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 $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ 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.López-Núñez, E.; Pérez 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

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