ABSTRACT
The Darboux transformation method is an effective method to obtain explicit
solutions of soliton equations from a trivial solution. In this paper, by using Darboux
transformation method, we construct explicit solutions of the Boussinesq system
Boussinesq equation is a model equation which described shallow water wave motion,
and it is of great significance to study explicit solutions of the Boussinesq system.
There are three sections in this article.
The first part is to sketch the development of the soliton theory and soliton types,
followed by introduction of the method of solving soliton equations, as well as the main
research contents and innovation points of the article.
In the second part, under the action of the first function transformation
and combined with the existing literatures, we generalize the N-fold Darboux
transformation of the corresponding matrix spectral problem. Selecting different
parameters, we obtain more explicit solutions. When by Mathematica
software, we get a new phenomenon of the Boussinesq system, which describing
elastic-fusion-coupled interaction. Meanwhile, by auxiliary spectral problems, we
derive (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation according two (1+1)-
dimensional soliton equations. And with the help of the constructed Darboux
transformation , we obtain new bell soliton solutions of the KP equation.
In section three, under the action of the second function transformation
the corresponding spectral problem is converted to matrix spectral
problem. In order to avoid the tedious iteration of the one-fold Darboux transformation,
we constructs two new N-fold Darboux transformations of the Boussinesq system. Since
we introduce the vector the elements of the constructed Darboux matrix
need certain constraints. It is great different from obtained forms, and relevant research
is still few. At the same time, we also obtain the explicit solutions of the Boussinesq
system.
Key word: soliton equations, Boussinesq equation, KP equation,