Abstract
Abstract: This thesis consists of four papers related to various aspects of water waves with vorticity.
Paper I: Symmetry of steady periodic gravity water waves with vorticity.
We prove that steady periodic twodimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.
Paper II: Spatial dynamics methods for solitary gravitycapillary water waves with an arbitrary distribution of vorticity.
We present existence theories for several families of smallamplitude solitarywave solutions to the classical twodimensional waterwave problem in the presence of surface tension and vorticity. The established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinitedimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A centremanifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.
Paper III: A Hamiltonian formulation of water waves with constant vorticity.
We show that the governing equations for twodimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation.
Paper IV: Hamiltonian longwave approximations of water waves with constant vorticity
Starting with the Hamiltonian formulation in Paper III we derive several longwave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory.
Paper I: Symmetry of steady periodic gravity water waves with vorticity.
We prove that steady periodic twodimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid.
Paper II: Spatial dynamics methods for solitary gravitycapillary water waves with an arbitrary distribution of vorticity.
We present existence theories for several families of smallamplitude solitarywave solutions to the classical twodimensional waterwave problem in the presence of surface tension and vorticity. The established local bifurcation diagram for irrotational solitary waves is shown to remain qualitatively unchanged for any choice of vorticity distribution. The hydrodynamic problem is formulated as an infinitedimensional Hamiltonian system in which the horizontal spatial direction is the timelike variable. A centremanifold reduction technique is employed to reduce the system to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. Homoclinic solutions to the reduced system, which correspond to solitary water waves, are detected by a variety of dynamical systems methods.
Paper III: A Hamiltonian formulation of water waves with constant vorticity.
We show that the governing equations for twodimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the surface elevation.
Paper IV: Hamiltonian longwave approximations of water waves with constant vorticity
Starting with the Hamiltonian formulation in Paper III we derive several longwave approximations. These approximate models are also Hamiltonian and the connection between the symplectic structures is described by a simple transformation theory.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  2008 Mar 19 
ISBN (Print)  9789162873974 
Publication status  Published  2008 
Bibliographical note
Defence detailsDate: 20080319
Time: 13:15
Place: Aula C, Centre for Mathematical Sciences, Sölvegatan 18, Lund
External reviewer(s)
Name: Escher, Joachim
Title: Professor
Affiliation: Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1 30167 Hannover, Germany

Subject classification (UKÄ)
 Mathematics