Persistent Homoclinic Orbits for Nonlinear Schroedinger Equation Under Singular Perturbation

TL;DR

This paper proves the existence of homoclinic orbits in the cubic nonlinear Schrödinger equation under singular perturbations, emphasizing semigroup regularity and correcting previous work with a normal form approach.

Contribution

It provides a significant generalization of prior results on homoclinic orbits in nonlinear Schrödinger equations, including correction of earlier mistakes and analysis of multiple unstable modes.

Findings

Existence of homoclinic orbits under singular perturbations.

Analysis of semigroup regularity at zero perturbation.

Correction of previous errors using normal form transform.

Abstract

Existence of homoclinic orbits in the cubic nonlinear Schr\"odinger equation under singular perturbations is proved. Emphasis is placed upon the regularity of the semigroup $e^{\e t \pa_x^2}$ at $\e = 0$. This article is a substantial generalization of \cite{LMSW96}, and motivated by the effort of Dr. Zeng \cite{Zen00a} \cite{Zen00b}. The mistake of Zeng in \cite{Zen00b} is corrected with a normal form transform approach. Both one and two unstable modes cases are investigated.