 There will be later, right? So as it is right now, for this session, it is the last talk, right? So by then, the audience is spare and have sleepy, and, you know, it's a good thing, right? It will not make any trouble. And then the chair is more relaxed, right, looking already friendly, right? So it will not push me to finish in time. I will promise you to be so polite, but anyway, our next speaker is Melyvod Belich. So I'll be relaxed and I'll proceed forward in time, just to be sure. And I'll try to make it sort of clear and interesting and understandable, because I feel like a guest here. So I don't do turbulence, I don't do mixing, so I must be coming from beyond. And so what I do is nonlinear optics and nonlinear dynamics of optical systems. So at least I hope that I come from another world, and so that in the end, you will just at least learn something about rogue waves and cardboard carpets. So this is research done by my postdoc, Nikolich Stanko, by Omar Ashur, who is an undergraduate student, soon to be graduate student at Berkeley, and Jiang Yiqi, who is my collaborator from Xi'an Jiatong University in China. And so the title is Rogue Waves and Towel Carpets. It is a bit longer, but I made it short so that we'll be dealing mostly with these two things. Research is sponsored by the Qatar National Research Foundation. This is done at Texas A&M University at Qatar. And so this is how the campus looks like. And so let's now proceed, and let me just give you an outline. So I'll introduce rogue waves first, which are intimately connected, which are, as a matter of fact, solutions of the nonlinear Schrodinger equation. I'll try to make it as simple as possible, but not too simple. Then I'll introduce towel boat carpets as something that I believe not many people in this audience have heard before about. And then I'll just connect with the modulation instability and the homo-clinic chaos. This is one of the few instances in nonlinear optics in which you deal with things that are sort of approaching turbulence, just a simple, relatively simple notion of homo-clinic chaos. And in the abstract, just there is the shorted definition of what these things are. So rogue waves are giant waves that sporadically appear and disappear in oceans and in optics. Towel boat carpets are elaborate recurrent images of light and of plasma waves, achieved by some devices. And I'll try to bring the two together and discuss the role of modulation instability and homo-clinic chaos in their generation. So let's just proceed. So what is a rogue wave? The simplest definition here is just a giant solitary wave. The notion comes from the waves in the ocean. So it just looks as if it appears from nowhere and then recedes to nothing. This is an old by Hokusai, the Japanese wood block artist who just created this print some 200 years ago. And you can see really a giant wave, the great wave of Kanagawa, which by now I think is close to Fukushima some, no, not Fukushima. Anyway, close to Fuji Mountain, relatively close to Fuji Mountain. So it really looks big and threatening. Here is another example, a big wave. And if you can measure relative size of this one to the ships here, look at these two guys here, surface not seemingly oblivious to what is going to happen. And then there is another example here. Oops. And so here I'll present a few movies, a few examples of what a rogue wave probably is and what surely it is not. So this doesn't work this, oops, yeah, this doesn't work that way. I have to start it from here. So these two, that's not what I want. So these two examples are probably rogue waves and this is something else. So let's start with the first one. So with the volume turned on, it would look more dramatic, but this is what we have to stay with. Some of these waves look to be rogue waves, huge giant waves appearing suddenly on an ocean. So a working definition of a rogue wave is that its amplitude should be four and more times larger than the background waves. And this example here, which is not a rogue wave, I'm certain this audience will easily recognize. So it is not moving. Let's try to make it move for some reason. The one that you should recognize easily is not running. Okay. So it was suspicious right away that it says here atomcentral.com, which is a government agency which shows a nuclear explosion next to a ship. So it is an example of a shock wave. This one started on without purpose. So let's go to this example, which is an example of supposedly a rogue wave on a beach in Puerto Rico. So it is a beach which is shielded from the ocean. So the waves are coming, impinging on the wall. And just watch this lady. Now comes the real big one and she doesn't care at all. So obviously it happens fairly often that this big giant waves hit on this beach, but she's just enjoying her time. So that looks as a real rogue wave on an ocean. And the last example you will also, you should also recognize easily. This is the 2011 tsunami of Fukushima. So it is not a rogue wave. A tsunami is well defined. Wave, although it is rogue, but not a rogue wave. Now a bad thing about this movie is that actually in this accident, tsunami, more than 19, close to 20,000 people perished. And nowadays all you would hear from media when they speak of Fukushima disaster, it is just the nuclear incident at the Fukushima power plant. And there practically nobody died. Four casualties not even related to anything nuclear. So I think it is a sad thing to have this nowadays in news and nobody mentions 20,000 people that died, but they mentioned a huge nuclear catastrophe when there was none. So this just continues on and on. So let's get back to what we are here for, which is starting with an explanation of the nonlinear Schrodinger equation. And as I said, I'll be really simple. And just note a happy incident as far as optics are concerned, which is the paraxial wave equation in optics, which is something that we deal often with. He is actually equivalent to the Schrodinger equation in quantum, the usual Schrodinger equation in quantum mechanics. And here is the simplest form of the nonlinear Schrodinger equation, dimensionless form in which you have the time, the space. So here u, if it is quantum mechanics, is just the wave function. This is the kinetic energy p squared half. And this is like the potential. In this case, the potential is just proportional to the probability density. That's in the language of quantum mechanics. In the language of optics, u here is the optical envelope of the optical electric field. Bell x squared describes the fraction of that envelope. And this nonlinearity is the change of the index of refraction. So all the things are really measurable, defined nicely. And in this case, this is known as the Kerr nonlinearity, because it is proportional to the intensity of the wave. Of the electric field. And so if, as it happens, the diffraction which describes the spreading of the envelope is balanced by nonlinearity, then you can get a stable, bright solitons which move in a certain direction. And so the basic stable solutions are the bright and dark solitons, which you get when this nonlinearity coefficient is larger than zero or smaller than zero. And if you live in Middle East, as I do, right, then you can connect the bright soliton with the single-humped camels and the dark soliton, which has tangent hyperbolic square there as the double-humped solitons. What is important here is that actually rogue waves are also solutions of this simple nonlinear Schrodinger equation. As you probably all know, the nonlinear Schrodinger equation is written here is completely integral. So, but it contains very many solutions. And some of those are stable, like the bright solitons. But most of the solutions which write on a background, like the dark and other types of solitons, are unstable as such. And so among the basic rogue waves, we distinguish the peregrine soliton, kuznetso ma breeders, ahmedie breeders, and then out of these basic solitons or breeders higher order rogue waves are formed. So, this here, this is the transverse coordinate, this is the propagation distance, this is the equation written in a slightly different form just to account for these coordinates there. This is then the real peregrine soliton, and then this is the kuznetso ma, which is a breeder in the direction of propagation. And so how does this look like? Here it is a soliton on a finite background introduced by Nile Ahmediev. Much of the progress in this field is actually achieved by Ahmediev and his collaborators. So, this is a breeder that he introduced known by his name. It has this form and just know that it has three parameters a, b and omega, and actually b and omega are also given in terms of a. So, there is only one very important parameter here, which is a, and a starts from zero and goes to infinity, but between zero and one half, these solutions will describe Ahmediev breaders. At a equal one half exactly you'll get the peregrine soliton and then later on you'll get kuznetso ma. And so if we just look at how they look like, this is just given in this. Just look at how a changes as the wave appears. Oops, that's not what I wanted. I need the pointer. Here it is. Now it starts, so it starts with a changing from zero to an increasing and this is all Ahmediev breeder at exactly one half. There is the simple peregrine there and then the kuznetso ma breeder reappears. Just remember here that there is one only important parameter here, which is a, which describes completely an Ahmediev breeder and then higher order rogue waves you can get by scattering, by colliding to, for example, Ahmediev breaders. Then you'll get the second or third order rogue waves. And now what is a Talbot effect? So here it is a text from Wikipedia. This is Henry for Talbot, Henry Fox Talbot. So Talbot effect is a near field diffraction effect observed in 1836 by Henry Fox Talbot. When a plane wave is incident upon a periodic diffraction grading as it is here, this is the illuminating plane wave, here is the grating. The image of the grating, here it is, the near field image of the grating, is repeated at regular distances away from the grating plane and here it is. This is the first time and this is known as the primary Talbot image, which is actually repeating what you see close to the grating. The regular distance is called the Talbot length for some reason here it is called 2ZT, right then this is ZT but actually this is the half of the Talbot length. And the repeated images are called self images or Talbot images. Furthermore, at half of the Talbot length a self image is also, it also occurs, which is given here and as you can see it is shifted by pi relative to the primary or to the initial image. A self image also occurs but it is phase shifted by half a period. At smaller regular fractions, fractions of the Talbot length, some images can also be observed. So here is the secondary Talbot image, this is the double frequency fractional image, this is the triple etc. So you see also very many images as you put the screens and this is how Talbot really saw this effect. So he was not a physicist or engineer, he was just a photographer. So he took a piece of paper very close to the grating and then there was moving piece of paper which played the role of a screen and then he saw the appearance of this. It was usually blurred because the white light is not monochromatic so it was, he sees different colors but once he saw the full Talbot image, yes, he saw that it is reoccurring. And there is another sort of nice article which mentions for the first time I believe quantum carpets, carpets of light and so these are Talbot carpets and being a photographer here is another image of heavy folks Talbot. So he really liked to take selfies of himself back then. So you can read it there but what is sort of new here, this is the same experiment, the incident light and then you see the distributions. You see fractional as well as fractal Talbot images. And the theory just comes from the Fresnel diffraction theory so you need to get the diffractive field amplitude. You define it in terms of the amplitude transmission of the object a of x right here. There is the phase factors there then the coherent amplitude of the source. This is how the source is distributed and then is obtained from this formula provided you add another propagation constant there and this s which is for some reason called t here could be symmetric or asymmetric so these are the Talbot carpets coming from up and going down which could be symmetric or asymmetric as it is. And what is the use of this relatively unknown and exoteric effect. Well actually this is surface plasma polaritons that you get on a film metallic film by just drilling holes in here and then this will appear as a surface plasma polariton surface wave on the surface of the metal and this obviously is a diffraction but on a nano scale so you might use it for lithography. And there are other examples you may just note that in experiment these are all experiments actually you cannot see very many images repeating because you always start with some finite window and finite window cannot contain more than very few generations. Now this one here is relevant to what we are I'm going to talk about and the time is pressing I'm not even at the half of the talk so let's just speed up bring the two together and when you try to bring two together try to represent by propagating here something which will develop into reoccurring images so we just try to put here a simple peregrine soliton and then did the numerics in that direction so what would you expect if your numerics is really good that there will be only initial initial peregrine and nothing else because the peregrine has infinite length right however any all each and every numerical scheme that you apply something appears here and that's the consequence of the modulation instability and you just cannot get away from it you can change the numerical scheme then this thing will move towards the end but there will something always appear because the modulation instability is forcing you to go into some of the modes are being amplified and there is something that after finite time appears this is a homo I mean chaos homo clinic chaos and there is no way around it but we ask ourselves okay if it doesn't go with Talbot let's put Akhmediyal breeder here so this was an unexpected phenomenon but easily understand once you include a modulation instability an expected phenomenon is that Akhmediyal among other breeders that he has found he found also doubly periodic Akhmediyal which looks as if you start with the Akhmediyal and then it just produces a primary secondary secondary primary images along the way if you propagate it correctly and this is actually what happens so if you just take as an initial wave tal Akhmediyal breeder and you propagate it what happens here you'll get no linear Talbot effect but there are no fractional fractal images there is only the secondary and then primary and secondary and primary etc image and I cut it here because sooner or later if you propagate modulation instability will force in and you will not see such nice Talbot images now of course if you take away non-linearity here from the equation and you still solve it numerically you'll get the linear Talbot effect which is which is presented here and now what about higher order always since I'm sort of running of time let me just keep a number of of so it is now that we have to discuss dynamic modulation instability or it goes also under the name of Benjamin Fair instability it is a complex process and blah blah blah now the major difficulty is how to distinguish rogue waves from numerical artifacts and so what is the solution is the homoclinic chaos which appears there but numerical results are very often wrong and so here is one example which is published which contains higher order rogue wave but if you take exactly the same mode of solution the rogue wave and just step half then the row this higher the rogue wave disappears so the published results are very often wrong and so this is what we discovered harmonic cascade row to row to homoclinic chaos let me just go further down there this is totally oops this is totally not too many but let me just keep all the way so if you try to put second order rogue waves and second orders by their appearance look like butterflies then yes they'll they'll just be arranged as a Talbot carpet for only a few generations but then the chaos homoclinic chaos takes over if you do stabilize this propagation by decimating some unstable modes then you can get the stabilized perfect carpet from random rogue waves and this is the second order and if you continue you can do much with the third order but you can also get it in other equations for example in Hirata's equation which has higher order terms in there with stabilization it is perfect without stabilization modulation instability takes over but what is the message here by propagating higher order rogue waves which usually are thought to be appear spontaneously and at random actually you can build them one after the other right in a regular Talbot carpet thank you