1 Introduction

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together with they scalar and three-dimensional reductions. Variants of the Kadomtsev-Petviashvili type and Korteweg-de Vries type equations are represented among them. Our algorithm is based on the combination of two Frobenius type reductions and special differential reduction imposed on the matrix fields of integrable PDEs. It is shown that the derived four-dimensional nonlinear PDEs admit arbitrary functions of two variables in their solution spaces which clarifies the integrability degree of these PDEs.

Differential reductions of the Kadomtsev-Petviashvili equation and associated higher dimensional nonlinear PDEs.

A. I. Zenchuk

Institute of Chemical Physics RAS, Acad. Semenov av., 1 Chernogolovka, Moscow region 142432, Russia

e-mail: zenchuk@itp.ac.ru

July 27, 2019

1 Introduction

The problem of construction of such multidimensional nonlinear partial differential equations (PDEs) which are either completely integrable or, at least, possess a big manifold of particular solutions is very attractive problem of integrability theory. First of all, one should emphasize several remarkable classical works regarding the completely integrable models such as [1] (where the Korteweg-de Vries equation (KdV) has been first time studied), [2, 3] (where the dressing method for a big class of soliton and instanton equations has been formulated), [4] (where the so-called Sato approach to the integrability is applied to Kadomtsev-Petviashvili equation (KP)). A big class of the first order systems of quasilinear PDEs is integrated in [5, 6] using the generalized hodograph method. However, most of the above integrable nonlinear PDEs are (2+1)- and/or (1+1)-dimensional, excepting the Selfdual-type PDEs [7, 8] and the equations assotiated with commuting vector fields [9, 10, 11, 12, 13, 14].

In this paper we use modification of the algorithm represented in [15, 16] allowing one to construct new multidimensional partially integrable nonlinear PDEs. It is shown in [16] that Frobenious reduction of the matrix fields of the nonlinear PDEs integrable either by the Hoph-Cole substitution [17] (-integrable PDE [18, 19, 20, 21, 22, 23]) or by the method of characteristics [24, 25] (-integrable PDE) leads to one of two big classes of the nonlinear PDEs integrable by the inverse spectral transform method (ISTM) [26, 27, 28, 29] (-integrable PDEs [18]). These classes are () soliton equations, such as KdV [1, 30], the Nonlinear Schrödinger equation (NLS) [31], the Kadomtsev-Petviashvili equation (KP) [32], the Deavi-Stewartson equation (DS) [33], and () instanton equations, such as the Self-dual Yang-Mills equation (SDYM) [7, 8].

The natural question is whether the Frobenius reduction (or its modification) can be used for construction new types of integrable (or at least partially integrable) systems starting with any known integrable system or this method works only for derivation of soliton and instanton PDEs from - and -integrable ones?

At first glance the answer is negative. In fact, one can verify that Frobenius reduction applied to such matrix -integrable PDE as the GL() SDYM, the -wave equation and the KP produces the same -integrable PDE. However, there is a method to generate new higher dimensional partially integrable systems of nonlinear PDEs using Frobenius type reduction after the appropriate differential reduction imposed on the matrix fields of the above -integrable nonlinear PDEs.

Such combination of reductions has been already used in [34]. It is shown there that GL() SDYM supplemented by the pair of reductions, namely, differential reduction relating certain blocks of the matrix field and Frobenius reduction of these blocks, produces a new five-dimensional system of matrix nonlinear PDEs (with three-dimensional solution space), whose scalar reduction results in the nonlinear PDE assotiated with commuting vector fields [13, 14].

Following the strategy of ref.[34], we consider an algorithm for construction new partially integrable PDEs starting with the matrix KP (although this algorithm may be applied to any -integrable model). We will derive two representative of matrix systems whose scalar reductions yield the following four-dimensional equations:


Another reduction,


reduces these PDEs into the following three-dimensional ones


Here we take and as evolutionary parameters (times) while , and are taken as space parameters. Since the linear parts of eqs.(2) and (5) coincide with the linear parts of the KP and the KdV respectively, eqs.(2) and (5) may be treated as new variants of KP- and KdV-type equations respectively. The feature of these equations is that the derivative with respect to appears only in the nonlinear parts. For this reason, these equations may be referred to as dispersionless ones.

Remember, that the KP originates from the (1+1)-dimensional -integrable Bürgers hierarchy due to the Frobenius reduction [16]. Thus the complete set of transformations leading to eqs.(1) and (2) is following (see also Fig.1). We start with the -integrable Bürgers hierarchy of nonlinear PDEs with independent variables and , . Frobenius reduction of this hierarchy [16] yields the proper hierarchy of discrete chains of nonlinear PDEs, which is equivalent to the chains obtained in the Sato approach to the integrability of (2+1)-dimensional KP [4]. The later may be derived as an intermediate result of our algorithm after eliminating all extra fields using combination of the first and the second representatives of the constructed discrete hierarchy. Next, apply the differential reduction introducing one more independent variable . This step yields a three-dimensional system of nonlinear PDEs, i.e. dimensionality coincides with that of KP. Finally, the Frobenius type reduction applied to the matrix fields of the latter nonlinear PDEs results in a four-dimensional system of matrix PDEs whose scalar versions yield eqs.(1,2). It will be shown that the derived four-dimensional nonlinear PDEs may not be completely integrated by our method because the available solution spaces to them are restricted.

Fig.1 The chain of transformations from the (1+1)-dimensional -integrable Bürgers hierarchy to the four-dimensional PDEs (1) and (2). Here , are matrix fields, , , and are blocks, and are - and -dimensional identity matrices respectively. Blocks and are defined by eqs.(26). Here , and are arbitrary positive integer parameters.

This paper is organized as follows. In Sec.2 we briefly recall some results of [16] and derive the discrete chains of nonlinear PDEs produced by the (1+1)-dimensional Bürgers hierarchy (with independent variables and , ) supplemented by the Frobenius reduction. Using a few equations of theses chains we eliminate all extra fields and derive the matrix KP. In Sec.3 we suggest the special type differential reduction imposed on the blocks of the matrix fields of the above chains. This reduction introduces a new independent variables and and allows one to generate three-dimensional matrix PDEs. Frobenius type reduction of the above PDEs results in the four-dimensional systems of matrix PDEs, Sec.4. Scalar versions of these PDEs result in eqs.(1) and (2) which, in particular, may be reduced to eqs.(4) and (5) respectively. Solution spaces to the nonlinear PDEs derived in Secs.3 and 4 will be studied in Sec.5. The obstacles to the complete integrability of eqs.(1), (2), (4) and (5) as well as the obstacles to their integrability by the ISTM are briefly discussed in Sec.6. Conclusions are given in Sec.7.

2 Relation between the (1+1)-dimensional -integrable Bürgers hierarchy and the matrix KP

2.1 The (1+1)-dimensional -integrable Bürgers hierarchy

Here we use the Bürgers hierarchy as a simplest example of -integrable hierarchies linearizable by the matrix Hopf-Cole substitution. Namely, let be solution of the following linear algebraic matrix equation:


Here and are matrix functions, are commuting constant matrices. To anticipate, the matrices are introduced in order to establish the reductions eliminating derivatives with respect to from the nonlinear PDEs, see, for example, reduction (3). The compatibility conditions of eqs. (6) and (7) yield the linearizable Bürgers hierarchy. We will need only the first and the second representatives of this hierarchy below, i.e. in eqs.(7):


2.2 The Frobenius reduction and assotiated chains of nonlinear PDEs

Introduce the Frobenius reduction [16]:


where and are identity and zero matrices respectively, are matrix functions. Substituting eq.(10) into eq.(6) we obtain the following block structure of :


where are matrix functions. Thus, eqs.(6) and (7) may be written as follows:


where are some commuting constant matrices. The meaning of the integer parameters and is clarified in Fig.1. The compatibility conditions of eqs.(12) and (13) (with ) yield the following discrete chains:


Alternatively, these chains may be derived substituting eq.(10) into eqs.(8) and (9).

2.3 Matrix KP and its scalar reduction

The matrix KP is represented by the system of three equations involving fields , : eq.(15) with and eq.(16) with . Eliminating and from this system one gets the following nonlinear PDE for the field :


where square parenthesis mean matrix commutator. In the scalar case this equation reduces to the following one, :


which is the scalar potential KP.

3 Differential reduction of the matrix KP

Let matrices have the following block structure:


where are matrix functions. We introduce -dependence of by the following second order PDE:


Here , and are diagonal constant matrices. The block structure of (19) suggests us the relevant block structure of :


where , , and are matrix functions. Now matrix equations (12) may be written as two equations:




The compatibility condition of eqs.(22) and (23),


yields the expressions for and in terms of and :


Thus only two blocks of are independent, i.e. and . Matrix equation (22) may be considered as the uniquely solvable system of linear matrix algebraic equations for the matrix functions and , , while eq.(23) is the consequence of eq.(22).

Eqs.(13) yield:


where are commuting constant matrices. Eq.(20) allows us to introduce one more set of parameters as follows:


Before proceeding further we select the following three equations out of system (27,28):


We also assume that , and are scalars, i.e.


In order to derive the nonlinear PDEs for and , we must consider the conditions providing the compatibility of eq.(22) and eqs.(29-31):


These equations generate the following chains of nonlinear PDEs for and , :


In addition, we must to take into account the compatibility condition of eqs.(20) and (22),


which gives us the following system of non-evolutionary nonlinear chains:


One can show that the derived chains (36-3,43,44) generate three compatible three-dimensional systems of nonlinear PDEs as follows:

We do not represent these equations explicitely since they are intermediate equations in our algorithm.

4 The Frobenius type reduction, new four-dimensional systems of matrix nonlinear PDEs and their scalar reductions

Three systems (3-3) represent three commuting flows with times , and respectively. In this section we follow the strategy of ref.[16] and show that hierarchy of nonlinear chains (36-3,43,44) supplemented by the Frobenius type reduction of the matrix fields and generates the hierarchy of four-dimensional systems of nonlinear PDEs, see eqs.(65-68) and text thereafter.

This is possible due to the remarkable property of chains of nonlinear PDEs (36-3,43,44). Namely, these chains admit the following Frobenious type reduction:


where is an arbitrary positive integer parameter, and are arbitrary positive integer functions of positive integer argument, and are matrix fields. In particular, if , then this reduction becomes Frobenius one [16], which is shown in Fig.1.

Eqs.(48) require the following block-structures of the functions and of the constant matrices :


Here are matrix functions and are commuting constant matrices. In turn, eq.(22) reduces to the following one:

while eqs.(29-31,20) yield


Then the chains of nonlinear PDEs (36-3,43,44) get the following block structures: