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

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.