Hybrid Solutions of (3 + 1)‐Dimensional Jimbo‐Miwa Equation

Abstract

The rational solutions, semirational solutions, and their interactions to the (3+1)-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at t≪0 and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

A Study on Lump Solutions to a Generalized Hirota‐Satsuma‐Ito Equation in (2+1)‐Dimensions

“…Many nonlinear equations possess lump solutions, which include (2+1)-dimensional generalized KP, BKP, KP-Boussinesq, Sawada-Kotera, and Bogoyavlensky-Konopelchenko equations [32][33][34][35][36]. Some recent studies also demonstrate the strikingly high richness of lump solutions to linear partial differential equations [37] and nonlinear partial differential equations in (2+1)-dimensions (see, e.g., [38][39][40][41]) and (3+1)-dimensions (see, e.g., [42][43][44][45][46][47][48]). Diversity of lump solutions supplements exact solutions generated from different kinds of combinations (see, e.g., [49][50][51][52]) and can yield various Lie-Bäcklund symmetries, which can be used to determine conservation laws by symmetries and adjoint symmetries [53][54][55].…”

Solitary Wave Solutions for the Higher Dimensional Jimo-Miwa Dynamical Equation via New Mathematical Techniques

“…Runfa Zhang et al found the exact solutions for (3 + 1)-dim JM equation named cross-kink wave, periodic wave with generalized bilinear method in [36] and demonstrated the periodic lump wave and interaction solutions via the Hirota bilinear method in [37]. Yong Zhang et al found the rational and semirational solutions named lump, breather, and rogue wave [38] for (3 + 1)-dim JM equation via the Hirota bilinear method. Bintao Cao calculated the two families of explicit exact solutions with the help of two methods named stable-range and logarithmic stable-range methods in [39][40][41][42][43][44].…”

Lump and Interaction Solutions of Linear PDEs in (3 + 1)-Dimensions

“…There are many other integrable equations with lump solutionse.g. three-dimensional three-wave resonant interaction [8], BKP equation [5,38] [22,40,47] and of the (3 + 1)-dimensional Jimbo-Miwa like equation in [6] are not rationally localised in all space directions, either. Therefore, in (3 + 1)-dimensions, lump and interaction solutions of PDEs are interesting objects to study.…”