2021/01/07 by Yiling Yang, Yang, Yiling, Engui Fan +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2101.02489
openalex publication_date 2021/01/07 · openalex created_date 2021/05/10 · openalex updated_date 2026/07/28
In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region amp;mt+(m(u2-ux2))x+κux=0, m=u-ux x, amp;u(x, 0)=u0(x),where κ is a positive constant. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a Riemann-Hilbert (RH) problem. Further using the ∂ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution u(x,t) in different space-time solitonic region of x/t. These asymptotic approximations can be characterized with an N(Λ)-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region with diverse residual error order from ∂ equation: O(|t|-1+2ρ) for ξ=(y)/(t)∈(-∞,-0.25)∪(2,+∞) and O(|t|-3/4) for ξ=(y)/(t)∈(-0.25,2). Our results also confirm the soliton resolution conjecture and asymptotically stability of N-soliton solutions for the mCH equation.