105 research outputs found
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
Real-valued algebro-geometric solutions of the two-component Camassa-Holm hierarchy
We provide a construction of the two-component Camassa-Holm (CH-2) hierarchy
employing a new zero-curvature formalism and identify and describe in detail
the isospectral set associated to all real-valued, smooth, and bounded
algebro-geometric solutions of the th equation of the stationary CH-2
hierarchy as the real -dimensional torus . We employ
Dubrovin-type equations for auxiliary divisors and certain aspects of direct
and inverse spectral theory for self-adjoint singular Hamiltonian systems. In
particular, we employ Weyl-Titchmarsh theory for singular (canonical)
Hamiltonian systems.
While we focus primarily on the case of stationary algebro-geometric CH-2
solutions, we note that the time-dependent case subordinates to the stationary
one with respect to isospectral torus questions.Comment: 35 pages. arXiv admin note: substantial text overlap with
arXiv:nlin/020802
Classification of integrable hydrodynamic chains and generating functions of conservation laws
New approach to classification of integrable hydrodynamic chains is
established. Generating functions of conservation laws are classified by the
method of hydrodynamic reductions. N parametric family of explicit hydrodynamic
reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new
hydrodynamic chains are found
Cnoidal Waves on Fermi-Pasta-Ulam Lattices
We study a chain of infinitely many particles coupled by nonlinear springs,
obeying the equations of motion [\ddot{q}_n = V'(q_{n+1}-q_n) -
V'(q_n-q_{n-1})] with generic nearest-neighbour potential . We show that
this chain carries exact spatially periodic travelling waves whose profile is
asymptotic, in a small-amlitude long-wave regime, to the KdV cnoidal waves. The
discrete waves have three interesting features: (1) being exact travelling
waves they keep their shape for infinite time, rather than just up to a
timescale of order wavelength suggested by formal asymptotic analysis,
(2) unlike solitary waves they carry a nonzero amount of energy per particle,
(3) analogous behaviour of their KdV continuum counterparts suggests long-time
stability properties under nonlinear interaction with each other. Connections
with the Fermi-Pasta-Ulam recurrence phenomena are indicated. Proofs involve an
adaptation of the renormalization approach of Friesecke and Pego (1999) to a
periodic setting and the spectral theory of the periodic Schr\"odinger operator
with KdV cnoidal wave potential.Comment: 25 pages, 3 figure
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