Critical Half-Wave Problems

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Critical Half-Wave Problems
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In this talk, I will a give a survey of recent results about the mass-critical nonlinear half-wave equation on the line. Furthermore, I will discuss work in progress on the energy-critical half-wave maps equation, which posesses some intriguing connections to completely integrabe spin chains and the theory of minimal surfaces
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has and if you intend to do that and here and thank you so much and thanks to the organizers for giving me the opportunity to give a talk and action my original plan was said and the time to talk about critical half with problems which would mean to the 0 2 critical problem that particular always talking about on Monday and another new problem which I will now try to focus on because but they did such a great job it's hard to really had something to celebrate its let me let me explain what I want cast is what phrase is a halfway flaps problem OK so what's that so it's an equation that is extremely easy to write down so let's see and so I was very different words so the which means the cross product in all 3 and you know use unknown functions depending on the time when in trouble said from 0 to capital T and say defined on the body the values next to which we always want things being embedded in office now so this is a reason quite some ' simple-minded at 1st glance generalization of the shredding of MEPs problem and so you just take the sale under Livshits equation the this would be the last class you know it would be minus the Lofoten but minus signs can be easily so to say absorbed by redefining the direction of time this isn't Hambletonian question by the way would show to you soon so I might get some very assigned mistakes in my formulas but they don't matter so that's the only way that matters I think from rock C OK so this is the half wave MEPs equation and so there are 2 reasons why has recently started to work on that 1st of all I've seen this equation before but did so only recently I realized that this has a really somewhat very nice physical interpretation that's the 1 bonus thing about it which I try to make a somewhat clear to you here and the 2nd thing as which I was always also afraid about is that you cannot do much with that that is rather quiet on explicit what you can do compared to what lecturing on maps wave MEPs sequentially a nice explicit structures sales grew variant of a wave maps and so on and study stability and real problems like that but it turns out to be wrong the statement that I was show to you that you can do a lot of the explicitly because I will have my Maine as result today discussed that in the 1 dimensional case which is the energy critical I was shown to you hope to completely classify all traveling solitary wave solutions and an explicit form in terms of so-called blushed products cocaine so this is something that I recently found into something that amazed me and gives me hope to continue this study of this energy critical problem talking so as I said this is an Hambletonian equation so it's an energy functional attached to it which of course but it looks like this some of guess what this is going to be made this half-year if you like that and violated purposes will just sort of congestive erected that this is of course expressible as a double into which will assist the make a connection to a very interesting physical application of this model and the for song bracket attached to their salute you can see that you can't write this equation of as Hambletonian equation of motion is the 1 that is given by functions with values and has to state the components and X and Y satisfy bread structure which is given up this expression this is the year and major latitude symbol and the Delta should so if you if you use the structure of this fall will you see other half with maps equation is formally equivalent to this evolution equation given by wasn't Reckitt given with H became so particulates as a beast of formerly homes of quantity so let me now start and discussed the motivation behind this equation itself 1st of all the idea is to get you to the ii a few years here I mean you have to plug in what this possible structures and you work out what this is by planning the same and then you finally see that this is the question but you I mean if you take for instance here the Dirichlet Intergraph you get the land ownership the question of possible structure the structure some universal thing just the energy function determines whether you have halfway admits equation is the quality of the initial under the militia to give the shredding on maps equation I don't have to rush to get shredding Amabel unleash it's coming from physics likely continuous version of the Heisenberg equations for pheromone system yet so deep 2 1 by the way would correspond to the energy critical cases everything about equal to 2 and higher costs and energy subcritical supercritical sought by the way and the energy critical case also physically most relevant as I will explain 2nd and there is also another thing attached to it it has a conformal invariance property OK so this is the case I will focus on so the motivation as 1 motivation comes on differential geometry only
recently there was some growing interest in what is called a half half harmonic man Back at the same time it is tools OK so that was correspond of course obviously too static solutions to my fellow equation so in particular these static solutions will play also important role for this time-dependent equation of course you might also think about something that if half the money met heat flow which would be a parabolic and to this equation which I discussed here so this is something that was recently introduced the
media and the DeLeo and when particular the
regularity series for these equations like say if you the critical dimension that marriage one-half maps I was actually a smooth that like an analog of analysts regularity theory for the harmonic maps into the consent but it's a different thing because the media use some kind of commutator type arguments to get this regularity result and also from a mall geometric point of view the very reason work the players Shane it is seen that these equations corresponds to a free boundary minimal disk I would come to the plate that is to say it is also naturally connected that you can look for something which is a minimal surface the inside wall the unit ball with a certain condition how the boundary of this minimal surface hits the boundary of the ball which is to and in and the setting tested perpendicular lines you can lead if you see that this problem is equivalent to this to study the free boundary minimal to say about it yes this is a boundary questions will come to this point we see that as a completely natural connection to the theory of minimal surface can I come to this and B. This is the physical motivation behind that of course I will later for traveling solitary waves which will be generalizing these half harmonic maps and the kid in the sense that there is another None 0 right inside the camp and I would classify holds all the solutions and the use of physics this is a difficult thing you have to know the right names to look for because physicist like these names the week watering it stated if you don't know what we give sources are hard to find out in the literature what's this about once you have the right called works it works so the 1st 2 names that someone playing a role in this business holding interest to resistance who wrote down the Hambletonian for Spain chain and I will be the final of course which is some object OK the suspend operators I won't go into any details and bares soul to say interaction among spends this is on a discrete letters in 1 dimension fixed ladders science and X K X K plus prominence for all of the letters sides and to each letter cited teacher quantity and I don't want it to be grouped into details what this is about and they interact with certain was called long-range potential problems that might be assigned squared of the difference of the letters sites or there is also the soldiers under bright mobile where this square off this so missiles rule this and the point about this these are exactly solvable Model and sense you can calculate the spectrum and so forth and you can also find lacks Pearson's for on the on-the-spot long-range quantum chains which have very Nice particular structure on the physical point of view so it's hard to draw these quantum operators do actually but I install plants constantly adjust to indicate that in the southern limiting regime which you can think of to send classical limit also large spending limits you at least formally led to what is called a system of classical spins so then this Hambletonian becomes something which you could face as a color Gerald type Hambletonian with spin which would look like this that formerly at least the spin operators scared replaced there is something of this fall where the best case and now the unit vectors and 3 space cancel in the classical reading spins are given by client arrow which lies on the units in three-dimensional space that's that least for a physicist at typical procedure to go from a quantum spin system to a classical spins so if you just had just nearest neighbor interaction so this wasn't this function but just on the 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 0 would be the classical and trading on equation however ever here you can of course somewhat anticipate that in the next step of taking this limit you get that there's some leads to a double integral of that form that you have a continuous spin variables which are called and asked his this form and we once space dimension making this rigorous makes it of course is 1 step is to study action that the dynamics convergence 7 cents and of course and 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 what unit length the of crossed provided estimates which is the Hambletonian array of left and I want this to match the Kent again this callejera malls that tied the also shares the magic property that you have lacked spare have very explicit structure him and you might wonder what if this survive 64 this limiting Pt there is a strong indication of that because I was show to you an explicit classification of all solitary waves it might be a hint that this is completely integrated system which is yet the energy critical switch shows a criticality which in principle can also say that you had fun at Hambro consultants that so now I come to them all the time so that the producers of the yes exactly in but dimension to enlarge its supercritical and in 1 dimension is energy critical and in principle you can think like foreign energy critical shredding and that you have also brought solutions that in this case you might have because of an integral structure too many conservation laws which forbid such a technique but it could also be like it for the CU cheekily creation that they only live up to a certain level of regularity and you still see some kind of soul left type growth phenomenon which are a little bit counterintuitive 1st if you think of a competition I think difficult to prove system so what I'm here to interested in us to consider traveling solitary waves for this model and this is interesting in 1 respect 1st that the fall the number equation such a thing does not exist any mention their only static final energy solutions they don't cannot create and that traveling solitary waves for the the a question you might think you can apply a living was but this is not working but it's not but of course for the wave MEPs equation you can do with Lawrence post get traveling on wake Wakeman me if you wish now but here this is not the uh Lorentz invariant system so we cannot apply something but you you will see that we get a rather explicit way of constructing these things and I show to you that this is the only way to to get OK so you make an arms out so close that you say I the velocity given a real number you and you say I look for solutions of the frog is equal to inspect some profile UVA X minus the and this is it's simple to see that this not satisfying and questions of this fall at the end of this reminds him in inessential and I have to be a little bit more precise what mainly I mean a solution which is an H . 1 half going from the realigned units failed
him and there's a certain boundary conditions at infinity fixed South Pole if you wish to have this is always understood implicitly 1 thing which I will completely skippers that you might ask for a regularity the reformers equation of course we want ultimately that any solution is a smooth solution because the arguments of represent you will in some way need this kind of a property but I will completely skip this and just mentioned that joint work with my former pollster ,comma Jakarta because this month naturally at of course it's not obvious at all because the perturbations so to say that you had also 1st solid operator and this is a massive to say change of the equation we know already smoothness for DEC equal which is 0 case but that's if this issue aside from so what you can do 1st of course it can try to cook up the solution of How could you maybe try to do that 1st as 1 way of course you can think you check this variation of this is critical points all the at the calling on the energy and then you have to find another functional which is the site conditions may be as such that the left side comes from that site conditions views allotments multiplied on him but but call this functional he used the concert which in a sense would correspond to a linear momentum but this is it tricky business in this setting I will not write down you can infect find the function at these formally so that you get those the problem is as you will see is that both functional also show a conformal invariance property and hence uh discussing minimizing say sequence is not so simple it's a little bit like the couple problems so it might be a little bit painful to attack this problem variation only but in principle it should be to attend 1 thing so we won't follow this road 2nd raucous applause an implicit function theorem at least 4 small movie you might be able to construct in the neighborhood of these aware the the half harmonic met at least slow moving traveling but we won't do that so there's a lucky punch here which does the job working at least you get solutions you make an undeveloped so you start with a powerful monochromatic which corresponds to the static solution you can find such things like just living standards make it a crate of playing some of the 3rd component is 0 of course then we have to make sure that this point also license plate but its and now you make an answer notes that for UV and that sort of magic thing you know but you will see a geometric musical that later Of course they made them much more general that's 1st because I was thought fault but was thinking about kind of mimicking Lawrence whose while the alpha and beta largest and if and only if are found under the vessels there's sign so if all it's one-liners the square for this of course assumes that these less than unity which means the solitary waves cannot propagate faster than the speed 1 and my units OK look let's assume that people will want NBC to controversy you get a solution that's quite fascinating and how to get an explicit Cheap Trick to construct so to say boosted solutions but that is the increase in the budget said the list I don't think so and I think there's no Lorentz invariance there's no Galilean invariant and the funny thing is also that this transform which comes from the static to a traveling 1 is just acting sources say from the outset on the target I mean you don't have to do anything in the arguments of places of unit of still I mean you could mean the OK let me draw pictures beaches this you have just got OK I change the boundary conditions but they can rotate back so I can always that's everything I say is module rotation on the target that's true I violate the boundary condition but they could retain and then I would still get the same conditions school and gentle so so what I did is that this half a morning Mass and of course Levangie of certain special functions but it's it's a certain kind of parameterization of the threat of a great circle gets in a sense was that To some other circled the ministry and in the extreme case when the state tends to 1 just on the better the constant which lives on the tip of the state of OK so the theorem that I'm going to talk about now says this is the only way to get the solutions but can't boast solutions and it's an explicit characterization of all the possible pharmacies and you can have but if you prefer and you can't let's do the media will man and QVC all of this equation brakes perfect sense of it's an one-half function in his solution since some of them then you have to cases if the gene is bigger and modulus than 1 equal to 1 the only way is that have a trivial solution it's a constant so nothing else can have and if but this has this then what rotations of France on the sphere on the target you have to have the view of use of X of that formed Major rolled down there and now what is anything do you have where did people up good is the real party St. of financial capital and includes the imaginary part it's the real part of of the whole market functions effect on the quality of living on the upper half plane and on the boundary you get a little less ingeniously imaginary part of the function and after but is a form of the as for what is called a final lashed
product this caper running from 1 to the CD and there's maybe along the K C minus K there's a shift of course that this may be possible plus and East London cases in a real numbers net 0 and a case arbitrary .period an and the D is a the national the head of the yes for of the crew 0 you can characterize that the but the point is
and this is the points that you can really reduce such to the case of the secret 0 so I will explain that the proof is really an advancement of understanding the case when equal to the so in the next few years old stations social go call on case when I look at the stationary solution half a monarch maps this is not completely easy to say but the doable came with what's known in the literate in the recent letter said the point is that you can't sort of also untrue the Bush transformation I will explain how can this be Is this is the year yeah it is still in particular characterization of all have a monarch maps because it's a case of music with 0 OK and there is of course a saying there's no traveling wave speed larger than 1 hurricane this alter 1 .period OK and an interesting fact is that the energy of these guys because of this fall that is a conversation of course involved like 400 maps the daily the index of the blast the products however by tuning the V you can make this arbitrary small so it's like in 2 critical half wave equation with particular way have a L to criticality but you've a solitary waves which have arbitrarily small L 2 mass and now we have a solitary waves was arbitrary small energy Energy is not the critical thing so it's completely different savings Lipschitz when you have don't have that kind of all right so I have roughly 20 minutes from the 2 17 clock counts OK let me explain to you what proved OK this is actually 2 statements a and B. How can I will along says something about 80 I want but I will of course focus on the more interesting part B so what's the initial situations so thing 1st that we're given a solution you and think it's moved cocaine use of what many issues so that the image of course traces also closed curve on the units the market and Exley would we up Osiris see that these are just great sockets or in the case when these equal to 0 or when the is not equal to 0 it's it's a circle OK but 1st we don't know anything about this moment it's so if you take story right for simplicity Lilly unit is equal to you VII 1 escape that index that because writings but the 1st thing of course that we do is also to see that you have a conformal invariance is of course you consider this as a two-dimensional problem in a sense because this arises as a boundary of something also you take the you to be the harmonic extension of little you know that this capital you a matter from March to France which you can also be identified the complex of a plane In 2 or 3 it's a component wise harmonic function yet and the boundary conditions start is a boundary condition and because it's you know it's a classic effect that operated just becomes the normal directed respect to this extension variable -minus the normative so effectively his study the problem minus the year of the excess capital you and as a minor so it will change here but as in inessential of GUY this my new variable the efforts to my space but missed on the boundary of the time so so by the maximum principal that Unisys strong maximum principle that inside strictly less than 1 what you see out that you and the image of capital you it's something inside the Sphere Of course because of this and the boundaries so to say given by this little you the voluntary code profile of the solitary way OK now it will be opened the 1st step in that sense of technology you'll parameterization office surface module on there might be branching points in these things is really not in bad that we just say it's a minimal provides a minimum surface it's the 1st step it's hard to see the 1st ideas to use something which is called a half differential so it's a function that you could up depending on the same Susie is just notation cost for this and Daisy is the village in the and that's that's a fairly standard step step here what I do but it just 1 explained so I remember there should be really afford to have everything nice here so now you take the derivatives of capital you with respect to disease so it's actually just this you combine them so that still ahead c 3 valued function it's a 3 vector with complex entries and to take the scale-up products but the Hermitian products so take this product is a systemic thing and minimal surface here what you get is this tacitly warned him nest nothing to do really with the questions the climate here working on so that the
upshot is that this is a because you was a harmonic function this is a hollow Norfolk function of came so you the check that the Zebari weekends Gooding a derivative of this 0 hands it's a whole Norfolk which it's easy to see because the zebari times Desi is just in time to constantly look possible direct OK so far so good and now comes the point that look at that functions and to consider the imaginary part of this whole differential on the boundary McCain so when you use the boundary question him and now we have to explicitly work out what this means actually this means in terms of real hearts pledges for this -minus maybe too I am not 100 per cent sure whether that's a plus or minus but it doesn't matter here that's opinion so the imaginary part on the real axis of that whole Norfolk function in the upper Plains is the scale of product of the X you times scale product you want you but now look at the the questioning of immediately tells you that at least 1 it's not equal to 0 this has to be perpendicular to that victory and you see that the imaginary part of that whole Norfolk function the realigned 0 you can be extended to the whole the complex plane plane the imaginary part but all of reflection plan the imaginary part of finding because of the boundaries H 1 have so this is actually an H . 1 function so this is this times this season 0 1 function integral when you can use the concludes that the imaginary part to constants identically 0 OK so because it's a whole market function the real part has to be a constant a real constant however it's also no one's soul it is it's also identically 0 so this whole differential it's actually identities you'll find so that tells you this is always equal to this and this is always 0 also it's right angle so you Capital USA conformal met and this is just saying it's and harmonic function with which is conformal hence it's minimal surface which now as the next step this as a next step but there could be lots of minimal surfaces inside the human ball of course they're very funny ones but they have to meet the boundary in a certain specific way they have to respect of course this boundary conditions and when these equal to 0 this is something which I 1st referred to as the 3 minimal disk which was studied by Frazer and shame and they classified that this can only be a plane disk and then you're on the situations but the boundaries of great and then you are in good shape to to get a classification that you have however that proved I could not see how to make it really work in the case when he's not to 0 because you have more complicated boundary condition but I show you an argument which certainly includes the phrase attention result by considering the following things so what you learned from that 1st step that's the conformal that he also learned that you can rewrite the boundary equation on right union the very interesting way a people from harmonic maps Noel such a thing uh that you would maybe expecting square but this is not true and the fractional case it's it's not the square 1 and there's a deeper reason for that because if you test the Sequoyah equation against you In this time of course drops out and you times you scalar why this 1 so you get this OK what is this geometrically the right inside corresponds to the length of the curve and this if you use the harmonic extension lift corresponds to the area spent care and you will see that it will actually satisfying as a parametric inequality set is saturated nicer parametric is satisfied with questions of primitive the quality of would be better uh saturates and from that you could also conclude that you have to have claimed disks but you don't know yet what the value of the left scientists but this is just a society so the next step this and this is standing in the invasion and is to consider another hopes differential differential well what is that take note of this you can now I leave the details because this is maybe too much on the blackboard write down what that is again as the whole market function you consider considered study what's on the boundary you wanna make my maybe run trick like that but of course you have to work a little bit harder but using this formulation of the other branch equation you also see this as a whole Norfolk function which is identically sick but that's too much and minimal surface which of course satisfies the 1st thing a lot of them it also has to satisfy this weekend and then you can use what's called the highest costs and upon representation of a minimal service just once In these things so that you can use them in which 1 of its principal will so what's the answer so it
means that you can use classical thing parametrized the built-in a derivative of this you will in terms of the whole market function fundamental attention which satisfies competently condition but if you plug this into this equation you work a little bit you will see ultimately that you with the image of you can only lie in the fixed playing in all 3 of them can only do what you get from finally only the details of course that the image size in the plane and in all 3 of the flats and flat minimal so it's a disk so you will ultimately sees that this is a disk now and of course the boundary of the disk is a sucker on the the now it can undo this transformation of course and go back to this case was unmoved suitcases of and then you almost done because now you can invoke some complex analysis because in the UN boosted cases way transformative like this that it's in the X White Plains then he was see with this analysis that s and only have to be say the real and the imaginary part of a complex functions have about half complex plane trees complex values and because you will lies on the unit's Fiore the modulus squared off Air Force One on the boundary so that is very special homomorphic 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 plastic products and because also ever it's an H . 1 it's not party it's that's all been Yukon conclude that this is actually a finite Lashkar problem which I will down so there only a finitely many factors otherwise he would have infinite energy which will the form if I called hence you get the complete classification of you and I finally should say that the last idea of using these blush products for this kind of problem is also something that all ready pool and kiss-and-tell used for something which is related to where you look for what's "quotation mark maybe half harmonic maps from this 1 into a swamp of conventional consecrated to the work some of so what I want say is now OK if you work out these real imagine parts of his final blast opposes just kind of rational functions you have so it's very explicit you can now studied stability or instability of very explicit formulas for your solitary waves came and this is also the point elected and because it's freaking won him his 2nd thank you allusions under the question that was also the year he has the solution to the all in all this is what I have to say this is a byproduct of the pool so if you have a stationary solution right terminology these equal to 0 and it has should have finite energy on all Al-Qaeda module rotations on the field ,comma but they all have to do it all again the of all is this is that no that's a byproduct actually and and I should maybe say about 8 absolutely opposed sees this morning the soldiers who were yet but I'll cannot locate what I show finally is that I'm allowed to do this because they're always Equatorial disks so I can always rotated that the last is identically 0 with him and I should say is is a funny proved you test the equation against the Hilbert transform of human to some magic in the in the L 2 critical half case we cannot actually will solitary waves which have speech more than what we don't want to do this in this model it's magic thing to really get the show limit for the introduction of 2 if you think that this is only a few days all appeared on the wish that you see is using his son the yesterday I have no not talked about the existence of the fuel you have caught him and that's a lot of things to do this the stability I would expect started with a phone call away from the clothing for this I would expect that some also because it formally you see if you think of the trading on that brought you have an unstable 1 because the office slow decay of residents don't mean a facility came the linear reservation 0 mode which is very slow decay and I think he it does not happen Jefferson I don't say I don't think so but I would not say that much might have to look into after the euro the head of the man who took it to the principle of the various minutes we got time the 1st is it's only if Mama and that is a good question it's a bit like this I federal picture so far the momentum of a configuration you closed curve is the about solid angle that this curve so to send the units here gives
you but this is of course not well-defined up tool for pies because of ambiguities so um so in the physically literature and there's a kind of a dispute about what is whether there is a true momentum or not for such a model but you have to have a conserved quantity module 0 value of for part you could consider an explanatory but it's not it's it's the essence of funny functional momentum is system really not that straightforward say for the small of course also you might wonder whether the traveling solitary waves to get all the existing in the discrete models and this is always a delicate question for analysts there might some traveling waves in the discrete letters model or not and so on and futile to it's clear what survive so what gets created by by this continue perhaps going attention he is the structure of of the review of the structure and it does come from the United electoral Systems plan also will be used sustainably I would speculate itself completely and but I mean I try to to cook expense it's not mine prime education so take some time the other day I don't know maybe there's good maybe I'm wrong maybe you you serious and blocked him I don't that yeah yeah yeah that may be in some weird way you see like for the we question might have infinitely many conservation laws in abortion up to a certain kind of regularity and in and higher regularity you might as head as least grow for infant and in some small kind of infinite time was something of a response to the views of the reform of the EU's investors-mostly mostly users of this in no way of knowing what the outcome you will teach me and I know there was no difference on how you define the please go ahead the From operations the idea of a few B