The soliton resolution conjecture and the asymptotic stability of soliton solutions have long been one of the hot research topics in the mathematical field. The soliton resolution conjecture states that when time t is sufficiently large, the solution can be decomposed into the sum of a finite number of separated soliton solutions, a radiative part, and related errors. From Zabusky and Kruskal in the 1960s, to Martel and Merle, experts in the field of solitons in 2008, and Terence Tao, a famous mathematician and Fields Medal winner, many experts and scholars have been committed to the research on the soliton resolution conjecture and the asymptotic stability of soliton solutions. However, for most dispersive equations, the soliton resolution conjecture remains an open problem. Meanwhile, the asymptotic stability of soliton solutions for nonlinear differential equations with negative-order flow Lax pairs is still a major challenge.

Recently, Professor Tian Shoufu from the School of Mathematics of our university, together with his doctoral students Li Zhiqiang and Yang Jinjie, has made important progress in proving the soliton resolution conjecture and the asymptotic stability of soliton solutions for the Wadati-Konno-Ichikawa (WKI) equation with negative-order flow Lax pairs under the condition of finite density initial values. The relevant results have been published in Advances in Mathematics, an international top mathematics journal and a T1-class journal of the Chinese Mathematical Society. This work mainly develops the Dbar steepest descent method to study the Cauchy problem of integrable systems in weighted Sobolev spaces. Taking the Complex Short Pulse (CSP) equation as an example, the long-time asymptotic behavior of the solution is derived in a fixed space-time cone, and the soliton resolution conjecture of the CSP equation is proved. This conjecture includes soliton terms corresponding to N(I)-solitons in the discrete spectrum and t⁻¹/²-order terms in the continuous spectrum (with a residual error of O(t⁻¹)). For the modified Camassa-Holm (mCH) equation, the team studied the long-term asymptotic behavior of its Cauchy problem under non-zero boundary conditions and found that the solution of the equation can be expressed as soliton solutions in the discrete spectrum, leading terms in the continuous spectrum, and residuals. In addition, the research group explored the Riemann problem of the Fokas-Lenells equation and, through Whitham modulation theory, classified the solutions of the Riemann problem for the first time under clockwise (negative velocity) and counterclockwise (positive velocity) cases.

This paper, together with another paper completed by Professor Tian Shoufu and his two doctoral students Li Zhiqiang and Yang Jinjie, "Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data" (Ann. Henri Poincaré 23 (2022) 2611–2655, 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.