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DOI: 10.23952/jnfa.2026.7
Received August 21, 2025; Accepted January 6, 2026; Published March 8, 2026
Abstract. This paper is devoted to the study of a two-dimensional Schr\”{o}dinger equation with a general nonlinear term and a large forcing term. It is proved that the equation admits a Whitney smooth family of small amplitude quasi-periodic solutions which are partially hyperbolic for the given frequency vector (non-external parameters) and the large forcing term. Firstly, by introducing a symplectic change of coordinates, the Hamiltonian of the equation is transformed into a linear autonomous system plus higher order term perturbation (non-small perturbation), that is, the reducibility of the non-autonomous linear part is realized. Secondly, by introducing a symplectic change of coordinates, and action-angle variables, the Hamiltonian is transformed into a small perturbation of nonlinear integrable normal form that depends on angle variables. Then, by introducing a new symplectic change of coordinates, the Hamiltonian is transformed into a small perturbation of linear integrable normal form. Finally, the existence of invariant tori of the Hamiltonian system associated with the equation is proved by constructing an infinite-dimensional Kolmogorov-Arnold-Moser (KAM) theorem.
How to Cite this Article:
X. Liu, M. Zhang, Invariant tori for a two-dimensional Schrödinger equation with a general nonlinear term and a large forcing term, J. Nonlinear Funct. Anal. 2026 (2026) 7.