Soliton solutions to integrable equations.

摘要

In recent years, integrable systems and soliton theory play an important role in the study of nonlinear water wave equations. In this thesis, we will focus on the procedure of how to get soliton solutions for integrable equations. The fundamental idea is to use the traveling wave setting to convert a partial differential equation to an ordinary differential equation and to solve ordinary differential equations yields soliton solutions for the integrable equations under certain boundary conditions at both negative and positive infinities. In our work, we will consider five integrable equations and present their solitons solutions, one of which will be solved using the so-called bilinear approach. All the solutions will be given in either an explicit or an implicit form, and we will show how their graphs look like. In our future work, we will work on multi-solitons and Lax pair scheme.