Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type*

Abstract

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

“…In the meanwhile, interactions between solitons and other types of waves can be constructed by the symmetry reductions related to nonlocal symmetry. Besides the classical integrable systems, the study of nonlocal symmetry systems has become an important subject in nonlinear science [57,58]. ese aspects of the nonlocal CID system are worthy of study in the future.…”

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

“…[2] Nonlinear partial differential equations (NPDEs) may model various physical phenomena in optics, biology, plasma, and other research fields. [6][7][8] However, for some events such as the motion of one-dimensional particles, the propagation of pulses in biological chains and transmission of electrical signals in nonlinear inductance-capacitance circuits, [1,[8][9][10][11] the discrete nonlinear equations serve as more appropriate descriptions compared with NPDEs. The main motivation for studying the discrete nonlinear equation is that they can keep some characteristics of the original continuous equations unaltered.…”

“…[27][28][29] Generally speaking, the discrete N-fold DT can only obtain solitons, while the discrete generalized (r, N − r)-fold DT can obtain more types of exact solutions, including soliton solutions, rational, semi-rational and their mixed interaction solutions. At present, while the discrete generalized (r, N − r)-fold DT has been primarily utilized for addressing the discrete integrable equations related to 2 × 2 Lax pair, [11,16,[30][31][32] the research on discrete equations pertaining to 4 × 4 Lax pair is still rare. [33,34] Therefore, it is significant to extend this method from 2 × 2 Lax pair to 4 × 4 Lax pair.…”

Darboux transformation and soliton solutions of a nonlocal Hirota equation

“…[11][12][13][14][15] On the other hand, some new nonlocal nonlinear equations have also been proposed by using symmetry reduction method. [16][17][18][19][20] For these new nonlocal equations, the integrability and exact solutions are studied with the method of inverse scattering, Darboux transformation and the Hirota bilinear transformation, etc., and the results show that the nonlocal equation possesses new features which are different from the corresponding local equation. This paper, inspired by the study of nonlocal Schrödinger equation in Ref.…”