On well-posedness results for the cubic-quintic NLS on \mathbbT³

2023/01/31 by Luo Yongming, Xueying Yu, Luo, Yongming +5
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.2301.13433

Abstract

We consider the periodic cubic-quintic nonlinear Schrödinger equation (i∂_t +Δ)u=μ₁ |u|² u+μ₂ |u|⁴ u on the three-dimensional torus \mathbbT³ with μ₁,μ₂∈ ℝ ∖\0\. As a first result, we establish the small data well-posedness of \eqrefcqnls_abstract for arbitrarily given μ₁ and μ₂. By adapting the crucial perturbation arguments in \citezhang2006cauchy to the periodic setting, we also prove that \eqrefcqnls_abstract is always globally well-posed in H¹(\mathbbT³) in the case μ₂>0.

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