Water-wave model equations

Water waves are the most common observable kind of waves. Understanding their structure and mechanisms leading to phenomena like traveling waves or wave-breaking is of great importance for scientific advances in fields like for instance oceanography. The evolution of water-waves is described by the free surface Euler equations.

While the Euler equations capture the essence of fluid behavior, they are often too complex to be of practical use for numerical implementation and simulations. Focusing on a certain regime, such as for instance shallow water, the Euler equations can be reduced to so-called water-wave model equations. Those are less complex, but still take into account nonlinear and dispersive effects. The competition of these effects leads to wave phenomena such as traveling waves or wave-breaking.

By now there exists very larges classes of surface wave model equations in one and two space dimensions. The most famous being the Korteweg-de Vries equation, which models the surface elevation of small-amplitude shallow water-waves. Many of these model equations are of nonlinear dispersive form and can be written abstractly as

$\left(u_t+u^2+Lu\right)_x=0$

where $u=u\left(t,x\right)$ is the unknown depending on time and space, and $L$ is a Fourier multiplier operator with real and even symbol. In the case of the Korteweg-de Vries equation it is $L=\frac{d^2}{dx^2}$ .

Possible degree projects include

Symmetry and decay of solitary solution for the Whitam equation with surface tension/vorticity
Existence of periodic traveling waves and properties for surface wind waves
Existence and decay of fully localized traveling waves for the fractional Zakharov-Kutznetsov equation (higher dimensional equation)

If you are interested in writing a degree project in water-wave model equations, please contact me (Gabriele Brüll: gabriele.brull@math.lth.se) and we can discuss a suitable problem.

Starting 2025, there will be a DFG (German Research Foundation) Network Program between the Lund, Stuttgart, Saarbrücken, and Bonn on Instability Phenomena in Asymptotic Models in Fluid Dynamics, where water-wave model equations build a subproject. There might be the possibility to include the degree project into the study group of the network program, which enables cooperation with researchers from another network university including a research visit.

For more information on the network, check the webpage. Links to an external site.