A Degenerate Edge Bifurcation in the 1D Linearized Nonlinear Schrodinger Equation

TL;DR

This paper investigates how the spectrum of the linearized operator around solitons in the 1D focusing nonlinear Schrödinger equation changes near the cubic nonlinearity, revealing a degenerate bifurcation of resonances into eigenvalues.

Contribution

It provides a rigorous analysis of the degenerate bifurcation of resonances into eigenvalues for the linearized operator near cubic nonlinearity, confirming previous numerical results.

Findings

Resonances at the spectrum edges bifurcate into eigenvalues as nonlinearity deviates from cubic.

The leading-order eigenvalue expressions match prior numerical predictions.

The work characterizes the spectral transition in the linearized operator near the critical nonlinearity.

Abstract

This work deals with the focusing Nonlinear Schrodinger 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.