2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61711Existence of homoclinic orbits in the cubic nonlinear Schrödinger 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.43 pagesAnalysis of PDEsDynamical Systemstext