Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations

A study of high-order solitons in three nonlocal nonlinear Schrodinger equations is presented. These include the \(\mathcal {PT}\)-symmetric, reverse-time, and reverse-space-time nonlocal nonlinear Schrodinger equations. General high-order solitons in three different equations are derived from the same Riemann–Hilbert solutions of the AKNS hierarchy, except for the difference in the corresponding symmetry relations on the “perturbed” scattering data. Dynamics of general high-order solitons in these equations is further analyzed. It is shown that the high-order fundamental-soliton is moving on several different trajectories in nearly equal velocities, and they can be nonsingular or repeatedly collapsing, depending on the choices of the parameters. It is also shown that the high-order multi-solitons could have more complicated wave structures and behave very differently from high-order fundamental-solitons. More interestingly, via the combinations of different size of block matrix in the Riemann–Hilbert solutions, high-order hybrid-pattern solitons are found, which describe the nonlinear interaction between several types of solitons.