%0 Report
%D 2010
%T Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
%A Simonetta Abenda
%A Tamara Grava
%A Christian Klein
%X The small dispersion limit of solutions to the Camassa-Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture....
%G en_US
%U http://hdl.handle.net/1963/3840
%1 487
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-02-05T10:20:56Z\\nNo. of bitstreams: 1\\n0909.1020v1.pdf: 613403 bytes, checksum: be892250a6d664faff51d74b323fea67 (MD5)
%0 Journal Article
%J J. Phys. A 40 (2007) 10769-10790
%D 2007
%T Reciprocal transformations and flat metrics on Hurwitz spaces
%A Simonetta Abenda
%A Tamara Grava
%X We consider hydrodynamic systems which possess a local Hamiltonian structure of Dubrovin-Novikov type. To such a system there are also associated an infinite number of nonlocal Hamiltonian structures. We give necessary and sufficient conditions so that, after a nonlinear transformation of the independent variables, the reciprocal system still possesses a local Hamiltonian structure of Dubrovin-Novikov type. We show that, under our hypotheses, bi-hamiltonicity is preserved by the reciprocal transformation. Finally we apply such results to reciprocal systems of genus g Whitham-KdV modulation equations.
%B J. Phys. A 40 (2007) 10769-10790
%G en_US
%U http://hdl.handle.net/1963/2210
%1 2034
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-12T09:26:01Z\\nNo. of bitstreams: 1\\n0704.1779v2.pdf: 286756 bytes, checksum: 3d2d03a6f16be9191b242adb35638601 (MD5)
%R 10.1088/1751-8113/40/35/004
%0 Journal Article
%J Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834
%D 2005
%T Modulation of the Camassa-Holm equation and reciprocal transformations
%A Simonetta Abenda
%A Tamara Grava
%X We derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.
%B Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834
%G en_US
%U http://hdl.handle.net/1963/2305
%1 1711
%2 Mathematics
%3 Mathematical Physics
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-29T10:50:29Z\\nNo. of bitstreams: 1\\n0506042v2.pdf: 305542 bytes, checksum: 045f6c919e0338003f17f6827528ad0d (MD5)
%0 Thesis
%D 1994
%T Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems
%A Simonetta Abenda
%K Hamiltonian systems
%I SISSA
%G en
%U http://hdl.handle.net/1963/5685
%1 5534
%2 Mathematics
%3 Mathematical Physics
%4 -1
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