Part I
The forced Korteweg-de Vries model has been found satisfactory in predicting the periodic generation of upstream-advancing solitary waves by a bottom topography moving in a layer of shallow water with a steady transcritical velocity. It is also known that with certain characteristic forcing distributions, there exist waves, according to the fKdV model, which can remain steady in accompanying the characteristic forcing, provided such a wave exists initially, whereas for a different initial condition the phenomenon of periodic generation can still manifest itself. The stability of two such transcritically forced steady solitary waves is investigated, with their bifurcation diagrams determined with respect to the velocity and the amplitude of the forcing as parameters. The linear stability analysis is first carried out; it involves solving a singular, non-self-adjoint eigenvalue problem, which is examined by applying techniques of matched asymptotic expansions with suitable multiscales for singular perturbations, about the isolated bifurcation points of the parameters. The eigenvalues and eigenfunctions for the full range of the parameters are then obtained by numerically summing a power series expansion for the solution. The numerical results, which accurately match with the local analysis, show that the eigenvalues have only four branches σ = ±σr ±iσi. The real part σr is nonvanishing for the velocity less than a certain supercritical value and for the amplitude greater than a certain marginal bound except at a single point in the parametric plane at which the external forcings vanish, reducing the forced waves to the classical free solitary wave. Within this parametric range, the real part of the four eigenvalues is algebraically two to five orders smaller than the imaginary part σi, wherever σi exists; such a small σr indicates physically a weak exponential growth rate of perturbed solutions and mathematically the need of a very accurate numerical method for its determination. Beyond this parametricrange, linear stability theory appears to fail because no eigenvalues can there be found to exist. In this latter case a non linear analysis based on the functional Hamiltonian formulation is found to prevail, and our analysis predicts stability. Finally, extensive numerical simulations using various finite difference schemes are pursued, with results providing full confirmation to the predictions made in various regimes by the analysis.
We consider the Korteweg-de Vries equation in the semi-infinite real line with a boundary condition at the origin. The numerical investigations of Chu et al.[2], are revisited and different new forms for the boundary forcing are assumed. In order to provide some qualitative description for the numerical simulations we develop a simple model based on the IST formalism. It is found that the model is also able to provide some quantitative predictions in agreement with the numerical results.
Part II
There has been considerable interest recently on chaotic advection, for the first time explored in the context of Rayleigh-Benard roll (2D) convection by the experimental work of J. Gollub and collaborators. When the Rayleigh number increases across a (supercritical) value, depending on the wavelength of the rolls, an oscillatory instability sets in. The flow near the onset of the instability can still be modelled by a stream function, which can be split into a time independent part plus a small time dependent perturbation. The motion of fluid particles can therefore be regarded as the flow for a near integrable, "one-and a half" degree of freedom Hamiltonian vector field, with the phase space corresponding to the physical domain. In absence of molecular diffusivity, the evolution of a certain region of phase space can thus be viewed as the motion of a dyed part of fluid, when the tracer is perfectly passive. The most important objects for a theory of transport are the invariant manifolds for the Poincare map of the flow homoclinic to fixed points, which physically correspond to the stagnation points. As fluid particles cannot cross invariant lines, these curves constitute a sort of "template" for their motion. For the time independent flow, the invariant manifolds connect the stagnation points and define the roll boundaries. Thus, no transport from roll to roll can occur in this case. Switching the perturbation on, these connections are broken and the manifolds are free to wander along the array of rolls. We use segments of stable and unstable manifold to define the time dependent analogue of the roll boundaries. Transport of fluid across a boundary can then be attributed to the way a region bounded by segments of stable and unstable manifold, or "lobe," is evolving under map iterations. This allows us to write explicit formulae for describing the fluid transport in terms of a few of these lobes, for a general cross section defining the Poincare map. Using the symmetries of special cross sections, we are able to further reduce the number of necessary lobes to just one. Furthermore, these symmetries allow us to derive analytically a lower and upper bound for the first time tracer invades a roll, and a lower bound on the stretching of the interface between dyed and clear fluid. These results are independent of the fact that the perturbation is small. When this is the case however, the analytical tools of the Melnikov and subharmonic Melnikov functions are available, so that an approximation to the lobe areas and location and size of the island bands can be determined analytically. It turns out that in our case these approximations are quite good, even for relatively large perturbations. The results we have produced regarding the strong dependence of transport on the period of the oscillation suggest an effect for which no experimental verification is currently available. The presence of molecular diffusivity introduces a (long) time scale into the problem. We discuss the applicability of the theory in this situation, by introducing a simple rule for determining when the effects of diffusivity are negligible, and perform numerical simulations of the flow in this case to provide an example.
