The Modified Camassa–Holm Equation

Przemysław Górka , Enrique G. Reyes

Abstract

The Camassa–Holm (CH) equation describes pseudo-spherical surfaces and therefore its integrability properties can be studied by geometrical means [Reyes, “Geometric integrability of the Camassa–Holm equation.” Letters in Mathematical Physics 59 (2002): 117–31]. Using this fact, we introduce a “Miura transform” and a “modified” Camassa–Holm (mCH) equation, in analogy with the Korteweg–de Vries theory. We obtain an infinite number of local conservation laws for CH from these data and also an infinite number of (non)local symmetries. We then compute conservation laws for mCH and also show that it describes pseudo-spherical surfaces, so that, in particular, it is the integrability condition of an -valued linear problem. Finally, we investigate mCH analytically: we define weak solutions and prove their existence and uniqueness.
Author Przemysław Górka (FMIS / DPDE)
Przemysław Górka,,
- Department of Partial Differential Equations
, Enrique G. Reyes - [Universidad de Santiago de Chile]
Enrique G. Reyes,,
-
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Journal seriesInternational Mathematics Research Notices, ISSN 1073-7928, (A 30 pkt)
Issue year2011
Vol2011
No12
Pages2617-2649
Publication size in sheets1.6
ASJC Classification2600 General Mathematics
DOIDOI:10.1093/imrn/rnq163
URL http://imrn.oxfordjournals.org/content/2011/12/2617
Languageen angielski
Score (nominal)30
Publication indicators WoS Citations = 17; Scopus Citations = 19; Scopus SNIP (Source Normalised Impact per Paper): 2011 = 1.373; WoS Impact Factor: 2011 = 1.014 (2) - 2011=1.087 (5)
Citation count*12 (2014-02-15)
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