Long-time asymptotics for the $N_{\infty}$-soliton solution to the KdV equation with two types of generalized reflection coefficients
Guoqiang Zhang, Zhenya Yan
TL;DR
The paper develops long-time asymptotics for N∞-soliton solutions of the KdV equation under two generalized reflection coefficients on a spectral interval, using a sequence of Riemann-Hilbert transforms (Y→T→S→E) and a Deift-Zhou steepest descent approach. Central to the analysis are region-specific g- and f-functions, plus local parametrices built from modified Bessel and confluent hypergeometric functions (and Airy in auxiliary steps) to handle endpoints and a singular interior point η0. The main results deliver region-wise leading terms: in certain regions the solution approximates a cnoidal wave u0,c with phase determined by a Whitham-type relation, while in others the solution decays exponentially or as a refined O(1/t) correction, with explicit phase corrections φ0,c(α) and a Whitham map ξ↔α. The work extends prior soliton-gas asymptotics by incorporating zeros and jumps in the generalized reflection coefficients, providing a richer description of soliton gas dynamics and paving the way for multi-singularity generalizations. Overall, the findings deepen the understanding of how spectral weights and singularities shape the macroscopic behavior of integrable soliton gases for the KdV equation.
Abstract
We systematically investigate the long-time asymptotics for the $N_{\infty}$-soliton solution to the KdV equation in the different regions with the aid of the Riemann-Hilbert (RH) problems with two types of generalized reflection coefficients on the interval $\left[η_1, η_2\right]\in \mathbb{R}^+$: $r_0(λ,η_0; β_0, β_1,β_2)=\left(λ-η_1\right)^{β_1}\left(η_2-λ\right)^{β_2}\left|λ-η_0\right|^{β_0}γ\left(λ\right)$, $r_c(λ,η_0; β_1,β_2)=\left(λ-η_1\right)^{β_1}\left(η_2-λ\right)^{β_2}χ_c\left(λ, η_0\right)γ\left(λ\right)$, where the singularity $η_0\in (η_1, η_2)$ and $β_j>-1$ ($j=0, 1, 2$), $γ: \left[η_1, η_2\right] \to\mathbb{R}^+$ is continuous and positive on $\left[η_1, η_2\right]$, with an analytic extension to a neighborhood of this interval, and the step-like function $χ_c$ is defined as $χ_c\left(λ,η_0\right)=1$ for $λ\in\left[η_1, η_0\right)$ and $χ_c\left(λ,η_0\right)=c^2$ for $λ\in\left(η_0, η_2\right]$ with $c>0, \, c\ne1$. A critical step in the analysis of RH problems via the Deift-Zhou steepest descent technique is how to construct local parametrices around the endpoints $η_j$'s and the singularity $η_0$. Specifically, the modified Bessel functions of indexes $β_j$'s are utilized for the endpoints $η_j$'s, and the modified Bessel functions of index $\left(β_0\pm 1\right)\left/\right.2$ and confluent hypergeometric functions are employed around the singularity $η_0$ if the reflection coefficients are $r_0$ and $r_c$, respectively. This comprehensive study extends the understanding of generalized reflection coefficients and provides valuable insights into the asymptotics of soliton gases.