Multiloop soliton solutions and compound WKI–SP hierarchy

Abstract

AbstractIn this paper, a compound equation which is a mix of the Wadati–Konno–Ichikawa (WKI) equation and the short‐pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compound WKI–SP equation into the mKdV–SG (modified Korteweg–de Vries and sine‐Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the ‐loop soliton solution may be decomposed exactly into separate soliton elements by using a Moloney–Hodnett‐type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of and are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI–SP‐type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time‐varying coefficients and the corresponding dispersion relation will have a time‐dependent velocity. The whole hierarchy of equations which include the WKI‐type equations, the SP‐type equations, and the compound generalized WKI–SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as equation so that its Lax pairs can be directly written out with the help of and . A unified hodograph transformation is established to relate the compound WKI–SP hierarchy with the mKdV–SG hierarchy.