Studies on a Three-Field Lattice System: N-Fold Darboux Transformation, Conservation Laws and Analytic Solutions

Researches on the nonlinear lattice equations are active, with the applications in nonlinear optics, condensed matter physics, plasma physics, etc. What we study in this paper is a three-field lattice system, which can be reduced to a modified Toda lattice system and a coupled lattice system. Based on a known Lax pair, we present an N-fold Darboux matrix, and then construct an N-fold Darboux transformation for that system, where N is a positive integer. The first three conservation laws of that system are determined via the Lax pair. Utilizing that N-fold Darboux transformation with N = 1 and 2, we obtain the one-fold solutions and two-fold solutions of that system. Those solutions can be used to describe the discrete solitons. Via the one fold solutions, we present a combination of the kink-shaped discrete one soliton and bell-shaped discrete one soliton. Amplitude, shape and velocity of that combination remain unchanged during the propagation.

Automated summary

Research on nonlinear lattice equations is active in various fields such as nonlinear optics, condensed matter physics, and plasma physics. This paper focuses on a three-field lattice system, which can be simplified into a modified Toda lattice system and a coupled lattice system. By utilizing a known Lax pair, an N-fold Darboux matrix is presented, and then an N-fold Darboux transformation is constructed for the system, where N is a positive integer. The first three conservation laws of the system are determined using the Lax pair. One-fold and two-fold solutions of the system are obtained by applying the N-fold Darboux transformation with N = 1 and 2. These solutions can describe discrete solitons. A combination of a kink-shaped discrete one soliton and a bell-shaped discrete one soliton is presented using the one-fold solutions, where the amplitude, shape, and velocity of the combination remain unchanged during propagation.