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
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.
PDF file (280 kB)
MSC: 35Q15, 35B40
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
\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.
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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
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
A. Minakov, “Asymptotics of step-like solutions for the Camassa-Hohn equation”, J. Differ. Equ., 261:11 (2016), 6055–6098
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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