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Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 2012, Volume 8, Number 1, Pages 38–62
(Mi jmag524)
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This article is cited in 9 scientific papers (total in 9 papers)
Step-initial function to the mKdV equation: hyper-elliptic long-time asymptotics of the solution
V. Kotlyarov, A. Minakov Mathematical division, B.I. Verkin Institute for Low Temperature Physics and Engineering,
47 Lenin Avenue, 61103 Kharkiv, Ukraine
Abstract:
The modified Korteveg–de Vries equation on the line is considered. The initial function is a discontinuous and piece-wise constant step function, i.e. $q(x,0)=c_r$ for $x\geq0$ and $q(x,0)=c_l$ for $x<0$, where $c_l$, $c_r$ are real numbers which satisfy $c_l>c_r>0$. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as $t\to\infty$. Using the steepest descent method we deform the original oscillatory matrix Riemann–Hilbert problem to explicitly solvable model forms and show that the solution of the initial-value problem has different asymptotic behavior in different regions of the $xt$ plane. In the regions $x<-6c_l^2t+12c_r^2t$ and $x>4c_l^2t+2c_r^2t$ the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $(-6c_l^2+12c_r^2)t<x<(4c_l^2+2c_r^2)t$ the asymptotics of the solution takes the form of a modulated hyper-elliptic wave generated by an algebraic curve of genus 2.
Key words and phrases:
modified Korteweg–de Vries equation, step-like initial value problem, Riemann–Hilbert problem, steepest descent method, modulated hyper-elliptic wave.
Received: 07.11.2011
Citation:
V. Kotlyarov, A. Minakov, “Step-initial function to the mKdV equation: hyper-elliptic long-time asymptotics of the solution”, Zh. Mat. Fiz. Anal. Geom., 8:1 (2012), 38–62
Linking options:
https://www.mathnet.ru/eng/jmag524 https://www.mathnet.ru/eng/jmag/v8/i1/p38
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