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Homoclinic jumping in the perturbed nonlinear Schrödinger equation
Author(s) -
Haller G.
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199901)52:1<1::aid-cpa1>3.0.co;2-s
Subject(s) - homoclinic orbit , mathematics , jumping , forcing (mathematics) , mathematical analysis , plane (geometry) , nonlinear system , periodic boundary conditions , boundary (topology) , plane wave , mode (computer interface) , boundary value problem , physics , bifurcation , geometry , quantum mechanics , physiology , biology , computer science , operating system
We study the damped, periodically forced, focusing NLS equation with even, periodic boundary conditions. We prove the existence of complicated solutions that repeatedly leave and come back to the vicinity of a quasi‐periodic plane wave with two time scales. For pure forcing, we prove the existence of a complicated, self‐similar family of homoclinic bifurcations. For mode‐independent damping, we construct “jumping” transients. For mode‐dependent damping, we find generalized Šilnikov‐type solutions that connect a periodic plane wave to itself through repeated jumps. We also study the breakdown of the unstable manifold of plane waves through repeated jumping. Our results give a direct explanation for the numerical observations of Bishop et al. © 1999 John Wiley & Sons, Inc.