The modified Hunter-Saxton equation

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• 作者：Przemys艂aw Gó ; rka^a ; ^b ; Enrique G. Reyes^c ; ^{ereyes@fermat.usach.cl} ; ^{symplectic2001@yahoo.ca}
• 关键词：Hunter&ndash ; Saxton equation ; Equations of pseudo-spherical type ; Nonlocal symmetries ; Virasoro algebra ; Modified Hunter&ndash ; Saxton equation
• 刊名：Journal of Geometry and Physics
• 出版年：2012
• 期刊代码：29_03930440
• 类别：ph
• 出版时间：August, 2012
• 卷：62
• 期：8
• 页码：1793-1809
• 文件大小：307 K

摘要

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 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 鈥渕odified鈥?Hunter-Saxton (mHS) equation connected with HS via a 鈥淢iura transform鈥? We observe that this new equation describes pseudo-spherical surfaces (and that therefore it is the integrability condition of an -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 B盲cklund transformation for HS. Thus, our result on existence and uniqueness of solutions really is a rigorous analytic statement on B盲cklund transformations.