The soliton resolution conjecture has long been one of the hot research topics in the mathematical field. In the 1960s, Zabusky and Kruskal first experimentally observed that any solution to the Korteweg–de Vries equation eventually decomposes into the sum of several soliton solutions. In 2008, the famous mathematician and Fields Medalist Terence Tao proved that solutions to the nonlinear Schrödinger equation split into a linearly dispersive part in the far field and a localized part belonging to a compact attractor in radial and high-dimensional cases. Currently, experts and scholars generally agree that for general dispersive equations, under sufficiently long-time conditions, they exhibit the same "universal behavior". Mathematically, this is the well-known "soliton resolution conjecture" problem. The soliton resolution conjecture states that when time tt is sufficiently large, the solution can be decomposed into the sum of a finite number of separated soliton solutions and a radiative part. However, for most dispersive equations, the soliton resolution conjecture remains an open problem. 

Recently, Professor Tian Shoufu from the School of Mathematics of our university, leading his doctoral students Li Zhiqiang and Yang Jinjie, has made important progress in proving the soliton resolution conjecture for the Wadati-Konno-Ichikawa (WKI) equation. The relevant results were published in the international top mathematical physics journal Annales Henri Poincaré, whose origins can be traced back to Annales de l'Institut Henri Poincaré, physique théorique (1930) and Helvetica Physical Acta (1928). This work mainly develops the Dbar steepest descent method to study the Cauchy problem of the WKI equation with initial conditions belonging to weighted Sobolev spaces. The paper first presents the long-time asymptotic behavior of solutions to the WKI equation on a spacetime cone, and then based on the long-time asymptotic results, provides a rigorous proof of the soliton resolution conjecture for the Cauchy problem of the WKI equation. Furthermore, the error of the long-time asymptotic results obtained in the paper reaches O(t^{-3/4})O(t −3/4 ). 

This paper, together with another paper completed by Professor Tian Shoufu and his two doctoral students Li Zhiqiang and Yang Jinjie, "On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions" (Advances in Mathematics, 409 (2022) 108639, with a total of 110 pages for both papers), comprehensively analyzes the long-time asymptotic behavior of solutions, the soliton resolution conjecture, and the asymptotic stability of solutions for the WKI equation under zero boundary conditions and finite density initial conditions.