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

by Alexander Minakov (SISSA Trieste)

Location: E161, Louvain-la-Neuve
Time: Wednesday March 7, 2018 at 15:00

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 \(4c^2 t - \epsilon t < x < 4 c^2 t - \beta t^\sigma \log t\) for \(\epsilon > 0\), \(\sigma \in (0,1)\) and \(\beta > 0\). 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 \(4 c^2 t - \rho \log t\).