Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time

TL;DR

This paper investigates the long-time behavior of solutions to the Novikov equation, revealing soliton interactions and asymptotic expansions in different space-time regions using Riemann-Hilbert and ar steepest descent methods.

Contribution

It introduces a novel analysis of the Novikov equation's asymptotics via a new scale and ar steepest descent, characterizing soliton resolution and interactions.

Findings

Asymptotic solutions are characterized in different solitonic regions.

Residual error orders are orrelated with the ar equation analysis.

Soliton interactions are described by modulated parameters in the asymptotics.

Abstract

In this paper, we study the long time asymptotic behavior for the Cauchy problem of the Novikov equation with $3\times 3$ matrix spectral problem \begin{align} &u_{t}-u_{txx}+4 u_{x}=3uu_xu_{xx}+u^2u_{xxx}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $u_0(x)$ $u_0(x)\rightarrow \kappa>0, \ x\rightarrow \pm \infty$ and $u_0(x)-\kappa$ 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 $$y=x-\int_{x}^{\infty}\left( (u-u_{xx}+1)^{2/3} -1\right) ds.$$ In different space-time solitonic regions of $\xi=y/t\in (-\infty,-1/8)\cup(1,+\infty) $ and $\xi \in(-1/8,1)$, we apply $\overline\partial$ steepest descent method to obtain the different long time asymptotic expansions of the solution $u(y,t)$. The corresponding residual error order is $\mathcal{O}(t^{-1+\rho})$ and $\mathcal{O}(t^{-3/4})$ respectively from a $\overline\partial$-equation. Our result implies that soliton resolution can be characterized with an $N(\Lambda)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the regions.