Persistent Homoclinic Orbits for Nonlinear Schroedinger Equation Under Singular Perturbation

Details Persistent Homoclinic Orbits for Nonlinear Schroedinger Equation Under Singular Perturbation Persistent Homoclinic Orbits for Nonlinear Schroedinger Equation Under Singular Perturbation

2001-06-22

arxiv.org/abs/math/0106194v1

Mathematics

Analysis of PDEs

43 pages

Scientific paper

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.

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