A nonisospectral Ablowitz–Ladik equation: Double Casoratian solutions and bilinearization reduction scheme

Abstract

In this paper, we study double Casoratian solutions for a nonisospectral Ablowitz–Ladik equation. Through the bilinearization reduction scheme, we present double Casoratian solutions for a real nonisospectral semi-discrete modified Korteweg–de Vries equation and a complex nonisospectral semi-discrete modified Korteweg–de Vries equation. Soliton-like solutions and Jordan-block solutions for these two equations are constructed by solving the constraint matrix equations. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis, where solutions of the complex nonisospectral semi-discrete modified Korteweg–de Vries equation exhibit space-time localized wave behavior.