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

Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for... We consider the compressive wave for the modified Korteweg–de Vries equation with background constants $$c>0$$ c > 0 for $$x\rightarrow -\infty $$ x → - ∞ and 0 for $$x\rightarrow +\infty $$ x → + ∞ . We study the asymptotics of solutions in the transition zone $$4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma }\ln t$$ 4 c 2 t - ε t < x < 4 c 2 t - β t σ ln t for $$\varepsilon >0,$$ ε > 0 , $$\sigma \in (0,1),$$ σ ∈ ( 0 , 1 ) , $$\beta >0.$$ β > 0 . In this region we have a bulk of nonvanishing oscillations, the number of which grows as $$\frac{\varepsilon t}{\ln t}.$$ ε t ln t . Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain $$4c^2t-\rho \ln t<x<4c^2t$$ 4 c 2 t - ρ ln t < x < 4 c 2 t with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann–Hilbert problem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

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

Analysis and Mathematical Physics , Volume 9 (4) – Dec 12, 2018

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-018-0273-1
Publisher site
See Article on Publisher Site

Abstract

We consider the compressive wave for the modified Korteweg–de Vries equation with background constants $$c>0$$ c > 0 for $$x\rightarrow -\infty $$ x → - ∞ and 0 for $$x\rightarrow +\infty $$ x → + ∞ . We study the asymptotics of solutions in the transition zone $$4c^2t-\varepsilon t<x<4c^2t-\beta t^{\sigma }\ln t$$ 4 c 2 t - ε t < x < 4 c 2 t - β t σ ln t for $$\varepsilon >0,$$ ε > 0 , $$\sigma \in (0,1),$$ σ ∈ ( 0 , 1 ) , $$\beta >0.$$ β > 0 . In this region we have a bulk of nonvanishing oscillations, the number of which grows as $$\frac{\varepsilon t}{\ln t}.$$ ε t ln t . Also we show how to obtain Khruslov–Kotlyarov’s asymptotics in the domain $$4c^2t-\rho \ln t<x<4c^2t$$ 4 c 2 t - ρ ln t < x < 4 c 2 t with the help of parametrices constructed out of Laguerre polynomials in the corresponding Riemann–Hilbert problem.

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Dec 12, 2018

References