New mixed solutions generated by velocity resonance in the $$(2+1)$$-dimensional Sawada–Kotera equation

Zequn Qi,Miaomiao Wang,Qingqing Chen,Biao Li

doi:10.1007/s11071-022-07248-2

Abstract

Based on the N-soliton solutions of the $(2+1)$-dimensional Sawada–Kotera equation, the collisions among lump waves, line waves, and breather waves are studied in this paper. By introducing new constraints, the lump wave does not collide with other waves forever, or stays in collision forever. Under the condition of velocity resonance, the soliton molecules consisting of a lump wave, a line wave, and any number of breather waves are derived for the first time. In particular, the interaction of a line wave and a breather wave will generate two breathers under certain conditions, which is worth exploring, and the method can also be extended to other $(2+1)$-dimensional integrable equations.

Highlights

Soliton molecules are a hot topic and have been studied extensively in recent years [1,2,3,4,5,6,7,8,9]
We already know that the soliton molecules are the stable bound state of solitons generated by velocity resonance [7], which breaks the previous understanding of the nature of traveling wave motion, that is, the speed of high wave is slower than the short wave
It is currently known that multiple soliton solutions can be derived by the Hirota bilinear method [3, 4] and the Darboux transformation method [5, 8, 11, 12], and soliton molecules can be obtained by constraints on parameters

Summary

Introduction

Soliton molecules are a hot topic and have been studied extensively in recent years [1,2,3,4,5,6,7,8,9]. The main point of this paper is as follows: First, using the Hirota bilinear method to obtain the form of the N-soliton solutions of the p2 ` 1q-dimensional Sawada-Kotera equation; summarizing the velocity formulas of lump waves, line waves and breathers respectively; proposing different velocity constraints to obtain lump molecules, breather molecules and the nonlinear superposition of a line wave, a lump wave, and breathers at the same speed. Based on the N-soliton solutions, a new type of mixed solutions is derived using the partial long wave limit method and the new constraints mentioned in this paper, in which the lump wave will never collide with other waves or always collide with other waves The discovery of this new type of nonlinear superposition will enrich the interaction solutions in fluid systems. Lump wave can be obtained by long wave limit method and module resonance [10, 14,15,16], and based on formula Eq(3), let N “ 2, k1 “ K1 ̊ ǫ, k2 “ K2 ̊ ǫ, p1 “ P1 ̊ ǫ, p2 “ P2 ̊ ǫ, φ1 “ φ2 “ πi, K1 “ K2 ̊, P1 “ P2 , ǫ Ñ 0. the velocity of a lump wave can be expressed in the following form: Vlump lump, lump

The new nonlinear superposition of a lump wave and line waves

P1P2k3 K1 K2

The new nonlinear superposition of a lump wave and breathers