Resonance-induced nonlinear bound states

TL;DR

This paper extends the understanding of nonlinear bound states in 1D NLS/GP equations by showing they can bifurcate from scattering and transmission resonances of the linear operator, not just from bound states.

Contribution

It proves that nonlinear bound states can bifurcate from resonance states associated with poles and zeros of the reflection coefficient, extending previous results.

Findings

Nonlinear bound states arise from scattering resonance states.

Resonance bifurcations occur at a positive $L^2$ threshold.

Resonance states are non-decaying and only locally $L^2$.

Abstract

We study nonlinear bound states -- time-harmonic and spatially decaying ($L^2$) solutions -- of the nonlinear Schr\"odinger / Gross--Pitaevskii equations (NLS/GP) with a compactly supported linear potential. Such solutions are known to bifurcate from the $L^2$ bound states of an underlying Schr\"odinger operator $H_V=-\partial_x^2+V$. In this article we prove an extension of this result: for the 1D NLS/GP, nonlinear bound states also arise via bifurcation from the scattering resonance states and transmission resonance states of $H_V$, associated with the poles and zeros, respectively, of the reflection coefficients, $r_\pm(k)$, of $H_V$. The corresponding resonance states are non-decaying and only $L^2_{\rm loc}$. In contrast to nonlinear states arising from $L^2$ bound states of $H_V$, these resonance bifurcations initiate at a strictly positive $L^2$ threshold which is determined by the position of the complex scattering resonance pole or transmission resonance zero.