Periodic solutions for a class of nonlinear partial differential equations in higher dimension

TL;DR

This paper proves the existence of periodic solutions for various nonlinear partial differential equations in higher dimensions, including cases with infinite-dimensional bifurcation equations, expanding understanding of wave packet solutions.

Contribution

It introduces new existence results for periodic solutions in higher-dimensional nonlinear PDEs, especially for equations with infinite-dimensional bifurcation scenarios.

Findings

Existence of periodic solutions for nonlinear Schrödinger, wave, and beam equations in higher dimensions.

Solutions include wave packets at leading order for zero-mass Schrödinger equations.

Results applicable to equations with infinite-dimensional bifurcation equations.

Abstract

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases where the bifurcation equation is infinite-dimensional, such as the nonlinear Schroedinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.