A KAM theorem for 2-dimensional nonlinear Schrödinger equations with forcing terms and (2p+1)-nonlinearities

Abstract

In this paper, we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrödinger equations with different large forcing terms and (2p + 1)-nonlinearities

　　　　　 \mathrm{i}{u}_t-\mathrm{\Delta }u+{\phi }_1\left({\overline{\omega }}_{1}t\right)u+{\phi }_2\left({\overline{\omega }}_{2}t\right)|u{|}^{2p}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2

under periodic boundary conditions. As a result, the existence of a Whitney smooth family of small-amplitude reducible quasi-periodic solutions is obtained.

Key words： Schrödinger equation; reducible KAM tori; small divisor; quasi-periodic solution

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