English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper, we investigate the Painlev\'e asymptotics in a transition zonefor the solutions to the Cauchy problem of the Novikov equation under a nonzerobackground \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 $\overline\partial$-steepestdescent analysis to a Riemann-Hilbert problem associated with the the Cauchyproblem 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 thatthe leading order term of the asymptotic approximation comes from thecontribution of solitons, while the sub-leading term is related to the solutionof the Painlev\'e \uppercase\expandafter{\romannumeral2} equation.n.

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