Nonlinear self-dual network equations: modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours

Abstract

The nonlinear self-dual network equations that describe the propagations of electrical signals in nonlinear LC self-dual circuits are explored. We firstly analyse the modulation instability of the constant amplitude waves. Secondly, a novel generalized perturbation ( M , N − M )-fold Darboux transform (DT) is proposed for the lattice system by means of the Taylor expansion and a parameter limit procedure. Thirdly, the obtained perturbation (1, N − 1)-fold DT is used to find its new higher-order rational solitons (RSs) in terms of determinants. These higher-order RSs differ from those known results in terms of hyperbolic functions. The abundant wave structures of the first-, second-, third- and fourth-order RSs are exhibited in detail. Their dynamical behaviours and stabilities are numerically simulated. These results may be useful for understanding the wave propagations of electrical signals.

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

“…In particular, if we take C 1 = −C 2 = 1/ε, c = 3/4 and expand the vector function φ in equation (25) as two Taylor series at ε = 0, let ξ = 64n + 675t, and then, we can obtain…”

“…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.…”

“…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. Recently, a discrete generalized ðm, N − mÞ-fold DT method has been proposed by one of the present authors of this paper [24,25]. 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].…”

“…Recently, a discrete generalized ðm, N − mÞ-fold DT method has been proposed by one of the present authors of this paper [24,25]. 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.…”

“…Besides, if 2 < m < N, one can use the generalized ðm, N − mÞ-fold DT to give complex mixed interaction soliton solutions. For more details of Theorem 1, the reader can refer to [23][24][25] and the references therein, so we omit the detailed proof here. In the previous section, we have constructed the discrete generalized ðm, N − mÞ-fold DT of equation (1).…”