he Cauchy problem for the Novikov equation under a nonzero background: Painlev\'e asymptotics in a transition zone

In this paper, we investigate the Painlev\'e asymptotics in a transition zone for the solutions to the Cauchy problem of the Novikov equation under a nonzero background \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 $u_0(x)\rightarrow \kappa>0, \ x\rightarrow \pm \infty$ and $u_0(x)-\kappa$ is assumed in the Schwarz space. This result is established by performing the $\overline\partial$-steepest descent analysis to a Riemann-Hilbert problem associated with the the Cauchy problem in a new spatial scale \begin{equation*} y = x - \int_{x}^{\infty} \left((u-u_{xx}+1)^{2/3}-1\right)ds, \end{equation*} for large times in the transition zone $y/t \approx -1/8$. 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 Painlev\'e \uppercase\expandafter{\romannumeral2} equation.n.