 Thank you so much and thanks to the organizers for giving me the opportunity to give a talk and actually my Original plan was as said in the title to talk about critical half-wave problems, which would mean to the L2 critical problem that Patrick Girard was talking about on Monday and Another new problem, which I will now try to focus on because Patrick did such a great job It's hard to really add something to this. So let me let me Let me explain what I want to discuss is what I phrase is a half-wave maps problem Okay, so what's that? So it's an equation that is extremely easy to write down so let's see and I will say a Few words so the wedge means the cross product in R3 and U is our unknown function Depending on time on an interval say from zero to capital T and say is defined on Rd with values and as to which we always of course think being embedded in R3 Yeah, so this is a it's a quite I mean Simple-minded at first glance generalization of the Schrödinger maps problem. Yeah, so you just take the say Landau-Liffchitz equation This would be the Laplacian or it would be minus the Laplacian But minus signs can be easily so to say absorbed by redefining Direction of time. This is in Hamiltonian equation by the ways I will show to you soon So I might get some various sign mistakes in my formulas, but they don't matter So that's only where it matters. I think I have them right. So let's see okay, so this is the half-wave maps equation and And so there are two reasons why it's recently started to work on that first of all I've seen this equation before but so only recently I realized that this has a really somewhat very nice Physical interpretation. That's the one bonus thing about it, which I tried to make somewhat clear to you here and the second thing is which I was always also afraid about is that you cannot do much With that that is rather quite unexplosive What you can do compared to what like Schrodinger maps or wave maps equation where you have a nice explicit structure say of Equivariant wave maps and so on it studies stability and blow up problems like that But it turns out to be wrong this statement that I will show to you that you can do a lot of it explicitly because I will as my main Result today discuss that in the one dimensional case, which is the energy critical case I will show to you how to completely Classify all traveling solitary wave solutions in an explicit form in terms of so-called Blush-Cup-Products. Okay, so this is something I recently found and Something that amazes me and gives me hope to continue this study of this energy critical problem Okay, so as I said, this is an Hamiltonian equation. So There's an energy functional attached to it, which of course Looks like this Some more guess what this is going to be. So maybe there's a half here if you like that and For later purposes, I will just Do this here for record that this is of course Expressible as a double integral which will as you see make a connection to a very interesting physical application of this model and the Poisson bracket attached to that so that you can see that you can write this Equation of as a Hamiltonian equation of motion is the one that is given by Functions with values in s2 say the components at x and y satisfy Poisson bracket structure which is given by this expression. So this is the anti-symmetric Levy-Givita symbol and The delta function so if you if you use this structure at least formally you see that the half wave map's equation is formally equivalent to this evolution equation given by Poisson bracket given with H. Okay So in particular H is at least a formally conserved quantity So let me now start and discuss the motivation behind this equation. So first of all Yeah, if you are clear, I mean you have to plug in what this Poisson structure is and You work out what this is by plugging this in and then you finally see that this is the equation that you get I mean if you take for instance Here the Dirichlet integral you get the Landau-Liffchitz equation with that Poisson structure the Poisson structure is some universal thing It's just the energy function of it determines whether you have the half wave map's equation as I call it or the Landau-Liffchitz equation At the Landau-Liffchitz is the Schrödinger maps equation. I don't have to write it Yeah, Schrödinger map or Landau-Liffchitz Coming from physics like the continuous version of the Heisenberg equation for a ferromagnetic system Okay Yeah, so D equal to 1 by the way would correspond to the energy critical case Everything above D equal to 2 and higher is of course an energy subcritical a supercritical So by the way and the energy critical case. It's also Physically most relevant As I will explain in a second and there's also another thing attached to it. It has a conformal invariance property So this is the case I will focus on so the motivation So one motivation comes from differential geometry. I mean recently there was some Growing interest in what is called a half harmonic map with target s2 Okay, so that was correspond of course obviously to static solutions to my flow equation Huh, so in particular these static solutions will play also an important role for this time dependent equation Of course, you might also think about something like a half harmonic map heat flow Which would be a parabolic analog to this equation, which I don't discuss here So this is something that was recently introduced Riviere and Dalio in particular the Regularity theory for these equations like say if you are in the critical dimension that an H1 half map is Actually a smooth map like an analog of a lance regularity theory for Harmonic maps in two dimensions. Yeah, but it's a different thing because Riviere used some kind of commutator type arguments to get this regularity result and also from a more geometric point of view in a very recent work by Fraser and Shane It is seen that this equation Corresponds to a free boundary minimal disc problem. I Will come to this later That is to say it is also naturally connected that you could look for something which is a minimal surface say inside the ball The unit ball with a certain condition how the boundary of this minimal surface Hits the boundary of the ball, which is s2 and in a in this setting it has to hit perpendicular And you can actually see that this problem is equivalent to this to study the free boundary minimal Yes, this is a boundary equation I will come to this point where you see that it's a completely natural connection to the theory of minimal surfaces Okay, I come to this and Be this is the physical motivation behind that Of course, I will later look for traveling solitary waves Which will be Generalizing these half harmonic maps in the case in the sense that there's another non-zero right-hand side Okay, and I will classify also all these solutions and B is physics And this is typical thing You have to know the right names to look for because physicists like these names You see week ordering give state if you don't know what who week was who gives was it's all hard to find out in the Literature what's this about once you have the right code words? It works. So the first two names that Somewhat play a role in these business a Haldane and Chastry to physicists Who wrote down a Hamiltonian for Spin chain and I will be a little bit formal here, of course, which is some Object like that Okay, these are spin operators. I won't go into any details and And there is a so-to-say interaction among spins. This is on a discreet Letters say In one dimension So you have fixed lattice sites and xk Xk plus one and fourth are the lattice sites and to each lattice sites you attach a quantum spin I don't want it to be go to into details what this is about and they interact With certain what's called long-range potentials That might be a sine squared of the difference of the lattice sites Or there's also the so to say algebraic model where this is a square of this and so on There's a whole zoo of this and the point about this these are exactly solvable models in the sense You can calculate the spectrum and so forth and you can also find lax pairs and so on these are long range Quantum chains, whichever very nice Particular structure from the physical point of view So it's hard to draw what these quantum operators do actually but I installed planks constant here Just to indicate that in a certain limiting regime, which you can think of the semi classical limit or also a large spin limit You're at least formally led to what is called a system of classical spins So then this Hamiltonian becomes something which you could phrase as a calogero Mosa type Hamiltonian with spin which would look like this that formally at least The spin operators get replaced By something of this form where the sk's are now Unit vectors in three space, okay So in the classical regime the spins are given by By an arrow which lies on the unit sphere in three-dimensional space That's a at least for a physicist a typical procedure to go from a quantum spin system to a classical spin system So if you just had just nearest neighbor Interactions so this was in this function But just only nearest neighbors couple you get the Heisenberg chain and the continuum limit of the Heisenberg chain in The sense that this parameter a shrinks to zero would be the classical Landau-Lyschitz Schrodinger map's equation however here you can of course somewhat anticipate that in the next step of taking this limit You get that this sum Leads to a double integral of that form that you have a continuous spin variable, which I call S which is of this form And we are here on one space dimension Making this rigorous makes it of course one step is to study actually that the dynamics Converge in a certain sense and of course in higher dimensions might be even more difficult in the critical case. I think it's still Doable, but it's it's it's not so simple and on account of the fact that this is of unit lengths You can of course rewrite it As this which is the Hamiltonian I wrote above in one space dimension, okay Again, this Kalogero-Mosa type Hamiltonian Also shares the magic property that you have lax pairs You have a very explicit structure here and you might wonder if this survives actually for this limiting PDE There's a strong indication for that because I would show to you an explicit classification of all solitary waves that might be an hint that this is a completely integrable system, which is yet Energy critical, so it shows a criticality, which in principle can also say that you have Final-time blow-up and so on so let's see So now I come to the more One Yes, exactly in Dimension two and larger supercritical and in one dimension is energy critical and in principle you can think like for Energy critical Schrodinger maps you have also blob solutions Now that in this case you might have because of an integrable structure too many conservation laws which forbid such a thing But it could also be like for the cubic chigo equation that they only live up to certain level of regularity and you still see some kind of Sobo-Lev type growth phenomena, which are a little bit counter to it the first if you think of a completely into if I think of a completely System So what I'm here interested in is to consider the traveling solitary waves For this model and this is interesting in one respect first that for the Landau-Levchitz equation Such a thing does not exist in the dimension They are only static finite energy solutions. You don't cannot create a Traveling solitary waves for the Landau-Levchitz equation You might think you can apply a Galilean boost, but this is not working It's not but of course for the wave maps equation you can do a Lorentz boost to get a traveling Wave map if you wish now, but here this is not a Lorentz invariant system So we cannot apply something but you will see that we get a rather explicit way of constructing these things And I'll show to you that this is the only way to get them Okay, so you make an ansatz Of course that you say I have a velocity given a real number V And you say I look for solutions of the form u t x is equal to say some profile u v of x minus v t And this is simple to see that this now satisfies an equation of this form Okay, there's a minus here, but it's Inessential and I have to be a little bit more precise what I mean here I mean a solution which is an h dot One half going from the real line To the unit sphere, okay, and there's a certain boundary condition At infinity I fixed Say this south pole if you wish maybe, huh? Okay, this is always understood implicitly One thing which I will completely skip is that you might ask for a regularity theory for this equation Of course, we want to ultimately that any solution is a smooth solution because the arguments I will present you Will in some way need this kind of property, but I will completely skip this and just mention that Joint work with my former post doc. I'm in chicora you get with this smooth Okay Naturally, of course, it's it's not obvious at all because the perturbation So to say that you add is also first order operator and this is a massive So to say change of the equation we know already smoothness for the v is equal to zero case, okay But let's leave this issue aside for a moment. So What you can do first is of course you can try to cook up a Solution, okay, how could you maybe try to do that first? There's one way of course you can think you attract this variation only this is critical points of the at least formally Of the energy And then you have to find another functional which is a side condition Maybe it's such that the left hand side comes From that side condition in v is a Lagrange multiplier. Okay Call this functional p of u v Constant which in a sense would correspond to a linear momentum, but this is a tricky business in this setting I will not write down you can in fact find a function at least formally so that you get this the problem is as you will see Is that both functionals show a conformal invariance property and hence Discussing minimizing say sequences is not so simple. It's a little bit like the plateau problem So it might be a little bit painful to attack this problem variationally, but in principle it should be doable Okay, one thing. So we won't follow this road Second road is of course an implicit function theorem at least for small v You might be able to construct in the neighborhood of these Whatever are the the half harmonic maps at least slow moving traveling waves, okay, but We won't do that. So there's a lucky punch here Which does the job Okay, at least you get solutions You make an ansatz so you start with a half harmonic map which corresponds to static solution and You can find such things like just living say in the equator plane. So the third component is zero, of course then I have to make sure that this point also lies on that plane, but it's fine. Okay And now you make an ansatz that for u v And that's sort of a magic thing here But you will see a geometric reason for that later Of course, I made a much more general ansatz first because I was I was thinking about kind of mimicking Lorentz boosts However alpha and beta are just constants And if and only if alpha And that it's not true. There's a there's a sign. So if alpha is one minus v squared So this of course assumes that v is less than unity Which means the solitary waves cannot propagate faster than the speed one in my units. Okay What let's assume that for a moment and b is equal to v You get a solution It's quite Fascinating that you get an explicit cheap trick to construct. So to say boosted solutions So it's it's even I don't think so. I don't think so There's no Lorentz invariance. There's no Galilean invariance And the funny thing is also that this Transform which comes from the static to a traveling one is just acting so to say from the outside on the target I mean, you don't have to do anything In the arguments of x. Well, it's just one but yeah, but still I mean you could I mean Okay, let me draw a picture The picture is this Don't you have to adjust the boundary conditions? Huh Okay, I changed the boundary conditions, but I can rotate back so I can always That's everything I say is modular rotation on the target. That's true I I violate the boundary condition, but I could rotate back and then I would Get the same boundary condition, but I I'm gentle about that. So so what I did Is that this half harmonic map and of course f and g are certain special functions? But it's it's a certain kind of parametrization of the great of a great circle Gets in a sense boosted To some other circle Which lives here and in the extreme case when v say tends to one You're just on the get a constant which just lives on the tip of the sphere. Okay Okay So the theorem that I'm going to talk about now says this is the only way to get the solutions Okay, the boosted solutions And gives an explicit characterization of all the possible functions f and g you can have here So theorem so let you Let v be a real number And u v solve this equation makes perfect sense if it's an h dot One-half function and a distributional sense solve that Then you have two cases If v Is bigger and modulus than one or equal to one The only way is that you have a trivial solution. It's a constant. So it's Nothing else can happen And if v satisfies this then What rotations of course On the sphere on the target You have to have that u v of x Is of that form which I wrote Down and now what is f and g? A v sorry this Where f is the real part Say Of a function capital f And g is the imaginary part Is the real part of a holomorphic function in fact On the boundary living on the upper half plane And on the boundary you get the little f and g is the imaginary part Of that function And f Is of the form f of z Is equal to what is called a finite Blush product. So there is a k Running from one to say d And there's maybe a lambda k z minus ak there's a shift of course That is maybe possible plus i And these lambda k's Are real numbers Not zero And the ak's are arbitrary points In r and d is a Natural number of course Yes for v equal to zero you can characterize that but the point is and this is the Point that you can really reduce it to the case v is equal to zero So I will explain that the proof is Really an advancement of understanding the case when v is equal to zero, okay But so can this be understood directly on the on stationary solution of the half harmonic map or unique okay When I look at the stationary solution the half harmonic maps This is not completely easy to see but Doable okay with what's known in the literature in the recent literature say The point is that you can so to say also undo the boost transformation. I will explain that okay So this will give you a characterization of quantization Yeah, yeah, it gives you in particular characterization of all half harmonic maps because it's a case v is equal to zero, okay And there's of course a saying there's no traveling way full speed larger than one. Okay. This is also one point Okay, and an interesting fact is that the energy of these guys is of this form There's a quantization of course involved like for harmonic maps The d the index of the Blaschke product. However by tuning the v you can make this arbitrarily small So it's like the ill too critical half wave equation with patrick girand where you have L2 criticality, but you have solitary waves which have arbitrarily small l2 mass and now you have Solitary waves with arbitrary small energy. Energy is now the critical thing So it's completely different say from the olive shits where you have don't have that. Okay. All right, so I have Roughly 20 minutes That's a true 17 clock counts Okay, let me explain to you about the proof Okay, this is actually two statements a and b. Okay. I will along say something about a And but I will of course focus on the more interesting part b So what's the Initial situation So think first that we are given a solution uv and think it's smooth. Okay to avoid any issues here So that the image Of course, it raises us a closed curve on the unit sphere. Okay, and actually what we Apostrarily see that these are just great circles or in the case when v is equal to zero Or when v is not equal to zero it's it's a circle on the sphere. Okay, but first we don't know anything about this More than this So if you take So rewrite for simplicity little u Is equal to uv. I want to skip that index little v because writing is a bit difficult Okay First thing of course that we do is also to see that you have a conformal invariance Here is of course you consider this as a two-dimensional problem in a sense Because this arises as a boundary of something. Yeah, so you take the u to be the harmonic extension of little u that is capital u is a map from r2 plus Which you can also identify with a complex alpha half plane Into r3. It's a component wise harmonic function Yeah, and the boundary condition Star is a boundary condition then because as you know, it's a classical fact That operator just becomes the Normal derivative with respect to this extension variable or minus the normal derivative So effectively you study the problem minus v of dx capital u And there's a minus so it will change here, but it's Inessential of dy. This is my new variable that I add to my space Like this on the boundary Okay Fine so far so good So by the maximum principle Actually, there's a strong maximum principle that inside is strictly less than one Well, you see out that you the image of capital u is Something inside the sphere of course because of this And the boundary is so to say given by this little u the boundary curve Profile of the solitary wave Okay, so now Comes the first step That says okay capital u is a Parameterization of a surface Modular there might be branching points and these things it's maybe not embedded, but we just say it's a minimal surface Provides a minimal surface That's the first step. So how to see that? The first idea is to use something which is called a hopf differential So it's a function that you cook up Depending on z say So z is just a notation Of course for this And tz Is the wilting derivative That's that's a fairly standard step step here what I do, but I just want to explain it So I remember there should be maybe a four to have everything nice here. So now you take The derivative of capital u with respect to the z. So it's actually just this you combine them. So that's a C three-valued function. It's a three vector with complex entries and you take the scalar product Not the Hermitian product. So Take this product. This is a standard Thing in minimal surface theory So what you get is this it's a norm here There's nothing to do really with the questions. It's the climate here. Okay Okay, so the the upshot is that this is a Because u is a harmonic function. This is a holomorphic function. Okay, so you either check that d z bar The anti wilting derivative of this Is zero hence. It's a holomorphic function. It's easy to see because d z bar times d z is just Times the constant the Laplace operator. Okay, so far so good and now comes the point that You look at that function And you consider the imaginary part of this hopf differential On the boundary, okay So then you use the boundary equation. Okay And now you have to explicitly work out what this Means actually It means In terms of real parts That you have this Minus maybe 2i. I'm not 100% sure whether there's a plus or a minus, but it doesn't matter here This okay, so the imaginary part On the real axis of that holomorphic function in the upper half plane is the scalar product of dx u times Scalar product dy u but now look at the Equation here this immediately tells you that at least when v is not equal to zero This has to be perpendicular to that vector and You see that the imaginary part of that holomorphic function on the real line is zero You can extend it to the whole Complex plane plane the imaginary part by odd reflection. However, the imaginary part Of phi Because on the boundary it's h1 half. So this is actually an h dot one function. So this is This times this is an l1 function. It's integrable So then you can easily conclude that the imaginary part Is a constant it's identically zero. Okay, so because it's a holomorphic function The real part has to be a constant a real constant However, it's also an l1. So it is also identically zero So this hopf differential is actually identically zero fine So that tells you this is always equal to this and this is always zero. So it's Right angle. So u capital u is a conformal map And this is just saying It's an harmonic function with which is conformal hence. It's a minimal surface what you get Now as a next step Yes as a next step But there could be lots of minimal surfaces inside the unit ball Of course, they are very funny ones But they have to meet the boundary In a certain specific way. They have to respect of course this boundary condition here Yeah And when v is equal to zero This is something which I first referred as a free minimal disc Which was studied by Fraser and Shane and they classified that this can only be a plain disc And then you are in the situation That the boundary is a great circle And then you are in good shape to get to get your classification theorem. However, they prove I could not see how to make it really work in the case when v is not equal to zero because you have a more complicated boundary condition, okay But I show you an argument which sort of includes the Fraser and Shane result by considering the following thing So what you learn from that first step? That's a conformal map You also learn that you can rewrite The boundary equation or write little u here In a very interesting way Are people from harmonic maps know such a thing? That you would maybe expect here a square, but this is not true in the fractional case. It's it's not a square It's a one And there's a deeper reason for that because if you test this equation Against u then this term of course drops out and u times u scalar wise is one So you get this Okay, what is this geometrically the right hand side corresponds to the length of the curve And this if you use the harmonic extension the lift corresponds to the area you Spend okay, and you will see that it will actually satisfy an isoparametric inequality Uh, set a set rate an isoparametric. It's a satisfies of course the isoparametric inequality otherwise would be bad Saturates and from that you could also conclude that you have to have plain discs But you don't know yet what the value of the left hand side is but this is just as a side remark So the next step Is and this is then the little innovation maybe is to consider another hopf differential differential Well, what is that take Now this You can now I I leave the details because this is maybe too much on the blackboard Write down what that is again. It's a holomorphic function. You consider Study what's on the boundary. You want to make May maybe run a trick like that, but of course you have to Work a little bit harder, but using this formulation of the Euler Lagrange equation. You also see That this is a holomorphic function, which is identically zero But that's too much a minimal surface, which Of course satisfies the first thing. There are lots of them. However, it also has to satisfy this okay, and then You can use what's called the Weierstrass in a representation of a minimal surface Just want to Say these things here So that you can Did use One gets things right So what's that so it means that you can this is classical thing parametrized the Wiltinger derivative of this u in terms of a holomorphic function f and the meromorphic function which satisfies some Compatibility condition, but if you plug this into this equation you work a little bit You will see ultimately that u the image of u can only lie in a fixed plane in r3 Okay, so so what you get from that finally, I leave the details, of course that the image Lies in the plane In r3, so it's a flat really flat minimal surface So it's a disc so you will ultimately see that this is a disc Huh, and of course the boundary of the disc is a circle on the sphere Now you can undo this transformation, of course and go back to this case where it's the unboosted case Okay, and then you're almost done because now you can invoke some complex analysis Because in the unboosted case say Transform it like this that it's in the xy plane Then you will see with this analysis that F and g have to be say the real and the imaginary part of a complex function F i g on the upper half complex plane with complex values and because U lies on the unit sphere the modulus squared of f is 1 on the boundary Huh, so that's a very special holomorphic function on the upper half plane such that the modulus is 1 identically on the boundary and there's a classification for that in terms of Blaschke products and because also F is In h dot 1. It's not hardy. It's a sobolev space here You can conclude that this is actually a finite Blaschke product Which I wrote down so they only have finitely many factors. Otherwise you would have Infinite energy, which is a bit formal, okay Hence you get the complete classification. I wrote to you and I finally should say that The last idea of using these Blaschke products for this kind of problem is also something that already Mironescu and Pisante Used for something which is related where you look for what's called maybe a half harmonic map from s1 Into s1. Okay, I should pay also credit to this worker somewhat. Okay So what I want to say is now Okay, if you work out these real imagining parts of these finite Blaschke products, it's just kind of rational functions you have So it's very explicit. You can now Study stability or instability. You have very explicit Formulas for your solitary waves, okay And this is also the point I like to end because it's freaking warm here You do have time for questions So the questioner is asking before so can't can't there be as a stationary solution because it's a equatorial one Oh, no, this is what I this is a by-product of the proof So if you have a stationary solution in my terminology v is equal to zero And it has should have finite energy other of these special Okay modular rotations on the sphere, of course, but they all have to Yeah, yeah, yes, yeah This is a Yeah, that's a by-product actually so and and I should maybe say about a I said you're looking for Things of the special form, right? That's why I'm asking the question Who's beyond that I can't have the last component. Yeah, but I cannot okay what I show finally is that I'm Allowed to do this because they're always equatorial discs So I can always rotate it that the last is identically zero Okay, and I should say a is is a funny proof You test the equation against the hillbill transform of you and do some magic And the in the l2 critical half wave case we cannot actually rule out Solitary waves which have speed more than one We don't know how to do this in this model. It's a it's a magic thing to really get the sharp limit for the Do you have two questions? Take me so I can understand. Yes. Okay. Now we have done the job. Okay. Oh, okay. I go home. The next question And just existence of the flow, right? Yes, I have not talked about existence of the flow. Yes, of course Yeah, that's that's a lot of things to do. Yes. Yes, but as a form of level stability Stability, I would expect I would expect that but Also, because formally you see if you think of the Schrödinger map blow up You have an unstable one Because you have a slow decay of a resonance I mean if a slow decay in the linearization a zero mode, which is very slow decay And I think here it does not happen that you have I don't I don't think so, but I would not bad much my I have to look into after the euro Look into the things more closely. Yeah, maybe yes, maybe next week I have time More questions please. So what is your momentum function? Oh, yeah, that is a good question It's a bit like this I draw a picture So the momentum of a configuration u a closed curve Is the solid angle That this curve so to say in the unit sphere Gives you but this is of course not well defined up to four pies because you have ambiguities so so in the physical literature, um, there is a kind of a Dispute about what is whether there is a true momentum or not for such a model But you have a you have a conserved quantity Modular a value of four pie you could consider an exponential of that, but it's not it's a It's a it's a funny functional the momentum is Really not that straightforward say for these models, of course Also, you might wonder whether the traveling solitary waves you get are also existing in the discrete models And this is always a delicate question for nls There are might some traveling Waves in the discrete lattice model or not and so on and here it's also not so clear What survives or what gets created by this continuum limit? Sorry, I was going off the tension Yes, there's there's so much structure. It's actually very beautiful There's a lot of structure and it does come from not just one iterable system, but a class Yes, so what would you speculate I would speculate it's also completely integral, but I don't I mean, I try to to cook up lex pairs, but it's not my Prime education Take some time maybe but I don't know. Maybe there's maybe I'm wrong. Maybe you see really Serious blow up. Yeah I don't know Yeah, yeah, that may be in some weird way. You see like for the chego equation. You might have infinitely many conservation laws In evolution up to a certain kind of regularity and in higher regularity You might as have at least grow up for infant I mean some small kind of infinite type blow up or something like that. I don't know The conservation laws are needed to be Needed to be coercive. So for example, bluzanesca is nicely integral that solutions just don't exist In compatibility, okay. Yeah, okay. You will teach me more than I know it's good Yeah, yeah, it depends on how you define this. I don't know