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau),$
(3.28)

with $\chi$ and $\phi$ satisfying the Equations (2.7)-(2.8). The dynamics of
these solutions are shown in Figure 3.

####  3.1.4. Case $a=-(1+t^{2})\cos t/2,\ b=2a,\ c=4a,\ d=2a,\ f=g=0,\
h=\frac{2a}{\cosh\theta-\sqrt{2}\sinh\theta}$

Consider the system of the nonlinear Schr$\ddot{\mbox{o}}$dinger equations

$i\psi_{t}=(1+t^{2})\cos
t\left(\dfrac{\psi_{xx}}{2}+(i-x^{2})\psi+2ix\psi_{x}-\frac{\left(|\varphi|^{2}+|\psi|^{2}\right)\psi}{\cosh\theta-\sqrt{2}\sinh\theta}\right),$
(3.29) $i\varphi_{t}=(1+t^{2})\cos
t\left(\dfrac{\varphi_{xx}}{2}+(i-x^{2})\varphi+2ix\varphi_{x}-\frac{\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi}{\cosh\theta-\sqrt{2}\sinh\theta}\right),$
(3.30)

with $\theta=\sqrt{8}t\cos t+\sqrt{2}(t^{2}-1)\sin t$. In this case, the
solution of the Riccati system has the form

$\displaystyle\alpha(t)=\dfrac{1}{-2+\sqrt{2}\coth\theta},\quad\quad\quad\quad\quad\beta(t)=\dfrac{1}{\cosh\theta-\sqrt{2}\sinh\theta},$
$\displaystyle\gamma(t)=\dfrac{1}{4-\sqrt{8}\coth\theta},\quad\quad\quad\quad\quad\delta(t)=\dfrac{1}{\cosh\theta-\sqrt{2}\sinh\theta},$
$\displaystyle\
\varepsilon(t)=\dfrac{1}{-2+\sqrt{2}\coth\theta},\quad\quad\quad\quad\kappa(t)=\dfrac{1}{-4+\sqrt{8}\coth\theta},$
$\displaystyle\mu(t)=\cosh\theta-\sqrt{2}\sinh\theta.\quad\quad\quad\quad\quad\quad\quad\quad$

(a) Dynamics of $|\psi|^{2}$.

(b) Dynamics of $|\varphi|^{2}$.

(c) Difference $|\psi-\varphi|$.

Figure 4. Solutions for the system (3.29)-(3.30) with parameters $d_{2}=1,$
$q=-1$.

Then, by the assumption that $\chi$ and $\phi$ satisfy equations (2.7)-(2.8),
Theorem 1 provides the following solutions:

$\psi(x,t)=\dfrac{1}{\sqrt{\cosh\theta-\sqrt{2}\sinh\theta}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.31)
$\varphi(x,t)=\dfrac{1}{\sqrt{\cosh\theta-\sqrt{2}\sinh\theta}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau).$
(3.32)

Figure 4 shows the profiles of $|\psi|^{2},|\varphi|^{2}$ for these particular
solutions.

###  3.2. Dark-bright soliton-type solutions

This part of the work is concerned with the construction of solutions for the
generalized coupled NLS system with soliton-type properties. As we will see,
(3.1)-(3.2) admits DB type-solitons.

####  3.2.1. Case $a=2,\ b=\frac{t+1}{2},\ c=-2,\ d=-1,\ f=g=0,\
h=\frac{-4}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}$

The system of the nonlinear Schr$\ddot{\mbox{o}}$dinger equations

$i\psi_{t}=-2\psi_{xx}+\frac{t+1}{2}x^{2}\psi+i\psi+2ix\psi_{x}-\frac{4\left(|\varphi|^{2}+|\psi|^{2}\right)\psi}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},$
(3.33)
$i\varphi_{t}=-2\varphi_{xx}+\frac{t+1}{2}x^{2}\varphi+i\varphi+2ix\varphi_{x}-\frac{4\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},$
(3.34)

(a) Profile of $|\psi|^{2}$.

(b) Profile of $|\varphi|^{2}$.

Figure 5. Solutions for the system (3.33)-(3.34) for the parameters $e=-1,$
$C=1,$ $a_{3}=1$ and $b_{3}=1$.

admits the functions

$\displaystyle\alpha(t)=\dfrac{1}{4}+\dfrac{1}{8t}+\dfrac{1}{-8t{}_{0}F_{1}\left(2/3,-4t^{3}/9\right){}_{0}F_{1}\left(4/3,-4t^{3}/9\right)+16t^{2}{}_{0}F_{1}^{2}\left(4/3,-4t^{3}/9\right)}-\dfrac{t^{2}{}_{0}F_{1}\left(7/3,-4t^{3}/9\right)}{8{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},$
$\displaystyle\
\beta(t)=\dfrac{1}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},\quad\quad\gamma(t)=\dfrac{2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},$
$\displaystyle\
\delta(t)=\dfrac{1}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},\quad\quad\varepsilon(t)=\dfrac{-4t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},$

$\displaystyle\kappa(t)=\dfrac{-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)},\quad\quad\mu(t)={}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right).$

In these terms, explicit solutions are given by the expressions

$\psi(x,t)=\dfrac{1}{\sqrt{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.35)
$\varphi(x,t)=\dfrac{1}{\sqrt{{}_{0}F_{1}\left(2/3,-4t^{3}/9\right)-2t{}_{0}F_{1}\left(4/3,-4t^{3}/9\right)}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau),$
(3.36)

with $\chi$ and $\phi$ satisfying Equations (2.3)-(2.4).

####  3.2.2. Case $a=1,\ b=\frac{\sin^{2}t-\cos t}{4},\ c=-\sin t,\ d=\sin t,\
f=g=0,\ h=-2e^{3-3\cos t}$

Consider the system of nonlinear Schr$\ddot{\mbox{o}}$dinger equations defined
for $t>0,$

$i\psi_{t}=-\psi_{xx}+\frac{\sin^{2}t-\cos t}{4}x^{2}\psi-i\sin t\psi+ix\sin
t\psi_{x}-2e^{3-3\cos t}\left(|\varphi|^{2}+|\psi|^{2}\right)\psi,$ (3.37)
$i\varphi_{t}=-\varphi_{xx}+\frac{\sin^{2}t-\cos t}{4}x^{2}\varphi-i\sin
t\varphi+ix\sin t\varphi_{x}-2e^{3-3\cos
t}\left(|\varphi|^{2}+|\psi|^{2}\right)\varphi.$ (3.38)

It has the explicit soliton-type solutions

$\psi(x,t)=\dfrac{1}{\sqrt{e^{3-3\cos
t}}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\chi(\xi,\tau),$
(3.39) $\varphi(x,t)=\dfrac{1}{\sqrt{e^{3-3\cos
t}}}\exp\left[i\left(\alpha(t)x^{2}+\delta(t)x+\kappa(t)\right)\right]\phi(\xi,\tau),$
(3.40)

(a) Evolution of $|\psi|^{2}$.

(b) Evolution of $|\varphi|^{2}$.

Figure 6. Solutions for the system (3.37)-(3.38) for the parameters $e=-1,$
$C=2,$ $a_{3}=-1$ and $b_{3}=1$.

where $\chi$ and $\phi$ satisfy equations (2.3)-(2.4), and

$\displaystyle\alpha(t)=\dfrac{\sin
t}{4},\quad\quad\quad\beta(t)=1,\quad\quad\quad\gamma(t)=t,\quad\quad\quad\delta(t)=1,$
$\displaystyle\
\varepsilon(t)=-2t,\quad\quad\quad\kappa(t)=-t,\quad\quad\quad\mu(t)=e^{3-3\cos
t}.$

In Figure 6, we display the profiles of $|\psi|^{2},|\varphi|^{2}$ by using
the values $e=-1,$ $C=2,$ $a_{3}=-1$, and $b_{3}=1$.

###  3.3. Bending dynamics

The ability to manipulate the dynamics of solutions is extremely valuable in
practice. In this sense, the Riccati approach used here allows the solutions
to be controlled. To demonstrate this point, consider the system (3.37)-(3.38)
again, but this time change the initial conditions for $\delta(t)$ and
$\varepsilon(t)$, i.e., $\delta(0)=0.8,$ $\varepsilon(0)=-1$; and
$\delta(0)=1.5,$ $\varepsilon(0)=-4.5$. The $\delta(t)$, $\varepsilon(t)$, and
$\kappa(t)$ functions are now:

$\displaystyle\delta(t)=0.8,\quad\quad\quad\varepsilon(t)=-1-1.6t,\quad\quad\quad\kappa(t)=-0.64t;$
$\displaystyle\
\delta(t)=1.5,\quad\quad\quad\varepsilon(t)=-4.5-3t,\quad\quad\quad\kappa(t)=-2.25t.$

(a) Profile of $|\psi|^{2}$ bended to the left: $\delta(0)=0.8$ and
$\varepsilon(0)=-1$.

(b) Profile of $|\varphi|^{2}$ bended to the left: $\delta(0)=0.8$ and
$\varepsilon(0)=-1$.

(c) Profile of $|\psi|^{2}$ bended to the right: $\delta(0)=1.5$ and
$\varepsilon(0)=-4.5$.

(d) Profile of $|\varphi|^{2}$ bended to the right: $\delta(0)=1.5$ and
$\varepsilon(0)=-4.5$.

Figure 7. Control of the dynamics of the system’s solution (3.37)-(3.38).

Figure 7, shows how these new parameters influence the system’s nonlinear
dynamics: The propagation axis of the solutions is bended.

##  4\. Blowup Solutions of the Generalized Coupled NLS System

The current section is focused on the study of singular solutions of the
coupled VCNLS. More precisely, we demonstrate by explicit construction the
existence of finite-time singular solutions in the $L^{\infty}$ norm.

######  Theorem 2 (Solutions with singularity in finite time with $L^{\infty}$
norm).

Consider the generalized coupled nonlinear Schrödinger system

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\psi_{xx}+(b\left(t\right)x^{2}-id(t)-xf(t))\psi+i(g(t)-c\left(t\right)x)\psi_{x}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\psi,$
(4.1) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-a(t)\varphi_{xx}+(b\left(t\right)x^{2}-id(t)-xf(t))\varphi+i(g(t)-c\left(t\right)x)\varphi_{x}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\varphi,$
(4.2)

where $a(t),b(t),c(t),d(t),f(t),g(t),h(t)$ are time-dependent functions and
$s>0.$ Then, the system (4.1)-(4.2) admits the $L^{\infty}$ singular solutions

$\psi(x,t)=\dfrac{1}{\sqrt{\mu(t)}}\exp\left[i\left(\alpha(t)x^{2}+\beta(t)xy+\gamma(t)y^{2}+\delta(t)x+\varepsilon(t)y+\kappa(t)\right)\right],$
(4.3)
$\varphi(x,t)=\dfrac{1}{\sqrt{\mu(t)}}\exp\left[i\left(\alpha(t)x^{2}+\beta(t)xz+\gamma(t)z^{2}+\delta(t)x+\varepsilon(t)z+\kappa(t)\right)\right],$
(4.4)

where $y,z$ are real parameters and
$\alpha(t),\beta(t),\gamma(t),\delta(t),\varepsilon(t),\kappa(t),\mu(t)$
satisfy the modified Riccati system (4.5)-(4.10).

###### Proof.

We will proceed as in [12, 16]. Substituting (4.3)-(4.4) into equations
(4.1)-(4.2) we get the modified Riccati system:

$\dfrac{d\alpha}{dt}+b(t)+2c(t)\alpha+4a(t)\alpha^{2}=0,$ (4.5)
$\dfrac{d\beta}{dt}+(c(t)+4a(t)\alpha(t))\beta=0,$ (4.6)
$\dfrac{d\gamma}{dt}+a(t)\beta^{2}(t)=0,$ (4.7)
$\dfrac{d\delta}{dt}+(c(t)+4a(t)\alpha(t))\delta=f(t)+2\alpha(t)g(t),$ (4.8)
$\dfrac{d\varepsilon}{dt}=(g(t)-2a(t)\delta(t))\beta(t),$ (4.9)
$\dfrac{d\kappa}{dt}=g(t)\delta(t)-a(t)\delta^{2}(t)-\dfrac{2h(t)}{\mu^{s}(t)}.$
(4.10)

Here the previous system (4.5)-(4.10) differs from (3.5)-(3.10) by the extra
term $-2h(t)/\mu^{s}(t)$ in equation (4.10). Now, considering the substitution

$\alpha=\left(\dfrac{1}{4a(t)}\dfrac{\mu^{\prime}(t)}{\mu(t)}-\dfrac{d(t)}{2a(t)}\right),$
(4.11)

it follows that the Riccati equation (4.5) becomes

$\mu^{\prime\prime}-\eta(t)\mu^{\prime}+4\sigma(t)\mu=0,$ (4.12)

with

$\eta(t)=\frac{a^{\prime}}{a}-2c+4d,\hskip 28.45274pt\sigma(t)=ab-
cd+d^{2}+\frac{d}{2}\left(\frac{a^{\prime}}{a}-\frac{d^{\prime}}{d}\right).$
(4.13)

Using Equations (7.1)-(7) in the Appendix with $l_{0}=1$, (4.5)-(4.9) can be
solved, but (4.10) must be solved separately. Let $I$ be an interval such that
$\mu_{0}(t)\not=0$ for all $t\in I.$ Now, since the functions $\mu_{0}$ and
$\mu_{1}$ are linearly independent on an interval $I^{\prime}$, we have

$\gamma_{0}^{\prime}(t)=\frac{W\left[\mu_{0}(t),\mu_{1}(t)\right]}{2\mu_{0}^{2}(t)}\not=0,\quad
t\in I\cap I^{\prime},$ (4.14)

and by $\mu(t)$ given by (7.1), see [13], functions (4.3)-(4.4) will have a
finite time blowup in the $L^{\infty}$ norm at
$T_{b}=\gamma_{0}^{\prime}(-\alpha(0))\in I\cap I^{\prime}$, i.e.,

$|\psi(x,t)|,|\varphi(x,t)|\to\infty\quad\textrm{as}\quad t\to T_{b}.$ (4.15)

∎

####  4.0.1. Case $a=1/2,\ b=c=d=f=g=0,$ and arbitrary $h$

Consider the coupled VCNLS system

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\psi_{xx}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\psi,$
(4.16) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\varphi_{xx}+h(t)(\left|\varphi\right|^{2s}+\left|\psi\right|^{2s})\varphi.$
(4.17)

Then, according to Theorem 2, the system admits the explicit solutions given
by the Equations (4.3)-(4.4) with the functions

$\displaystyle\alpha(t)=\frac{\alpha(0)}{1+2\alpha(0)t},\quad\quad\quad\beta(t)=\frac{\beta(0)}{1+2\alpha(0)t},\quad\quad\quad\gamma(t)=\gamma(0)-\frac{\beta^{2}(0)t}{2+4\alpha(0)t},$
$\displaystyle\
\delta(t)=\frac{\delta(0)}{1+2\alpha(0)t},\quad\quad\quad\kappa(t)=\kappa(0)+\frac{\delta^{2}(0)t}{4\alpha(0)(1+2\alpha(0)t)}-\frac{2}{\mu^{s}(0)}\int_{0}^{t}\frac{h(t^{\prime})}{(1+2\alpha(0)t^{\prime})^{s}}dt^{\prime},$
$\displaystyle\
\varepsilon(t)=\varepsilon(0)-\frac{\beta(0)\delta(0)t}{1+2\alpha(0)t},\quad\quad\quad\quad\quad\quad\mu(t)=\mu(0)(1+2\alpha(0)t).$

Therefore, these solutions blow up in the $L^{\infty}$ norm at the instant
$T_{b}=-\frac{1}{2\alpha(0)}$, provided $\alpha(0)<0.$

In this context, this last result allows us to extend the singular solutions
established in Theorem 1 in [16] (see also Example 1 in the same reference).

##  5\. An n-dimensional Generalized Coupled NLS System

In Section 3, we established a relationship between the one-dimensional
coupled VCNLS (3.1)-(3.2) and the well-known Manakov equations (2.1)-(2.2),
provided the coefficients satisfy the Riccati system (3.5)-(3.10). In the
subsequent Section 4 and by “exploiting” the potential of the Riccati system,
the construction of explicit singular solutions for (3.1)-(3.2) was shown. In
this section, we reveal that a similar Riccati system (see Equations
(5.7)-(5.12)) allows us to link an n-dimensional coupled NLS system with a
generalized Manakov system. It will be useful in constructing soliton
solutions for the high-dimensional system.

######  Theorem 3.

The nonautonomous coupled NLS n-dimensional

$i\psi_{t}=-a(t)\Delta\psi+b(t)|\boldsymbol{x}|^{2}\psi+\lambda
a(t)\beta^{2}(t)\mu^{s}(t)(|\varphi|^{2s}+|\psi|^{2s})\psi,$ (5.1)
$i\varphi_{t}=-a(t)\Delta\varphi+b(t)|\boldsymbol{x}|^{2}\varphi+\lambda
a(t)\beta^{2}(t)\mu^{s}(t)(|\varphi|^{2s}+|\psi|^{2s})\varphi,$ (5.2)

with $t\in\mathbb{R}$ and $\boldsymbol{x}\in\mathbb{R}^{n}$ can be transformed
into the standard coupled NLS

$iu_{\tau}=\Delta_{\boldsymbol{\xi}}u-\lambda(|v|^{2s}+|u|^{2s})u,$ (5.3)
$iv_{\tau}=\Delta_{\boldsymbol{\xi}}v-\lambda(\left|v\right|^{2s}+|u|^{2s})v,$
(5.4)

by the transformations

$\psi(\boldsymbol{x},t)=\dfrac{1}{\sqrt{\mu(t)}}e^{i(\sum_{i=1}^{n}\alpha(t)x_{i}^{2}+\delta_{i}(t)x_{i}+\kappa_{i}(t))}u(\boldsymbol{\xi},\tau),$
(5.5)
$\varphi(\boldsymbol{x},t)=\dfrac{1}{\sqrt{\mu(t)}}e^{i(\sum_{i=1}^{n}\alpha(t)x_{i}^{2}+\delta_{i}(t)x_{i}+\kappa_{i}(t))}v(\boldsymbol{\xi},\tau),\qquad$
(5.6)

with $\xi_{i}=\beta(t)x_{i}+\varepsilon_{i}(t)$ for $i=1,2,...,n$ and
$\tau=\gamma(t).$ Additionally, the following Riccati system is satisfied

$\dfrac{d\alpha}{dt}+b(t)+4a(t)\alpha^{2}=0,$ (5.7)
$\dfrac{d\beta}{dt}+4a(t)\alpha(t)\beta=0,$ (5.8)
$\dfrac{d\gamma}{dt}+a(t)\beta^{2}(t)=0,$ (5.9)
$\dfrac{d\delta_{i}}{dt}+4a(t)\alpha(t)\delta_{i}=0,$ (5.10)
$\dfrac{d\varepsilon_{i}}{dt}+2a(t)\beta(t)\delta_{i}(t)=0,$ (5.11)
$\dfrac{d\kappa_{i}}{dt}+a(t)\delta_{i}^{2}(t)=0,$ (5.12)

$i=1,...,n.$ Considering the substitution

$\alpha=\dfrac{\mu^{\prime}(t)}{4na(t)\mu(t)},$ (5.13)

it follows that the Riccati equation (5.7) becomes

$\mu^{\prime\prime}-\frac{a^{\prime}}{a}\mu^{\prime}+4nab\mu+\left(\frac{1-n}{n}\right)\frac{(\mu^{\prime})^{2}}{\mu}=0.$
(5.14)

In general, the study of n-dimensional coupled NLS systems is carried out by
degenerating them into the one-dimensional coupled NLS, see for example [23,
27] for the two-dimensional case, or directly dealing with the high-
dimensional system as can be seen in [22] for $n=3$. In contrast, our
transformation does not reduce the spatial dimension of the system, and then,
to find soliton solutions for (5.1)-(5.2) in terms of Proposition 1, we used
the ideas given in the works [22, 23, 27].

####  5.0.1. Case $a=2,\ b=-2,\ h=-8$

The two-dimensional coupled NLS system

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-2(\psi_{xx}+\psi_{yy})-2(x^{2}+y^{2})\psi-8(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\psi,$
(5.15) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-2(\varphi_{xx}+\varphi_{yy})-2(x^{2}+y^{2})\varphi-8(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\varphi$
(5.16)

admits the DB soliton solution given by equations (5.5)-(5.6) with

$\displaystyle\alpha(t)=\frac{1}{2}\tanh(4t),\quad\quad\quad\quad\quad\beta(t)=\operatorname{sech}(4t),\quad\quad\quad\quad\gamma(t)=-\frac{1}{2}\tanh(4t),$
$\displaystyle\
\delta_{i}(t)=\operatorname{sech}(4t),\quad\quad\kappa_{i}(t)=-\frac{1}{2}\tanh(4t),\quad\quad\varepsilon_{i}(t)=-\tanh(4t),\quad\quad\mu(t)=\cosh^{2}(4t),$

and the functions [23, 27]:

$\displaystyle u(\xi_{1},\xi_{2},\tau)=\chi(\xi_{1}+\xi_{2},-2\tau),$ (5.17)
$\displaystyle v(\xi_{1},\xi_{2},\tau)=\phi(\xi_{1}+\xi_{2},-2\tau),$ (5.18)

with $\chi$ and $\phi$ given by the Equations (2.3)-(2.4).

####  5.0.2. Case $a=2,\ b=-\frac{1+2t^{2}}{4},\ h=-12e^{t^{2}}$

Consider the three-dimensional coupled NLS system

$\displaystyle i\psi_{t}$ $\displaystyle=$
$\displaystyle-2(\psi_{xx}+\psi_{yy}+\psi_{zz})-\frac{1+2t^{2}}{4}(x^{2}+y^{2}+z^{2})\psi-12e^{t^{2}}(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\psi,$
(5.19) $\displaystyle i\varphi_{t}$ $\displaystyle=$
$\displaystyle-2(\varphi_{xx}+\varphi_{yy}+\varphi_{zz})-\frac{1+2t^{2}}{4}(x^{2}+y^{2}+z^{2})\varphi-12e^{t^{2}}(\left|\varphi\right|^{2}+\left|\psi\right|^{2})\varphi.$
(5.20)

Then, such a system admits the type I Rogue wave solution given by the
Equations (5.5)-(5.6) with

$\displaystyle\alpha(t)=\frac{t}{4},\quad\quad\quad\quad\quad\beta(t)=e^{-t^{2}},\quad\quad\quad\quad\gamma(t)=-\sqrt{\frac{\pi}{2}}\mbox{Erf}(\sqrt{2}t),$
$\displaystyle\
\delta_{i}(t)=e^{-t^{2}},\quad\quad\kappa_{i}(t)=-\sqrt{\frac{\pi}{2}}\mbox{Erf}(\sqrt{2}t),\quad\quad\varepsilon_{i}(t)=-\sqrt{2\pi}\mbox{Erf}(\sqrt{2}t),\quad\quad\mu(t)=e^{3t^{2}},$

and the functions

$\displaystyle
u(\xi_{1},\xi_{2},\xi_{3},\tau)=\chi(\xi_{1}+\xi_{2}+\xi_{3},-3\tau),\quad\quad
v(\xi_{1},\xi_{2},\xi_{3},\tau)=\phi(\xi_{1}+\xi_{2}+\xi_{3},-3\tau),$ (5.21)

with $\chi$ and $\phi$ given by equations (2.5)-(2.6).

##  6\. Conclusions and Final Remarks

Through this current work, we proposed and showed the integrability of a
generalized coupled VCNLS using the similarity transformation established in
Theorem 1 and Theorem 3. Here, the Riccati system played a crucial role in the
explicit construction of soliton-type solutions, rogue waves, and singular
solutions for these kinds of equations. The explicit construction of soliton
and Rogue wave-type solutions for an n-dimensional coupled NLS system was
shown as well. This work should motivate further analytical and numerical
studies looking to clarify the connections between the dynamics of variable-
coefficient NLS and the dynamics of Riccati systems. We’re hoping this fresh
approach to dealing with VCNLS systems will provide new insights into the
physics of such models and aid in comprehending real-world applications.

######  Acknowledgement 1.

We thank the anonymous reviewers for their valuable time spent reading our
work.

##  7\. Appendix: Solution of the Riccati System

In this appendix, we provide a solution of the Riccati system (3.5)-(3.10),
and we point out that all the formulas involved in such a solution have been
verified previously in [21]. A solution of the Riccati system with
multiparameters is given by the following expressions (with the respective
inclusion of the parameter $l_{0}$) [12, 16, 38, 42]:

$\mu\left(t\right)=2\mu\left(0\right)\mu_{0}\left(t\right)\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right),$
(7.1)
$\alpha\left(t\right)=\alpha_{0}\left(t\right)-\frac{\beta_{0}^{2}\left(t\right)}{4\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)},$
(7.2)
$\beta\left(t\right)=-\frac{\beta\left(0\right)\beta_{0}\left(t\right)}{2\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)}=\frac{\beta\left(0\right)\mu\left(0\right)}{\mu\left(t\right)}w\left(t\right),$
(7.3)
$\gamma\left(t\right)=l_{0}\gamma\left(0\right)-\frac{l_{0}\beta^{2}\left(0\right)}{4\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)},$
(7.4)
$\delta\left(t\right)=\delta_{0}\left(t\right)-\frac{\beta_{0}\left(t\right)\left(\delta\left(0\right)+\varepsilon_{0}\left(t\right)\right)}{2\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)},$
(7.5)
$\varepsilon\left(t\right)=\varepsilon\left(0\right)-\frac{\beta\left(0\right)\left(\delta\left(0\right)+\varepsilon_{0}\left(t\right)\right)}{2\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)},$
(7.6)
$\kappa\left(t\right)=\kappa\left(0\right)+\kappa_{0}\left(t\right)-\frac{\left(\delta\left(0\right)+\varepsilon_{0}\left(t\right)\right)^{2}}{4\left(\alpha\left(0\right)+\gamma_{0}\left(t\right)\right)},$
(7.7)

subject to the initial arbitrary conditions $\mu\left(0\right),$
$\alpha\left(0\right),$ $\beta\left(0\right)\neq 0,$ $\gamma(0),$ $\delta(0),$
$\varepsilon(0)$ and $\kappa(0)$. $\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$,
$\delta_{0}$, $\varepsilon_{0}$ and $\kappa_{0}$ are given explicitly by

$\alpha_{0}\left(t\right)=\frac{1}{4a\left(t\right)}\frac{\mu_{0}^{\prime}\left(t\right)}{\mu_{0}\left(t\right)}-\frac{d\left(t\right)}{2a\left(t\right)},$
(7.8)
$\beta_{0}\left(t\right)=-\frac{w\left(t\right)}{\mu_{0}\left(t\right)},\quad
w\left(t\right)=\exp\left(-\int_{0}^{t}\left(c\left(s\right)-2d\left(s\right)\right)\
ds\right),$ (7.9)
$\gamma_{0}\left(t\right)=\frac{1}{2\mu_{1}\left(0\right)}\frac{\mu_{1}\left(t\right)}{\mu_{0}\left(t\right)}+\frac{d\left(0\right)}{2a\left(0\right)},$
(7.10)
$\delta_{0}\left(t\right)=\frac{w\left(t\right)}{\mu_{0}\left(t\right)}\ \
\int_{0}^{t}\left[\left(f\left(s\right)-\frac{d\left(s\right)}{a\left(s\right)}g\left(s\right)\right)\mu_{0}\left(s\right)+\frac{g\left(s\right)}{2a\left(s\right)}\mu_{0}^{\prime}\left(s\right)\right]\
\frac{ds}{w\left(s\right)},$ (7.11)
$\displaystyle\varepsilon_{0}\left(t\right)$ $\displaystyle=$
$\displaystyle-\frac{2a\left(t\right)w\left(t\right)}{\mu_{0}^{\prime}\left(t\right)}\delta_{0}\left(t\right)+8\int_{0}^{t}\frac{a\left(s\right)\sigma\left(s\right)w\left(s\right)}{\left(\mu_{0}^{\prime}\left(s\right)\right)^{2}}\left(\mu_{0}\left(s\right)\delta_{0}\left(s\right)\right)\
ds$
$\displaystyle+2\int_{0}^{t}\frac{a\left(s\right)w\left(s\right)}{\mu_{0}^{\prime}\left(s\right)}\left[f\left(s\right)-\frac{d\left(s\right)}{a\left(s\right)}g\left(s\right)\right]\
ds,$ $\displaystyle\kappa_{0}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{a\left(t\right)\mu_{0}\left(t\right)}{\mu_{0}^{\prime}\left(t\right)}\delta_{0}^{2}\left(t\right)-4\int_{0}^{t}\frac{a\left(s\right)\sigma\left(s\right)}{\left(\mu_{0}^{\prime}\left(s\right)\right)^{2}}\left(\mu_{0}\left(s\right)\delta_{0}\left(s\right)\right)^{2}\
ds$
$\displaystyle\quad-2\int_{0}^{t}\frac{a\left(s\right)}{\mu_{0}^{\prime}\left(s\right)}\left(\mu_{0}\left(s\right)\delta_{0}\left(s\right)\right)\left[f\left(s\right)-\frac{d\left(s\right)}{a\left(s\right)}g\left(s\right)\right]\
ds,$

with
$\delta_{0}\left(0\right)=g_{0}\left(0\right)/\left(2a\left(0\right)\right),$
$\varepsilon_{0}\left(0\right)=-\delta_{0}\left(0\right),$
$\kappa_{0}\left(0\right)=0.$ Here $\mu_{0}$ and $\mu_{1}$ represent the
fundamental solution of the characteristic equation subject to the initial
conditions $\mu_{0}(0)=0$, $\mu_{0}^{\prime}(0)=2a(0)\neq 0$ and
$\mu_{1}(0)\neq 0$, $\mu_{1}^{\prime}(0)=0$.

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