Modulational instability and dynamics of discrete rational soliton and mixed interaction solutions for a higher-order nonlinear self-dual network equation

Abstract

In this paper, the higher-order nonlinear self-dual network equation is investigated. Firstly, an integrable lattice hierarchy associated with this equation is constructed from a discrete matrix spectral problem without the denominator. Secondly, the condition of modulational instability for this equation is given. Thirdly, the infinitely many conservation laws are constructed on the basis of its new Lax representation. Finally, the discrete generalised $$(m, N-m)$$ -fold Darboux transformation (DT) with non-zero constant as seed solution is used to derive new rational soliton (RS) and mixed interaction solutions. As an application, RS solutions can be obtained by the discrete generalised $$(1, N-1)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=1$$ ), and mixed interaction solutions of the usual sech-type soliton (US) and RS can be derived by the discrete generalised $$(2, N-2)$$ -fold DT (i.e., generalised $$(m, N-m)$$ -fold DT with $$m=2$$ ). We also perform the numerical simulations for such soliton solutions to explore their dynamical behaviours. Results given in this paper may have some prospective applications for understanding the propagation of electrical signals.