Infinite energy quasi-periodic solutions to nonlinear Schr\"odinger equations on $\mathbb R$

TL;DR

This paper constructs smooth, infinite energy, space-time quasi-periodic solutions with two frequencies for nonlinear Schrödinger equations on the real line, extending previous time-only quasi-periodic results and adapting Bourgain's method to non-compact settings.

Contribution

It introduces a novel approach to find infinite energy quasi-periodic solutions on or nonlinear PDEs on ree space, expanding the scope of Bourgain's semi-algebraic set method.

Findings

Constructed smooth infinite energy solutions with two frequencies.

Extended Bourgain's method to non-compact, real-line setting.

Demonstrated solutions are space-time quasi-periodic without spatial symmetry.

Abstract

We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schr\"odinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous results [B2,1], and their generalizations [W2-4], are quasi-periodic in time, but periodic in space. This paper generalizes Bourgain's semi-algebraic set method [B3] to analyze nonlinear PDEs, in the non-compact space quasi-periodic setting on $\Bbb R$.