Generalized Harry-Dym Type And Short Pulse Equations: Darboux Transformation And Multi-Soliton Solutions

Darboux transformation is an effective method to study the exact solutions of soliton equations.However,it is not easy to construct the Darboux transformation for CH type and WKI type spectral problems directly.In this paper,the exact solutions of the generalized Harry-Dym type and the generalized short pulse equations are discussed by combing Darboux transformation with reciprocal transformation.First,resorting to a reciprocal transformation,the generalized Harry-Dym type equation is related to the Kd V equation.With the help of this reciprocal transformation and Darboux transformation for the Kd V equation,we obtain some parametric multi-soliton solutions of the generalized Harry-Dym type equation,including smooth and loop solitons.Second,we construct infinitely many conservation laws of the generalized short pulse equation using its Lax pairs.Based on this,an appropriate reciprocal transformation is introduced,which linked the generalized short pulse equation with the first negative flow of Sawada-Kotera hierarchy.Applying the Darboux and reciprocal transformations,we arrive at some exact solutions of the generalized short pulse equation.