Darboux transformation-based LPNN generating novel localized wave solutions

Darboux transformation method is one of the most essential and important methods for solving localized wave solutions of integrable systems. In this work, we introduce the core idea of Darboux transformation of integrable systems into the Lax pairs informed neural networks (LPNNs), which we proposed earlier. By fully utilizing the data-driven solutions, spectral parameter and spectral function obtained from LPNNs, we present the novel Darboux transformation-based LPNN (DT-LPNN). The notable feature of DT-LPNN lies in its ability to solve data-driven localized wave solutions and spectral problems with high precision, and it also can employ Darboux transformation formulas of integrable systems and non-trivial seed solutions to discover novel localized wave solutions that were previously unobserved and unreported. The numerical results indicate that, by utilizing the single-soliton solutions as the non-trivial seed solutions, we obtain novel localized wave solutions for the Kraenkel–Manna–Merle (KMM) system by employing DT-LPNN, in which solution $u$ changes from original bright single-soliton on zero background wave to new dark single-soliton dynamic behavior on a variable non-zero background wave. Moreover, by treating the two-soliton solutions as the non-trivial seed solutions, DT-LPNN generates novel localized wave solutions for the KMM system that exhibit completely different dynamic behaviors from prior two-soliton solutions. DT-LPNN combines the Darboux transformation theory of integrable systems with deep neural networks, offering a new approach for generating novel localized wave solutions using non-trivial seed solutions.