# Jonatan´s research focuses on the solution of so-called nonlinear PDEs

We are surrounded by waves: Water waves, sound waves, electromagnetic waves, and gravitational waves are a few examples. In mathematics, the motion of a wave is described by a partial differential equation (PDE).

My research focuses on the solution of so-called nonlinear PDEs. Although linear PDEs are sufficient to understand some wave phenomena, many physical systems are inherently nonlinear. When two nonlinear waves meet, the result can be more than the sum of the individual parts. An extreme example is the sudden creation of a monster wave on the ocean which alone can demolish a large ship.

Two phenomena that I have looked at more closely are electromagnetic waves in a fiber optic cable and the propagation of waves. I have also used Einstein’s theory of relativity to study the collision of gravitational waves and to study the curvature of spacetime in the vicinity of a rotating black hole.

Of special interest to me are boundary value problems and integrable systems. A boundary value problem for a wave equation consists of determining the wave profile throughout some given domain provided that it is known on the boundary of that domain. By enhancing our understanding of boundary value problems, we can send information in fiber optic cables more efficiently and predict the propagation of water waves (for example, tsunami waves) with higher precision.

Integrable systems make up a class of equations which, to a large extent, can be solved exactly. This makes them unique in the world of nonlinear equations. Indeed, most nonlinear systems are too complicated to be well-understood even when employing the most modern methods or computers. The mathematics of integrable systems is full of surprising results, intriguing connections, and beautiful formulas. I am particularly interested in understanding geometric and asymptotic properties of the solution and its interplay with any existing boundaries.