Nonlocal Symmetries, Conservation Laws and Interaction Solutions of the Generalised Dispersive Modified Benjamin-Bona-Mahony Equation

We consider the generalised dispersive modified Benjamin-Bona-Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painleve expansion method, we derive its non-local symmetry and Backlund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.