229 research outputs found
Variational derivation of the Camassa-Holm shallow water equation
We describe the physical hypothesis in which an approximate model of water
waves is obtained. For an irrotational unidirectional shallow water flow, we
derive the Camassa-Holm equation by a variational approach in the Lagrangian
formalism.Comment: 10 page
Blowup issues for a class of nonlinear dispersive wave equations
In this paper we consider the nonlinear dispersive wave equation on the real
line,
,
that for appropriate choices of the functions and includes well known
models, such as Dai's equation for the study of vibrations inside elastic rods
or the Camassa--Holm equation modelling water wave propagation in shallow
water. We establish a local-in-space blowup criterion (i.e., a criterion
involving only the properties of the data in a neighbourhood of a single
point) simplifying and extending earlier blowup criteria for this equation. Our
arguments apply both to the finite and infinite energy case, yielding the
finite time blowup of strong solutions with possibly different behavior as
and
The periodic b-equation and Euler equations on the circle
In this note we show that the periodic b-equation can only be realized as an
Euler equation on the Lie group Diff(S^1) of all smooth and orientiation
preserving diffeomorphisms on the cirlce if b=2, i.e. for the Camassa-Holm
equation. In this case the inertia operator generating the metric on Diff(S^1)
is given by A=1-d^2/dx^2. In contrast, the Degasperis-Procesi equation, for
which b=3, is not an Euler equation on Diff(S^1) for any inertia operator. Our
result generalizes a recent result of B. Kolev.Comment: 8 page
Variational derivation of two-component Camassa-Holm shallow water system
By a variational approach in the Lagrangian formalism, we derive the
nonlinear integrable two-component Camassa-Holm system (1). We show that the
two-component Camassa-Holm system (1) with the plus sign arises as an
approximation to the Euler equations of hydrodynamics for propagation of
irrotational shallow water waves over a flat bed. The Lagrangian used in the
variational derivation is not a metric.Comment: to appear in Appl. Ana
Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms
We study an equation lying `mid-way' between the periodic Hunter-Saxton and
Camassa-Holm equations, and which describes evolution of rotators in liquid
crystals with external magnetic field and self-interaction. We prove that it is
an Euler equation on the diffeomorphism group of the circle corresponding to a
natural right-invariant Sobolev metric. We show that the equation is
bihamiltonian and admits both cusped, as well as smooth, traveling-wave
solutions which are natural candidates for solitons. We also prove that it is
locally well-posed and establish results on the lifespan of its solutions.
Throughout the paper we argue that despite similarities to the KdV, CH and HS
equations, the new equation manifests several distinctive features that set it
apart from the other three.Comment: 30 pages, 2 figure
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