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Bahri, Yakine; Ibrahim, Slim; Kikuchi, Hiroaki

Remarks on solitary waves and Cauchy problem for half-wave-Schrödinger equations. (English) Zbl 1479.35260

Commun. Contemp. Math. 23, No. 5, Article ID 2050058, 31 p. (2021).
The authors investigate solitary wave solutions of the Cauchy problem for the half-wave-Schrödinger equation in the plane. First, they show the existence and orbital stability of the ground states. Second, they show that given any speed \(v\), traveling wave solutions exist and converge to the zero wave as the velocity tends to 1. Finally, the authors study the Cauchy problem for initial data in \(L^2_ xH^s_y(\mathbb R^2)\) with \(s > 1/2\), but the critical case \(s = 1/2\) still remains as an interesting open problem.
Reviewer: Xiaoming He (Beijing)

Cited in 1 Review
Cited in 6 Documents

MSC:

35J10 Schrödinger operator, Schrödinger equation
35B09 Positive solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

Schrödinger and half-wave equation; solitary wave; stable and unstable standing wave
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References:

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