Long time dynamics for semirelativistic NLS and Half Wave in arbitrary\n dimension

2016/11/15 by Jacopo Bellazzini, Bellazzini, Jacopo, Vladimir Georgiev +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Waves and Solitons #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.1611.04823

Abstract

We consider the Cauchy problems associated with semirelativistc NLS (sNLS)\nand half wave (HW). In particular we focus on the following two main questions:\nlocal/global Cauchy theory; existence and stability/instability of ground\nstates. In between other results, we prove the existence and stability of\nground states for sNLS in the L² supercritical regime. This is in sharp\ncontrast with the instability of ground states for the corresponding HW, which\nis also established along the paper, by showing an inflation of norms\nphenomenon. Concerning the Cauchy theory we show, under radial symmetry\nassumption the following results: a local existence result in H¹ for energy\nsubcritical nonlinearity and a global existence result in the L² subcritical\nregime.\n

Citations

• Orbital stability of standing waves for some nonlinear Schr�dinger equations
• Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems
• Existence of solitary waves in higher dimensions
• A relation between pointwise convergence of functions and convergence of functionals

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