General lump solutions, lumpoff solutions, and rogue wave solutions with predictability for the (2+1)-dimensional Kortew
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(2019) 38:164
General lump solutions, lumpoff solutions, and rogue wave solutions with predictability for the (2+1)-dimensional Korteweg-de Vries equation Hui Wang1 · Shou-Fu Tian1 · Tian-Tian Zhang1 · Yi Chen1 · Yong Fang2 Received: 21 December 2018 / Revised: 11 July 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract In this work, the (2+1)-dimensional Korteweg-de Vries equation is investigated, which can be used to represent the amplitude of the shallow–water waves in fluids or electrostatic wave potential in plasmas. By employing the properties of Bell’s polynomial, we obtain bilinear representation of the equation with the aid of an appropriate transformation. Based on the obtained Hirota bilinear form, its lump solutions with localized characteristics are constructed in detail. We then derive the lumpoff solutions of the equation by studying a soliton solution generated by lump solutions. Furthermore, special rogue wave solutions with predictability are well presented, and the time and place of appearance are also derived. Finally, some graphic analysis is represented to better understand the propagation characteristics of the obtained solutions. It is hoped that our results provided in this work can be used to enrich the dynamic behaviors of the equation. Keywords The (2+1)-dimensional Korteweg-de Vries equation · Bilinear form · Lump solutions · Lumpoff solutions · Rogue wave solutions Mathematics Subject Classification 35Q51 · 35Q53 · 35C99 · 68W30 · 74J35
Communicated by Corina Giurgea.
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Shou-Fu Tian [email protected]; [email protected] Tian-Tian Zhang [email protected] Yong Fang [email protected]
1
School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China
2
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
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1 Introduction It is well known that nonlinear evolution equations (NLEEs) play a pivotal role in the field of mathematical physics. The research of constructing exact solutions for NLEEs is also one of the central themes for scholars all the time. In the past few decades, all kinds of powerful methods, such as Lie group method (Bluman and Kumei 1989), inverse scattering transformation (Ablowitz and Clarkson 1991), Darboux transformation method (Matveev and Salle 1991), Bäcklund transformation (Kaup 1981), and the Hirota bilinear method (Hirota 2004), etc, have been applied to find exact solutions for NLEEs. It is highly necessary to state that the Hirota bilinear method is one of the most direct and convenient methods to find exact solutions of NLEEs. It was introduced by Hirota in 1971 to construct multi-soliton solutions for integrable NLEEs (Hirota 1971). Hiertarinta discussed in detail how this works for equations in Hietarinta (1997). More recently, the lump solutions with localized property attract the attention of ma
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