Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data
(2019) Analysis and Mathematical Physics — Vol. 9, n° 4, p. 1761-1818 (2019)
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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 \(4c^2t-\varepsilon t < x < 4c^2t-\beta t^\sigma\) 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 \({\varepsilon t \over \mbox{ln} t}\). 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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Minakov, O., & Bertola, M. (2019). Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data. Analysis and Mathematical Physics, 9(4), 1761-1818. https://doi.org/10.1007/s13324-018-0273-1 (Original work published 2019)