---
abstract: 'The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE).  Method is based on the specific link between algebraic matrix equations and PDE.  Admittable solutions depend on arbitrary functions of $n$ variables.' 
author:
- |
    Alexandre I. Zenchuk\
    Center of Nonlinear Studies of\
    L.D.Landau Institute for Theoretical Physics\
    (International Institute of Nonlinear Science)\
    Kosygina 2, Moscow, Russia 119334\
    E-mail: zenchuk@itp.ac.ru\
title: '**On integration of some classes of $(n+1)$ dimensional nonlinear Partial Differential Equations** '
---

=-27mm =-20mm

Introduction
============

Many different methods have been developed for analytical investigation of nonlinear PDE during last decades.  Especially attractive are methods for study of so-called completely integrable systems.  The particular interest to these equations is enhanced due to their wide range of application in physics.  We emphasize different dressing methods, which are based on fundamental properties of linear operators, either differential or integral: Zakharov-Shabat dressing method [@ZSh1; @ZSh2], $\bar{\partial}$-problem [@ZM; @BM; @K; @Z7], Sato theory [@OSTT; @Z4]. 

We suggest the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear PDE, $n\ge 2$. It is based on general properties of linear [*algebraic*]{} matrix equations.  Essentially we develop some ideas represented in the ref.[@OSTT] and recently in the ref. [ @Z4; @Z8].  In general, for $(n+1)$ dimensional PDE this method supplies solutions, depending on the set of arbitrary functions of $n$ variables.  The represented method works also for classical $(2+1)$-dimensional PDE integrable by the Inverse Scattering Technique (IST).  In this case our algorithm is similar to the algorithm represented in refs.[@OSTT; @GM1]. 

The structure of the paper is following.  First, we discuss general algorithm relating linear algebraic equation with nonlinear PDE.  Then we show that these PDE are compatibility condition for some overdetermined linear system of equations having different structure in comparison with linear system associated with completely integrable nonlinear PDE.  We give an example of (2+1)-dimensional system which can not be described in frames of classical dressing methods. 

General algorithm
=================

As mentioned above, our algorithm is based on the fundamental properties of linear matrix algebraic equation $$\begin{aligned}
\label{MATR}
\Psi U = \Phi,\end{aligned}$$ where $\Psi=\{\psi_{ij}\}$ is $N\times N$ nondegenerate matrix, $U$ and $\Phi$ are $N\times M$ matrices $M<N$. Namely, the solution of this equation is unique, $U=\Psi^{-1}\Phi$, and consequently the homogeneous equation with matrix $\Psi$ has only the trivial solution.  Thus, if we find transformation $T$ which maps the nonhomogeneous equation (\[MATR\]) into the homogeneous equation $\Psi \tilde U(U) =0
$, then $\tilde U(U)=0$.

Let us show that such transformations can be performed by means of differential operators having special structure.  For this purpose let us introduce two types of additional parameters $x=(x_1,\dots, x_n)$ ($n={\mbox{dim}}(x)$) and $t=(t_1,t_2,\dots)$ with the following systems: $$\begin{aligned}
\label{x}
\Psi_{x_i} = \Psi B_i + \Phi C_i,\;\;i=1,\dots,n\end{aligned}$$$
(B_i$ and $C_i$ are constant $N\times N$ and $M\times N$ matrices respectively) and $$\begin{aligned}
\label{t}
{\cal{ M}}_i\Psi =0,\;\;\; {\cal{M}}_i\Phi = 0,\;\;
{\cal{M}}_i=\partial_{t_i} + L_i,\;\;\end{aligned}$$ where $L_i$ are arbitrary linear differential operators having derivatives with respect to variables $x_j$ and constant scalar coefficients, so that the system (\[x\]) is compatible with the system (\[t\]).  For the sake of simplicity in this paper we use only one parameter $t$, omit subscripts in the eq.(\[t\]) and use $n$-dimensional Laplacian for $L$: $$\begin{aligned}
\label{Mi}
%M=\partial_{t}+\partial_{x_i}^2\\
% \label{Mlapl}
 {\cal{M}}=\partial_{t}+\sum_{k=1}^n\alpha_k\partial_{x_k}^2\end{aligned}$$ Hereafter indexes $i$, $j$ and $k$ run values from $1$ to $n$ unless otherwise specified. 

Let us study compatibility conditions for the system (\[x\]) itself, which has the following form: $$\begin{aligned}
\label{comp}
(\Psi B_j + \Phi C_j)) B_i + \Phi_{x_j} C_i =
(\Psi B_i + \Phi C_i)) B_j + \Phi_{x_i} C_j.\end{aligned}$$ Require that matrices $B_i$ and $C_i$ satisfy two conditions: $$\begin{aligned}
\label{matrcomp1}\label{matrcomp2}
C_j B_i-C_i B_j=0,\;\; B_jB_i-B_i B_j=0,\;\;i\neq j,\end{aligned}$$ and matrices $C_j$ have the following structure: $C_j=\left[\begin{array}{c|c}
 P_{j}&0_{M,N-R}
\end{array}\right]$, $R\le M
$, where $P_{j}$ are $M\times R$ matrices with rang $R$ and $0_{A,B}$ means $A\times B$ zero matrix.  Then equation (\[comp\]) is reduced to the next one: $$\begin{aligned}
\label{Phi}
 \Phi_{x_i} P_{j}- \Phi_{x_j} P_{i}=0,
 \end{aligned}$$ which results in the first nonlinear equation for $U$ owing to the eq.(\[MATR\]): $$\begin{aligned}
\label{comp1}
(B_j+UC_j) UP_{i} + U_{x_j} P_{i} =
(B_i+UC_i) UP_{j} + U_{x_i} P_{j}.\end{aligned}$$

Let us show that another nonlinear matrix equation can be derived using operator ${\cal{M}}$. For this purpose we apply operator ${\cal{M}}$ to both sides of the eq. ( \[MATR\]) and use eqs. ( \[x\]) and (\[t\]): $$\begin{aligned}
\label{MPsi1} 0={\cal{M}}\Phi =  ({\cal{M}} \Psi) U+ \Psi
{\cal{M}} U + 2 \sum_{k=1}^n \alpha_k\Psi_{x_k} U_{x_k}
=  \Psi \left(
{\cal{M}} U + 2 \sum_{k=1}^n \alpha_i (B_k+ U C_k) U_{x_k} \right) =0 \end{aligned}$$ Since ${\mbox{det}}(\Psi)\neq 0$ one has the second nonlinear equation for the matrix $U$: $$\begin{aligned}
\label{flow2}
  U_{t} + \sum_{k=1}^n \alpha_k \left(U_{x_kx_k} +2 (B_k+ U
C_k) U_{x_k} \right)  =0.\end{aligned}$$ Having eqs.(\[matrcomp1\],\[comp1\],\[flow2\]) one can derive the complete system of equations for elements of the matrix $V$ composed of first $R$ rows of the matrix $U
$.  First equation exists if $n>2$. In this case we can write the matrix equation without operators $B_i$ using any three equations (\[comp1\]) with pairs of indexes $(i,j)$, $(j,k)$ and $(k,i)$ and relations (\[matrcomp1\]) ($U_i=V P_i$): $$\begin{aligned}
\label{nonlin1}
\sum_{perm} P_i \left({U_j}_{x_k} - {U_k}_{x_j} +
[U_k,U_j]\right)=0,\end{aligned}$$ where sum is over cycle permutation of indexes $i$, $j$ and $k$. In general, this equation is not complete system for all elements of the matrix $V$. To complete it we derive additional matrix equation, which can be done for any $n$. Eliminate operators $B_i
 $ from the eq. ( \[comp1\]) ($i=1$, $j=2$) using eqs. ( \[matrcomp1\]) and (\[flow2\]): $$\begin{aligned}
\label{nored}
P_i{U_j}_t - P_j {U_i}_t+
2 \sum_{k=1,k\neq i}^n\alpha_k(P_i U_k-P_kU_i){U_j}_{x_k}-
2 \sum_{k=1,k\neq j}^n\alpha_k(P_j U_k-P_kU_j){U_i}_{x_k}+\\\nonumber
\sum_{k=1}^n\alpha_k \left(P_i{U_j}_{x_kx_k}-P_j {U_i}_{x_kx_k}+
2 P_k({U_i}_{x_j} - {U_j}_{x_i})_{x_k}-2 P_k ({U_i}_{x_k} U_j-
{U_j}_{x_k} U_i)
\right)=0\end{aligned}$$ In particular, if $\alpha_j=0$, $j>1$, $\alpha_1=1$, then eq.(\[nored\]) reduces in $$\begin{aligned}
\label{red1}
P_1 \left({U_2}_{t} + 2 {U_1}_{x_1x_2} - {U_2}_{x_1x_1}+
2 (U_2U_1)_{x_1} - 2 {U_1}_{x_1} U_2
\right)-\\\nonumber
P_2 \left({U_1}_{t} +  {U_1}_{x_1x_1} + 2 U_1 {U_1}_{x_1}
\right)=0.\end{aligned}$$ Thus equations (\[nonlin1\],\[nored\],\[red1\]) do not depend on both parameter $N$ (which characterizes dimensions of the matrices in the eq.(\[MATR\])) and matrices $B_i$, i.e. $N$ is arbitrary positive integer, $B_i$ are arbitrary $N\times N$ matrices fitting relations (\[matrcomp1\]). 

We point on two trivial reductions of the eq.(\[red1\]). 

1\. Let matrix $A$ exist such that $A P_1=0$ and $AP_2=I_R$ ($I_R$ is $R\times R$ identity matrix).  Multiplying eq.(\[red1\]) by $A$ from the right one receives matrix Burgers equation for $U_1$.

2\. If $R=M=2$ and $$\begin{aligned}
\label{P}
 P_1=I_2, \;\;
P_2=\left[\begin{array}{cc}
0&1\cr1&0\end{array}\right],\;\; V=
\left[\begin{array}{cc}
u_1&v_1\cr u_2&v_2\end{array}\right]
,
 \end{aligned}$$ then eq.(\[red1\]) is reduced to the next system: $$\begin{aligned}
\label{DS}
 r_{t} - r_{x_1x_2} -2 r w_{x_1x_2} =0,\;\;\;
 q_{t} + q_{x_1x_2} +2 q w_{x_1x_2} =0 ,\;\;\;
 w_{x_1x_1} - w_{x_2x_2} = qr ,
 \end{aligned}$$ where functions $r$ and $q$ are related with elements of the matrix $V$ by the formulae $$\begin{aligned}
u_1=\frac{1}{4} (r-q+2 w_{x_1}),\;\;u_2=\frac{1}{4} (r+q+2
w_{x_2}),\;\;\\\nonumber
v_1=\frac{1}{4} (-r-q+2 w_{x_2}),\;\;v_2=\frac{1}{4} (-r+q+2
w_{x_1}).\;\;\end{aligned}$$ Eq.(\[DS\]) becomes Devi-Stewartson equation (DS) after reduction $r=\psi$, $q=\bar\psi$, $t_1 = i t$, where $i^2=-1$, bar means complex conjugated value. 

We will see in the next section, that the case $ R=M$ does always correspond to the classical completely integrable $(2+1)$-dimensional systems. 

Arbitrary functions of variables $x_j$ ($j=1,\dots,n$) appear in the solution $V$ due to the matrix function $\Phi$, defined by the system (\[Phi\]).  The number of arguments in the arbitrary functions as well as the number of these functions is defined by particular choice of the matrices $P_j$ and dimension $n$ of $x$-space.  If $n=2$, then one has at most $N$ arbitrary functions of two variables, see [**Example**]{}.  In general, for $n$-dimensional $x$-space and $R<M$ we are able to represent examples with $N$ functions of $n$ variables.  If $R=M$ then $\Phi$ may depend at most on $N\times M$ arbitrary scalar functions of [*single*]{} variable, which is in accordance with [@OSTT].  Detailed discussion of this problem is left beyond the scope of this paper. 

On the operator representation of PDE
-------------------------------------

We derive the overdetermined linear system of PDE with compatibility condition in the form of eqs.(\[nonlin1\]) and (\[nored\]).  First, introduce arbitrary $R\times N$ matrix function ${\bf R} (\lambda)$ of the additional parameter $\lambda$. Multiply eqs.(\[comp1\]) and (\[flow2\]) by ${\bf R} (\lambda)\exp(\eta) $ from the left and introduce function $\hat \Psi = {\bf R}  e^{\eta} U$ $$\begin{aligned}
\label{newfunc}\label{eta}
\eta= \sum_{k=1}^n B_k x_k -
 \left(\sum_{k=1}^n  \alpha_k B_k^2\right) t.\end{aligned}$$ We get after transformations: $$\begin{aligned}
\label{comp1lin}
  \hat\Psi_{x_j} P_{i}- \hat\Psi_{x_i} P_{j}=
\hat\Psi P_i U_j-\hat\Psi P_j U_i,\\
\label{flow2lin}
  \hat\Psi_{t} + \sum_{k=1}^n \alpha_k \left(\hat\Psi_{x_kx_k} +2
  \hat\Psi P_k V_{x_k} \right)=0.\end{aligned}$$ If $R=M$, t.e.  all $P_j$ are square nondegenerate matrices, then the system (\[comp1lin\]), (\[flow2lin\]) is equivalent to the classical $M\times M$ overdetermined linear system for correspondent (2+1)-dimensional integrable system.  In fact, one can express all derivatives of $\hat\Psi$ with respect to $x_j$, $j>1$ through the derivatives of $\hat\Psi$ with respect to $x_1$ using equation (\[comp1lin\]).  Both equations (\[comp1lin\]) and (\[flow2lin\]) are $M\times M$ matrix equations for $M\times M$ matrix function $\hat\Psi$. Thus eq. ( \[comp1lin\]) can be taken for the spectral problem while eq. ( \[flow2lin\]) represents evolution part of the overdetermined linear system.  For instance, if $M=R=n=2$, $\alpha_1=1$, $\alpha_k=0$, $k>1$, and $P_i$ have the form (\[P\]), then the compatibility condition of the linear system (\[comp1lin\],\[flow2lin\]) is given by the eqs.(\[DS\]). 

In the case $R<M$ situation is different.  $\hat \Psi$ is $R\times M$ matrix function, while (\[comp1lin\]) is $R\times R$ matrix equation.  Thus it can not be taken for the spectral problem.  Also, eq.(\[flow2lin\]) involves all derivatives which form operator $M$. So, we have ($n+1$) dimensional equations.  Below we represent example of $(2+1)$-dimensional system of this type. 

Example
=======

Let $M=3$, $R=2$, $Q=2$, $\alpha_2=\alpha_3=0$, $\alpha_1=1$, $N=3 k + 2$, $k=1,2,\dots$. Everywhere indexes $i$ and $j$ take values $1$ and $2$. Let $$\begin{aligned}
\nonumber
P_1=\left[
\begin{array}{cc}
1&0\cr
0&1\cr
0&0
\end{array}
\right],\;\;P_2=\left[
\begin{array}{cc}
0&0\cr
1&0\cr
0&1
\end{array}
\right],\;\;
V=\left[
\begin{array}{ccc}
u_1&v_1&w_1\cr
u_2&v_2&w_2
\end{array}
\right],
\;\;\\\nonumber
B_j=\left[\begin{array}{ccc}
0_{2,2}&b_{j1}&0_{2,N-5}\cr
0_{3 (k-1) ,2}&0_{3(k-1),3}&b_{j2}\cr
0_{3,2}&0_{3,3}&0_{3,N-5}
\end{array}\right],
b_{11}=\left[\begin{array}{ccc}
1&1&1\cr
0&0&0
\end{array}\right],
\;\;\\\nonumber
b_{21}=\left[\begin{array}{cccc}
0&0&0\cr
1&1&1
\end{array}\right],\;\;
b_{j2}={\mbox{diag}}(\underbrace{A_j,A_j,\dots}_{k-1}),\;\;
j=1,2,\;\;\\\nonumber
A_1=\left[\begin{array}{rrr}
1&0&-1\cr
-1&1&0\cr
0&-1&1
\end{array}
\right],\;\;A_2=\left[\begin{array}{rrr}
1&-1&0\cr
0&1&-1\cr
-1&0&1
\end{array}
\right],\end{aligned}$$ $\Phi=[\phi_1 \;\;\phi_2\;\;\phi_3]$, where $\phi_k$ are $N$-dimensional columns.  Then equation (\[Phi\]) can be written in the form ${\phi_1}_{x_2}={\phi_2}_{x_1}$, ${\phi_2}_{x_2}={\phi_3}_{x_1}$, i.e. $\phi_1=S_{x_1x_1}$, $\phi_2=S_{x_1x_2}$, $\phi_3=S_{x_2x_2}$, where $S$ is arbitrary function of variables $x_1$ and $x_2$. In view of eq.(\[t\]), we can write for $\Phi$: $\Phi=\int_{-\infty}^\infty c(k_1,k_2) [1\;\;\;k_2/k_1\;\;\;
 k_2^2/k_1^2] \exp[k_1 x_1 + k_2
x_2 -k_1^2 t] dk_1 dk_2$, where $c(k_1,k_2)$ is arbitrary column of $N$ elements.  Function $\Psi$ solves system (\[x\],\[t\]) and can be represented in the form $
\Psi=(\Psi_0+\tilde \Psi ) e^\eta,
$ where $\eta$ is given by the eq.(\[eta\]), $
\tilde \Psi =\partial_{x_1}^{-1} \Phi C_1 e^{-\eta} 
$ and $\Psi_0$ is arbitrary constant $N\times N$ matrix.  Then eq.(\[MATR\]) can be solved for the function $U$, its first $R$ rows define matrix $V$. Elements of this matrix satisfy the system (\[red1\]) which gets the following form: $$\begin{aligned}
\nonumber 
{u_1}_t +{u_1}_{x_1x_1}- 2
{u_2}_{x_1x_2} + 2 {v_2}_{x_1x_1}+ 2 u_1 {u_1}_{x_1} -
 2 (v_2 {u_1})_{x_1} +&&\\\nonumber 
 2 v_1
{u_2}_{x_1} - 2 (w_2 {u_2})_{x_1} +
 2 (u_2 {v_1})_{x_1} + 4 v_2 { v_2}_{x_1}  &=&0
\\\nonumber 
{v_1}_t-
v_{x_1x_1}+2 {u_1}_{x_1x_2}  + 2 ({w_1} {u_2})_{x_1} + 2 u_1 {v_1}_{x_1}-
2 v_2 {v_1}_{x_1}  &=&0\\\nonumber 
{w_1}_t -
{w_1}_{x_1x_1} + 2 {v_1}_{x_1x_2} - 2 w_1 {u_1}_{x_1}+4 v_1 {v_1}_{x_1} -
2 w_2 {v_1}_{x_1}
+2 (w_1{v_2})_{x_1} &=&0\\\nonumber 
{u_2}_t +{u_2}_{x_1x_1}+ 2 u_2 {u_1}_{x_1} + 2 v_2
{u_2}_{x_1}\;=\;0,\;\;\;
{v_2}_t +{v_2}_{x_1x_1}+ 2 u_2 {v_1}_{x_1} + 2 v_2
{v_2}_{x_1}&=&0\\\nonumber 
{w_2}_t-{w_2}_{x_1x_1} +
2 {u_1}_{x_1x_2}  + 2 {v_2}_{x_1x_2} - 2
{v_1}_{x_1x_1}+ 2 u_2 {w_1}_{x_1} + 2 v_2
{w_2}_{x_1} &=&0\\\nonumber \end{aligned}$$

Conclusions
===========

The represented version of the dressing method serves for wide class of $(n+1)$-dimensional PDE.  It supplies solutions depending on arbitrary functions of $n$ variables provided $R<M$. In the last case solution depends on the functions of single variables.  By construction, equations have infinite number of commuting flows and are compatibility conditions for some specific linear overdetermined systems, which is equivalent to the classical linear problem if only $R=M$.

The work is supported by RFBR grants 01-01-00929 and 00-15-96007.  Author thanks Prof. S.V.Manakov and Dr.Marikhin for useful discussions. 

[30]{}

V.E.Zakharov and A.B.Shabat, Funct. Anal. Appl. [** 8**]{}, 43 (1974)

V.E.Zakharov and A.B.Shabat, Funct. Anal. Appl. [** 13**]{}, 13 (1979)

V.E.Zakharov and S.V.Manakov, Funct. Anal. Appl. [** 19**]{}, 11 (1985)

L.V.Bogdanov and S.V.Manakov, J.Phys. A:Math. Gen. [**21**]{}, L537 (1988)

B.Konopelchenko,[*Solitons in Multidimensions*]{} (World Scientific, Singapore, 1993)

A.I.Zenchuk, J.Math. Phys., [** 41**]{}, 6248 (2000)

Y.Ohta, J.Satsuma, D.Takahashi and T. Tokihiro, Progr.  Theor. Phys.  Suppl.,  No.94, p.210 (1988). 

A.I.Zenchuk, arXiv:nlin. SI/0202053 v1 (2002)

A.I.Zenchuk, arXiv:nlin. SI/0210031 v1 (2002)

F.Guil, M.Mañas and G.Álvarez, Phys. Lett. A, [**190**]{}, 49 (1994)
