Alexander Minakov (SISSA Trieste) : Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data

E161 (MdeHemptinne)



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 $4c^2t-\varepsilon t < x< 4c^2t -\beta t^{\sigma}\log t$ for $\varepsilon>0$, $\sigma\in(0,1)$, $\beta>0.$ In this region we have a bulk of nonvanishing oscillations, the number of which grows as $\frac{\varepsilon t}{\log t}$. Also we show how to obtain Khruslov-Kotlyarov's asymptotics in the domain $4c^2t-\rho\log t
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