`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 2991 - 3009, Volume 36, Issue 6, June 2016      doi:10.3934/dcds.2016.36.2991

 
       Abstract        References        Full Text (433.4K)       Related Articles       

Matt Coles - Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2, Canada (email)
Stephen Gustafson - Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2, Canada (email)

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.

Keywords:  Nonlinear Schrödinger equation, linearized operator, edge bifurcation, Birman-Schwinger formulation, resolvent expansion, Lyapunov-Schmidt reduction
Mathematics Subject Classification:  35P15, 35Q55.

Received: July 2015;      Revised: September 2015;      Available Online: December 2015.

 References