Maximal dimension of invariant subspaces to systems of nonlinear evolution equations

Abstract

In this paper, the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator \mathbb{F} = (F^1 ,F^2 ) with orders {k ₁, k ₂} (k ₁ ≥ k ₂) preserves the invariant subspace W_{n_1 }^1 \times W_{n_2 }^2 (n_1 \geqslant n_2 ), then n ₁ − n ₂ ≤ k ₂, n ₁ ≤ 2(k ₁ + k ₂) + 1, where W_{n_q }^q is the space generated by solutions of a linear ordinary differential equation of order n _q (q = 1, 2). Several examples including the (1+1)-dimensional diffusion system and Itô’s type, Drinfel’d-Sokolov-Wilson’s type and Whitham-Broer-Kaup’s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.

Keywords

Invariant subspace / Nonlinear PDEs / Exact solution / Symmetry / Dynamical system

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