2021/05/21 by Yiling Yang, Yang, Yiling, Engui Fan +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Differential Equations and Numerical Methods #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2105.10085
openalex publication_date 2021/05/21 · openalex created_date 2021/06/07 · openalex updated_date 2026/07/28
In this paper, we study the long time asymptotic behavior for the Cauchy problem of the Novikov equation with 3× 3 matrix spectral problem amp;ut-utxx+4 ux=3uuxuxx+u2uxxx, amp;u(x, 0)=u0(x),where u0(x) u0(x)→ κ>0, x→ ± ∞ and u0(x)-κ is assumed in the Schwarz space. It is shown that the solution of the Cauchy problem can be characterized via a Riemann-Hilbert problem in a new scale (y,t) with y=x-∫x∞( (u-uxx+1)2/3 -1) ds. In different space-time solitonic regions of ξ=y/t∈ (-∞,-1/8)∪(1,+∞) and ξ∈(-1/8,1), we apply ∂ steepest descent method to obtain the different long time asymptotic expansions of the solution u(y,t). The corresponding residual error order is O(t-1+ρ) and O(t-3/4) respectively from a ∂-equation. Our result implies that soliton resolution 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 regions.