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This article is cited in 5 scientific papers (total in 5 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.
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MSC: 35Q15, 35B40 Received: 07.11.2011
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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
Citation in format AMSBIB
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\by V. Kotlyarov, A. Minakov
\paper Step-initial function to the mKdV equation: hyper-elliptic long-time asymptotics of the solution
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2012
\vol 8
\issue 1
\pages 38--62
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2963009}
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Citing articles on Google Scholar:
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This publication is cited in the following articles:
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V. P. Kotlyarov, E. A. Moskovchenko, “Matrix Riemann–Hilbert Problems and Maxwell–Bloch Equations without Spectral Broadening”, Zhurn. matem. fiz., anal., geom., 10:3 (2014), 328–349
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Kotlyarov V., Minakov A., “Modulated Elliptic Wave and Asymptotic Solitons in a Shock Problem To the Modified Korteweg-de Vries Equation”, J. Phys. A-Math. Theor., 48:30 (2015), 305201
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A. Minakov, “Asymptotics of step-like solutions for the Camassa-Hohn equation”, J. Differ. Equ., 261:11 (2016), 6055–6098
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I. Egorova, Z. Gladka, G. Teschl, “On the form of dispersive shock waves of the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 12:1 (2016), 3–16
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Iryna Egorova, Johanna Michor, Gerald Teschl, “Long-time asymptotics for the Toda shock problem: non-overlapping spectra”, Zhurn. matem. fiz., anal., geom., 14:4 (2018), 406–451
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