| This thesis considers two-dimensional stratified water waves propagating under the force of gravity over an impermeable at bed and with a free surface. In the absence of surface tension, it is proved that there exists of a global continuum of classical solutions that are periodic and traveling. These waves, moreover, can exhibit large density variation, speed, and amplitude. When the motion is assumed to be driven by capillarity on the surface and a gravitational force acting on the body of the fluid, it is shown that there exists global continua of such solutions. In both regimes, this is accomplished by first constructing a 1-parameter family of laminar flow solutions, then applying bifurcation theory methods to obtain local curves of small amplitude solutions branching from the laminar curve at an eigenvalue of the linearized problem. Each solution curve is then continued globally by means of a degree theoretic argument in the spirit of Rabinowitz. We also provide an alternate global bifurcation theorem via the analytic continuation method of Dancer.;Finally, we consider the question of symmetry for two-dimensional stably stratified steady periodic gravity water waves with surface profiles monotonic between crests and troughs. We provide sufficient conditions under which such waves are necessarily symmetric. We do this by first exploiting some elliptic structure in the governing equations to show that, in certain size regimes, a maximum principle holds. This then forms the basis for a method of moving planes argument. |
