Darboux transformations and exact solutions of nonlocal Kaup–Newell equations with variable coefficients

Abstract

This paper investigates an integrable nonlocal Kaup–Newell (NKN) equation with variable coefficients. Utilizing Lax pair theory, the construction of the variable coefficient NKN equation is presented for the first time, alongside a systematic analysis employing the Darboux transform technique. This approach explicitly derives the form of the nth-order Darboux transform, which is presented for the first time. The article offers a thorough explanation of the derivation process for the second-order Darboux transform using Cramer’s rule, further extending this to propose a general formula for the nth Darboux transform applicable to multi-parameter scenarios. By applying a zero-seed solution, the exact solution of the variable coefficient NKN equation is obtained. To explore the influence of different coefficient functions on the solutions, specific coefficient functions are selected, and their corresponding graphical representations are analyzed, uncovering a range of solution types, including single soliton solutions, multi-solitons, rogue wave solutions, mixed twisted soliton solutions and breather wave solutions. Through the comprehensive analysis of these solutions, the study underscores the significant enhancement in modeling accuracy when time- and space-dependent coefficients are incorporated into the NKN equations, particularly in the context of simulating the dynamic behavior of nonlinear waves in real-world applications.