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Embedded Eigenvalues and the Nonlinear Schrodinger Equation
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Embedded Eigenvalues and the Nonlinear Schrodinger Equation

Reza Asad and Gideon Simpson
arXiv.org, v 52(3), pp 033511-033511-26
31 Jan 2011
DOI: https://doi.org/10.1063/1.3567152
url
http://arxiv.org/abs/1101.2485View
SubmittedOpen Access (License Unspecified),  Open

Abstract

Eigen values Eigenvalues Mathematical analysis Nonlinear equations Schrodinger equation Solitary waves Source code Stability
A common challenge to proving asymptotic stability of solitary waves is understanding the spectrum of the operator associated with the linearized flow. The existence of eigenvalues can inhibit the dispersive estimates key to proving stability. Following the work of Marzuola & Simpson, we prove the absence of embedded eigenvalues for a collection of nonlinear Schrodinger equations, including some one and three dimensional supercritical equations, and the three dimensional cubic-quintic equation. Our results also rule out nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances. The proof is computer assisted as it depends on the sign of certain inner products which do not readily admit analytic representations. Our source code is available for verification at http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.

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Physics, Mathematical
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