Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation

Abstract

In this paper, we investigate the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.

Data-driven fusion and fission solutions in the Hirota–Satsuma–Ito equation via the physics-informed neural networks method

“…This equation was introduced by Hirota and Satsuma to model unidirectional shallow water waves [32]. Due to its importance and wide applications, the HSI equation has been studied extensively and various exact solutions have been constructed, such as complexiton solutions [33], semi-rational solutions [34], rational localized waves and their absorb-emit interactions [35], multi-wave, breather-wave and hybrid solutions [36], resonant multi-soliton solutions [37], and symmetric invariant solutions [38]. Unlike the above work, our aim is to construct data-driven fusion and fission solutions for the HSI equation usingthe PINN method.…”

Multiple Exp-Function Solutions, Group Invariant Solutions and Conservation Laws of a Generalized (2+1)-dimensional Hirota-Satsuma-Ito Equation

“…In recent decades, many nonlinear evolution equations (NLEEs) have been studied using efficient analytical techniques to find closed-form analytic solutions [10,11,13]. Furthermore, using computerised symbolic computation and numerical simulation software such as Maple and Mathematica, many effective methods for solving exact solutions of NLEEs have been discovered, including the Darboux transformation, Hirota method [16,17,21], and Lie symmetry approach [5,12].…”

X-type Soliton, Resonance Y-type Soliton and Hybrid Solutions of a (2+1)-dimensional Hirota-Satsuma-Ito System for the Shallow Water Waves

“…For the shallow water waves, Refs. [37][38][39][40][41][42][43][44] have considered the following generalization of Eqn. (1) called the Hirota-Satsuma-Ito (HSI) system:…”

“…with N being a positive integer, ı, , ℓ = 1, 2, • • • , N , k ı 's, p ı 's and ζ ı 's being the complex constants, N ı< implying the summation over all possible combinations of N elements under the condition ı < , and N ı< meaning the product over all possible combinations of N elements under the condition ı < . There have been some works closely related to System (2): complexiton solutions [37], localized wave interaction solutions [38], rational localized waves and their Absorb-Emit interactions [39], multi-wave, breather wave and hybrid solutions [40], resonant multi-soliton solutions [41], higher-order breathers, lumps and semi-rational solutions [42], lump and lump-soliton solutions [43], Alice-Bob HSI system and its Bäcklund transformation, bilinear form, lump and breather solutions [44].…”