Recently, Professor Tian Shoufu from the School of Mathematics of CUMT, together with young teachers Yang Jinjie and Li Zhiqiang, published a long academic paper (97 pages in total) titled "The modified Camassa-Holm equation with nonzero background: Soliton resolution conjecture and asymptotic stability of N-soliton solutions" in the international top mathematics journal Advances in Mathematics. In this paper, the authors developed inverse scattering theory combined with the Dbar problem to solve the long-time asymptotic behavior of solutions to the modified Camassa-Holm (CH) equation under finite density initial values. They proved the famous soliton resolution conjecture and solved the challenging problem of asymptotic stability of N-soliton solutions, providing a theoretical framework for studying finite density initial value problems of negative-order integrable systems.
Introduction to Advances in Mathematics
Founded in 1961, Advances in Mathematics is a T1 journal in mathematics recognized by the Chinese Mathematical Society and a Top Journal in JCR/SCI Q1 (as classified by the Chinese Academy of Sciences). The journal is dedicated to publishing groundbreaking and important results across all fields of mathematics and is widely regarded as a top-tier journal in the mathematics community. The successful publication of this paper marks a new and significant breakthrough of our school in the field of integrable systems.
Background of the Soliton Resolution Conjecture
The soliton resolution conjecture originated from the experimental observations of Zabusky and Kruskal on the KdV equation in the 1960s. This phenomenon profoundly reveals the dynamic behavior of nonlinear wave equations and has since continuously attracted extensive attention from the mathematics and physics communities. Many renowned scholars, including Terence Tao (a Fields Medalist), have engaged in related research. The conjecture is generally described as follows: Under well-posedness and sufficiently large time conditions, the solution of a nonlinear dispersive equation can decompose into a finite number of solitons and a radiative component. However, for most dispersive equations, this conjecture remains an open problem.
Core Contributions of the Research
This study successfully overcame the soliton resolution conjecture for the modified CH equation under non-zero boundary conditions by innovatively developing the nonlinear steepest descent method and the Dbar problem.
1. Mapping Relationship Between Initial Values and Reflection Coefficients
The paper first established a strict mapping relationship between initial value conditions and reflection coefficients, and systematically demonstrated the well-posed properties of this mapping in the corresponding function space.
2. Asymptotic Mechanism and Structural Framework
Based on the above, the paper characterized the asymptotic mechanism of the initial value problem for the equation and established four asymptotic regions with different stationary phase points, providing a clear structural framework for long-term behavior analysis.
3. Proof of the Soliton Resolution Conjecture
For the first time, the paper rigorously proved that under initial value conditions in a given weighted Sobolev space, as time tends to infinity, the solution of the modified CH equation can decompose into:
An N-soliton solution;
A dominant radiative term with a decay order of t⁻¹/²;
An exponentially decaying error term of O(t⁻³/⁴).
This result not only fully confirms the soliton resolution conjecture in this case but also reveals the decay law of the radiative component and error control in a quantitative manner—a key feature and innovation of this work.
Further Achievement: Asymptotic Stability of N-Soliton Solutions
In addition, the research further established the asymptotic stability theory corresponding to N-soliton solutions. This achievement not only completely solves the theoretical problem of the modified CH equation in this classic context but also lays a solid analytical foundation for the dynamic system theory of related integrable systems and nonlinear wave equations.
