Abstract

In our recent works, dispersive shock waves (DSWs) in the (2+1) and (3+1) dimensional Kadomtsev- Petviashvili type equations (KP, modified KP and Gardner-KP equations) were studied with a step-like initial condition along some chosen special fronts. By using a similarity reduction, the problem of studying DSWs in the multidimensional equations reduced to finding DSW solution of a (1+1) dimensional equations. Whitham modulation equations were derived which describes DSW evolution in the reduced equations by using the method of multiple scales. These equations were written in terms of appropriate Riemann type variables to obtain the Whitham systems of the reduceded (1+1) dimensional equations. DSW solutions which were obtained from the numerical solutions of the Whitham systems and the direct numerical solution of the reduced (1+1) dimensional equations were compared. In this comparison, an agreement was found between these solutions. Also, some physical qualitative results about DSWs in the reduced equations were presented. DSW solutions in the reduced equations provide some information about DSW behavior along the intial fronts in the reduced equations. In this talk, I will compare the similarities and distinctions of DSW solutions of these reduced (1+1) dimensional equations. I will also discuss how DSW solutions of these reduced equations can be used to construct DSW solutions of the original mutidimensional equations.