Homoclinic breather waves, rouge waves and multi-soliton waves for a (2+1)-dimensional Mel’nikov equation

Abstract

Purpose The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets. Design/methodology/approach The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method. Findings The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures. Research limitations/implications This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations. Originality/value The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

Dynamics of breather waves, lump-kink solutions and interaction solutions for a (3+1)-dimensional generalized shallow water waves equation

“…Various types of analytic solutions have been obtained, including multiple soliton solutions, similarity solutions, rational solutions, rogue solutions and others. Thus, several powerful approaches have been proposed to construct such solutions, including the inverse scattering transform (Novikov et al, 1984;Ablowitz and Clarkson, 1991;Ablowitz et al, 2004), the Darboux transformation method (Gu et al, 2005;Guo et al, 2013;Matveev and Salle, 1991;Zhang et al, 2017;Zhou, 2018), the Hirota bilinear method (Gao et al, 2021;Hirota, 2004;Hu et al, 2016;Ma and Abdeljabbar,2012;Ma and You, 2005), the Riemann-Hilbertapproach (Geng and Wu, 2016;Guo et al, 2018;Ma, 2020;Yang, 2019), the homoclinic test approach (Dai et al, 2009;Guo et al, 2009;Kang and Xia, 2020), the parameter limit method (Liu, 2021;Tan, 2021;Tan et al, 2018) and so on. Among these methods, Hirota's bilinear method stands out as a straightforward tool for constructing interaction solutions among the lump, rouge and different solitary waves.…”

Novel dynamical behaviors of interaction solutions of the new (3+1)-dimensional integrable fourth-order nonlinear equation

“…Recently, the interaction solutions among the lump waves, rouge waves and other solitary waves of NLPDEs have attracted the attention of many scholars. Various powerful methods have been proposed to construct these solutions of NLPDEs, such as the inverse scattering transformation (Ablowitz and Clarkson 1991), the Darboux transformation method (Imai and Nozaki 1996;Wang et al 2015), Hirota's bilinear method (Hirota 2004;Xu and Deng 2016;Zhou et al 2019), the quadratic function method (Ma 2015;Tan et al 2020;Ma 2016;Kaur and Wazwaz 2019;Liu and Sun 2018;Tan 2018), the extended homoclinic test approach (Li and Dai 2011;Liu 2021;), the long-wave limit method ) and other methods (Seadawy and Tariq 2021;Liu 2018;Kaur and Wazwaz 2018).…”