Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators

Abstract

In this paper, the dimension of invariant subspaces admitted by m-component nonlinear vector differential operators is estimated. It is shown that if the m-component nonlinear vector differential operators of order k preserves the invariant subspace \documentclass[12pt]{minimal}\begin{document}$W^1_{n_1}\times \cdots \times W^m_{n_m}$\end{document}Wn11×⋯×Wnmm, where \documentclass[12pt]{minimal}\begin{document}$W_{n_q}^q$\end{document}Wnqq is the space generated by solutions of linear ordinary differential equations of order nq (q = 1, …, m), then max {n1, …, nm} ⩽ 2mk + 1. To illustrate the approach, examples of nonlinear vector differential operators admitting invariant subspace \documentclass[12pt]{minimal}\begin{document}$W^1_{n_1}\times W^2_{n_2}$\end{document}Wn11×Wn22 with max {n1, n2} = 9 and three-component nonlinear vector differential operators admitting invariant subspace \documentclass[12pt]{minimal}\begin{document}$W^1_{n_1}\break\times W^2_{n_2}\times W^3_{n_3}$\end{document}Wn11×Wn22×Wn33 with max {n1, n2, n3} = 13 are presented.