General $N$-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations

Abstract

General $N$-solitons in three recently-proposed nonlocal nonlinear Schrödinger equations are presented. These nonlocal equations include the reverse-space, reverse-time, and reverse-space-time nonlinear Schrödinger equations, which are nonlocal reductions of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. It is shown that general $N$-solitons in these different equations can be derived from the same Riemann-Hilbert solutions of the AKNS hierarchy, except that symmetry relations on the scattering data are different for these equations. This Riemann-Hilbert framework allows us to identify new types of solitons with novel eigenvalue configurations in the spectral plane. Dynamics of $N$-solitons in these equations is also explored. In all the three nonlocal equations, a generic feature of their solutions is repeated collapsing. In addition, multi-solitons can behave very differently from fundamental solitons and may not correspond to a nonlinear superposition of fundamental solitons.

11 pages, 6 figures

References in corpus (4)

• Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations
• Nonstandard bilinearization of $\mathcal{P}\mathcal{T}$-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions
• Solutions of local and nonlocal equations reduced from the AKNS hierarchy
• Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems