01048nas a2200121   4500008004300000245012100043210006900164520059600233100002200829700001800851700002100869856003600890       2010                        en_Ud  00aNumerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions0 aNumerical Solution of the Small Dispersion Limit of the CamassaH3 aThe 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....1 aAbenda, Simonetta1 aGrava, Tamara1 aKlein, Christian  uhttp://hdl.handle.net/1963/384000973nas a2200109   4500008004300000245006600043210006600109520061200175100002200787700001800809856003600827       2007                        en_Ud  00aReciprocal transformations and flat metrics on Hurwitz spaces0 aReciprocal transformations and flat metrics on Hurwitz spaces3 aWe 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.1 aAbenda, Simonetta1 aGrava, Tamara  uhttp://hdl.handle.net/1963/221001078nas a2200109   4500008004300000245007500043210006900118520070500187100002200892700001800914856003600932       2005                        en_Ud  00aModulation of the Camassa-Holm equation and reciprocal transformations0 aModulation of the CamassaHolm equation and reciprocal transforma3 aWe 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.1 aAbenda, Simonetta1 aGrava, Tamara  uhttp://hdl.handle.net/1963/230500392nas a2200109   4500008004100000245008000041210006900121260001000190653002400200100002200224856003600246       1994                        en  d00aAnalysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems0 aAnalysis of Singularity Structures for QuasiIntegrable Hamiltoni  bSISSA10aHamiltonian systems1 aAbenda, Simonetta  uhttp://hdl.handle.net/1963/5685