• Abstract

We develop an inverse scattering method for an integrable higher-order nonlinear Schrödinger equation (NLSE) with the zero boundary condition at the infinity. An appropriate Riemann-Hilbert problem is related to two cases of scattering data—viz. for $N$ simple poles and a one higher-order pole. This allows to obtain the exact formulae of $N$-th order position and soliton solutions in the form of determinants. In addition, special choices of free parameters allow to determine remarkable characteristics of these solutions and to discuss them graphically. The results can be also applied to other types of NLSEs such as the standard NLSE, Hirota equation, and complex modified KdV equation. They can help to further explore and enrich related nonlinear wave phenomena.

• Abstract

This paper aims to explore dispersion-managed lump waves in a spatial symmetric KP model. Negative second-order linear dispersive terms play an important role in creating lump waves with the nonlinearity in the model. The starting point is a Hirota bilinear form with an ansatz on quadratic function solutions to the corresponding Hirota bilinear equation. Symbolic computation with Maple is conducted to determine lump waves, and characteristic behaviors are analyzed for the resulting lump wave solutions.