Soliton resolution, asymptotic stability and Painlev\'e transcendents in the combined Wadati-Konno-Ichikawa and short-pulse equation

In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the combined Wadati-Konno-Ichikawa and short-pulse (WKI-SP) equation. The solution of the Cauchy problem is first expressed in terms of the solution of a RH problem with direct scattering transform based on the Lax pair. Further through a series of deformations to the RH problem by using the $\bar{\partial}$-generalization of Deift-Zhou steepest descent method, we obtain the long-time asymptotic approximations to the solution of the WKI-SP equation under a new scale $(y,t)$ in three kinds of space-time regions. The first asymptotic result from the space-time regions $ \xi:=y/t <-2\sqrt{3\alpha\beta}, \alpha\beta>0$ and $|\xi|<\infty,\alpha\beta<0$ with saddle points on $\mathbb{R}$, is characterized with solitons and soliton-radiation interaction with residual error $\mathcal{O}(t^{-3/4})$. The second asymptotic result from the region $ \xi >-2\sqrt{3\alpha\beta}, \alpha\beta>0$ without saddle point on $\mathbb{R}$, is characterized with modulation-solitons with residual error $\mathcal{O}(t^{-1})$; These two results above are a verification of the soliton resolution conjecture for the WKI-SP equation. The third asymptotic result from a transition region $\xi \approx -2\sqrt{3\alpha\beta},\alpha\beta>0$ can be expressed in terms of the solution of the Painlev\'{e} \uppercase\expandafter{\romannumeral2} equation with error $\mathcal{O}(t^{-1/2})$. This is a new phenomena that the long-time asymptotics for the solution to the Cauchy problem of the WKI equation and SP equation don't possesses.