2025/05/06 by Yidan Zhang, Zhang, Yidan, Engui Fan +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2505.03144
openalex publication_date 2025/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∂-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 ξ:=y/t <-2√(3αβ), αβ>0 and |ξ|<∞,αβ<0 with saddle points on ℝ, is characterized with solitons and soliton-radiation interaction with residual error O(t-3/4). The second asymptotic result from the region ξ>-2√(3αβ), αβ>0 without saddle point on ℝ, is characterized with modulation-solitons with residual error 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 ξ≈ -2√(3αβ),αβ>0 can be expressed in terms of the solution of the Painlevé \uppercase\expandafter\romannumeral2 equation with error 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.