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.