Abstract
The existences of new important supersolitary and periodic solutions are obtained using the expansion method of Jacobian elliptic function (EMJEF) for the MKdV equation. This equation can be describe the critical case for vanishing the nonlinearity of the well-known KdV equation. This robust method is appropriate for production of new super waves which can confirm the new super pulse observations. It is established that some new solitary waves as periodic, solitons, shock, cnoidal, soliton-shock like, supersoliton and localized soliton waves are obtained. Also, these new forms which may be applicable in fluids and physics applications dependence on the system and Jacobian parameters.