Soliton resolution for the Hirota equation with weighted Sobolev initial data

2021/01/15 by Jin‐Jie Yang, Yang, Jin-Jie, Shou‐Fu Tian +3
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 Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2101.05942

openalex publication_date 2021/01/15 · openalex created_date 2021/04/13 · openalex updated_date 2026/07/28

Abstract

In this work, the ∂ steepest descent method is employed to investigate the soliton resolution for the Hirota equation with the initial value belong to weighted Sobolev space H^1,1(ℝ)=\f∈ L²(ℝ): f',xf∈ L²(ℝ)\. The long-time asymptotic behavior of the solution q(x,t) is derived in any fixed space-time cone C(x₁,x₂,v₁,v₂)=\(x,t)∈ ℝ×ℝ: x=x₀+vt ~with~ x₀∈[x₁,x₂]\. We show that solution resolution conjecture of the Hirota equation is characterized by the leading order term \mathcal O(t^-1/2) in the continuous spectrum, \mathcal N(\mathcal I) soliton solutions in the discrete spectrum and error order \mathcal O(t^-3/4) from the ∂ equation.

Citations

• Long-time Asymptotics for the NLS equation via dbar methods
• Soliton resolution for the modified KdV equation
• Soliton Resolution for the Short-pluse Equation
• Global well-posedness for the derivative nonlinear Schr "odinger\n equation

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