Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Nonlinear surface waves on the plasma-vacuum interface


Author: Paolo Secchi
Journal: Quart. Appl. Math. 73 (2015), 711-737
MSC (2010): Primary 76W05; Secondary 35Q35, 35L50, 76E17, 76E25, 35R35, 76B03
DOI: https://doi.org/10.1090/qam/1405
Published electronically: September 15, 2015
MathSciNet review: 3432280
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Abstract: In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagates along the plasma-vacuum interface when it is linearly weakly stable.

Following the approach of Ali and Hunter (2003), we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.


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Additional Information

Paolo Secchi
Affiliation: DICATAM, Mathematical Division, University of Brescia, Via Branze 43, 25123 Brescia, Italy
Email: paolo.secchi@unibs.it

DOI: https://doi.org/10.1090/qam/1405
Received by editor(s): March 25, 2014
Published electronically: September 15, 2015
Additional Notes: The author is supported by the national research project PRIN 2012 “Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and applicative aspects”.
Article copyright: © Copyright 2015 Brown University


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