The modified Korteweg-de Vries equation $q_t+ 6q^2q_x+ q_{xxx}= 0$ on the line is considered. The initial data is the pure step function, i.e. $q(x, 0) = c_r$ for $x>0$ and $q(x, 0) = c_l$ for $x < 0$, where $c_l > c_r> 0$ are arbitrary real numbers. The goal of this paper is to study the asymptotic behavior of the solution of initial-value problem as $t\to-infty$, i.e. to study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann-Hilbert problem to the 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 $x < 6c_l^2 t$ and $x > 6c_r^2 t$, the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $6c_l^2 t < x < 6c_r^2 t$, the asymptotics of the solution tends to $\sqrt{\frac{x}{6t}}$. A. Minakov, 2011.

Asymptotics of rarefaction wave solution to the mKdV equation / Minakov, O.. - In: ŽURNAL MATEMATIčESKOJ FIZIKI, ANALIZA, GEOMETRII. - ISSN 1812-9471. - 7:1(2011), pp. 59-86.

### Asymptotics of rarefaction wave solution to the mKdV equation

#### Abstract

The modified Korteweg-de Vries equation $q_t+ 6q^2q_x+ q_{xxx}= 0$ on the line is considered. The initial data is the pure step function, i.e. $q(x, 0) = c_r$ for $x>0$ and $q(x, 0) = c_l$ for $x < 0$, where $c_l > c_r> 0$ are arbitrary real numbers. The goal of this paper is to study the asymptotic behavior of the solution of initial-value problem as $t\to-infty$, i.e. to study the long-time dynamics of the rarefaction wave. Using the steepest descent method and the so-called g-function mechanism we deform the original oscillatory matrix Riemann-Hilbert problem to the 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 $x < 6c_l^2 t$ and $x > 6c_r^2 t$, the main term of asymptotics of the solution is equal to $c_l$ and $c_r$, respectively. In the region $6c_l^2 t < x < 6c_r^2 t$, the asymptotics of the solution tends to $\sqrt{\frac{x}{6t}}$. A. Minakov, 2011.
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Minakov, O.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/62155
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