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arXiv · 1711.02362

Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data

Marco Bertola·Alexander Minakov
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Abstract

We consider the compressive wave for the modified Korteweg--de Vries equation with background constants $c>0$ for $x\to-\infty$ and $0$ for $x\to+\infty.$ We study the asymptotics of solutions in the transition zone $4c^2t-\varepsilon t 0,$ $σ\in(0,1),$ $β>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\varepsilon t}{\ln t}.$ Also we show how to obtain Khruslov--Kotlyarov's asymptotics in the domain $4c^2t-ρ\ln t<x<4c^2t$ with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem.

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Marco Bertola, Alexander Minakov. 2017-11-07. Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data. https://arxiv.org/abs/1711.02362

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