New maximal dimension of invariant subspaces to coupled systems with two-component equations

Abstract

In this paper, new maximal dimension of invariant subspaces to coupled systems with two-component equations is estimated under certain conditions. It is shown that if the really coupled operator F=(F1,F2) with orders {k1,k2}(k1⩾k2) preserves the invariant subspace Wn11×Wn22(0<n1<n2), then there holds n2-n1⩽k1, n2⩽2k1+k2+1, where F2∈F is a nonlinear differential operator and Wnqq is the space generated by solutions of a linear ordinary differential equation of order nq,(q=1,2). Several concrete examples are presented to illustrate the result.