Asymptotic Stability of multi-solitons for $1$d Supercritical NLS

Abstract

We consider the one-dimensional $L^2$-supercritical nonlinear Schr\"odinger equation \[ i\partial_t \psi + \partial_x^2 \psi + |\psi|^{2k}\psi = 0, \qquad k>2. \] In this regime solitary waves are spectrally unstable and dispersion is weak. In the pioneering work of Krieger and Schlag~\cite{KriegerSchlag}, asymptotic stability of a single soliton was established on a codimension-one center-stable manifold. We prove asymptotic stability of well-separated multi-solitons on a finite-codimension center-stable manifold. Specifically, for $k>\frac{11}{4}$, perturbations lying on a codimension-$m$ Lipschitz manifold around a superposition of $m$ solitons with distinct velocities converge in $H^1(\mathbb{R})$ to a sum of modulated solitons plus dispersive radiation. Due to the comparatively weak dispersion in one dimension, the analysis of multi-soliton dynamics is considerably more delicate. Existing full-line asymptotic stability results for multi-solitons in non-integrable dispersive equations treat only two solitons and rely on strong relative velocity assumptions together with additional structural conditions on the nonlinearity. Our proof combines a modulation analysis, a refined linear theory for one-dimensional matrix charge transfer models developed in our earlier works~\cite{dispanalysis1,dispanalysis2} together with carefully designed norms that capture the interactions of multiple moving solitons. Our result applies to arbitrarily many solitons with the natural power-type nonlinearity in the $L^2$-supercritical regime under the sole requirements of distinct velocities and sufficient spatial separation.