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Inverse Scattering Transform For The Nonlocal Nonlinear Schrodinger Equation With Nonzero Boundary Conditions

Title: Inverse Scattering Transform For The Nonlocal Nonlinear Schrodinger Equation With Nonzero Boundary Conditions.
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Name(s): Ablowitz, Mark J., author
Luo, Xu-Dan, author
Musslimani, Ziad H., author
Type of Resource: text
Genre: Journal Article
Text
Journal Article
Date Issued: 2018-01
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: In 2013, a new nonlocal symmetry reduction of the well-known AKNS (an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, and Alan C. Newell et al. (1974)) scattering problem was found. It was shown to give rise to a new nonlocal PT symmetric and integrable Hamiltonian nonlinear Schrodinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one-soliton solutions was found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem, which arises from a novel nonlocal system, is developed via a left-right Riemann-Hilbert problem in terms of a suitable uniformization variable and the time dependence of the scattering data is obtained. This leads to a method to linearize/solve the Cauchy problem. Pure soliton solutions are discussed, and explicit 1-soliton solution and two 2-soliton solutions are provided for three of the four different cases corresponding to two different signs of nonlinearity and two different values of the phase difference between plus and minus infinity. In another case, there are no solitons. Published by AIP Publishing.
Identifier: FSU_libsubv1_wos_000424017000001 (IID), 10.1063/1.5018294 (DOI)
Publication Note: The publisher's version of record is available at https://doi.org/10.1063/1.5018294
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_libsubv1_wos_000424017000001
Owner Institution: FSU
Is Part Of: Journal of Mathematical Physics.
0022-2488
Issue: iss. 1, vol. 59

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Ablowitz, M. J., Luo, X. -D., & Musslimani, Z. H. (2018). Inverse Scattering Transform For The Nonlocal Nonlinear Schrodinger Equation With Nonzero Boundary Conditions. Journal Of Mathematical Physics. Retrieved from http://purl.flvc.org/fsu/fd/FSU_libsubv1_wos_000424017000001