Soliton resolution for the complex short pulse equation with weighted Sobolev initial data

2021/01/28 by Li, Zhi-Qiang, Tian, Shou-Fu, Yang, Jin-Jie
#Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2101.12697

Abstract

We employ the ∂-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space H^1,1(ℝ)=\f∈ L²(ℝ): f',xf∈ L²(ℝ)\. The long time asymptotic behavior of the solution u(x,t) is derived in a fixed space-time cone S(x₁,x₂,v₁,v₂)=\(x,t)∈ℝ²: y=y₀+vt, ~y₀∈[y₁,y₂], ~v∈[v₁,v₂]\. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by N(I)-soliton on discrete spectrum and the t^-(1)/(2) order term on continuous spectrum with residual error up to O(t^-1).

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