Dynamics of the rogue lump in the asymmetric Nizhnik–Novikov–Veselov system

Abstract

The (2 + 1)‐dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) equation describes an incompressible fluid such as waves with weakly nonlinear restoring forces, long internal waves in a density‐stratified ocean, or acoustic waves on a crystal lattice. In this paper, we construct a new kind of solutions doubly localized in both time and space, that is, a rogue lump wave, of the ANNV equation using the binary Darboux transformation (BDT). This solution is expressed by a class of semirational functions, in which lumps appear from one line soliton and then rapidly disappear into another line soliton. Phase parameter rj$r_j$ newly introduced in eigenfunctions plays a vital role in controlling the distinct dynamical behaviors of these rogue lumps. The explicit formulas of the approximate locations and heights of rogue waves, lumps, and line solitons are given by asymptotic analysis. Their collision process is discussed in detail. These results further help us to understand the extreme and transient waves in a variety of physical systems.

Prediction of general high-order lump solutions in the Davey–Stewartson II equation

“…[13] also derived several new non-local reduced wave equations, such as non-local nonlinear Schrödinger equations with arbitrary even order, Sasa–Satsuma type matrix integrable hierarchies [14] and non-local nonlinear Schrödinger hierarchies of type false(−λ,λfalse) [15] and matrix-type mKdV equation [16], which have the potential to explore novel type basic solitary solutions such as N-soliton solutions. Some other integrable equations include the Boussinesq equation [17], false(2+1false)-dimensional NLS equation [18] and false(2+1false)-dimensional asymmetric Nizhnik–Novikov–Veselov equation [19]. Furthermore, there are plenty of methods for studying nonlinear integrable equations, such as inverse scattering transformation [20], Darboux transformation [21], Bäcklund transformation [22], Hirota’s bilinear method [23], Lie group symmetry [24,25] and so on.…”