Asymptotic stability of small solitons to 1D NLS with potential

Abstract

We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equations $$i{u}_t+u_{xx}=Vu\pm |u{|}^{p-1}u\text{for (x,t)∈R×R,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L_t^2(0,\infty ;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that the local smoothing effect of Kato type holds global in time and combine this estimate with the Strichartz estimate to show $\Vert (1+x^2{)}^{-3/4}v{\Vert }_{L_x^{\infty }{L}_t^2}<\infty$, which implies the asymptotic stability of a solitary wave.

24 pages, no figure. To appear in Journal of Mathematics of Kyoto University

References in corpus (3)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Selection of the ground state for nonlinear Schroedinger equations
• Asymptotic stability of N-soliton states of NLS

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• A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II
• The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states
• On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential
• On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D
• An invariant set in energy space for supercritical NLS in 1D
• Asymptotic stability of small solitary waves for nonlinear Schrödinger equations with electromagnetic potential in ${\mathbb{R}}^3$
• On instability of some approximate periodic solutions for the full nonlinear Schrödinger equation