期刊
NONLINEARITY
卷 29, 期 3, 页码 915-946出版社
IOP PUBLISHING LTD
DOI: 10.1088/0951-7715/29/3/915
关键词
integrable nonlocal NLS equation; left-right Riemann-Hilbert problem; PT symmetry
资金
- NSF 253 [DMS-1310200]
- U.S. Air Force Office 254 of Scientific Research [FA9550-12-1-0207]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1310200] Funding Source: National Science Foundation
A nonlocal nonlinear Schrodinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inverse scattering theory is developed by using a novel left-right Riemann-Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg-de Vries and sine-Gordon equations are made.
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