2,192 research outputs found
Multi-Lagrangians for Integrable Systems
We propose a general scheme to construct multiple Lagrangians for completely
integrable non-linear evolution equations that admit multi- Hamiltonian
structure. The recursion operator plays a fundamental role in this
construction. We use a conserved quantity higher/lower than the Hamiltonian in
the potential part of the new Lagrangian and determine the corresponding
kinetic terms by generating the appropriate momentum map. This leads to some
remarkable new developments. We show that nonlinear evolutionary systems that
admit -fold first order local Hamiltonian structure can be cast into
variational form with Lagrangians which will be local functionals of
Clebsch potentials. This number increases to when the Miura
transformation is invertible. Furthermore we construct a new Lagrangian for
polytropic gas dynamics in dimensions which is a {\it local} functional
of the physical field variables, namely density and velocity, thus dispensing
with the necessity of introducing Clebsch potentials entirely. This is a
consequence of bi-Hamiltonian structure with a compatible pair of first and
third order Hamiltonian operators derived from Sheftel's recursion operator.Comment: typos corrected and a reference adde
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB)
model of plasma physics. This model consists of the pressureless gas dynamics
equations coupled with the Poisson equation and where the Boltzmann relation
relates the potential to the electron density. If the quasi-neutral assumption
is made, the Poisson equation is replaced by the constraint of zero local
charge and the model reduces to the Isothermal Compressible Euler (ICE) model.
We compare a numerical strategy based on the EPB model to a strategy using a
reformulation (called REPB formulation). The REPB scheme captures the
quasi-neutral limit more accurately
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
On a Lagrangian reduction and a deformation of completely integrable systems
We develop a theory of Lagrangian reduction on loop groups for completely
integrable systems after having exchanged the role of the space and time
variables in the multi-time interpretation of integrable hierarchies. We then
insert the Sobolev norm in the Lagrangian and derive a deformation of the
corresponding hierarchies. The integrability of the deformed equations is
altered and a notion of weak integrability is introduced. We implement this
scheme in the AKNS and SO(3) hierarchies and obtain known and new equations.
Among them we found two important equations, the Camassa-Holm equation, viewed
as a deformation of the KdV equation, and a deformation of the NLS equation
A brief summary of nonlinear echoes and Landau damping
In this expository note we review some recent results on Landau damping in
the nonlinear Vlasov equations, focusing specifically on the recent
construction of nonlinear echo solutions by the author [arXiv:1605.06841] and
the associated background. These solutions show that a straightforward
extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces
on is impossible and hence emphasize the
subtle dependence on regularity of phase mixing problems. This expository note
is specifically aimed at mathematicians who study the analysis of PDEs, but not
necessarily those who work specifically on kinetic theory. However, for the
sake of brevity, this review is certainly not comprehensive.Comment: Expository note for the Proceedings of the Journees EDP 2017, based
on a talk given at Journees EDP 2017 in Roscoff, France. Aimed at
mathematicians who study the analysis of PDEs, but not necessarily those who
work specifically on kinetic theory. 16 page
The local Gromov-Witten theory of CP^1 and integrable hierarchies
In this paper we begin the study of the relationship between the local
Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line
and the theory of integrable hierarchies. We first of all construct explicitly,
in a large number of cases, the Hamiltonian dispersionless hierarchies that
govern the full descendent genus zero theory. Our main tool is the application
of Dubrovin's formalism, based on associativity equations, to the known results
on the genus zero theory from local mirror symmetry and localization. The
hierarchies we find are apparently new, with the exception of the resolved
conifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For this
example the relevant dispersionless system turns out to be related to the
long-wave limit of the Ablowitz-Ladik lattice. This identification provides us
with a complete procedure to reconstruct the dispersive hierarchy which should
conjecturally be related to the higher genus theory of the resolved conifold.
We give a complete proof of this conjecture for genus g<=1; our methods are
based on establishing, analogously to the case of KdV, a "quasi-triviality"
property for the Ablowitz-Ladik hierarchy at the leading order of the
dispersive expansion. We furthermore provide compelling evidence in favour of
the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing
it successfully in the primary sector for g=2.Comment: 30 pages; v2: an issue involving constant maps contributions is
pointed out in Sec. 3.3-3.4 and is now taken into account in the proofs of
Thm 1.3-1.4, whose statements are unchanged. Several typos, formulae,
notational inconsistencies have been fixed. v3: typos fixed, minor textual
changes, version to appear on Comm. Math. Phy
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