Partial Differential Equations in Applied Mathematics (Sep 2024)
Integration of the Kaup–Boussinesq system via inverse scattering method
Abstract
In this paper, we investigate the Cauchy problem for the KB system in the class of rapidly decreasing functions and give an algorithm of the constructing a solution to the Cauchy problem of the KB system in the class of rapidly decreasing functions via IST method when the scattering coefficient a(k) of the problem (6) has multiple zeros. In this case, the class of initial functions will be expanded by removing the assumption about simplicity of the scattering coefficient a(k) of the problem (6). We also present an efficient method to obtain the time evolution of scattering data. The advantage of this method is its reliability and the possibility of using other soliton equations to obtain the time evolution of scattering data. The resulting equalities completely determine the scattering data at any t, which allows applying the IST method to solve the Cauchy problem for the KB system. The results obtained will be useful in carrying out further analysis in the context of shallow water waves that arises in the context of oceanography.
