An algebraic method with computerized symbolic computation and new families of travelling wave solutions for the Hirota-Satsuma coupled KdV equation

A new algebraic method for constructing multiple travelling wave solutions of nonlinear equations is applied to the Hirota-Satsuma coupled KdV equation. As a result, many new families of travelling wave solutions including soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions are found. It is shown that the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The proposed method, which can be implemented in a computer with help of symbolic computation software like Mathematica or Maple, not only picks up new and more general solutions than the typical hyperbolic-function method, but also provides a guideline to classify the various types of the solutions according to some parameters. A large number of nonlinear equations may be studied and solved in this simple and systematic way.