# American Institute of Mathematical Sciences

June  2016, 36(6): 2991-3009. doi: 10.3934/dcds.2016.36.2991

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

Received  July 2015 Revised  September 2015 Published  December 2015

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.
Citation: Matt Coles, Stephen Gustafson. A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2991-3009. doi: 10.3934/dcds.2016.36.2991
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