22 research outputs found
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators
We present a stability theory for kink propagation in chains of coupled
oscillators and a new algorithm for the numerical study of kink dynamics. The
numerical solutions are computed using an equivalent integral equation instead
of a system of differential equations. This avoids uncertainty about the impact
of artificial boundary conditions and discretization in time. Stability results
also follow from the integral version. Stable kinks have a monotone leading
edge and move with a velocity larger than a critical value which depends on the
damping strength.Comment: 11 figure
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
Large time existence for 3D water-waves and asymptotics
We rigorously justify in 3D the main asymptotic models used in coastal
oceanography, including: shallow-water equations, Boussinesq systems,
Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre
approximation and full-dispersion model. We first introduce a ``variable''
nondimensionalized version of the water-waves equations which vary from shallow
to deep water, and which involves four dimensionless parameters. Using a
nonlocal energy adapted to the equations, we can prove a well-posedness
theorem, uniformly with respect to all the parameters. Its validity ranges
therefore from shallow to deep-water, from small to large surface and bottom
variations, and from fully to weakly transverse waves. The physical regimes
corresponding to the aforementioned models can therefore be studied as
particular cases; it turns out that the existence time and the energy bounds
given by the theorem are always those needed to justify the asymptotic models.
We can therefore derive and justify them in a systematic way.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and
remarks added) To appear in Inventione
Effect of the horizontal aspect ratio on thermocapillary convection stability in annular pool with surface heat dissipation
[EN] A linear stability analysis of the thermoconvective problem of a thin liquid film contained in an annular domain has been conducted. The influence of the horizontal aspect ratio on the solution has been considered by keeping a fixed external wall while the internal radius of the annular domain was modified. The parameter used in the study, Gamma(h), has been defined as the ratio of the internal radius to the domain depth. The other control parameter of the study is the Prandtl number ranging from 0.7 to 50, i.e. characteristic of fluids as air to n-butanol. The study has been performed for different Bond (Bo) regimes ranging from 0.0 for surface tension dominated flows to 67 for buoyancy dominated ones. Three different kind of bifurcations are found in the Gamma(h) - Pr plane for large Bonds, while for low Bonds only two of them appear. In the case of pure buoyancy or surface tension flows, for every Gamma(h) there exists a Prandtl number such that oscillatory and stationary coexist in a co-dimension two bifurcation point. These transitions show a strong dependency with the Bond number. Indeed, the lower transition disappears for low Bo and the upper one disappears with intermediate Bo values. Furthermore, there is a non-linear dependency of the number of structures of the growing bifurcation with Gamma(h). These co-dimension two lines show a strong dependency with Bo. Firstly, looking at the frontier between HWI and LR regions, for large Bo numbers, Pr increases with Gamma(h), while for low Bo the trend is reversed. Additionally, this transition only appears in the extreme Bo cases, for the central values of the considered, no transition is found. Similarly, the second transition found only appears for Bo larger than 30.SH and MJPQ work have been supported by project RTI2018-102256-B-I00 of Mineco/FEDER. PF work has been partially supported by the Spain's National Research and Development Plan (Project ESP2016-75887) and by the CHEOPS project (Grant Agreement 730135). This work was supported by a generous grant of computer time from the supercomputing center of the UPV.LĂłpez-NĂșñez, E.; PĂ©rez Quiles, MJ.; Fajardo, P.; Hoyas, S. (2020). Effect of the horizontal aspect ratio on thermocapillary convection stability in annular pool with surface heat dissipation. International Journal of Heat and Mass Transfer. 148:1-8. https://doi.org/10.1016/j.ijheatmasstransfer.2019.119140S1814
Methods of measuring rheological properties of interfacial layers (Experimental methods of 2D rheology)
On the completeness of solutions of Boussinesq, Timpe, Love, and Michell in axisymmetric elasticity
The Norton dome and the nineteenth century foundations of determinism
The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome