Solitary wave interactions modelled using conserved and empirically-conserved quantities discovered with machine learning

Conservation laws play an important role in analysing physical systems and solving partial differential equations (PDEs). In this paper, we utilise the associated conserved quantities to construct a simple and robust way to approximate the interaction between surface solitary waves – a common nonlinear phenomenon in many research fields and engineering applications. One major constraint of such an approach is how many conserved quantities are available from the governing equation. In this study, we start with the Korteweg-De Vries (KdV) equation, which has an infinite number of conserved quantities. We show that these conserved quantities can be used to predict soliton interaction accurately. The Regularised Long Wave (RLW) equation – an equivalent reduction form as KdV, however, has insufficient conserved quantities to apply the approach. To address this problem, we use a new symbolic-based machine-learning approach to search for any higher-order empiricallyconserved quantities, which allows the proposed approach to be applied directly to RLW equations and leverages the constraints on the number of conserved quantities available. This opens a new opportunity for more complex systems governed by a variety range of PDEs to be modelled with such a conserved quantity-based approach.