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Please use this identifier to cite or link to this item: http://dspace.utalca.cl/handle/1950/9158

Title: The modified Hunter-Saxton equation
Authors: Gorka, P.
Reyes, E.G.
Keywords: Hunter-Saxton equation
Equations of pseudo-spherical type
Nonlocal symmetries
Virasoro algebra
Modified Hunter-Saxton equation
Issue Date: Aug-2012
Publisher: ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS
Citation: JOURNAL OF GEOMETRY AND PHYSICS Volume: 62 Issue: 8 Pages: 1793-1809
Abstract: We introduce a quadratic pseudo-potential for the Hunter-Saxton equation (HS), as an application of the fact that HS describes pseudo-spherical surfaces. We use it to compute conservation laws and to obtain a full Lie algebra of nonlocal symmetries for HS which contains a semidirect sum of the loop algebra over sl(2, R) and the centerless Virasoro algebra. We also explain how to find families of solutions to HS obtained using our symmetries, and we apply them to the construction of a recursion operator. We then reason by analogy with the theory of the Korteweg-de Vries and Camassa-Holm equations and we define a "modified" Hunter-Saxton (mHS) equation connected with HS via a "Miura transform". We observe that this new equation describes pseudo-spherical surfaces (and that therefore it is the integrability condition of an sl(2, R)-valued over-determined linear problem), we present two conservation laws, and we solve an initial value problem with Dirichlet boundary conditions. We also point out that our mHS equation plus its corresponding Miura transform are a formal Backlund transformation for HS. Thus, our result on existence and uniqueness of solutions really is a rigorous analytic statement on Backlund transformations. (C) 2012 Elsevier B.V. All rights reserved.
Description: Gorka, P (Gorka, Przemyslaw)2,3. 2. Univ Talca, Inst Matemat & Fis, Talca, Chile 3. Warsaw Univ Technol, Dept Math & Informat Sci, PL-00661 Warsaw, Poland
URI: http://dspace.utalca.cl/handle/1950/9158
ISSN: 0393-0440
Appears in Collections:Artículos en publicaciones ISI - Universidad de Talca

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