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

  • M. Bertola
  • A. MinakovEmail author


We consider the compressive wave for the modified Korteweg–de Vries equation with background constants \(c>0\) for \(x\rightarrow -\infty \) and 0 for \(x\rightarrow +\infty \). We study the asymptotics of solutions in the transition zone \(4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma }\ln t\) 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 \(\frac{\varepsilon t}{\ln t}.\) Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain \(4c^2t-\rho \ln t<x<4c^2t\) with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann–Hilbert problem.


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Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada
  2. 2.International School for Advanced Studies (SISSA)TriesteItaly
  3. 3.Institut de Recherche en Mathématique et Physique (IRMP)Université catholique de Louvain (UCL)Louvain-la-NeuveBelgium

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