Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schrödinger equations with a trapping potential

Abstract

In this paper, we analyze the long-time dynamics of small solutions to the $1d$ cubic nonlinear Schrödinger equation (NLS) with a trapping potential. We show that every small solution will decompose into a small solitary wave and a radiation term which exhibits the modified scattering. In particular, this result implies the asymptotic stability of small solitary waves. Our analysis also establishes the long-time behavior of solutions to a perturbation of the integrable cubic NLS with the appearance of solitons.

69 pages

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Cited by in corpus (2)

• Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode
• Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation