Resonance-induced nonlinear bound states

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.