Exact periodic cross-kink wave solutions for the (2+1)-dimensional Korteweg-de Vries equation
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Exact periodic cross-kink wave solutions for the (2+1)-dimensional Korteweg-de Vries equation Jian-Guo Liu1 · Qing Ye1 Received: 31 July 2018 / Revised: 25 August 2020 / Accepted: 3 October 2020 © Springer Nature Switzerland AG 2020
Abstract The movement of any object has a certain natural law, and the studies and solutions to many natural laws boil down to the problem of mathematical physics equations. Many important physical situations such as fluid flows, plasma physics, and solid state physics have been described by the Korteweg-de Vries (KdV)-type models. In this article, the (2+1)-dimensional KdV equation is presented. By using the Hirota’s bilinear form and the extended Ansätz function method, we obtain new exact periodic cross-kink wave solutions for the (2+1)-dimensional KdV equation. With the aid of symbolic computation, the properties for these exact periodic cross-kink wave solutions are shown with some figures. Keywords Hirota’s bilinear form · Extended Ansätz function method · KdV Equation · Cross-kink wave solutions · Symbolic computation Mathematics Subject Classification 35C08 · 68M07 · 33F10
1 Introduction Nonlinear evolution equations(NLEEs) describe a variety of complex scientific phenomena and dynamic processes in marine engineering, fluid dynamics, plasma physics, chemistry, physics and so on. Solving exact solutions in particular some types of soliton or solitary wave solutions of NLEEs has been attractive in nonlinear physical phenomena. Based on symbolic computation [1–8], many methods have been proposed, such as Hirota direct method [9], homogeneous balance method [10–12], F−expansion method [13], the similarity transformation method [14], three-wave approach [15–20] and etc.
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Qing Ye [email protected] College of Computer, Jiangxi University of Traditional Chinese Medicine, Nanchang 330004, Jiangxi, China 0123456789().: V,-vol
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J.-G. Liu, Q. Ye
In this paper, with the help of Hirota’s bilinear form, extended Ansätz function method and symbolic computation [21–38], we will discuss the following (2+1)dimensional KdV equation [39]: u t + 3(uv)x + u x x x = 0, u x = v y ,
(1)
where u = u(x, y, t). Eq. (1) was derived by Boiti et al. in Ref. [39] by employing the idea of the weak Lax pair. Equation (1) is also named as Boiti–Leon–Manna– Pempinelli equation [40] and read as the ubiquitous KdV equation in dimensionless variables in v = u and y = x [41]. Lou [42,43] and Wazwaz [44] revealed the rich dromion structures and localized structures for Eq. (1). Liu [41] obtained exact periodic solitary wave solutions and Jacobi elliptic function double periodic solutions of Eq. (1) by using the trial function method and the extended Ansätz function method. Wang [45] presented the periodic type of three-wave solutions including periodic two-solitary solution, doubly periodic solitary solution and breather type of two-solitary solution of Eq. (1) using Hirota’s bilinear form and generalized three-wave type of Ansätz approach. The extended Ansätz function method is used instead o
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