Differential and Integral Equations

On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities

Victor A. Galaktionov, Sergey A. Posashkov, and Sergey R. Svirshchevskii

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In this paper we introduce a technique to construct new explicit solutions of nonlinear partial differential equations of mathematical physics with quadratic nonlinearities. We show that under certain assumptions the nonlinear operators under consideration admit so-called invariant sets on a two-dimensional linear functional subspace that makes it possible to rewrite the corresponding equation as an overdetermined finite-dimensional dynamical system. Some classes of such systems are shown to have nontrivial solutions. Examples of linear invariant subspaces for nonlinear quadratic operators from parabolic and hyperbolic equations are also given. In this case, equations on the corresponding invariant subspaces are reduced to finite-dimensional dynamical systems. Several generalizations including a similar analysis on three-dimensional subspaces, $N$-dimensional operators and equations with cubic nonlinearities are discussed.

Article information

Differential Integral Equations, Volume 8, Number 8 (1995), 1997-2024.

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q58
Secondary: 35C05: Solutions in closed form 35K55: Nonlinear parabolic equations 35L70: Nonlinear second-order hyperbolic equations


Galaktionov, Victor A.; Posashkov, Sergey A.; Svirshchevskii, Sergey R. On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities. Differential Integral Equations 8 (1995), no. 8, 1997--2024. https://projecteuclid.org/euclid.die/1369056137

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