Multiloop soliton solutions and compound WKI–SP hierarchy

In 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 $N$-loop soliton solution may be decomposed exactly into $N$ separate soliton elements by using a Moloney–Hodnett-type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of $N=2$ and $N=3$ 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 $\mathrm{WKI}\text{--}{\mathrm{SP}}^{(n,m)}$ equation so that its Lax pairs can be directly written out with the help of $n$ and $m$. A unified hodograph transformation is established to relate the compound WKI–SP hierarchy with the mKdV–SG hierarchy.