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- W4310825101 abstract "In this paper, we study the long time asymptotic behavior to the Cauchy problem of the Degasperis-Procesi (DP) equation with $3times3$ matrix Lax pair begin{align} &u_t-u_{txx}+3kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, nonumber &u(x,0)=u_{0}(x),nonumber end{align} where $kappa$ is a positive parameter. It is shown that the solution of the Cauchy problem can be characterized via a $3times3$ matrix Riemann-Hilbert (RH) problem in a new scale $(y,t)$. We divide the upper half-plane $(y,t)in mathbb{R}times mathbb{R}^+$ into three kinds of space-time regions: I. solitonic regions $xi <-3/8,$ and $ xi>3$, II. solitonless region $ -3/8<xi< 3$ and III. transition regions $xi approx -3/8$ and $ xiapprox 3$.With $overlinepartial$ steepest descent analysis and double limit technique, we obtain a complete long-time asymptotics for the solution $u(x,t)$ in three different space-time regions. The corresponding residual error functions come from singularities and a $overlinepartial$-equation respectively. Our first asymptotic result from soliton region I is characterized with a sum of single solitons with different velocity. This is a verification of soliton resolution conjecture for DP equation. Our second asymptotic result from the solitonless region II is characterized with parabolic cylinder function. Our third asymptotic result from the transition region III can be expressed in terms of the Painlev'{e} II equation." @default.
- W4310825101 created "2022-12-18" @default.
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- W4310825101 date "2022-12-04" @default.
- W4310825101 modified "2023-10-12" @default.
- W4310825101 title "The Degasperis-Procesi equation on the line: Soliton resolution, asymptotic stability of $N$-soliton solutions and Painlev'e asymptotics" @default.
- W4310825101 doi "https://doi.org/10.48550/arxiv.2212.01765" @default.
- W4310825101 hasPublicationYear "2022" @default.
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