TY - RPRT
T1 - Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
Y1 - 2010
A1 - Simonetta Abenda
A1 - Tamara Grava
A1 - Christian Klein
AB - 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....
UR - http://hdl.handle.net/1963/3840
U1 - 487
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Reciprocal transformations and flat metrics on Hurwitz spaces
JF - J. Phys. A 40 (2007) 10769-10790
Y1 - 2007
A1 - Simonetta Abenda
A1 - Tamara Grava
AB - 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.
UR - http://hdl.handle.net/1963/2210
U1 - 2034
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Modulation of the Camassa-Holm equation and reciprocal transformations
JF - Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834
Y1 - 2005
A1 - Simonetta Abenda
A1 - Tamara Grava
AB - 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.
UR - http://hdl.handle.net/1963/2305
U1 - 1711
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - THES
T1 - Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems
Y1 - 1994
A1 - Simonetta Abenda
KW - Hamiltonian systems
PB - SISSA
UR - http://hdl.handle.net/1963/5685
U1 - 5534
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -