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Inverse scattering transform for the nonlocal reverse space–time nonlinear Schrödinger equation

M. J. Ablowitz^a, Bao-Feng Feng^b, X. Luo^c, Z. Musslimani<sup>d

a</sup> Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO, USA
^b School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, USA
^c Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, USA
^d Department of Mathematics, Florida State University, Tallahassee, FL, USA

Abstract: Nonlocal reverse space–time equations of the nonlinear Schrödinger (NLS) type were recently introduced. They were shown to be integrable infinite-dimensional dynamical systems, and the inverse scattering transform (IST) for rapidly decaying initial conditions was constructed. Here, we present the IST for the reverse space–time NLS equation with nonzero boundary conditions (NZBCs) at infinity. The NZBC problem is more complicated because the branching structure of the associated linear eigenfunctions is complicated. We analyze two cases, which correspond to two different values of the phase at infinity. We discuss special soliton solutions and find explicit one-soliton and two-soliton solutions. We also consider spatially dependent boundary conditions.

Keywords: inverse scattering transform, nonlocal RST NLS equation.

MSC: 37K15; 35Q51; 35Q15.

Received: 24.08.2017

DOI: 10.4213/tmf9449

 English version: Theoretical and Mathematical Physics, 2018, 196:3, 1241–1267

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