8,009 research outputs found
Orbital stability: analysis meets geometry
We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system
Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups
We consider the orbital stability of relative equilibria of Hamiltonian
dynamical systems on Banach spaces, in the presence of a multi-dimensional
invariance group for the dynamics. We prove a persistence result for such
relative equilibria, present a generalization of the Vakhitov-Kolokolov slope
condition to this higher dimensional setting, and show how it allows to prove
the local coercivity of the Lyapunov function, which in turn implies orbital
stability. The method is applied to study the orbital stability of relative
equilibria of nonlinear Schr{\"o}dinger and Manakov equations. We provide a
comparison of our approach to the one by Grillakis-Shatah-Strauss
Invited review: KPZ. Recent developments via a variational formulation
Recently, a variational approach has been introduced for the paradigmatic
Kardar--Parisi--Zhang (KPZ) equation. Here we review that approach, together
with the functional Taylor expansion that the KPZ nonequilibrium potential
(NEP) admits. Such expansion becomes naturally truncated at third order, giving
rise to a nonlinear stochastic partial differential equation to be regarded as
a gradient-flow counterpart to the KPZ equation. A dynamic renormalization
group analysis at one-loop order of this new mesoscopic model yields the KPZ
scaling relation alpha+z=2, as a consequence of the exact cancelation of the
different contributions to vertex renormalization. This result is quite
remarkable, considering the lower degree of symmetry of this equation, which is
in particular not Galilean invariant. In addition, this scheme is exploited to
inquire about the dynamical behavior of the KPZ equation through a
path-integral approach. Each of these aspects offers novel points of view and
sheds light on particular aspects of the dynamics of the KPZ equation.Comment: 16 pages, 2 figure
Symmetries and Fixed Point Stability of Stochastic Differential Equations Modeling Self-Organized Criticality
A stochastic nonlinear partial differential equation is built for two
different models exhibiting self-organized criticality, the Bak, Tang, and
Wiesenfeld (BTW) sandpile model and the Zhang's model. The dynamic
renormalization group (DRG) enables to compute the critical exponents. However,
the nontrivial stable fixed point of the DRG transformation is unreachable for
the original parameters of the models. We introduce an alternative
regularization of the step function involved in the threshold condition, which
breaks the symmetry of the BTW model. Although the symmetry properties of the
two models are different, it is shown that they both belong to the same
universality class. In this case the DRG procedure leads to a symmetric
behavior for both models, restoring the broken symmetry, and makes accessible
the nontrivial fixed point. This technique could also be applied to other
problems with threshold dynamics.Comment: 19 pages, RevTex, includes 6 PostScript figures, Phys. Rev. E (March
97?
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
Self-Similar Dynamics of a Relativistically Hot Gas
In the presence of self-gravity, we investigate the self-similar dynamics of
a relativistically hot gas with or without shocks in astrophysical processes of
stellar core collapse, formation of compact objects, and supernova remnants
with central voids. The model system is taken to be spherically symmetric and
the conservation of specific entropy along streamlines is adopted for a
relativistic hot gas. In terms of equation of state, this leads to a polytropic
index . The conventional polytropic gas of ,
where is the thermal pressure, is the mass density, is the
polytropic index, and is a global constant, is included in our
theoretical model framework. Two qualitatively different solution classes arise
according to the values of a simple power-law scaling index , each of which
is analyzed separately and systematically. We obtain new asymptotic solutions
that exist only for . Global and asymptotic solutions in various
limits as well as eigensolutions across sonic critical lines are derived
analytically and numerically with or without shocks. By specific entropy
conservation along streamlines, we extend the analysis of Goldreich & Weber for
a distribution of variable specific entropy with time and radius and
discuss consequences in the context of a homologous core collapse prior to
supernovae. As an alternative rebound shock model, we construct an Einstein-de
Sitter explosion with shock connections with various outer flows including a
static outer part of a singular polytropic sphere (SPS). Under the joint action
of thermal pressure and self-gravity, we can also construct self-similar
solutions with central spherical voids with sharp density variations along
their edges.Comment: 21 pages, 15 figures, accepted for publication in MNRA
High-speed shear driven dynamos. Part 2. Numerical analysis
This paper aims to numerically verify the large Reynolds number asymptotic
theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper
Deguchi (2019). To avoid any complexity associated with the chaotic nature of
turbulence and flow geometry, nonlinear steady solutions of the
viscous-resistive magneto-hydrodynamic equations in plane Couette flow have
been utilised. Two classes of nonlinear MHD states, which convert kinematic
energy to magnetic energy effectively, have been determined. The first class of
nonlinear states can be obtained when a small spanwise uniform magnetic field
is applied to the known hydrodynamic solution branch of the plane Couette flow.
The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak
and the resonant layer at which strong vorticity and current sheets are
observed. These flow features, and the induced strong streamwise magnetic
field, are fully consistent with the vortex/Alfv\'en wave interaction theory
proposed in Deguchi (2019). When the spanwise uniform magnetic field is
switched off, the solutions become purely hydrodynamic. However, the second
class of `self-sustained shear driven dynamos' at the zero-external magnetic
field limit can be found by homotopy via the forced states subject to a
spanwise uniform current field. The discovery of the dynamo states has
motivated the corresponding large Reynolds number matched asymptotic analysis
in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory
have been solved numerically. The asymptotic solution provides remarkably good
predictions for the finite Reynolds number dynamo solutions
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