## Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations with time-dependent potential.(English)Zbl 1443.81029

Summary: In this paper, we consider the nonlinear Schrödinger equation $$i \partial_t \psi = - \frac{1}{2} \Delta \psi + V(t, x) \psi - F(| \psi |^2) \psi$$ with time-dependent potential in $$\mathbb{R}^3$$. We prove that the weakly interacting $$N$$-soliton is asymptotically stable in a Sobolev space $$H^1(\mathbb{R}^3)$$ under certain assumptions on the time dependent potential $$V(t, x)$$ and the spectral structures of the linearized Hamiltonian.
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### MSC:

 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q41 Time-dependent Schrödinger equations and Dirac equations 35C08 Soliton solutions 46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables 93B18 Linearizations
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### References:

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