474,111 research outputs found
Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism
Models incorporating delay have been frequently used to understand climate
variability phenomena, but often the delay is introduced through an ad-hoc
physical reasoning, such as the propagation time of waves. In this paper, the
Mori-Zwanzig formalism is introduced as a way to systematically derive delay
models from systems of partial differential equations and hence provides a
better justification for using these delay-type models. The Mori-Zwanzig
technique gives a formal rewriting of the system using a projection onto a set
of resolved variables, where the rewritten system contains a memory term. The
computation of this memory term requires solving the orthogonal dynamics
equation, which represents the unresolved dynamics. For nonlinear systems, it
is often not possible to obtain an analytical solution to the orthogonal
dynamics and an approximate solution needs to be found. Here, we demonstrate
the Mori-Zwanzig technique for a two-strip model of the El Nino Southern
Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The
resulting nonlinear delay model contains an additional term compared to
previously proposed ad-hoc conceptual models. This new term leads to a larger
ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure
On the validity of mean-field amplitude equations for counterpropagating wavetrains
We rigorously establish the validity of the equations describing the
evolution of one-dimensional long wavelength modulations of counterpropagating
wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We
consider both periodic amplitude functions and localized wavepackets. For the
localized case, the wavetrains are completely decoupled at leading order, while
in the periodic case the amplitude equations take the form of mean-field
(nonlocal) Schr\"odinger equations rather than locally coupled partial
differential equations. The origin of this weakened coupling is traced to a
hidden translation symmetry in the linear problem, which is related to the
existence of a characteristic frame traveling at the group velocity of each
wavetrain. It is proved that solutions to the amplitude equations dominate the
dynamics of the governing equations on asymptotically long time scales. While
the details of the discussion are restricted to the class of model equations
having a leading cubic nonlinearity, the results strongly indicate that
mean-field evolution equations are generic for bimodal disturbances in
dispersive systems with \O(1) group velocity.Comment: 16 pages, uuencoded, tar-compressed Postscript fil
Wave turbulence description of interacting particles: Klein-Gordon model with a Mexican-hat potential
In field theory, particles are waves or excitations that propagate on the
fundamental state. In experiments or cosmological models one typically wants to
compute the out-of-equilibrium evolution of a given initial distribution of
such waves. Wave Turbulence deals with out-of-equilibrium ensembles of weakly
nonlinear waves, and is therefore well-suited to address this problem. As an
example, we consider the complex Klein-Gordon equation with a Mexican-hat
potential. This simple equation displays two kinds of excitations around the
fundamental state: massive particles and massless Goldstone bosons. The former
are waves with a nonzero frequency for vanishing wavenumber, whereas the latter
obey an acoustic dispersion relation. Using wave turbulence theory, we derive
wave kinetic equations that govern the coupled evolution of the spectra of
massive and massless waves. We first consider the thermodynamic solutions to
these equations and study the wave condensation transition, which is the
classical equivalent of Bose-Einstein condensation. We then focus on nonlocal
interactions in wavenumber space: we study the decay of an ensemble massive
particles into massless ones. Under rather general conditions, these massless
particles accumulate at low wavenumber. We study the dynamics of waves
coexisting with such a strong condensate, and we compute rigorously a nonlocal
Kolmogorov-Zakharov solution, where particles are transferred non-locally to
the condensate, while energy cascades towards large wave numbers through local
interactions. This nonlocal cascading state constitute the intermediate
asymptotics between the initial distribution of waves and the thermodynamic
state reached in the long-time limit
On universality of critical behaviour in Hamiltonian PDEs
Our main goal is the comparative study of singularities of solutions to the
systems of first order quasilinear PDEs and their perturbations containing
higher derivatives. The study is focused on the subclass of Hamiltonian PDEs
with one spatial dimension. For the systems of order one or two we describe the
local structure of singularities of a generic solution to the unperturbed
system near the point of "gradient catastrophe" in terms of standard objects of
the classical singularity theory; we argue that their perturbed companions must
be given by certain special solutions of Painleve' equations and their
generalizations.Comment: 59 pages, 2 figures. Amer. Math. Soc. Transl., to appea
The irreversible thermodynamics of curved lipid membranes
The theory of irreversible thermodynamics for arbitrarily curved lipid
membranes is presented here. The coupling between elastic bending and
irreversible processes such as intra-membrane lipid flow, intra-membrane phase
transitions, and protein binding and diffusion is studied. The forms of the
entropy production for the irreversible processes are obtained, and the
corresponding thermodynamic forces and fluxes are identified. Employing the
linear irreversible thermodynamic framework, the governing equations of motion
along with appropriate boundary conditions are provided.Comment: 62 pages, 4 figure
Conservation laws and normal forms of evolution equations
We study local conservation laws for evolution equations in two independent
variables. In particular, we present normal forms for the equations admitting
one or two low-order conservation laws. Examples include Harry Dym equation,
Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown
that for linear evolution equations all their conservation laws are (modulo
trivial conserved vectors) at most quadratic in the dependent variable and its
derivatives.Comment: 16 page
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