The general fifth-order nonlinear Schrödinger equation with nonzero boundary conditions: inverse scattering transform and multisoliton solutions. (English. Russian original) Zbl 1486.81103
Theor. Math. Phys. 210, No. 1, 8-30 (2022); translation from Teor. Mat. Fiz. 210, No. 1, 11-37 (2022).
Summary: We study the inverse scattering transform of the general fifth-order nonlinear Schrödinger (NLS) equation with nonzero boundary conditions (NZBCs), which can be reduced to several integrable equations. First, a matrix Riemann-Hilbert problem (RHP) for the fifth-order NLS equation with NZBCs at infinity is systematically investigated. Moreover, the inverse problems are solved by studying a matrix RHP. We construct the general solutions for reflectionless potentials. The trace formulas and theta conditions are also presented. In particular, we analyze the simple-pole and double-pole solutions for the fifth-order NLS equation with NZBCs. Finally, we discuss the dynamics of the obtained solutions in terms of their plots. The results in this work should be helpful in explaining and enriching the nonlinear wave phenomena in nonlinear fields.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81U40 | Inverse scattering problems in quantum theory |
| 35C08 | Soliton solutions |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
| 35G31 | Initial-boundary value problems for nonlinear higher-order PDEs |
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 81Q80 | Special quantum systems, such as solvable systems |
Keywords:
general fifth-order nonlinear Schrödinger equation; inverse scattering transform; multi-soliton solutions; Riemann-Hilbert problemReferences:
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