Lump and breather interaction with different soliton solutions for the generalized doubly dispersive equation by embedded the neural networks.

In this study, we present the neural networks generalized Kudryashov (NNGK) method for the first time to explore exact solutions of the generalized doubly dispersive equation. This is a novel analytical technique that combines neural networks (NNs) models with the generalized Kudryashov approach. The NNs are multilayer computational models composed of activation functions and weights that connect neurons in the input, hidden, and output layers. The generalized Kudryashov solutions are assigned to each neuron in the first hidden layer of the NNGK method. This is how the new trial functions are produced. A key novelty of this study is the construction of the novel activation functions from the Kudryashov approach solutions, which creates a new mathematical connection between differential equations theory and deep learning, which is a major innovation of this approach. There are different types of soliton solutions constructed, such as lump interaction, lump-singular and lump-dark, lump-bright, breather solitons, breather-kink, breather-antikink, and other interaction solitons. Some of these solutions are drawn in the form of 3D, 2D, and planar waves and corresponding contours for the physical interpretations of these solutions. This study offers a new methodological approach for dealing with NLPDEs that is widely applicable in engineering and science domains.