The mixed solutions of the (2+1)-dimensional Hirota–Satsuma–Ito equation and the analysis of nonlinear transformed waves

Abstract

In this paper, we obtain the N -soliton solution for the (2 + 1)-dimensional Hirota-Satsuma-Ito equation by the Hirota bilinear method. On this basis, the breathers and lumps can be obtained using the complex conjugate parameter as well as the long wave limit method, and the mixed solutions containing them are investigated. Then, different nonlinear transformed waves are obtained from breathers and lumps under specific conditions, which include quasi-anti-dark soliton, M -shaped soliton, oscillation Mshaped soliton, multi-peak soliton, quasi-periodic soliton and W -shaped soliton. Finally, on the basis of the two-breather solutions, we discuss in detail the mixed solutions consisting of one breather and one nonlinear transformed wave, and the mixed solutions formed by two nonlinear transformed waves.

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

“…However, a soliton solution is an analytic solution that is exponentially localized in all directions in space of 𝑥, 𝑦 and 𝑧 and time 𝑡. [6,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] Lump solutions arise when surface tension dominates the shallow water surface, as in plasmas, optical media and other physical applications. The basis of symbolic computation method, the generalized positive quadratic function, is a powerful technique to study lump solutions.…”

“…Many effective approaches [12][13][14][15][16][17][18][19][20][21][22][23][24] have been used to examine the complete integrability of nonlinear evolution equations, with elastic or nonelastic interactions, aiming to attain new results in scientific areas. Significant solutions in mathematical physics, such as breather, lump and rogue wave solutions, have been derived using a variety of efficient techniques, such as the algebraic-geometric method, the inverse scattering method, the Bäcklund transformation method, the Painlevé analysis, Lax integrability, the Darboux transformation method, [15][16][17][18][19][20][21][22] the Hirota bilinear method, [1][2][3][4][5][6][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and other powerful schemes. In Ref.…”

“…However, in Refs. [34,35], 𝑁 -soliton solutions, the breathers and lumps for the (2+1)-dimensional Hirota-Satsuma-Ito equation, were obtained by using the Hirota bilinear method, as well as the complex conjugate parameter and the long wave limit method.…”