# Master projects for Mats Ehrnstrom

I am a mathematician specialising in **partial differential equations using analytic methods** , but can also offer projects in **numerical applications** of the same, **harmonic analyis** , and **data-driven PDEs** , as well as **collaborative projects** with other researchers. My main strength is in analysis as applied to nonlocal and frequency models, and I have a research group in water waves and dispersive equations. Most of my students have gone on to pursue a PhD degree. If your basic interest is in industry, there are probably better supervisors, but if your are interested in analysis, programming or numerics, modelling or data-driven methods, chances are you could find a suitable project. Browse some of the earlier master projects here.

If you are interested in writing a master's thesis, don't hesitate to send an Email to mats [dot] ehrnstrom [at] ntnu [dot] no. Master students get meetings one hour weekly, and each student is given a secondary contact in the group, that can help with smaller details and discussions outside the ordinary supervision. Students are expected to take responsibility for the advancement of the project, and to be fairly independent.

## Water waves and dispersive equations

Surface water waves travelling in one direction may in general be described via a class of equations of the form

where **\(u(t,x)\) describes the horizontal velocity, \(L\) is a linear and symmetric differential or pseudo-differential operator like \(\partial_x^2\), and \(n(u)\) is a nonlinear term like \(u^2\)**. By changing the operator \(L\), one can describe different forms of dispersion — the speed at which small wave components of different Fourier frequencies spread out. The nonlinearity \(n\), on the other hand, accounts for concentration, wave-breaking and singularity formation in the wave. This gives a rich picture of different behaviours, see the image below.

**Travelling waves are special solutions of the form \(u(t,x) = \varphi(x - ct)\)**, where \(c\) is the speed of the wave; they are propagating with a fixed shape and speed, therefore also called steady solutions, as they appear such in a reference frame moving with the same speed as the wave. Steady waves are prototypical of water waves, as it is expected that over long times time-dependent solutions will split up into different packages of nearly steady waves. The study of steady waves include the **existence of solitary and periodic solutions** , spatial **symmetries**, and **regularity** of highest waves. The study of time-dependent solutions, on the other hand, include **well-posedness**, **stability**, **wave-breaking** and **long-term behaviour**.

In water waves I will have the following themes to start from, each of them adjustable.