Almost-periodic solutions for a quasi-periodically forced nonlinear Schrödinger equation

Shujuan Liu ()
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Shujuan Liu: Hunan Normal University

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 1, 10-31

Abstract: Abstract In this paper, we prove the existence of small amplitude solutions which are almost-periodic in time for a quasi-periodically forced nonlinear Schrödinger equation $$\begin{aligned} \sqrt{-1}u_{t}-u_{xx}+M_{\xi }u+f(\tilde{\omega }t)|u|^2u=0 \end{aligned}$$ - 1 u t - u xx + M ξ u + f ( ω ~ t ) | u | 2 u = 0 with periodic boundary conditions $$\begin{aligned} u(t,x+2\pi )=u(t,x),~~t\in \mathbb {R}, \end{aligned}$$ u ( t , x + 2 π ) = u ( t , x ) , t ∈ R , where $$M_{\xi }$$ M ξ is a real Fourier multiplier, $$f(\tilde{\theta })(\tilde{\theta }=\tilde{\omega }t)$$ f ( θ ~ ) ( θ ~ = ω ~ t ) is real analytic and the forced frequency vector $$\tilde{\omega }\in \mathbb {R}^{b}$$ ω ~ ∈ R b is fixed and Diophantine.

Keywords: Quasi-periodically forced; KAM theory; Schrödinger equation; Almost-periodic solution (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00098-5

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