Liouville Transformations For The 2-CH And Its Short-wave Limit And The WKI Hierarchy

Because the CH type equation do not possess the standard Painlevé property,the solution of the CH type equation can not be obtained by the general way we were used to solve the integrable system.In order to obtain some solutions of the CH type equation,We can map the CH type system onto classical system by the Liouville transformation or reciprocal transformation,then we can obtain the solution of the CH type equation by the inverse of Liouville transformation which map the solution of the classical system onto the solution of CH type equation.Firstly,2-CH equation and its short-wave limit can be mapped onto the first negative flow of AKNS system by introducing Liouville transformation which constructed by the conservation law of the 2-CH equation and its short-wave limit,and the link of the solutions between them and AKNS system can be found.Then we solve the AKNS equation by introducing Darboux transformation on appropriate initial solution,and obtain some solutions by conducting the inverse of Liouville transformation for the AKNS system.In addition to this,we discuss that WKI system can be transformed into AKNS system by Liouville transformation.