Soliton resolution for the Harry Dym equation with weighted Sobolev initial data

2021/03/18 by Lin Deng, Deng, Lin, Zhenyun Qin +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2103.10053

openalex publication_date 2021/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The soliton resolution for the Harry Dym equation is established for initial conditions in weighted Sobolev space H^1,1(ℝ). Combining the nonlinear steepest descent method and ∂-derivatives condition, we obtain that when (y)/(t)0) the long time asymptotic expansion of the solution q(x,t) in any fixed cone C(y₁, y₂, v₁, v₂)=\(y, t) ∈ R² | y=y₀+v t, y₀ ∈[y₁, y₂], v ∈[v₁, v₂]\ up to an residual error of order O(t^-1). The expansion shows the long time asymptotic behavior can be described as an N(I)-soliton on discrete spectrum whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the cone and the second term coming from soliton-radiation interactionson on continuous spectrum.

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