Persistent Homoclinic Orbits for Nonlinear Schroedinger Equation Under Singular Perturbation

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 $e^{\text{e}t{\text{pa}}_x^2}$ at $\text{e}=0$. 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.