The generalized N-coupled nonautonomous Gross–Pitaevskii equations: Darboux transformation and three new kinds of nonautonomous localized waves

The generalized N-coupled nonautonomous Gross–Pitaevskii (N-NGP) equations, which include damping term, linear and parabolic density profiles, are researched by Darboux transformation (DT). Firstly, the Lax pair of the nonautonomous system is given, which just admits the nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem. Based on the received Lax pair, we present a method for constructing DT of the nonisospectral AKNS systems and successfully apply it to the N-NGP system we investigated. Taking the 2-coupled Gross–Pitaevskii equation for example, and then we construct three kinds of nonautonomous localized waves through the resulted DT, i.e., the general soliton molecules, multi-breathers and higher-order rogue waves. Furthermore, the general soliton molecules, including m-soliton molecules and (M−m)-soliton (2≤m≤M), are skillfully derived from the zero background. Beginning from the nonzero background, multi-breathers on the curved background are generated. Combining both the limiting technique and the received DT, higher-order rogue waves are also excited on the curved background. And then, the underlying dynamics of these three received nonlinear waves are detailedly discussed. In recent published paper (Yang et al., 2023), the nonisospectral spectral parameter of the N-NGP equations is mistaken for owing the constant spectral parameter, and thus this initial error causes the wrong DT and further led to those incorrect localized wave solutions. Finally, we give a rigorous proof to verify the DT in the above-mentioned reference is incorrect.