Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation

Abstract

We consider the classical Boussinesq–Burgers (BB) equation, which describes the propagation of shallow water waves. Based on the truncated painleve expansion method and consistent Riccati expansion method, we successfully obtain its nonlocal symmetry and Backlund transformation. By introducing auxiliary-dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetries. By considering the consistent tanh expansion method, the interaction solution of soliton–cnoidal wave for the classical BB equation is studied by using the Jacobi elliptic function. The multi-solitary wave solutions are also obtained by introducing a linear combination of N exponential functions. Moreover, the conservation laws of the equation are successfully obtained with a detailed derivation.