Semi-rational solutions of the third-type Davey-Stewartson equation

The loop rogue wave solutions for the Wadati-Konno-Ichikawa equation

The first-order rogue wave solution with two arbitrary parameters of the Wadati-Konno-Ichikawa equation is generated based on the Darboux transformation and inverse hodograph transformation. The analyticity of first-order rogue wave solution is studied. A simple analysis shows that the parameter that denotes the amplitude of background wave plays an important role in controlling the analyticity of rogue wave solution. In particular, the rogue wave solution displays a loop-type profile when it is singular, and the general features of loop rogue waves are discussed in detail.

The n -component nonlinear Schrödinger equations: dark–bright mixed N - and high-order solitons and breathers, and dynamics

The general n -component nonlinear Schrödinger equations are systematically investigated with the aid of the Darboux transformation method and its extension. Firstly, we explore the condition of the existence for dark–bright mixed soliton solutions and derive an explicit formula of dark–bright mixed multi-soliton solutions in terms of the determinant. Secondly, we present the formula of dark–bright mixed high-order semi-rational solitons, and predict their general N th-order wave structures. Thirdly, we investigate the wing-shaped structures of breather. Finally, we perform the numerical simulations for some representative solitons to study their dynamical behaviours.

Dynamics of lumps and dark-dark solitons in the multi-component long-wave-short-wave resonance interaction system

General semi-rational solutions of an integrable multi-component (2+1)-dimensional long-wave-short-wave resonance interaction system comprising multiple short waves and a single long wave are obtained by employing the bilinear method. These solutions describe the interactions between various types of solutions, including line rogue waves, lumps, breathers and dark solitons. We only focus on the dynamical behaviours of the interactions between lumps and dark solitons in this paper. Our detailed study reveals two different types of excitation phenomena: fusion and fission. It is shown that the fundamental (simplest) semi-rational solutions can exhibit fission of a dark soliton into a lump and a dark soliton or fusion of one lump and one dark soliton into a dark soliton. The non-fundamental semi-rational solutions are further classified into three subclasses: higher-order, multi- and mixed-type semi-rational solutions. The higher-order semi-rational solutions show the process of annihilation (production) of two or more lumps into (from) one dark soliton. The multi-semi-rational solutions describe N(N≥2) lumps annihilating into or producing from N-dark solitons. The mixed-type semi-rational solutions are a hybrid of higher-order semi-rational solutions and multi-semi-rational solutions. For the mixed-type semi-rational solutions, we demonstrate an interesting dynamical behaviour that is characterized by partial suppression or creation of lumps from the dark solitons.