On the Integration of the Dullin–Gottwald–Holm Equation with a Self-Consistent Source in the Class of Rapidly Decreasing Functions

In this paper, we investigate the Cauchy problem for the Dullin–Gottwald–Holm equation with a self-consistent source in the class of rapidly decreasing functions and present an algorithm for constructing a solution via the IST method. Physically, sources arise in solitary waves with variable speed and lead to a variety of dynamics in physical models. Such systems are commonly used to describe interactions between different solitary waves. We also present an efficient method to obtain the time evolution of scattering data. The advantage of this method lies in its reliability and applicability to other soliton equations with sources. The resulting equalities fully determine the scattering data at any time \( t \), enabling the application of the IST method to solve the Cauchy problem for the Dullin–Gottwald–Holm equation with a self-consistent source. An illustrative example of a one-soliton solution is provided.

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