Thank you very much, Marius and Enrique for the opportunity to share some of my research with you.

Let me share my light window. Okay, so now you should be able to see my slides.

The talk is entitled numerical simulation of nonlinear Schrödinger equations.

I assume that many of you are not so much into numerical methods, but maybe more into modeling and analysis.

But I hope that nonetheless there will be some interesting aspects for you.

This work is a joint work with Robert Altman who just submitted his habitation thesis in Augsburg and Professor Patrick Henning from KTH in Stockholm.

So I kind of got interested in these nonlinear Schrödinger equations in a very specific physical context, which is called Bose-Einstein condensation.

Bose-Einstein condensate, this is an extreme state of matter.

You can in which kind of separate atoms or subatomic particles, they coalesce into a single quantum mechanical entity, so they are not distinguishable anymore for us.

And it happens, for example, if you take a dilute gas and cool it down to absolute zero and then such a condensate will form.

Historically, this was predicted by Albert Einstein based on work by Bose in the early 20th century.

And it took quite a while, but in the meantime, in the late 1990s, people were actually able to experimentally kind of create these condensates.

So by the two groups that are kind of referenced here.

And since then, this is kind of a very active field of modern physics and people are trying to get these kind of condensates with all sorts of particles.

And one example, for example, is they are also doing this condensation of light particles, so of photons, which was kind of more like a recent observation by colleagues in Bonn.

And during my time in Bonn, I kind of entered the topic because they asked me to do some numerical simulation on the topics.

So the practical relevance, just as a background information of these condensates is, or maybe the most prominent feature is super fluidity, which is really on an, happens on an observable scale.

And the application that some people have in mind nowadays is more like to build super accurate lasers by releasing particles from such a condensate.

And this would create some sort of lasers with very, very, very high frequency, much higher than what you can have with nowadays laser technologies.

This is just the background. Now, the question is, what is the connection to like numerics of the partial differential equations?

And the connection is that actually because of this, the property that all particles can essentially be described by one single wave function, you can imagine that you can model the condensate once it has formed with a PDE equation for this wave function.

And this equation in this particular physical context is called the Ross-Pieter-Jewski equation.

It's some sort, it's one instance of a nonlinear Schrödinger equation, but it has of course, it has some special property or it's a special case of a nonlinear Schrödinger equation.

And from now on, I will more use the phrase Ross-Pieter-Jewski. I usually don't put it in titles of talks to mathematical audience because it's more catchy to put the word Schrödinger there.

Now, what is this Ross-Pieter-Jewski equation? It's some sort of wave equation for this wave function u that describes the condensate or its density.

And you can see it's familiar probably to many of you. It's a nonlinear evolution equation with a cubic nonlinearity.

That's essentially it. Now, there are many of these equations in this particular setting. What you should know about the parameters is that the parameter in front of the nonlinearity is non-negative or positive.

That's kind of the scenario or the regime we are working in here. That's called the Repulsive regime.

And then there are several more kind of terms here that I will explain in a second. Let me first say that the applicability of this model is, from a physical point of view, reasonable for kind of to study these condensates or these bosonic gases

once the condensate has formed. So if the gas, for example, has already been cooled down to almost absolute zero, and then you can study the dynamics of the condensate, for example, or you can study the ground state of the system.

What you cannot model, of course, with this mean field type of equation is the formation of the condensate. That would be too much to ask.

But it's kind of accepted in the community. If you want to do more fancy stuff or more recent stuff, for example, if you want to do Bose-Einstein condensation of light particles and temperature is not close to absolute zero and has to be taken into account,

and then you can start all your favorite modeling games and couple it to heat equation whatsoever. And this gives like interesting coupled systems of PDEs, but this will not be the topic of this talk.

In this talk, I will stay with this kind of Grosz-Pieterjewski equation. Now, let me explain the two terms here. The Laplacian is, I think, clear to everybody.

This is a cubic non-linearity. This is formally, of course, the wave function is complex valued. In most of the talk, actually, it will be only real valued because I will exclude certain things to make things easier, but in general, it's, of course, complex valued.

Now, there are two more terms here. One is a term that is related to an angular velocity or to rotation of the condensate.

So here you can see, for example, in the picture, a ground state of such a condensate, which is confined in a potential V.

So this V is just a function like a parabola harmonic type of potential, which means that it goes to infinity when the when X goes to infinity and the condensate is kind of located in the minimum values of this potential V or around the minimum value of the potential V.

And then this other term kind of rotates the whole thing, and this allows you to observe the super fluidity effect, for example, namely what you will observe is these quantized vortices here, namely regions where there is no condensate anymore.

And now the equation would allow you to study, for example, the dynamics of such a vortex pattern by, for example, you perturb the system a little bit by changing the potential and then you can kind of simulate how these vortex patterns would behave in the dynamics.

Okay, this is kind of modeled by this rotational term here. So this produces the vortices. Omega here is a parameter that is related to the angular velocity, and it has to be chosen appropriately, of course.

Maybe I should make before I continue a general statement about physical parameters. They are very important and of course they determine the regime and all the phenomena.

So in the mathematical presentation here, I will be very lousy about the parameters, so I boiled down everything already to essentially two.

And actually if a physicist would see the equation, he would miss one half in front of the Laplacian, but for the kind of numerical derivation, it's not super important.

But of course, if you want to study these effects parameters are very important and we should be very careful about the actual scaling machine.

Now, to make life easy, I will most of the talk or the mathematical derivation. So skip this rotational term here.

I will only come back to it in the numerical experiments, and I will mostly be concerned with kind of special types of this potential be here.

So as I told you just on the slide before, and like a classical choice would be to consider just harmonic potential or maybe aquatic potential, some smooth function that confines the condensate in the end.

But there are also more interesting cases, namely, for example, you could study disorder potentials that are created by lasers, so-called laser speckles.

So essentially, in this case, V would be a highly oscillatory function, as you can see, with kind of a very short length of oscillation or correlation in it, and high contrast in the values that the potential V takes.