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

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 = ( F 1 , F 2 ) with orders { k 1 , k 2 } ( k 1 ⩾ k 2 ) preserves the invariant subspace W n 1 1 × W n 2 2 ( 0 < n 1 < n 2 ) , then there holds n 2 - n 1 ⩽ k 1 , n 2 ⩽ 2 k 1 + k 2 + 1 , where F 2 ∈ F is a nonlinear differential operator and W n q q is the space generated by solutions of a linear ordinary differential equation of order n q , ( q = 1 , 2 ) . Several concrete examples are presented to illustrate the result.