he Cauchy problem for the Novikov equation under a nonzero background: Painlevé asymptotics in a transition zone

2023/10/30 by Zhaoyu Wang, Xuan Zhou, Wang, Zhaoyu +3
Mathematics · Physics and Astronomy · #35C20 #35Q15 #35Q51 #37K15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2310.19278

openalex publication_date 2023/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the Painlevé asymptotics in a transition zone for the solutions to the Cauchy problem of the Novikov equation under a nonzero background amp;u_t-u_txx+4 u_x=3uu_xu_xx+u²u_xxx, amp;u(x, 0)=u₀(x),where u₀(x)→ κ>0, x→ ± ∞ and u₀(x)-κ is assumed in the Schwarz space. This result is established by performing the ∂-steepest descent analysis to a Riemann-Hilbert problem associated with the the Cauchy problem in a new spatial scale y = x - ∫_x^∞ ((u-u_xx+1)^2/3-1)ds, for large times in the transition zone y/t ≈ -1/8 . It is shown that the leading order term of the asymptotic approximation comes from the contribution of solitons, while the sub-leading term is related to the solution of the Painlevé \uppercase\expandafter\romannumeral2 equation.n.

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