Riemann–Hilbert problem and interactions of solitons in the ‐component nonlinear Schrödinger equations
Yuan Li, Shou‐Fu Tian, Jin‐Jie Yang
Abstract
Abstract In this work, we systematically investigate the general ‐component nonlinear Schrödinger equations by using the Riemann–Hilbert method. Starting from the Lax pair associated with an matrix spectral problem, we construct a Riemann–Hilbert problem and obtain the compact ‐soliton solutions formula expressed by determinants. Taking two‐components nonlinear Schrödinger equations as an example, this work detailedly analyzes its multisoliton solutions by means of parameters modulation. Many interesting new phenomena are displayed including elastic collision, soliton reflection, parallel propagation, time‐periodic propagation and (space, time)‐periodic propagation. In addition, the interactions between solitons from weak to strong are presented. Finally, some conjectures about the dynamical behaviors of ‐soliton solutions are proposed. We hope that the present analysis would be useful for a better understanding of soliton interactions in related fields such as nonlinear optics, plasma physics, oceanography, and Bose–Einstein condensates.