Integrable Systems as Fluid Models with Physical Applications

Abstract

In this thesis we begin with the development and analysis of hydrodynamical models as they arise in the theory of water waves and in the modelling of blood flow within arteries. Initially we derive three models of hydrodynamical relevance, namely the KdV equation, the two component Camassa-Holm equation and the Kaup-Boussinesq equation. We develop a model of blood flowing within an artery with elastic walls, and from the principles of Newtonian mechanics we derive the two-component Burger's equation as our first integrable model. We investigate the analytic properties of the system briefly, with the aim of demonstrating the phenomenon of wave breaking for the system. In addition we construct a pair of diffeomorphisms which allow us to solve the system explicitly in terms of the initial data. Finally, we show that when we consider the dynamics of the arterial walls themselves, the pressure within the fluid is seen to satisfy the KdV equation. In the following chapter we investigate the trajectories followed by individual fluid particles in a fluid, as they are subject to the effects of an extreme Stokes wave. In the case of a regular stokes wave there are no stagnation points or apparent stagnation points, i.e. locations where the fluid velocity and wave velocity are equal, however this condition does no remain true in the context of extreme Stokes waves. The result for the regular Stokes wave then have to be extended to semi-infinite regions with corners, and in doing so we show that the horizontal component of the fluid velocity field is strictly increasing along any stream line, which in turn ensures the non-closure of particle trajectories over the course of a fluid wave. Next we begin with a review of the inverse scattering transform method of solving the Kortweg-de Vries equation. We construct the one-soliton solution explicitly. We then proceed to examine the Qiao equation, a non-linear partial differential equation with cubic non-linearities. We show that by a suitable change of variables and with a ACKNOWLEDGEMENTS iv change of the spectral parameter of its associated spectral problem that we transform it into the spectral problem of the KdV equation. Having already analysed this spectral problem, we then proceed to construct the 1-soliton solution of the Qiao equation with this modified spectral problem. The soliton solutions decay to a non-zero constant value asymptotically. We also investigate the peakon solutions of the Qiao equation, and construct the 1 and 2-peakon profiles, the latter being in the form of travelling M-wave profile. We then go on to the analysis of a class of equations whose spectral problem are more complicated in the sense that the spectral problem has an energy dependant potential. We develop the inverse scattering transformmethod for these spectral problems, and construct the one-soliton solution explicitly, which in fact turn out to be a breather type solution. The hydrodynamical relevance of this problem arises from the fact that by an appropriate choice of one of the physical parameters of the system, we obtain the Kaup-Boussinesq equation, a partial differential equation with quadratic and cubic nonlinearities which arises in the theory of water waves in shallow water.