On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities
type: Item
instance of: scholarly article
title: On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities
zbMATH Open document ID: 0843.35002
author: V. A. Galaktionov
author: S. A. Posashkov
author: Sergey R. Svirshchevskij
published in: Differential and Integral Equations
publication date: 1995-12-11T00:00:00Z
review text: PDEs of the form \(A(v)= R(v)+ p(x, t)\), where \[ A(v)= \sum_{i+ j\neq 0} a_{ij} {\partial^i v\over \partial t^i} {\partial^j v\over \partial t^j}+ \alpha vv_{xx}+ \beta(v_x)^2+ \gamma v^2+ \varepsilon vv_x, \] \[ R(v)= \sum^m_{k= 1} b_k {\partial^k v\over \partial t^k}+ \sum^k_{l= 0} d_l {\partial^l v\over \partial x^l}, \] and \(a_{ij}\), \(\alpha\), \(\beta\), \(\gamma\), \(\varepsilon\), \(b_k\), \(d_l\) are real numbers, are investigated. It is shown that under certain assumptions the operator \(A\) admits an invariant subset on a linear function space. This makes it possible to represent the initial PDE on the invariant subset as a finite-dimensional dynamical system obtaining nontrivial solutions. Thus a technique to construct new explicit solutions of nonlinear PDEs with quadratic nonlinearities is given in this paper. A generalization of this approach to equations with cubic nonlinearities are discussed, and examples illustrating the approach are given.
Mathematics Subject Classification ID: 35A25
zbMATH DE Number: 811757
zbMATH Keywords: nonlinear evolution equation
zbMATH Keywords: quadratic nonlinearity
zbMATH Keywords: invariant set
zbMATH Keywords: explicit solutions
reviewed by: Valeriy Yumaguzhin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811757