Degenerate solutions for the spatial discrete Hirota equation
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ORIGINAL PAPER
Degenerate solutions for the spatial discrete Hirota equation Meng Li · Maohua Li
· Jingsong He
Received: 9 July 2020 / Accepted: 19 September 2020 © Springer Nature B.V. 2020
Abstract In this paper, some degenerate solutions of the spatial discrete Hirota equation are constructed via the degenerate idea of positon solution. Under the zero seed solution, the n-positon is obtained by N -fold degenerate Darboux transformation (DT). The degenerate DT is taking the degenerate limit λ j → λ1 for the eigenvalues λ j ( j = 1, 2, 3, . . . , N ) of N -fold DT and then performing the high-order Taylor expansion near λ1 . Considering the universal Darboux transformation, breather is obtained from the nonzero seed. Then, a new type of breather solution can be produced by using the same degenerated method and higher-order Taylor expansion for eigenvalues in determinant expression of breather solution. The explicit determinants of breather-positon solution and positon solution are constructed, respectively, and the complicated and significant dynamics of low-order solution are also revealed. Keywords Spatial discrete Hirota equation · Positon solutions · Degenerate Darboux transformation · Breather solution · Breather-positon solution
M. Li · M. Li (B) School of Mathematics and Statistics, Ningbo University, Zhejiang 315211, People’s Republic of China e-mail: [email protected] J. He Institute for Advanced Study, Shenzhen University, Shenzhen 518060, Guangdong, People’s Republic of China
1 Introduction Soliton equations have been widely used and deeply studied in the fields of fluid mechanics, physics, mathematics, communication and other natural disciplines from the discovery of solitons to the present. On the one hand, the content of soliton theory is to develop systematic methods for solving nonlinear equations [1–6]. The other is to study the algebraic and geometric properties of integrable systems [7,8]. With the continuous development of integrable systems, the theory of integrable systems is also gradually applied to discrete systems. Then, the discrete integrable systems have attracted more and more attention because of their wide application in many fields in recent decades. The research of the discrete integrable systems can be traced back to the works of Ablowitz, Ladik [9,10] and Hirota in the 1970s. Hirota firstly discretized the nonlinear partial difference KdV equation [11], the discrete-time Toda equation [12] and other soliton equations. Then, the properties and exact solutions of some discrete equations have been discussed [13–16]. Date et al proposed a method of discrete soliton equation by means of the transformation group theory, which gives a great number of integrable discretizations of soliton equations [17]. A case in point is the quantum field theory, in which discretization provides a strong implement for building models of quantum gravity [18]. Suris developed a universal Hamiltonian method for integrable discretization [19]. After this pioneering work of inte-
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