Bell polynomials and superposition wave solutions of Hirota–Satsuma coupled KdV equations

Abstract

In this article, the bilinear form, the bilinear Bäcklund transformation, Lax pair, the integrability, infinite conservation laws, nonlinear superposition formula and superposition wave solutions of Hirota–Satsuma coupled KdV (HSCKdV) equations are studied, which can help us get more properties, increase the diversity of solutions and get more new phenomena. The bilinear form, the bilinear Bäcklund transformation and Lax pair of HSCKdV equations are constructed by using the Bell polynomials approach. Based on this, HSCKdV system is integrable in the sense of Lax pair. According to the obtained bilinear Bäcklund transformation, infinite conservation laws and nonlinear superposition formula are derived. And some superposition wave solutions of HSCKdV equations can be attained by the nonlinear superposition formula. With the help of the symbolic calculation system Mathematica, the properties of the superposition wave solutions are analyzed. What is more, in addition to many properties obtained, the relationship between the solutions is found via the derived nonlinear superposition formula, and mixed solutions containing different functions and arbitrary functions are constructed. By using the Bell polynomial method and studying the integrability and other properties of HSCKdV equations, as well as their exact solutions, it can be found that solitons maintain their shape unchanged during motion and are not destroyed. Such waves have very important applications as a communication method. Furthermore, it has been observed that the results attained in this work have not been presented before.