Inverse Scattering Transform for the Nonlocal Reverse Space–Time Nonlinear Schrödinger Equation
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.
Keywordsinverse scattering transform nonlocal RST NLS equation
Unable to display preview. Download preview PDF.
- 10a.S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Plenum, New York (1984).Google Scholar
- 13a.L. D. Faddeev and L. A. Takhtajan, Springer, Berlin (1987).Google Scholar
- 27.M. J. Ablowitz, B.-F. Feng, X.-D. Luo, and Z. H. Musslimani, “Reverse space–time nonlocal sine-Gordon/sinh-Gordon equations with nonzero boundary conditions,” Stud. Appl. Math., Online first DOI: 10.1111/sapm.12222 (2018); “Inverse scattering transform for the nonlocal reverse space–time sine-Gordon, sinh-Gordon, and nonlinear Schrödinger equations with nonzero boundary conditions,” arXiv:1703.02226v1 [math-ph] (2017).Google Scholar