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

Olexander Minakow
Institution: 
SISSA
Location: 
A-134
Schedule: 
Wednesday, February 28, 2018 - 14:30
Abstract: 

We consider the compressive wave for the modified Korteweg-de Vries equation with background constants c>0 for x teding to -infinity and 0 for x tending to +infinity. We study the asymptotics of solutions in the transition zone $4c^2t-\epsilon t < x< 4c^2t -\beta t^{\sigma}\log t$ for $\epsilon>0$, $\sigma\in(0,1)$, $\beta>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\epsilon t}{\log t}$. Also we show how to obtain Khruslov-Kotlyarov's asymptotics in the domain $4c^2t-\rho\log t<x<4c^2t$ with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann-Hilbert problem. This is a joint work with Marco Bertola.

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