Solitons, scattering and blow-up for the nonlinear Schrödinger equation with combined power-type nonlinearities on ℝ^d×\mathbbT

2024/09/24 by Luigi Forcella, Yongming Luo, Forcella, Luigi +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2409.15860

openalex publication_date 2024/09/24 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28

Abstract

We investigate the long time dynamics of the nonlinear Schrödinger equation (NLS) with combined powers on the waveguide manifold ℝ^d×\mathbbT. Different from the previously studied NLS-models with single power on the waveguide manifolds, where the non-scale-invariance is mainly due to the mixed nature of the underlying domain, the non-scale-invariance of the present model is both geometrical and structural. By considering different combinations of the nonlinearities, we establish both qualitative and quantitative properties of the soliton, scattering and blow-up solutions. As one of the main novelties of the paper compared to the previous results for the NLS with single power, we particularly construct two different rescaled families of variational problems, which leads to an NLS with single power in different limiting profiles respectively, to establish the periodic dependence results.

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