Darboux-Bäcklund transformation, breather and rogue wave solutions for the discrete Hirota equation

Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

“…Following the integrable discretization work of the two pioneers Ablowitz and Hirota [6,7], there are an abundant discrete soliton equations that are proposed and studied by some well-established methods [8][9][10][11]. Therefore, a series of exact solutions including the discrete soliton solution, breather, positon solution and the rogue wave solutions are investigated [12][13][14][15][16][17][18][19][20].…”

Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

“…Searching for explicit exact solutions, especially soliton solutions, is used for depicting and explaining such nonlinear phenomena described by the discrete nonlinear lattice models. Methods of constructing the soliton solutions of the discrete integrable models have been proposed and developed such as the discrete inverse scattering method [13], the discrete Hirota bilinear formalism method [14,15], and the discrete Darboux transformation (DT) method [16][17][18][19][20][21][22][23][24][25]. Among them, the discrete DT based on corresponding Lax representation is a useful technique to solve the discrete nonlinear models and its main idea is to keep the corresponding Lax pair of these discrete equations unchanged.…”

“…Compared with the common discrete DT which can only give the usual soliton (US) solutions, the discrete generalized ðm, N − mÞ-fold DT [24,25] is a generalization of the common discrete DT and the main advantage of this method is that it can give not only the US solution but also the rational solutions (e.g., rouge wave solutions and rational soliton (RS) solutions) and mixed interaction solutions of US and rational solution [24,25]. For the US solutions of discrete nonlinear models, there have been a lot of literature studies [16][17][18][19][20][21][22][23] but the study of rational solutions and mixed interaction solutions of US and rational solutions is still not sufficient and systematic. As far as we know, the asymptotic behaviors of RS solutions and mixed interaction soliton solutions of US and RS by using asymptotic analysis have not been studied yet.…”