Nonautonomous solitons and breathers for the coupled variable-coefficient derivative nonlinear Schrödinger equation

Nonautonomous solitons and breathers described by a two-component variable coefficient derivative nonlinear Schrödinger (vc-DNLS) system are investigated through the Darboux transformation (DT). We firstly construct the Lax pair and nth-order DT for the vc-DNLS system which are used to generate the nonautonomous bright-bright, bright-dark solitons and breathers. Various nonautonomous solitons are obtained and analyzed by selecting different variable coefficient. The elastic interaction between two-solitons is confirmed through asymptotic analysis. On the nonzero background, two kinds of nonautonomous breathers are classified according to whether the relative wave vectors are zero or not. Validity of the obtained solitons are verified by a trained Deep Operator Network (DeepONet), which produces predictive solutions that are in good agreement with the exact solutions.