69 research outputs found
Weakly nonlocal fluid mechanics - the Schrodinger equation
A weakly nonlocal extension of ideal fluid dynamics is derived from the
Second Law of thermodynamics. It is proved that in the reversible limit the
additional pressure term can be derived from a potential. The requirement of
the additivity of the specific entropy function determines the quantum
potential uniquely. The relation to other known derivations of Schr\"odinger
equation (stochastic, Fisher information, exact uncertainty) is clarified.Comment: major extension and revisio
On the supersymmetric nonlinear evolution equations
Supersymmetrization of a nonlinear evolution equation in which the bosonic
equation is independent of the fermionic variable and the system is linear in
fermionic field goes by the name B-supersymmetrization. This special type of
supersymmetrization plays a role in superstring theory. We provide
B-supersymmetric extension of a number of quasilinear and fully nonlinear
evolution equations and find that the supersymmetric system follows from the
usual action principle while the bosonic and fermionic equations are
individually non Lagrangian in the field variable. We point out that
B-supersymmetrization can also be realized using a generalized Noetherian
symmetry such that the resulting set of Lagrangian symmetries coincides with
symmetries of the bosonic field equations. This observation provides a basis to
associate the bosonic and fermionic fields with the terms of bright and dark
solitons. The interpretation sought by us has its origin in the classic work of
Bateman who introduced a reverse-time system with negative friction to bring
the linear dissipative systems within the framework of variational principle.Comment: 12 pages, no figure
Modeling water waves beyond perturbations
In this chapter, we illustrate the advantage of variational principles for
modeling water waves from an elementary practical viewpoint. The method is
based on a `relaxed' variational principle, i.e., on a Lagrangian involving as
many variables as possible, and imposing some suitable subordinate constraints.
This approach allows the construction of approximations without necessarily
relying on a small parameter. This is illustrated via simple examples, namely
the Serre equations in shallow water, a generalization of the Klein-Gordon
equation in deep water and how to unify these equations in arbitrary depth. The
chapter ends with a discussion and caution on how this approach should be used
in practice.Comment: 15 pages, 1 figure, 39 references. This document is a contributed
chapter to an upcoming volume to be published by Springer in Lecture Notes in
Physics Series. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Simulation of Flow of Mixtures Through Anisotropic Porous Media using a Lattice Boltzmann Model
We propose a description for transient penetration simulations of miscible
and immiscible fluid mixtures into anisotropic porous media, using the lattice
Boltzmann (LB) method. Our model incorporates hydrodynamic flow, diffusion,
surface tension, and the possibility for global and local viscosity variations
to consider various types of hardening fluids. The miscible mixture consists of
two fluids, one governed by the hydrodynamic equations and one by diffusion
equations. We validate our model on standard problems like Poiseuille flow, the
collision of a drop with an impermeable, hydrophobic interface and the
deformation of the fluid due to surface tension forces. To demonstrate the
applicability to complex geometries, we simulate the invasion process of
mixtures into wood spruce samples.Comment: Submitted to EPJ
Existence of superposition solutions for pulse propagation in nonlinear resonant media
Existence of self-similar, superposed pulse-train solutions of the nonlinear,
coupled Maxwell-Schr\"odinger equations, with the frequencies controlled by the
oscillator strengths of the transitions, is established. Some of these
excitations are specific to the resonant media, with energy levels in the
configurations of and and arise because of the interference
effects of cnoidal waves, as evidenced from some recently discovered identities
involving the Jacobian elliptic functions. Interestingly, these excitations
also admit a dual interpretation as single pulse-trains, with widely different
amplitudes, which can lead to substantially different field intensities and
population densities in different atomic levels.Comment: 11 Pages, 6 Figures, presentation changed and 3 figures adde
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
Water waves generated by a moving bottom
Tsunamis are often generated by a moving sea bottom. This paper deals with
the case where the tsunami source is an earthquake. The linearized water-wave
equations are solved analytically for various sea bottom motions. Numerical
results based on the analytical solutions are shown for the free-surface
profiles, the horizontal and vertical velocities as well as the bottom
pressure.Comment: 41 pages, 13 figures. Accepted for publication in a book: "Tsunami
and Nonlinear Waves", Kundu, Anjan (Editor), Springer 2007, Approx. 325 p.,
170 illus., Hardcover, ISBN: 978-3-540-71255-8, available: May 200
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
In this work, the semi-inverse method has been used to derive the Lagrangian
of the Korteweg-de Vries (KdV) equation. Then, the time operator of the
Lagrangian of the KdV equation has been transformed into fractional domain in
terms of the left-Riemann-Liouville fractional differential operator. The
variational of the functional of this Lagrangian leads neatly to Euler-Lagrange
equation. Via Agrawal's method, one can easily derive the time-fractional KdV
equation from this Euler-Lagrange equation. Remarkably, the time-fractional
term in the resulting KdV equation is obtained in Riesz fractional derivative
in a direct manner. As a second step, the derived time-fractional KdV equation
is solved using He's variational-iteration method. The calculations are carried
out using initial condition depends on the nonlinear and dispersion
coefficients of the KdV equation. We remark that more pronounced effects and
deeper insight into the formation and properties of the resulting solitary wave
by additionally considering the fractional order derivative beside the
nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure
Modulational Instability in Equations of KdV Type
It is a matter of experience that nonlinear waves in dispersive media,
propagating primarily in one direction, may appear periodic in small space and
time scales, but their characteristics --- amplitude, phase, wave number, etc.
--- slowly vary in large space and time scales. In the 1970's, Whitham
developed an asymptotic (WKB) method to study the effects of small
"modulations" on nonlinear periodic wave trains. Since then, there has been a
great deal of work aiming at rigorously justifying the predictions from
Whitham's formal theory. We discuss recent advances in the mathematical
understanding of the dynamics, in particular, the instability of slowly
modulated wave trains for nonlinear dispersive equations of KdV type.Comment: 40 pages. To appear in upcoming title in Lecture Notes in Physic
An overview of the MHONGOOSE survey: Observing nearby galaxies with MeerKAT
MHONGOOSE is a deep survey of the neutral hydrogen distribution in a representative sample of 30 nearby disk and dwarf galaxies with HI masses from 10^6 to ~10^{11} M_sun, and luminosities from M_R ~ -12 to M_R ~ -22. The sample is selected to uniformly cover the available range in log(M_HI). Our extremely deep observations, down to HI column density limits of well below 10^{18} cm^{-2} - or a few hundred times fainter than the typical HI disks in galaxies - will directly detect the effects of cold accretion from the intergalactic medium and the links with the cosmic web. These observations will be the first ever to probe the very low-column density neutral gas in galaxies at these high resolutions. Combination with data at other wavelengths, most of it already available, will enable accurate modelling of the properties and evolution of the mass components in these galaxies and link these with the effects of environment, dark matter distribution, and other fundamental properties such as halo mass and angular momentum. MHONGOOSE can already start addressing some of the SKA-1 science goals and will provide a comprehensive inventory of the processes driving the transformation and evolution of galaxies in the nearby universe at high resolution and over 5 orders of magnitude in column density. It will be a Nearby Galaxies Legacy Survey that will be unsurpassed until the advent of the SKA, and can serve as a highly visible, lasting statement of MeerKAT's capabilities
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