A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation

Matt  Coles; Stephen  Gustafson

doi:10.3934/dcds.2016.36.2991

Article Contents

2016, Volume 36, Issue 6: 2991-3009. Doi: 10.3934/dcds.2016.36.2991

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A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation

Matt Coles^1, and
Stephen Gustafson^1,

1.
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2

Received: June 30, 2015

Revised: August 31, 2015

Published: May 2017

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Abstract

This work deals with the focusing Nonlinear Schrödinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.

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Mathematics Subject Classification: 35P15, 35Q55.

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