Number of Solitons Emerged in the Initial Profile of Shallow Water Using Convolutional Neural Networks

Abstract

The soliton resolution conjecture proposes that the initial value problem can evolve into a dispersion part and a soliton part. However, the problem of determining the number of solitons that form in a given initial profile remains unsolved, except for a few specific cases. In this paper, the authors use the deep learning method to predict the number of solitons in a given initial value of the Korteweg-de Vries (KdV) equation. By leveraging the analytical relationship between Asech²(x) initial values and the number of solitons, the authors train a Convolutional Neural Network (CNN) that can accurately identify the soliton count from spatio-temporal data. The trained neural network is capable of predicting the number of solitons with other given initial values without any additional assistance. Through extensive calculations, the authors demonstrate the effectiveness and high performance of the proposed method.