Soliton solutions, lump solutions, mixed interactional solutions and their dynamical analysis of the $(2+1)$-dimensional Calogero–Degasperis system

Abstract

This paper focusses on a class of non-isospectral $(2+1)$-dimensional Calogero–Degasperis system, which is a more general form of the classical nonlinear Schrödinger equation. Primarily, according to the plane wave seed solutions, we analyse the modulational instability of this system and obtain the formation mechanism of different localised waves. Secondly, based on the known Lax pair, we construct the generalised $(n, N-n)$-fold Darboux transformation of this system. As an application of the resulting Darboux transformation, we not only show the interaction structures of the multisoliton solutions, but also analyse its long-time asymptotic behaviours and list the relevant physical properties. In order to explore the relationship with differential geometry, we also show the multisoliton surfaces. Subsequently, we give some higher-order lump solutions and analyse their large-parameter asymptotic states. We also give some mixed interactional solutions to better understand the interaction phenomena of different localised waves, whose propagation structures and characteristics are shown graphically. These results and phenomena may be helpful to understand some physical phenomena in nonlinear optics, fluids and Bose–Einstein condensates.