Invariant Subspaces of the Two-Dimensional Nonlinear Evolution Equations

Chunrong Zhu,Changzheng Qu

doi:10.3390/sym8110128

Abstract

In this paper, we develop the symmetry-related methods to study invariant subspaces of the two-dimensional nonlinear differential operators. The conditional Lie–Bäcklund symmetry and Lie point symmetry methods are used to construct invariant subspaces of two-dimensional differential operators. We first apply the multiple conditional Lie–Bäcklund symmetries to derive invariant subspaces of the two-dimensional operators. As an application, the invariant subspaces for a class of two-dimensional nonlinear quadratic operators are provided. Furthermore, the invariant subspace method in one-dimensional space combined with the Lie symmetry reduction method and the change of variables is used to obtain invariant subspaces of the two-dimensional nonlinear operators.

Highlights

The invariant subspace method is an effective one to perform reductions of nonlinear partial differential equations (PDEs) to finite-dimensional dynamical systems
It is noticed that a large number of exact solutions, such as N-solitons of integrable equations, similarity solutions of nonlinear evolution equations and the generalized functional separable solutions to nonlinear PDEs, can be recovered by the invariant subspace methods [1,21,22,23,24,25,26,27,28,29,30,31]
The Lie theory of the symmetry group plays an important role for differential equations, which is a useful method to explore various properties and obtain exact solutions of nonlinear PDEs

Summary

Introduction

The invariant subspace method is an effective one to perform reductions of nonlinear partial differential equations (PDEs) to finite-dimensional dynamical systems. A key point for the invariant subspace approach is the estimate of maximal dimension of the invariant subspaces [1,5,6,15,16] It was shown in [1,5] that for k-th order one-dimensional nonlinear operator of the form:. It is of great interest to develop the invariant subspace method to study the multi-dimensional nonlinear evolution equations. The purpose of this paper is to develop symmetry-related method to study invariant subspaces of nonlinear evolution equations in the two- or multi-dimensional case. Since the two-dimensional nonlinear evolution equations can be reduced to one-dimensional equations by the

Direct Extensions in R2

Invariant Subspaces of a Quadratic Operator in R2

The Space Wnn

Invariant Subspace in R and Lie’s Classical Symmetries

Concluding Remarks