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 Zh. Mat. Fiz. Anal. Geom., 2012, Volume 8, Number 1, Pages 38–62 (Mi jmag524)

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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Bibliographic databases:
MSC: 35Q15, 35B40
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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
\Bibitem{KotMin12} \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 \mathnet{http://mi.mathnet.ru/jmag524} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2963009} \zmath{https://zbmath.org/?q=an:06082844} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000301173600003} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. 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
2. 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
3. A. Minakov, “Asymptotics of step-like solutions for the Camassa-Hohn equation”, J. Differ. Equ., 261:11 (2016), 6055–6098
4. 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
5. 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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