45 research outputs found
Reciprocal transformations and local Hamiltonian structures of hydrodynamic type systems
We start from a hyperbolic DN hydrodynamic type system of dimension which
possesses Riemann invariants and we settle the necessary conditions on the
conservation laws in the reciprocal transformation so that, after such a
transformation of the independent variables, one of the metrics associated to
the initial system be flat. We prove the following statement: let in
the case of reciprocal transformations of a single independent variable or
in the case of transformations of both the independent variable; then
the reciprocal metric may be flat only if the conservation laws in the
transformation are linear combinations of the canonical densities of
conservation laws, {\it i.e} the Casimirs, the momentum and the Hamiltonian
densities associated to the Hamiltonian operator for the initial metric. Then,
we restrict ourselves to the case in which the initial metric is either flat or
of constant curvature and we classify the reciprocal transformations of one or
both the independent variables so that the reciprocal metric is flat. Such
characterization has an interesting geometric interpretation: the hypersurfaces
of two diagonalizable DN systems of dimension are Lie equivalent if
and only if the corresponding local hamiltonian structures are related by a
canonical reciprocal transformation.Comment: 23 pages; corrected typos, added counterexample in Remark 3.
Numerical Solution of the Small Dispersion Limit of the Camassa-Holm and Whitham Equations and Multiscale Expansions
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. We present a quantitative
numerical comparison between the CH and the asymptotic solution. The dependence
on the small dispersion parameter is studied in the interior and at
the boundaries of the Whitham zone. In the interior of the zone, the difference
between CH and asymptotic solution is of the order , at the trailing
edge of the order and at the leading edge of the order
. For the latter we present a multiscale expansion which
describes the amplitude of the oscillations in terms of the Hastings-McLeod
solution of the Painlev\'e II equation. We show numerically that this
multiscale solution provides an enhanced asymptotic description near the
leading edge.Comment: 25 pages, 15 figure
Closed geodesics and billiards on quadrics related to elliptic KdV solutions
We consider algebraic geometrical properties of the integrable billiard on a
quadric Q with elastic impacts along another quadric confocal to Q. These
properties are in sharp contrast with those of the ellipsoidal Birkhoff
billiards. Namely, generic complex invariant manifolds are not Abelian
varieties, and the billiard map is no more algebraic. A Poncelet-like theorem
for such system is known. We give explicit sufficient conditions both for
closed geodesics and periodic billiard orbits on Q and discuss their relation
with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip
Cayley-Type Conditions for Billiards within Quadrics in
The notions of reflection from outside, reflection from inside and signature
of a billiard trajectory within a quadric are introduced. Cayley-type
conditions for periodical trajectories for the billiard in the region bounded
by quadrics in and for the billiard ordered game within
ellipsoids in are derived. In a limit, the condition describing periodic
trajectories of billiard systems on a quadric in is obtained.Comment: 10 pages, some corractions are made in Section
A 3-component extension of the Camassa-Holm hierarchy
We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed
with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means
of a bi-Hamiltonian reduction, and its first nontrivial flow provides a
3-component extension of the CH equation.Comment: 15 pages; minor changes; to appear in Letters in Mathematical Physic
On a Camassa-Holm type equation with two dependent variables
We consider a generalization of the Camassa Holm (CH) equation with two
dependent variables, called CH2, introduced by Liu and Zhang. We briefly
provide an alternative derivation of it based on the theory of Hamiltonian
structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the
same algebra underlying the NLS hierarchy. We study the structural properties
of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and
provide its Lax representation. Then we explicitly discuss how to construct
classes of solutions, both of peakon and of algebro-geometrical type. We
finally sketch the construction of a class of singular solutions, defined by
setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte
Polarons as stable solitary wave solutions to the Dirac-Coulomb system
We consider solitary wave solutions to the Dirac--Coulomb system both from
physical and mathematical points of view. Fermions interacting with gravity in
the Newtonian limit are described by the model of Dirac fermions with the
Coulomb attraction. This model also appears in certain condensed matter systems
with emergent Dirac fermions interacting via optical phonons. In this model,
the classical soliton solutions of equations of motion describe the physical
objects that may be called polarons, in analogy to the solutions of the
Choquard equation. We develop analytical methods for the Dirac--Coulomb system,
showing that the no-node gap solitons for sufficiently small values of charge
are linearly (spectrally) stable.Comment: Latex, 26 page
Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians
The quartic H\'enon-Heiles Hamiltonian passes the Painlev\'e test for
only four sets of values of the constants. Only one of these, identical to the
traveling wave reduction of the Manakov system, has been explicitly integrated
(Wojciechowski, 1985), while the three others are not yet integrated in the
generic case . We integrate them by building
a birational transformation to two fourth order first degree equations in the
classification (Cosgrove, 2000) of such polynomial equations which possess the
Painlev\'e property. This transformation involves the stationary reduction of
various partial differential equations (PDEs). The result is the same as for
the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases,
a general solution which is meromorphic and hyperelliptic with genus two. As a
consequence, no additional autonomous term can be added to either the cubic or
the quartic Hamiltonians without destroying the Painlev\'e integrability
(completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200
On the superintegrable Richelot systems
We introduce the Richelot class of superintegrable systems in N-dimensions
whose n<=N equations of motion coincide with the Abel equations on n-1 genus
hyperellipic curve. The corresponding additional integrals of motion are the
second order polynomials of momenta and multiseparability of the Richelot
superintegrable systems is related with classical theory of covers of the
hyperelliptic curves.Comment: 13 pages, a talk given at the Conference "Symmetry Methods in
Physics" devoted to the memory of Professor Y.F.Smirnov, July 6-9, 200