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

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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Bibliographic databases:

Document Type: Article
MSC: 35Q15, 35B40
Received: 07.11.2011
Language: English

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  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. Minakov, “Asymptotics of step-like solutions for the Camassa-Hohn equation”, J. Differ. Equ., 261:11 (2016), 6055–6098  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref
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