Double Wronskian Representation of the Nth-Order Solutions to the Davey–Stewartson Equations

Abstract

Solutions to the Davey–Stewartson equations (DS) in terms of the determinants of order $2N$ depending on $4N$ real parameters are constructed. The Darboux transformation is used to construct solutions of order N to this equation. We obtain a representation of the solutions to the equations (DS) in terms of two Wronskians. We obtain what we call a double Wronskian representation of the solutions to the (DS) equations. With this method, some particular explicit solutions for the first orders are constructed.

1. Introduction

We consider the Davey–Stewartson system (DS) which can be written in the form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}i{v}_t-v_{xx}-{\alpha }^{-2}{v}_{yy}+2\kappa {\alpha }^{-2}{\left|v\right|}^{2}v-Sv=0,\hfill \\ {\alpha }^2{S}_{xx}-S_{yy}-{4\kappa \left(\right|v|}^2{)}_{xx}=0\hfill \end{array}\right.\end{array}$$

(1)

This system was derived by Benney and Reskes in 1969 [1], then by Davey and Stewartson in 1974 [2].

In the Equation (1), v is the amplitude of the wave, and the field S is related to the mean flow.

The case $\kappa =\pm 1$, $\alpha =1$ is usually called the DS-I system; the case $\kappa =\pm 1$, $\alpha =i$ is referred to as the DS-II system.

In the case $\alpha =1$ (DSI equation), surface tension is more important than gravity in wave derivation. Conversely, if $\alpha =-1$ (DSII equation), it is the contrary.

These equations are generalizations of the nonlinear Schrödinger (NLS) equation which describes the evolution of a wave packet on water of finite depth in fluid mechanics [3].

These equations are also used in other areas such plasma physics [4], nonlinear optics [5], Bose–Einstein condensate [6].

The (DS) equations have been studied in a lot of works.

Anker and Freeman [7] used the inverse scattering method to construct single and txo solutions in particular.

Nakamura [8] proposed in 1983 some solutions in terms of Hemite polynomials and Airy functions, representing exploding and decaying solutions.

Boiti et al. [9] constructed solitons and multisolitons solutions to the DS equation in simple cases, which exponentially decay in spatial dimensions.

The Hirota bilinear method was used in the paper [10] to construct solutions to this equation in terms of Wronskians that decay exponentially in all directions, called dromions. Some figures were presented without providing explicit solutions.

The Hirota bilinear method was also applied in [11] to get rational solutions to the (DS) equation, and some representations of the solutions in the $(x,y)$ plane for different values of time and parameters were given until order 3 but without giving explicit solutions.

An algebro-geometric approach was carried out by Malanyuk in the article [12], where finite gap solutions were constructed in terms of Riemann theta functions. No explicit solution was built.

More recently, an article [13] proposes doubly localized solutions highlighting rogue waves using the bilinear Hirota method. Here again, some figures were created without giving the explicit expressions of the solutions.

In the paper [14], the N-breather solution of the DSI and DSII equations is constructed in particular, describing the nonlinear interaction of N unstable modes over the constant background solution.

The purpose of the paper is to construct explicit solutions to the DS equation obtained by means of the Darboux transformation. This method initiated by V.B. Matveev is perfectly exposed in the article [15], for readers who would like to obtain details of the framework and the notations used.

We present an algebraic construction of the solutions of the (DS) system based on the results of Salle [16] using a generalized Darboux transformation. It is a method for generating solutions to linear systems associated with DS equations. The auxiliary linear system is defined by matrices of order 2. The Darboux transformation is applied to obtain solutions using covariance results.

Here we present solutions to (DS) equations of order N depending on $4N$ real parameters in terms of the determinants. We give the complete formulas of the solutions for the order N. The method presented in the article [16] is slightly different since it is the binary version which is used, and this method involves integrals. However, no general formulation of the solutions is given and, in particular, no explicit solution is constructed.

Here, we construct explicit solutions to the DSI equation for the first orders.

2. Generalized Darboux Transformation

2.1. Auxiliary Linear System

Following the ideas of [16], we consider the following 2-order matrices:

$$\begin{array}{c}\hfill J=\left(\begin{array}{cc}\alpha & 0\\ 0& -\alpha \end{array}\right),U=\left(\begin{array}{cc}0& u\\ v& 0\end{array}\right),\end{array}$$

(2)

$$\begin{array}{c}\hfill V=\left(\begin{array}{cc}{\displaystyle \frac{w}{2}}+{\displaystyle \frac{iQ}{2}}& i{\alpha }^{-2}(\alpha u_x+u_y)\\ i{\alpha }^{-2}(\alpha v_x-v_y)& {\displaystyle \frac{w}{2}}-{\displaystyle \frac{iQ}{2}}\end{array}\right),\Psi =\left(\begin{array}{cc}{\psi }_1& {\psi }_2\\ {\varphi }_1& {\varphi }_2\end{array}\right).\end{array}$$

(3)

We consider the following auxiliary linear system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\Psi }_y=J{\Psi }_x+U\Psi ,\hfill \\ {\Psi }_t=2i{\alpha }^{-1}J{\Psi }_{xx}+2i{\alpha }^{-1}U{\Psi }_x+V\Psi .\hfill \end{array}\right.\end{array}$$

(4)

It is clear that the consistency condition of the system (4), i.e., the equality ${\left({\Psi }_y\right)}_t={\left({\Psi }_t\right)}_y$, gives the Davey–Stewartson system (1).

In particular, we get the following relation between Q of V and S of one of the equations of (1).

$$\begin{array}{c}\hfill Q=-2\kappa {\alpha }^{-2}{\left|u\right|}^2+S.\end{array}$$

(5)

2.2. Darboux Dressing

We consider an arbitrary (matrix) function $\Psi$:

$$\begin{array}{c}\hfill \Psi =\left(\begin{array}{cc}{\psi }_1& {\psi }_2\\ {\varphi }_1& {\varphi }_2\end{array}\right)\end{array}$$

(6)

solution of the linear system (4).

We consider the following Darboux transformation which is different from the one proposed in [16]:

$$\begin{array}{c}\hfill {\Psi }_{\left[1\right]}={\Psi }_x-S_1\Psi .\end{array}$$

(7)

The matrix $S_1$ is defined by

$$\begin{array}{c}\hfill S_1=\left(\begin{array}{cc}{s}_1& s_2\\ t_1& t_2\end{array}\right),\end{array}$$

(8)

with elements

$$\begin{array}{c}\hfill s_1={\displaystyle \frac{{\left({\psi }_1\right)}_x{\varphi }_2-{\varphi }_1{\left({\psi }_2\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}},s_2={\displaystyle \frac{{\left({\psi }_2\right)}_x{\psi }_1-{\psi }_2{\left({\psi }_1\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}},\end{array}$$

(9)

$$\begin{array}{c}\hfill t_1={\displaystyle \frac{{\left({\varphi }_1\right)}_x{\varphi }_2-{\varphi }_1{\left({\varphi }_2\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}},t_2={\displaystyle \frac{{\left({\varphi }_2\right)}_x{\psi }_1-{\psi }_2{\left({\varphi }_1\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}}.\end{array}$$

(10)

Then, we get the following covariance result

Proposition 2.1.

The function ${\Psi }_{\left[1\right]}$ defined by (7)

$${\Psi }_{\left[1\right]}={\Psi }_x-S_1\Psi$$

is a solution of the linear system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\Psi }_y=J{\Psi }_x+U_{\left[1\right]}\Psi ,\hfill \\ {\Psi }_t=2i{\alpha }^{-1}J{\Psi }_{xx}+2i{\alpha }^{-1}{U}_{\left[1\right]}{\Psi }_x+V_{\left[1\right]}\Psi .\hfill \end{array}\right.\end{array}$$

(11)

with

$$\begin{array}{c}\hfill U_{\left[1\right]}=U+[J,S_1]and V_{\left[1\right]}=V+2i{\alpha }_{-1}(J{\left(S_1\right)}_x+{\left(S_1\right)}_y),\end{array}$$

(12)

and

$$\begin{array}{c}\hfill S_1=\left(\begin{array}{cc}{\displaystyle \frac{{\left({\psi }_1\right)}_x{\varphi }_2-{\varphi }_1{\left({\psi }_2\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}}& {\displaystyle \frac{{\left({\psi }_2\right)}_x{\psi }_1-{\psi }_2{\left({\psi }_1\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}},\\ {\displaystyle \frac{{\left({\varphi }_1\right)}_x{\varphi }_2-{\varphi }_1{\left({\varphi }_2\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}}& {\displaystyle \frac{{\left({\varphi }_2\right)}_x{\psi }_1-{\psi }_2{\left({\varphi }_1\right)}_x}{{\psi }_1{\varphi }_2-{\varphi }_1{\psi }_2}}.\end{array}\right),\end{array}$$

(13)

Remark 2.1.

As usual, $[A,B]$ is the commutator defined by $[A,B]=AB-BA$.

Proof. 

It is sufficient to replace the expression of ${\Psi }_{\left[1\right]}$ in the equations of the associated linear system (4) and to identify the terms in $\Psi$ and ${\Psi }_x$ to get the expressions of $U_{\left[1\right]}$ and $V_{\left[1\right]}$.

The expressions of the terms of $S_1$ come from the resolution of two systems of two equations coming from the fact that $\Psi$ solves the associated system (4). □

2.3. Iterated Darboux Transformation

We consider N matrices ${\Psi }_j$ of order 2 defined by

$$\begin{array}{c}\hfill {\Psi }_j=\left(\begin{array}{cc}{\psi }_{2j-1}& {\psi }_{2j}\\ {\varphi }_{2j-1}& {\varphi }_{2j}\end{array}\right),\end{array}$$

(14)

for $1\le j\le N$ solutions to the linear system (4).

We define the following $2N$-order determinant $\Delta$ by

$$\begin{array}{c}\hfill \Delta =\left|\begin{array}{ccccccc}{\partial }_x^{N-1}{\psi }_1& {\partial }_x^{N-1}{\varphi }_1& {\partial }_x^{N-2}{\psi }_1& {\partial }_x^{N-2}{\varphi }_1& \cdots & {\psi }_1& {\varphi }_1\\ {\partial }_x^{N-1}{\psi }_2& {\partial }_x^{N-1}{\varphi }_2& {\partial }_x^{N-2}{\psi }_2& {\partial }_x^{N-2}{\varphi }_2& \cdots & {\psi }_2& {\varphi }_2\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\partial }_x^{N-1}{\psi }_{2N}& {\partial }_x^{N-1}{\varphi }_{2N}& {\partial }_x^{N-2}{\psi }_{2N}& {\partial }_x^{N-2}{\varphi }_{2N}& \cdots & {\psi }_{2N}& {\varphi }_{2N}\end{array}\right|.\end{array}$$

(15)

The determinants ${\Delta }^{\left(k\right)}$ and ${\tilde{\Delta }}^{\left(k\right)}$ are the determinants constructed from $\Delta$ in which the column k has been replaced, respectively, by

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\partial }_x^N{\psi }_1\\ {\partial }_x^N{\psi }_2\\ ⋮\\ {\partial }_x^N{\psi }_{2N}\end{array}\right),\left(\begin{array}{c}{\partial }_x^N{\varphi }_1\\ {\partial }_x^N{\varphi }_2\\ ⋮\\ {\partial }_x^N{\varphi }_{2N}\end{array}\right).\end{array}$$

(16)

The matrices $S_j$ for $1\le j\le N$ are defined by

$$\begin{array}{c}\hfill S_j=\left(\begin{array}{cc}{s}_{2j-1}& s_{2j}\\ t_{2j-1}& t_{2j}\end{array}\right).\end{array}$$

(17)

The function $\Psi$ is a solution to the linear system (4).

We define the Nth iterated Darboux transformation by

$$\begin{array}{c}\hfill {\Psi }_{\left[N\right]}={\partial }_x^N\Psi -S_1{\partial }_x^N\Psi -\dots -S_{\left[N\right]}\Psi .\end{array}$$

(18)

Then, we get the following covariance result:

Theorem 2.1.

The function ${\Psi }_{\left[N\right]}$ defined by (18)

$${\Psi }_{\left[N\right]}={\partial }_x^N\Psi -S_1{\partial }_x^N\Psi -\dots -S_{\left[N\right]}\Psi$$

is a solution to the linear system (4)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\Psi }_y=J{\Psi }_x+U_{\left[N\right]}\Psi ,\hfill \\ {\Psi }_t=2i{\alpha }^{-1}J{\Psi }_{xx}+2i{\alpha }^{-1}{U}_{\left[N\right]}{\Psi }_x+V_{\left[N\right]}\Psi .\hfill \end{array}\right.\end{array}$$

(19)

with

$$\begin{array}{c}\hfill U_{\left[N\right]}=U+[J,S_1]and V_{\left[N\right]}=V+2i{\alpha }^{-1}({\left(J{S}_1\right)}_x+{\left(S_1\right)}_y),\end{array}$$

(20)

and

$$\begin{array}{c}\hfill S_1=\left(\begin{array}{cc}{s}_1& s_2\\ t_1& t_2\end{array}\right)\end{array}$$

(21)

with

$$\begin{array}{c}\hfill s_1={\displaystyle \frac{{\Delta }^{\left(1\right)}}{\Delta }},s_2={\displaystyle \frac{{\Delta }^{\left(2\right)}}{\Delta }},\end{array}$$

(22)

$$\begin{array}{c}\hfill t_1={\displaystyle \frac{{\tilde{\Delta }}^{\left(1\right)}}{\Delta }},t_2={\displaystyle \frac{{\tilde{\Delta }}^{\left(2\right)}}{\Delta }}\end{array}$$

(23)

Remark 2.2.

The general formulation of $V_N$ was not given in [16]. For this reason, we will give the corresponding proof.

Proof. 

It is sufficient to replace the expression of ${\Psi }_{\left[N\right]}$ in the equations of the associated linear system (4) and to identify the terms in ${\partial }_x^N\Psi$ and ${\partial }_x^{N-1}{\Psi }_x$.

More precisely, replacing ${\Psi }_{\left[N\right]}$ in the two equations of the system (4), we get the following relations:

For the first equation of (4), by identifying the terms in ${\partial }_x^N\psi$ and ${\partial }_x^{N-1}\psi$, we get

$$U_{\left[N\right]}=U+[J,S_1]$$

and

$$N{U}_x=[S_1,U]+[S_2,J]-[J,S_1]S_1-J{\left(S_1\right)}_x+{\left(S_1\right)}_y.$$

For the second equation of (4), by identifying the terms in ${\partial }_x^N\psi$, we obtain

$$V_{\left[N\right]}=V-2i{\alpha }^{-1}([U,S_1]+[J,S_1]S_1-S_{2}J+2J{\left(S_1\right)}_x+N{U}_x.$$

Replacing $N{U}_x$ by

$$N{U}_x=[S_1,U]+[S_2,J]-[J,S_1]S_1-J{\left(S_1\right)}_x+{\left(S_1\right)}_y,$$

we get the relation

$$V_{\left[N\right]}=V+2i{\alpha }_{-1}({\left(J{S}_1\right)}_x+{\left(S_1\right)}_y).$$

The $4N$ elements of the matrices $S_j$$1\le j\le N$ are determined by two systems of $2N$ equations due to the the fact that the functions ${\Psi }_j$ are solutions to the associated linear system (4).

By applying the Cramer rules, we get the expressions of the matrices $S_j$ for $1\le j\le N$, and, in particular, those of $S_1$, which proves the result. □

Remark 2.3.

In fact, for the resolution of the (DS) equations, we only need the knowledge of the matrix $S_1$.

3. Double Wronskian Representation of the Solutions to the (DS) Equation

From the previous study, we deduce solutions to the (DS) equations.

We consider N matrices ${\Psi }_j$ of order 2, defined by

$$\begin{array}{c}\hfill {\Psi }_j=\left(\begin{array}{cc}{\psi }_{2j-1}& {\psi }_{2j}\\ {\varphi }_{2j-1}& {\varphi }_{2j}\end{array}\right),\end{array}$$

(24)

for $1\le j\le N$ solutions to the linear system (4).

We use the notations defined in (15)

$$\begin{array}{c}\hfill W=\left|\begin{array}{ccccccc}{\partial }_x^{N-1}{\psi }_1& {\partial }_x^{N-1}{\varphi }_1& {\partial }_x^{N-2}{\psi }_1& {\partial }_x^{N-2}{\varphi }_1& \cdots & {\psi }_1& {\varphi }_1\\ {\partial }_x^{N-1}{\psi }_2& {\partial }_x^{N-1}{\varphi }_2& {\partial }_x^{N-2}{\psi }_2& {\partial }_x^{N-2}{\varphi }_2& \cdots & {\psi }_2& {\varphi }_2\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ {\partial }_x^{N-1}{\psi }_{2N}& {\partial }_x^{N-1}{\varphi }_{2N}& {\partial }_x^{N-2}{\psi }_{2N}& {\partial }_x^{N-2}{\varphi }_{2N}& \cdots & {\psi }_{2N}& {\varphi }_{2N}\end{array}\right|.\end{array}$$

(25)

The determinants $W^{\left(k\right)}$ and ${\tilde{W}}^{\left(k\right)}$ are the determinants constructed from W in which the column k has been replaced, respectively, by

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\partial }_x^N{\psi }_1\\ {\partial }_x^N{\psi }_2\\ ⋮\\ {\partial }_x^N{\psi }_{2N}\end{array}\right),\left(\begin{array}{c}{\partial }_x^N{\varphi }_1\\ {\partial }_x^N{\varphi }_2\\ ⋮\\ {\partial }_x^N{\varphi }_{2N}\end{array}\right).\end{array}$$

(26)

We start from the initial solutions with $U=0$, $V=0$. We restrict to the case where $\alpha =1$ and $\kappa =1$ in order to verify the second equation of (1).

So we get the following result:

Theorem 3.1.

The function v defined by

$$\begin{array}{c}\hfill v(x,y,t)=-2{\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}\end{array}$$

(27)

is a solution of order N to the (DSI) Equation (1) with S defined by

$$\begin{array}{c}\hfill S(x,y,t)=8{\left|{\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}\right|}^2+2{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_y-2{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{W}}\right)}_y+2{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_x+2{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{W}}\right)}_y\end{array}$$

(28)

Remark 3.1.

The determinants W appear as double Wronskians, so we get what we call the double Wronskian representation of the solutions to the (DS) equation. We recover this terminology given in [10] where the bilinear Hirota method was used to construct solutions to the (DS) equation.

Remark 3.2.

For v to be a solution of the (DSI) equation (1), an additional condition must also be imposed. The function u defined

$$\begin{array}{c}\hfill u(x,y,t)=2{\displaystyle \frac{W^{\left(2\right)}}{W}}\end{array}$$

(29)

where $u=\overline{v}$.

Proof. 

We have found that

$$u_N=U+[J,S_1].$$

With the starting initial condition $U=0$, it gives, in particular,

$$\begin{array}{c}\hfill v(x,y,t)=-2\alpha t_1=-2\alpha {\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}\end{array}$$

(30)

which gives, in particular, the expression of the solution with $\alpha =1$

$$v(x,y,t)=-2{\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}.$$

Morever, we have found that

$$V_{\left[N\right]}=V+2i{\alpha }^{-1}({\left(J{S}_1\right)}_x+{\left(S_1\right)}_y).$$

Starting from initial condition $V=0$, it can be written as

$$\begin{array}{c}\hfill V_{\left[N\right]}=\left(\begin{array}{cc}2i{\alpha }^{-1}{\left(s_1\right)}_y+2i{\left(s_1\right)}_x& 2i{\alpha }^{-1}{\left(s_2\right)}_y+2i{\left(s_2\right)}_x\\ 2i{\alpha }^{-1}{\left(t_1\right)}_y+2i{\left(t_1\right)}_x& 2i{\alpha }^{-1}{\left(t_2\right)}_y+2i{\left(t_2\right)}_x\end{array}\right)\end{array}$$

(31)

So we get the relations

$$\begin{array}{c}\hfill {\displaystyle \frac{w_N+i{Q}_N}{2}}=2i{\alpha }^{-1}{\left(s_1\right)}_y+2i{\left(s_1\right)}_x\end{array}$$

(32)

$$\begin{array}{c}\hfill {\displaystyle \frac{w_N-i{Q}_N}{2}}=2i{\alpha }^{-1}{\left(t_2\right)}_y-2i{\left(t_2\right)}_x\end{array}$$

(33)

This gives

$$\begin{array}{c}\hfill Q_N=2{\alpha }^{-1}{\left(s_1\right)}_y-2{\alpha }^{-1}{\alpha }^{-1}{\left(t_2\right)}_y+2{\left(s_1\right)}_x+2{\left(t_2\right)}_x.\end{array}$$

(34)

So we get, using the relation between S and Q,

$$\begin{array}{c}\hfill S(x,y,t)=Q_N+2\kappa {\alpha }^{-1}{\left|v(x,y,t)\right|}^2,\end{array}$$

(35)

which can be rewritten as

$$\begin{array}{c}\hfill S(x,y,t)=8\kappa {\alpha }^{-2}{\left|{\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}\right|}^2+2{\alpha }^{-1}{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_y-2{\alpha }^{-1}{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{\Delta }}\right)}_y+2{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_x+2{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{W}}\right)}_y,\end{array}$$

(36)

which gives, with $\kappa =\alpha$, $\alpha =1$,

$$\begin{array}{c}\hfill S(x,y,t)=8{\left|{\displaystyle \frac{{\tilde{W}}^{\left(1\right)}}{W}}\right|}^2+2{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_y-2{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{W}}\right)}_y+2{\left({\displaystyle \frac{W^{\left(1\right)}}{W}}\right)}_x+2{\left({\displaystyle \frac{{\tilde{W}}^{\left(2\right)}}{W}}\right)}_y.\end{array}$$

(37)

For the additional condition, one deduces, from the compatibility conditions of the initial system expressing ${\Psi }_y$ and ${\Psi }_t$, the following relations:

$$w_y=i\alpha Q_x-2i{\alpha }^{-1}{\left(uv\right)}_x,$$

and

$$w_x={\alpha }^{-1}{Q}_y+2i{\alpha }^{-3}{\left(uv\right)}_y.$$

We deduce the relation

$$w_{xy}=i\alpha Q_{2x}-2i{\alpha }^{-1}{\left(uv\right)}_{2x}={\alpha }^{-1}{Q}_{2y}+2i{\alpha }^{-3}{\left(uv\right)}_{2y}.$$

It can be written

$${\alpha }^2{Q}_{2x}-Q_{2y}-2{\left(uv\right)}_{2x}+2{\alpha }^{-2}{\left(uv\right)}_{2y}.$$

Using the relation between Q and S,

$$Q=-2\kappa {\alpha }^{-2}{\left|v\right|}^2+S,$$

which we can write, for $\alpha =\kappa =1$,

$$S_{2x}-S_{2y}-{2\left(\right|v|}^2{+\left(uv\right))}_{2x}+{2\left(\right|v|}^2{-\left(uv\right))}_{2y}=0.$$

We recover the classical (DSI) equation if $\overline{v}=u$. □

4. Some Explicit Solutions to the DSI Equation

4.1. Choice of the Generating Functions

Among different choices, we take the following simplest functions ${\psi }_k$ and ${\varphi }_k$ defined by

$$\begin{array}{c}\hfill {\psi }_k(x,y,t)=c_k{e}^{a_{k}x+a_{k}y+2i{a}_k^{2}t}\end{array}$$

(38)

$$\begin{array}{c}\hfill {\varphi }_k(x,y,t)=d_k{e}^{b_{k}x-b_{k}y-2i{b}_k^{2}t}\end{array}$$

(39)

With these choices, the previous determinants can be written as

$$\begin{array}{c}\hfill D=\left|\begin{array}{ccccccc}{a}_1^{N-1}{\psi }_1& b_1^{N-1}{\varphi }_1& a_1^{N-2}{\psi }_1& b_1^{N-2}{\varphi }_1& \cdots & {\psi }_1& {\varphi }_1\\ a_2^{N-1}{\psi }_2& b_2^{N-1}{\varphi }_2& a_2^{N-2}{\psi }_2& b_2^{N-2}{\varphi }_2& \cdots & {\psi }_2& {\varphi }_2\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{2N}^{N-1}{\psi }_{2N}& b_{2N}^{N-1}{\varphi }_{2N}& a_{2N}^{N-2}{\psi }_{2N}& b_{2N}^{N-2}{\varphi }_{2N}& \cdots & {\psi }_{2N}& {\varphi }_{2N}\end{array}\right|.\end{array}$$

(40)

The determinants $D^{\left(k\right)}$ and ${\tilde{D}}^{\left(k\right)}$ are the determinants constructed from $\Delta$ in which the column k has been replaced, respectively, by

$$\begin{array}{c}\hfill \left(\begin{array}{c}{a}_1^N{\psi }_1\\ a_2^N{\psi }_2\\ ⋮\\ a_{2N}^N{\psi }_{2N}\end{array}\right),\left(\begin{array}{c}{b}_1^N{\varphi }_1\\ b_2^N{\varphi }_2\\ ⋮\\ a_{2N}^N{\varphi }_{2N}\end{array}\right).\end{array}$$

(41)

Then we have the particular result.

Theorem 4.1.

The function v defined by

$$\begin{array}{c}\hfill v(x,y,t)=-2{\displaystyle \frac{{\tilde{D}}^{\left(1\right)}}{D}}\end{array}$$

(42)

is a solution of order N to the (mDSI) Equation (43)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}i{v}_t-v_{xx}-{\alpha }^{-2}{v}_{yy}+2\kappa {\alpha }^{-2}{\left|v\right|}^{2}v-Sv=0,\hfill \\ S_{2x}-S_{2y}-{2\left(\right|v|}^2+{\left(uv\right)}_{2x}{)+2(\left|v\right|}^2-{\left(uv\right)}_y)=0\hfill \end{array}\right.\end{array}$$

(43)

with S defined by

$$\begin{array}{c}\hfill S(x,y,t)=8{\left|{\displaystyle \frac{{\tilde{D}}^{\left(1\right)}}{D}}\right|}^2+2{\left({\displaystyle \frac{D^{\left(1\right)}}{D}}\right)}_y-2{\left({\displaystyle \frac{{\tilde{D}}^{\left(2\right)}}{D}}\right)}_y+2{\left({\displaystyle \frac{D^{\left(1\right)}}{D}}\right)}_x+2{\left({\displaystyle \frac{{\tilde{D}}^{\left(2\right)}}{D}}\right)}_y\end{array}$$

(44)

and v defined by

$$\begin{array}{c}\hfill u(x,y,t)=2{\displaystyle \frac{D^{\left(2\right)}}{D}}\end{array}$$

(45)

4.2. Expression of the Solutions of Order 1

Example 4.1.

The function $v(x,y,t)$ defined by

$$\begin{array}{c}\hfill v(x,y,t)=-{\displaystyle \frac{2{\mathrm{e}}^{\left(-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2\right)t+\left(b_1+b_2\right)\left(x-y\right)}{d}_1{d}_2\left(b_1-b_2\right)}{c_1{\mathrm{e}}^{2\mathrm{I}{a}_1^{2}t+\left(x+y\right)a_1-2b_2\left(\mathrm{I}t{b}_2-\frac{x}{2}+\frac{y}{2}\right)}{d}_2-d_1{\mathrm{e}}^{2\mathrm{I}{a}_2^{2}t+\left(x+y\right)a_2-2b_1\left(\mathrm{I}t{b}_1-\frac{x}{2}+\frac{y}{2}\right)}{c}_2}}\end{array}$$

(46)

with

$$u(x,y,t)={\displaystyle \frac{2{\mathrm{e}}^{\left(2\mathrm{I}{a}_1^2+2\mathrm{I}{a}_2^2\right)t+\left(a_1+a_2\right)\left(x+y\right)}{c}_1{c}_2\left(a_1-a_2\right)}{d_1{\mathrm{e}}^{2\mathrm{I}{a}_2^{2}t+\left(x+y\right)a_2-2b_1\left(\mathrm{I}t{b}_1-\frac{x}{2}+\frac{y}{2}\right)}{c}_2-c_1{\mathrm{e}}^{2\mathrm{I}{a}_1^{2}t+\left(x+y\right)a_1-2b_2\left(\mathrm{I}t{b}_2-\frac{x}{2}+\frac{y}{2}\right)}{d}_2}}$$

and

$$S(x,y,t)={\displaystyle \frac{nS(x,y,t)}{dS(x,y,t)}}$$

$$\begin{array}{c}nS(x,y,t)=-4d_1{d}_2(4{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+a_2+3b_1+3b_2)x+y(a_1+a_2-3b_1-3b_2)}{b}_1^2{c}_1{c}_2{d}_1^2{d}_2^2\hfill \\ +4{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+a_2+3b_1+3b_2)x+y(a_1+a_2-3b_1-3b_2)}{b}_2^2{c}_1{c}_2{d}_1^2{d}_2^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{a}_1^2{c}_1{c}_2^3{d}_1^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{a}_2^2{c}_1{c}_2^3{d}_1^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{b}_1^2{c}_1{c}_2^3{d}_1^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{b}_2^2{c}_1{c}_2^3{d}_1^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{a}_1^2{c}_1^3{c}_2{d}_2^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{a}_2^2{c}_1^3{c}_2{d}_2^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{b}_1^2{c}_1^3{c}_2{d}_2^2\hfill \\ +{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{b}_2^2{c}_1^3{c}_2{d}_2^2\hfill \\ -2{\mathrm{e}}^{(2a_2+4b_1+2b_2)x+(2a_2-4b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_1^2{c}_2^2{d}_1^3{d}_2\hfill \\ -2{\mathrm{e}}^{(2a_2+4b_1+2b_2)x+(2a_2-4b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_2^2{c}_2^2{d}_1^3{d}_2\hfill \\ -2{\mathrm{e}}^{(2a_1+2b_1+4b_2)x+(2a_1-2b_1-4b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_1^2{c}_1^2{d}_1{d}_2^3\hfill \\ -2{\mathrm{e}}^{(2a_1+2b_1+4b_2)x+(2a_1-2b_1-4b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_2^2{c}_1^2{d}_1{d}_2^3\hfill \\ -8{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+a_2+3b_1+3b_2)x+y(a_1+a_2-3b_1-3b_2)}{b}_1{b}_2{c}_1{c}_2{d}_1^2{d}_2^2\hfill \\ +2{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{a}_1{a}_2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ +2{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_1{b}_2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ +2{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_1{b}_2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ +2{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{a}_1{a}_2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -2{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{b}_1{b}_2{c}_1{c}_2^3{d}_1^2\hfill \\ -2{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(a_1+3a_2+3b_1+b_2)x+y(a_1+3a_2-3b_1-b_2)}{a}_1{a}_2{c}_1{c}_2^3{d}_1^2\hfill \\ -2{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{b}_1{b}_2{c}_1^3{c}_2{d}_2^2\hfill \\ -2{\mathrm{e}}^{(2\mathrm{I}{a}_1^2-2\mathrm{I}{b}_2^2-2\mathrm{I}{b}_1^2+2\mathrm{I}{a}_2^2)t+(3a_1+a_2+b_1+3b_2)x+3y(a_1+{\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{b_1}{3}}-b_2)}{a}_1{a}_2{c}_1^3{c}_2{d}_2^2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{a}_1^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{a}_2^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_1^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_2^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{a}_1^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{a}_2^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_1^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ -{\mathrm{e}}^{(2a_1+2b_2+2b_1+2a_2)x+(2a_1+2a_2-2b_1-2b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_2^2{c}_1^2{c}_2^2{d}_1{d}_2\hfill \\ +4{\mathrm{e}}^{(2a_1+2b_1+4b_2)x+(2a_1-2b_1-4b_2)y+4\mathrm{I}t(a_1+b_2)(a_1-b_2)}{b}_1{b}_2{c}_1^2{d}_1{d}_2^3\hfill \\ +4{\mathrm{e}}^{(2a_2+4b_1+2b_2)x+(2a_2-4b_1-2b_2)y+4\mathrm{I}(a_2+b_1)t(a_2-b_1)}{b}_1{b}_2{c}_2^2{d}_1^3{d}_2)\hfill \end{array}$$

$$\begin{array}{c}dS(x,yt)={(d_1{\mathrm{e}}^{2\mathrm{I}{a}_2^{2}t+(x+y)a_2-2b_1(\mathrm{I}t{b}_1-{\displaystyle \frac{x}{2}}+{\displaystyle \frac{y}{2}})}{c}_2-c_1{\mathrm{e}}^{2\mathrm{I}{a}_1^{2}t+(x+y)a_1-2b_2(\mathrm{I}t{b}_2-{\displaystyle \frac{x}{2}}+{\displaystyle \frac{y}{2}})}{d}_2)}^3\hfill \\ (d_1{\mathrm{e}}^{-2\mathrm{I}{a}_2^{2}t+(x+y)a_2+2b_1(\mathrm{I}t{b}_1+{\displaystyle \frac{x}{2}}-{\displaystyle \frac{y}{2}})}{c}_2-c_1{\mathrm{e}}^{-2\mathrm{I}{a}_1^{2}t+(x+y)a_1+2b_2(\mathrm{I}t{b}_2+{\displaystyle \frac{x}{2}}-{\displaystyle \frac{y}{2}})}{d}_2)\hfill \end{array}$$

is a solution to the (mDSI) Equation (43).

4.3. Expression of the Solutions of Order 2

Already in this case, the explicit expressions of the solutions become too long to be written in the text. In the following, we present solutions with particular values of the parameters. In this case, we choose $a_1=1,b_1=1,c_1=1,d_1=1,a_2=2,b_2=2,c_2=1,d_2=1$, $a_3=3,b_3=3,c_3=1,d_3=0,a_4=4,b_4=4,c_4=0,d_4=1$.

Example 4.2.

The function $v(x,y,t)$ defined by

$$\begin{array}{c}\hfill v(x,y,t)=-{\displaystyle \frac{12{\mathrm{e}}^{-24\mathrm{I}t+10x-4y}}{-4{\mathrm{e}}^{-20\mathrm{I}t+10x-2y}+3{\mathrm{e}}^{-8\mathrm{I}t+10x}}}\end{array}$$

(47)

with

$$u(x,y,t)=-{\displaystyle \frac{4{\mathrm{e}}^{-4\mathrm{I}t+10x+2y}}{-4{\mathrm{e}}^{-20\mathrm{I}t+10x-2y}+3{\mathrm{e}}^{-8\mathrm{I}t+10x}}}$$

and

$$S(x,y,t)={\displaystyle \frac{n(x,y,t)}{d(x,y,t)}}$$

$$\begin{array}{c}n(x,y,t)=-96{\left({\mathrm{e}}^{3x+3y+18\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{4x-4y-32\mathrm{I}t}\right)}^2(12{\mathrm{e}}^{-8\mathrm{I}t+10x}{\mathrm{e}}^{20\mathrm{I}t+10x-2y}{\mathrm{e}}^{x+y+2\mathrm{I}t}\hfill \\ {\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}-9{\mathrm{e}}^{-8\mathrm{I}t+10x}{\mathrm{e}}^{8\mathrm{I}t+10x}{\mathrm{e}}^{x+y+2\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}\hfill \\ {\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}+48{\mathrm{e}}^{-24\mathrm{I}t+10x-4y}{\mathrm{e}}^{24\mathrm{I}t+10x-4y}{\left({\mathrm{e}}^{x+y+2\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{2x-2y-8\mathrm{I}t}\right)}^2\hfill \\ -72{\mathrm{e}}^{-24\mathrm{I}t+10x-4y}{\mathrm{e}}^{24\mathrm{I}t+10x-4y}{\mathrm{e}}^{x+y+2\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}\hfill \\ +27{\mathrm{e}}^{-24\mathrm{I}t+10x-4y}{\mathrm{e}}^{24\mathrm{I}t+10x-4y}{\left({\mathrm{e}}^{x-y-2\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{2x+2y+8\mathrm{I}t}\right)}^2\hfill \\ -16{\mathrm{e}}^{-20\mathrm{I}t+10x-2y}{\mathrm{e}}^{20\mathrm{I}t+10x-2y}{\mathrm{e}}^{x+y+2\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}\hfill \\ +12{\mathrm{e}}^{-20\mathrm{I}t+10x-2y}{\mathrm{e}}^{8\mathrm{I}t+10x}{\mathrm{e}}^{x+y+2\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t})\hfill \end{array}$$

$$\begin{array}{c}d(x,y,t)=(-4{\mathrm{e}}^{-20\mathrm{I}t+10x-2y}+3{\mathrm{e}}^{-8\mathrm{I}t+10x})(4{\mathrm{e}}^{20\mathrm{I}t+10x-2y}-3{\mathrm{e}}^{8\mathrm{I}t+10x})\hfill \\ {\left({\mathrm{e}}^{3x+3y+18\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{4x-4y-32\mathrm{I}t}\right)}^2{(4{\mathrm{e}}^{x+y+2\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}-3{\mathrm{e}}^{x-y-2\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t})}^2\hfill \end{array}$$

is a solution to the (mDSI) Equation (43).

4.4. Expression of the Solutions of Order 3

We present solutions with particular values of the parameters. In this case, we choose $a_1=1,b_1=1,c_1=1,d_1=1,a_2=2,b_2=2,c_2=1,d_2=1$, $a_3=3,b_3=3,c_3=1,d_3=1,a_4=4,b_4=4,c_4=1,d_4=0$, $a_5=5,b_5=1,c_5=0,d_5=1,a_6=0,b_6=0,c_6=0,d_6=1$.

Example 4.3.

The function $v(x,y,t)$ defined by

$$\begin{array}{c}\hfill v(x,y,t)={\displaystyle \frac{72{\mathrm{e}}^{6\mathrm{I}t+11x-y}}{-36{\mathrm{e}}^{22\mathrm{I}t+11x+3y}+12{\mathrm{e}}^{42\mathrm{I}t+11x+5y}}}\end{array}$$

(48)

with

$$u(x,y,t)={\displaystyle \frac{24{\mathrm{e}}^{58\mathrm{I}t+11x+9y}}{-36{\mathrm{e}}^{22\mathrm{I}t+11x+3y}+12{\mathrm{e}}^{42\mathrm{I}t+11x+5y}}}$$

and

$$S(x,y,t)={\displaystyle \frac{n(x,y,t)}{d(x,y,t)}}$$

$$\begin{array}{c}n(x,y,t)=24{\left({\mathrm{e}}^{4x+4y+32\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{x+y+2\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{x-y-2\mathrm{I}t}\right)}^2(3{\mathrm{e}}^{-6\mathrm{I}t+11x-y}{\left({\mathrm{e}}^{3x+3y+18\mathrm{I}t}\right)}^2\hfill \\ {\left({\mathrm{e}}^{2x-2y-8\mathrm{I}t}\right)}^2{\mathrm{e}}^{6\mathrm{I}t+11x-y}-18{\mathrm{e}}^{-6\mathrm{I}t+11x-y}{\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{3x-3y-18\mathrm{I}t}\hfill \\ {\mathrm{e}}^{2x+2y+8\mathrm{I}t}{\mathrm{e}}^{6\mathrm{I}t+11x-y}-9{\mathrm{e}}^{22\mathrm{I}t+11x+3y}{\mathrm{e}}^{-22\mathrm{I}t+11x+3y}{\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{3x-3y-18\mathrm{I}t}\hfill \\ {\mathrm{e}}^{2x+2y+8\mathrm{I}t}+3{\mathrm{e}}^{22\mathrm{I}t+11x+3y}{\mathrm{e}}^{-42\mathrm{I}t+11x+5y}{\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{3x-3y-18\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}\hfill \\ +3{\mathrm{e}}^{42\mathrm{I}t+11x+5y}{\mathrm{e}}^{-22\mathrm{I}t+11x+3y}{\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{3x-3y-18\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}-{\mathrm{e}}^{42\mathrm{I}t+11x+5y}\hfill \\ {\mathrm{e}}^{-42\mathrm{I}t+11x+5y}{\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}{\mathrm{e}}^{3x-3y-18\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t}+27{\mathrm{e}}^{-6\mathrm{I}t+11x-y}{\left({\mathrm{e}}^{3x-3y-18\mathrm{I}t}\right)}^2\hfill \\ {\left({\mathrm{e}}^{2x+2y+8\mathrm{I}t}\right)}^2{\mathrm{e}}^{6\mathrm{I}t+11x-y})\hfill \end{array}$$

$$\begin{array}{c}d(x,y,t)=(3{\mathrm{e}}^{22\mathrm{I}t+11x+3y}-{\mathrm{e}}^{42\mathrm{I}t+11x+5y})(3{\mathrm{e}}^{-22\mathrm{I}t+11x+3y}-{\mathrm{e}}^{-42\mathrm{I}t+11x+5y}){\left({\mathrm{e}}^{x+y+2\mathrm{I}t}\right)}^2\hfill \\ {\left({\mathrm{e}}^{x-y-2\mathrm{I}t}\right)}^2{\left({\mathrm{e}}^{4x+4y+32\mathrm{I}t}\right)}^2{({\mathrm{e}}^{3x+3y+18\mathrm{I}t}{\mathrm{e}}^{2x-2y-8\mathrm{I}t}-3{\mathrm{e}}^{3x-3y-18\mathrm{I}t}{\mathrm{e}}^{2x+2y+8\mathrm{I}t})}^2\hfill \end{array}$$

is a solution to the (mDSI) Equation (43).

5. Conclusions

General solutions to the (DS) equation have been given in terms of the determinants of order $2N$ depending on $4N$ real parameters. This work gives families of solutions to the (DS) equation expressed in the variables x, y, and t depending on $4N$ real parameters.

Depending on the choices of the generating functions, it gives a large spectrum of solutions to the (DS) equation.

It will be relevant to explore more particular solutions beyond the particular case of this simplest choice of generating functions.

It would be important to carry out a classification of these solutions.

These solutions could find applications in fluid dynamics, in nonlinear optics, and in plasma physics, particularly, in network dynamics.