Data-driven discoveries of Bäcklund transformations and soliton evolution equations via deep neural network learning schemes

We introduce a deep neural network learning scheme to learn the B\"acklund transforms (BTs) of soliton evolution equations and an enhanced for data-driven equation discovery based on known BTs, respectively. The first takes advantage some solution (or equation) information study BT sine-Gordon equation, complex real Miura between defocusing (focusing) mKdV KdV as well via transforms. second uses explicit/implicit BTs generating higher-order solitons train equations, in which high-order informations are more powerful leaning with higher accurates.