Data-driven discoveries of B\"acklund transforms and soliton evolution equations via deep neural network learning schemes

TL;DR

This paper presents deep neural network schemes for learning Bäcklund transforms and discovering soliton evolution equations, enhancing data-driven analysis of integrable systems like KdV, mKdV, and sine-Gordon equations.

Contribution

It introduces novel deep learning methods to learn Bäcklund transforms and discover soliton equations from data, improving accuracy with higher-order solution information.

Findings

Successfully learned Bäcklund transforms for sine-Gordon and Miura transforms.

Enhanced soliton equation discovery using higher-order solution data.

Achieved higher accuracy in data-driven soliton equation modeling.

Abstract

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