The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polyg...

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration di erences. In the former case, and when thermodynamical coe cients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. In this pape...

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in Lð 1;1Þ is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which...

The 1877 assumption/conjecture of Boussinesq is that turbulent fluctuations have a dissipative effect on the mean flow. The eddy viscosity hypothesis is that this dissipative effect can be modelled by a turbulent viscosity term. Many computational and experimental data sets have shown that the eddy viscosity hypothesis is not generally correct. The inverse cascade of energy to larger scales in ...

We study the effects of severe non-Boussinesq conditions on thermal convection at the moderate Rayleigh numbers of Ra = 2× 108 and 2× 109 by resorting to direct numerical computations of the full governing equations. We illustrate the effects by considering low temperature gaseous helium. The properties of helium are allowed to depend on the temperature around the mean of 5.4K. The Nusselt numb...

An analytical solution of the Boussinesq equations for the motion of a viscous stably stratified fluid driven by a surface thermal forcing with large horizontal gradients (step changes) is obtained. This analytical solution is one of the few available for wall-bounded buoyancy-driven flows. The solution can be used to verify that computer codes for Boussinesq fluid system simulations are free o...

In this paper, we derive asymptotic models for the propagation of two and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume the thickness of the upper and lower fluids to be of comparable size, and small compared to the characteristic wavelength of the system (shallow water regim...

We report measurements of the Nusselt number, Nu, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was Ra=2× 10. Except for the last half a decade or so of Ra, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the logNu–logRa slope saturates at a value close to 1/3, as ...

This work deals with the local rapid exponential stabilization for a Boussinesq system of KdVKdV type introduced by J. Bona, M. Chen and J.-C. Saut. This is a model for the motion of small amplitude long waves on the surface of an ideal fluid. Here, we will consider the Boussinesq system of KdV-KdV type posed on a finite domain, with homogeneous Dirichlet–Neumann boundary controls acting at the...

In this paper, an improved scheme is presented to investigate the approximate solutions of (2+1)-dimensional Boussinesq equation with initial conditions based on the decomposition method. The improve scheme makes all components ui of u = ∑∞ i=0 ui be of the same degree in t. We choose the (2+1)-dimensional Boussinesq equation with initial conditions of doubly-periodic functions to illustrate th...