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

2017/11/07 by Marco Bertola, Bertola, Marco, А. А. Минаков +1 · 1 citation
Mathematics · Physics and Astronomy · #35B40 #35Q53 #37K40 #41A60 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Equations and Boundary Problems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1711.02362

openalex publication_date 2017/11/07 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We consider the compressive wave for the modified Korteweg--de Vries equation\nwith background constants c>0 for x\→-\∞ and 0 for x\→+\∞. We\nstudy the asymptotics of solutions in the transition zone 4c²t-\ε\nt<x<4c²t-\β t^\σ\ln t for \ε>0, \σ\∈(0,1),\n\β>0. In this region we have a bulk of nonvanishing oscillations, the\nnumber of which grows as \(\ε t)/(\ln t). Also we show how to\nobtain Khruslov--Kotlyarov's asymptotics in the domain 4c²t-\ρ\ln\nt<x<4c²t with the help of parametrices constructed out of Laguerre\npolynomials in the corresponding Riemann-Hilbert problem.\n

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