 Thank you all for being here and also would like to thank the organizers for inviting me to give these lectures. So the presentation will be slightly multimedia, mostly will be blackboard and chalk, sometimes just speaking. And there'll be minor, minor interludes with computer sites. One thing I'd like to say is that some of you may know I gave a 14-hour course on the same subject, which started in May with the floods, maybe that's what brought the floods. Anyway, the notes for these lectures are online at the IHES website, okay? So you will be able to read more if you want to in there. So the talk is, the series of talks is about the solid turn resolution for the energy critical wave equation. So let me start out by saying a few words about the solid turn resolution. For quite a long time, there's been a belief in the community in mathematical physics that one can understand the long time asymptotics of general solutions to dispersive equations in the following way that the long term asymptotics, thank you Fabrice, that the long term asymptotics is given as a superposition of modulated solitons, traveling wave solutions, radiation, which are linear solutions, and small errors. And this is not a theorem or a conjecture, it's more a belief. And this belief has been named the solid turn resolution conjecture. But it's not a conjecture, it's more of a collection of conjectures. In each particular instance, there's a different version of the conjecture. So this, I think, was first postulated in work of Kruskal and Sabuski in the mid-60s, who did some numerical experiments which led them to this realization in connection with the correct risk equation. And these numerical experiments were really motivated by the pioneering ones of Fermi Pasta Ulam at Los Alamos during the Second World War, which was at the beginning of when you could start doing meaningful numerical simulations. Now this conjecture seems to me to be a very remarkable assertion because you start with a nonlinear dispersive equation. And a priori there's no reason to believe that there's any way to tell what the long-time behavior will be. It could be completely chaotic. But this conjecture says that, in fact, there is a simplification. If you wait long enough, things sort of resolve. And then the long-time asymptotics looks like a superposition of these basic nonlinear objects, which are solitons, traveling waves, and a linear object, which is the radiation term, and something that goes to zero. Until not so long ago, the only cases in which this conjecture had been established rigorously for Hamiltonian systems was in integrable situations. So integrable nonlinear equations are nonlinear equations which somehow can be reduced to a collection of linear problems. And in those cases, there was a mathematical proof of this soliton resolution conjecture. In some examples, like the corrective risk equation, the modified corrective risk equation, and the cubic nonlinear Schrodinger equation on the line. So in the mid-2000s, Frank Merle and I developed an approach to study the long-time behavior of solutions to dispersive equations in critical problems, mainly below the ground state level. And it was when studying one example in this direction, namely the nonlinear wave equation, that we became persuaded that this was a good example in which to test soliton resolution. And so then we started a long-time project with Tomadou Kaer, and this led to a number of papers, and eventually to a very substantial result in this direction, which I'm going to talk about in these lectures. So let me start by describing what the energy-critical wave equation is. So I will work in R3 cross R. Recently, I was giving another mini course, and there was a moat under, and I kept losing the erasers until in the end I could no longer erase. Anyway, let's hope it doesn't happen here. So this is the energy-critical nonlinear wave equation. In the three-dimensional case, the results that I will be discussing in these lectures are mostly also true in dimensions 4, 5 and 6. I will not discuss that here. When there's something that's truly specific to R3, I will mention it. Now there's an associated linear equation, which is a linear wave equation, which is a classical equation. So we have a right-hand side H, an initial data V0 and V1. I should say what H1 dot is. H1 dot is the space of functions which have a gradient in L2. So it's a very simple space of functions, and L2 is L2. I don't think I want to define L2. Maybe one is in L2. So this linear wave equation, of course, can be solved explicitly by the Fourier method. And I'm using functional notation here. The sign of the square root of the Laplacian in Fourier variables is the multiplier sign of absolute value of Xe. So, of course, given this formula to solve U solves NLW by take H equal to U to the fifth. So this is the integral equation holds where we have a solution. And I will sometimes abbreviate things by calling this linear operator S of t of V0, V1. And the other thing that I will be doing many times is I will write, let's say, rho of t to be the vector, the t u of t. Now, one thing that is extremely useful in everything we will be doing is the finite speed of propagation for the linear wave equation. So I'll draw a picture, so finite speed. So suppose that support of V0, V1. And suppose that H is 0 in there, in the tent. Then V is 0 inside the tent. This is a standard finite speed of propagation. And this is true for any dimension. In three dimensions, there's the strong Huygens principle. So let me take, so this is D equals to 3. This will be X0. This will be R. And so in this case, we take that the support of V0, V1 is contained in Vx0, R. Then we take this cone. And the usual finite speed of propagation gives us that V is 0 out there. But the strong Huygens principle, I'm going to run out. Okay, let's pretend that this is a slope one. This is where the support is. That's what the strong Huygens principle tells me. So there's nothing here and there's nothing here. And that's only true in 3D. And the difference between 3D and 2D, you can imagine the following. When you throw a pebble in a pond, you see all the little ripples. And they continue to see little, little ripples forever. But when you're in 3D and you see a supersonic plane go by, you hear the sonic boom for one instant and then that's gone completely. So that's what's happening there. Okay, so the last thing, can people see at the bottom here? Yeah, okay. So the last tool that we need in order to give a meaning to these equations are the so-called triggered estimates. It says the following. So, plus. So you have a control of the energy norm as time goes on in terms of the energy of the initial data L0 and the H1 cross L2 norm, I'm sorry, and the L1, L2 norm of H. So this notation means I first take the norm in X and I take the L1 norm in T of that norm. Here I take the L10 norm in X and here and then the L5 norm in T. Okay, so this term plays the role of a solvable estimate in the theory. So you can think of it as some extra norm that you control that has the same homogeneity and it's just like a solvable type embedding. So now I'm going to just say a few words about the well-posedness theory for NLW. Suppose I take an interval I, there exists some delta positive such that if I is contained in R is an interval of time and look at the linear norm but I restrict the time to I is less than delta, then there exists a unique U solving NLW and the mapping from data to solution is continuous in the appropriate norms and there's uniqueness among those things for which the L5, L10 norm is finite. Moreover, the linear equation and the linear solution and the nonlinear solution are very close. Maybe soup in T, well for any T and this is for T in I. So you have a unique solution that depends continuously and remains close to the linear solution just by keeping this norm small. And this is, we refer to this norm as the dispersive norm. Okay, spacetime norm, scattering norm, dispersive norm, norm-strickers norm, we will call it the S of I norm. Now how do you prove this? You just do a standard fixed point argument. There's nothing more to do. So an immediate consequence of this are the following things. Small data, yield, global in time solutions. If you can't read my handwriting, please let me know. Okay, and I'll try to do better. And this global in time solutions scatter. That means that there is u0 plus minus u1 plus minus such that, and this is h1 cross alpha. Okay, so what scattering means is that eventually the behavior becomes the behavior of a linear solution. That's what the dynamics is, okay? Why is this true? Well, because of the strickers inequality, if this is small, then this is small, and therefore we can apply that theorem. Now, next, the other thing you can say is that for general data, which need not be small, there exists a u which is continuous, which is with values in h1 cross l2, solving and which is in l5 of i prime l10 of x for all i prime compactly contained in i, such that u solves the equation and i is maximal. So many times I will refer to the maximal interval of existence. Okay? Now, a very important tool in what follows will be played by something that we call the perturbation theorem. Long time perturbation theorem. Suppose 0 is in the interval i and we have a u bar, which is continuous from i with values in h1 cross l2, bounded the strickers norm. It solves some kind of inhomogeneous wave equation, possibly inhomogeneous. Then there exists an epsilon star which depends only on m with a property that f is small and I take u0, u1 close to u of 0 in h1 cross l2. Then there exists a unique solution u, little u of nl w in i, such that it's close to u in h1 cross l2 and in l5 l10 at time 0 it equals u0, u1. So this is called a long time perturbation theorem because this epsilon star does not depend on the length of the interval. So the interval could be as large as we want, okay? And this is a key technical tool. Is there some mistake here? What does capital U solve exactly? And so U solves the same equation? No, U solves the nl w with no f, okay? Yeah, that was a typo there. Okay, so it's a way to do small perturbations, but for a very long time. So the next thing I want to explain very briefly is why we call this the energy critical equation. And that's because it's critical with respect to the scaling. If U solves nl w, u lambda of xt, which is lambda to the minus a half, u of x over lambda, t over lambda, also solves nl w. And the norm of u lambda at 0 of h1 cross l2 is exactly the same as the norm of u at 0 in h1 cross l2. So this scaling doesn't affect the h1 cross l2 norm, okay? And that's why we call this equation energy critical. This is the energy space, and the scaling doesn't affect the norm of the initial data in the energy space, okay? So in particular we cannot make ourselves have small data by scaling because we can't change anything. All right, now I'm going to do slides for just a few minutes because there's some other object that I need to introduce. And that's what we call the profile decomposition, which is another tool for the nonlinear wave equation. So are we ready? Okay, so this is a tool that we use in connection with concentration compactness in the study of nonlinear wave equations. This profile decomposition for the nonlinear wave equation was introduced by Bahoury and Gerard in the late 90s in connection with the wave equation and by Merle and Vega in connection with the Schrodinger equation. This was more or less simultaneous. But we'll be concentrating on the wave equation here. And so this, you can think of this embedding here as some kind of sobular embedding and there's some lack of compactness in that embedding because there's an infinite dimensional group of transformations that leaves the quantity invariant. And this is a way to understand how this compactness fails exactly. What's the defect of compactness? So here we have the profile decomposition. We have a bounded sequence in h1 cross l2 and for each j we have a linear solution and then we have parameters. Lambda jn is a scaling parameter, xjn is a space translation, tjn is a time translation. So these are sequences of parameters and we call such a sequence orthogonal if somehow the parameters don't see each other as expressed by these conditions. I have to stress that all of this is in the notes that are online so you don't really have to take notes unless it helps you learn this stuff. But all of this is part of the stuff that's online. So we call this a profile decomposition of this sequence if the parameters are orthogonal, which means what we just saw when we look at the linear solution with data, the original sequence. Minus the sum of the modulated linear solutions. By modulated we mean here that they are translated in space and time and rescaled. And the rescaling is the one that we saw leaves the scaling variant. And now something good has to be happening here. Otherwise this is meaningless. The thing that is good is that when you take the difference between your linear solution and these particular linear solutions modulated then the error will be uniformly bounded but in the dispersive norm it will go to zero. So the error is small not in the h1 cross l2 norm that can't be done but in the dispersive norm. The l5 l10 norm. And the Bahoury and Gerard prove that for any bounded sequence some subsequence verifies a profile decomposition and the error verifies this smallness in the dispersive norm and you can get some other norms that also go to zero. For example something that's not the energy norm but it's the l6 norm and we know that l6 is controlled by the gradient being in l2 and 3D by the ordinary solve of inequality. So this is something again slightly weaker than the energy norm. But it's strictly weaker. Now how do we construct these profiles? You construct them as the thing that is over there. Is there a fly? No. There is a fly, okay. I'm not having visions, alright. So how do you construct these profiles? What you do is you take an orthogonal sequence of parameters and then you modulate your sequence of solutions you take the appropriately time-translated linear solution and that converges weakly to h1 cross l2 to each profile. And the fact that this holds can be seen to be equivalent to the fact that this sequence which went to zero in the dispersive norm also goes to zero weakly in h1 cross l2. And now a very important feature of this profile decomposition are these Pythagorean expansions of the energy. This gives us the linear energy can be expressed as a sum of the linear energies of the profiles plus the error. And the same can be said about the l6 norm and therefore also for the nonlinear energy you have a Pythagorean expansion, okay. I'm going to skip this. And now all of this is about linear equations but we're dealing with nonlinear equations so we need to introduce the notion of nonlinear profile. So what is a nonlinear profile associated to the linear profile ujl and sequence of parameters lambda jtj is a solution of the nonlinear wave equation such that minus tjn lambda jn belongs to the maximal interval of existence of uj and the difference between the linear and the nonlinear profile as the parameter n goes to infinity tends to zero. So that's what we call the nonlinear profile, okay. And now it's easy to see that there's always a nonlinear profile after extraction in n. And we can always, okay, so we're in a setting in which all limits exist. All real limits exist because they could be plus infinity or minus infinity and by existence of the limit we mean up to subsequence. So if I have a sequence of real numbers up to subsequence it will either go to plus infinity or to minus infinity or to some finite limit. So that's why all sequences of numbers converge. Don't tell that to your calculus students, please. Okay. So we can find the nonlinear profiles and in the case when the time of existence is minus infinity or plus infinity you can see from the construction of the linear profile that the nonlinear of the nonlinear profile that it must scatter forward in time. And now we will also consider modulated nonlinear profiles and this nonlinear profiles what are we going to use this for? They turn out to be building blocks for solutions of the nonlinear problem now. And there's a technical statement that I will flash once and I will then refer to as the approximation theorem. Okay. And from now on I will be speaking about blocks. And when I say block I mean a nonzero nonlinear profile. Okay. And if you really want to know what the meaning is you go to the notes. Okay. So this is the approximation theorem. I have a sequence. I've taken a sub sequence so I already admit the profile decomposition. Now I look at the nonlinear solution. And I'll consider first the easy case all the nonlinear profiles scatter forward. Okay. Then I call R remainder to be this difference. The nonlinear solution minus the sum of the blocks the modulated blocks minus the linear solution with data W and then this remainder goes to zero not only in the dispersive norm but in the energy norm. Okay. So for all intents and purposes I can think of this solution as being made out of this sum. So it breaks up my solution into blocks. And this holds in this case for all time. In particular all these solutions exist for all times. Okay. And the second part of the theorem is the one that you more often have to use unfortunately which is maybe not all of these guys scatter forward. Maybe some of them scatter and some of them don't. And so as soon as we pick a time such that the modulated solution the modulated nonlinear profile stays away from its final time and all the space-time norms remain uniformly bounded then the same thing is valid. So the conclusion of the approximation theorem is that you can go up to times that give you uniform boundedness of all the space-time norms of the modulated profiles. Okay. And this is how we will use this thing. And now let's make it go away and let me now claim that I've taught you about the approximation theorem and the nonlinear profiles. So this will be the last audio-visual presentation. Okay. Okay. So let's bring back my chicken scratches here. Okay. Is it okay? Now we turn to two important concepts here. The concept of focusing and defocusing. Now this nonlinear wave equation has an energy which is a constant of a motion. Now write down what the energy is. So this is the energy, the nonlinear energy which is an invariant of the motion. That means that for each T in the maximal interval of existence this energy is constant. Okay. So what's the matter with this energy? There's a minus sign, right? We could have negative energy here. Okay. So why is that? That's because the Laplacian which gives rise to the kinetic energy and the nonlinearity have opposing signs. The Laplacian is a negative operator and in front of the u to the 5 I have the plus sign. So the two things compete with each other. From the scaling point of view the Laplacian and the u to the 5 have the same strength. That's reflected in that scaling. And so there's a competition between the two in the case when there's a negative sign. And this is the focusing effect. Now we could also have defocusing. The defocusing sign, right? So what does that mean? Instead of having u to the 5 I put minus u to the 5 in the equation, right? Everything that I've said so far works the same way for that. And for the... So this is the focusing and for the defocusing you get the plus. That means that there's no competition between the two. They just help each other. So this equation, the defocusing equation was studied extensively in the 80s, 90s and beginning of the 2000s. So there were a huge number of works in this very important works. I'll mention some names starting with Juergens then Struve then Grilakis then Chata and Struve then Bahuri and Gerard and Bahuri and Chata. And the offshoot of all of these works Kapitanski also had an important work here. The offshoot of this huge collection of works is the following statement. For the defocusing equation if I take any data in H1 cross L2 smaller or large the solution exists for all times and scatters. So the behavior is linear at infinity. So that's the statement of soliton resolution in this case. There are no solitons. You just have the linear term, the radiation and that's the end of the story. There's no nonlinear dynamics. It's just linear dynamics. There's no blow-up in finite time. But of course it took a very, very long time to prove this, a lot of works. But those guys did it, so fine. We have to move on, right? That's what we do. So now let's start the... So now I'm going to never again mention the defocusing equation. Let's start with the focusing equation and let's see what happens here. Because as I said, everything up to now works the same for both cases, focusing and defocusing. So the first thing I'm going to show you is that for the focusing equation there's finite time blow-up. Or before I say this, I want to say something about the finite speed of propagation for the nonlinear wave equation. If you remember, I said that you construct the solution of the nonlinear wave equation by a fixed point argument. Therefore, it can be solved by Picari iteration. And if you think about what that means you can inductively prove that the nonlinear wave equation has finite speed of propagation. Because in the wave equation there's something with the right-hand side also. And we inductively prove that it's zero in there. So that finite speed of propagation still is true for the nonlinear wave equation in the interval in which it exists. Of course, if it doesn't exist you can't have finite speed. So we leave that. However, what's definitely not true is the Strong-Huygens principle for the nonlinear wave equation. Because that gets destroyed by the inhomogeneous term. So now I'm going to talk about finite time blow-up. The first thing that you do to understand finite time blow-up is I'm going to forget about the Laplacian. I'm just going to consider functions of t. So then I have the ODE and I can write u of xt to be, there's a number here that you have to write. So that function solves the nonlinear wave equation. Why? Because it solves the ODE dt squared u equals u to the fifth. What happens to this guy at t equal to one? It goes boom, right? It explodes. Now you could complain rightfully that this doesn't have finite energy so maybe that's the problem. No, because there's finite speed of propagation. I can chop it off at time zero. I can chop it off here and then up to time one in a very long bit I will be equal to that by finite speed and therefore by chopping it off I can make it to be in the energy space but I can't fix the blow-up by finite speed of propagation. And so there are solutions that have the property that the limit as t tends to t star of h1 cross l2 is infinity. And this we call the ODE blow-up or the type one blow-up. Before I forget, I'm sorry you may wonder why I put this perturbation theorem let's backtrack a little bit. Why did I put this perturbation theorem here? Because this approximation theorem that I flashed is proven using this perturbation theorem. This is what you really need to. So, okay, let me erase that. So what is very interesting in the energy critical case is that there's also something called type two blow-up. So what is a type two blow-up? These are solutions which after their final time remain bounded in h1 cross l2. But still t plus is finite. That means that the solution cannot be continued beyond t plus. Okay, so how is this possible? What is it that's happening? So what's happening is that the gradient of the solution squared is concentrating like a delta mass. Okay, so that's what the meaning of this type two blow-up. And so therefore the solution cannot be continued continuously across that blow-up time. Now I write the definition but I don't say, you can say to me why do you do that? I mean, do you know that this happens? And yes, I know that it happens. So examples of this type were proven first by Krieger, Schlag and Tataru in the four-dimensional case by Illeret and Raphael. Let me not put dates. In the fifth-dimensional case by Jasek Gendritsch who just defended his thesis last week. So there's already striking differences between the focusing and defocusing case. There's finite time blow-up and it can be of two different varieties. So what happens at time plus infinity? There are solutions which exist for all positive time, let's say but do not scatter to a linear solution. So the first examples are solutions to the nonlinear elliptic equation. So now instead of forgetting about d dt squared instead of forgetting about the Laplacian I forget about d dt squared. Now since I have a solution to the elliptic equation and I regard it as a function independent of T I get a solution of the nonlinear wave equation just automatically. Now who are these solutions? So let's consider nonzero solutions otherwise it's not an interesting class and we will call that q belongs to sigma. The first example is this guy. So they do exist. This is a well-known solution to the nonlinear elliptic equation. Now this nonlinear elliptic equation was very well studied in the late 80s beginning of the 90s maybe even early 80s, even 60s. Anyway in connection with the Yamawe problem in differential geometry. So the Yamawe problem is the problem of whether you're given a compact manifold in dimension three and higher whether by conformally deforming the metric you can find one of constant scalar curvature. So this problem was solved by Ovan in high dimensions and then in the lower dimensions by Rijkschen and in order to solve these problems this elliptic equation was crucial and in particular this solution was crucial. I think it's the set of nonzero solutions. And I made a mistake. It was three not six there. Okay, so there are a few things I want to say about W before we go on. First of all, of course there's also translates and scaling of W. Well, let me scale. That also solves the same equation. This is the scaling here. The first property that I want to mention about W is that plus and minus W lambda are the only radial solutions. These are the only radial nonzero solutions. And this is a combination of work of Pohorzajev and Giedes Nierenberg. You get that. The second thing I want to say is the translates and scalings of W are the only non-negative solutions. These are the only non-negative solutions. And this is a result of Giedes Nierenberg. Okay, you can tell me well maybe there's no others. There's no other solutions. Why are you bothering with the notation sigma? No, that's not the case. There's infinitely many, there's a whole continuum of variable sine solutions to this equation. They are not classified. Nobody can write them all down. But we know that the possible energies are a continuum. And this was proved first by Ding in the 80s. And more recently there are more explicit constructions due to Delpino, Mousseau, Pacar, and Wei. So there's a whole zoology of solutions here. And we don't know them all and we don't know what they look like. We do know some uniform decay bounds. For instance, this one is the one that decays the soulless. The other decay is also polynomial decay? Yes, they always have polynomial decay. There is a conformal, I don't want to talk about that, but there's an inversion like a Kelvin transform. And that shows that they are all polynomial. And there's a unique continuation for them. So they can't vanish in an open set. Okay, so this solution is called the ground state. Why do we call it the ground state? And there are two reasons. I don't know, I'm going to use this. Let's see how it goes. Yeah, but it's rough. It makes a weird sound. So I'm not sure, Frank. The only problem is I lost the other one. I have no choice. Okay? Yes. Okay, ground state. The two reasons that I'm going to give are connected. If q belongs to sigma, then the energy of q is bigger than or equal to energy of w, which is positive. So w has the smallest energy among all possible non-zero solutions. And the energy of w can be computed. It's a number you can express in terms of pi and gamma functions. Okay? I don't want to write it down. I don't remember it. Then there's another mentioned before, the shovel of embedding in three dimensions. If you have a gradient in L2, you're in L6. And that embedding has a best constant that we call C3. And w, and it's this family plus or minus, gives you the extremals, the unique extremals for that inequality. And that's due to oba and talenti. Very good. So let me now state in which way this ground state plays a role in determining the scattering behavior. Okay? And this is what we call the ground state conjecture or the ground state theorem or... So this is the theorem that Merle and I proved right now a long time ago. The paper up here, they know 08. Suppose that I look at data u0, u1, whose energy is strictly smaller than the energy of w. Of course, w is constant in time, so the d dt of w is 0. So that's why I write w, 0. Then three things can happen. First one is that the L2 norm of the gradient of u0 is smaller than the L2 norm of the gradient of w. Then the solution exists forever and use caters. The second thing that can happen is that the gradient is bigger than the gradient of w. Then the solution blows up on both sides. And the third thing that can happen does not happen. No u0, u1 such that you have equality. So the third possibility is not a possibility. Okay? That is ruled out by this assumption. So in the energy space, if you are below the energy of w, and that is a fixed number that we can write down. It's not some epsilon, okay? It's a number. Then you can decide whether you live forever and scatter in both times directions or you blow up in both times directions. And this is the result. So that's why this, we call the ground state conjecture because it tells you what happens at the level of energy below the ground state. Okay? Yes? Yes. If I don't write, I'm in L2. Okay? But don't think that I'm consistent. Okay? Okay, so we have a few more minutes. And the first thing I want to say is how do we know that this w does not scatter? Okay? You could have asked me that. You should have asked me that. Why doesn't w scatter? Because if you have a scattering solution and I integrate it over x is less than 1, I fix a box. This tends to 0 as t tends to infinity. If you scatters. Because for linear solutions, when you confine the energy in space, the result goes to 0 in time. Okay? And clearly w doesn't have this property because it's constant in t. It doesn't go anywhere. And w is non-zero. Positive object. And the same is true for any elliptic solution. No elliptic solution can scatter. No function constant in time can scatter. So I want to say a few more things. So I've lost all of my raises. So there's other objects. So let's go back to t plus infinity. There are other solutions which stay bounded in h1 cross l2 as t goes to infinity. And here let me mention some constructions of such things. There's examples due to Krueger and Schlag where you have solutions that, instead of scattering to a linear solution, they scatter to w. Okay? You start out there near w and near radial, and you do it in the right side of a certain hyper surface in function space, then this happens. You're late to man. Okay. So that's one example. There are other examples which we don't know where they're going or at what speed. And these are constructions of Dorniger and Krueger. Then there's constructions of Martel and Merle in 5D where you go to multiple bubbles. Instead of going to one bubble, you go to multiple bubbles. And then there's other constructions due to Gendrange in similar direction. And then if Jasek is here or not, you're here. Okay. You can ask him. Okay? Finally. I've got five minutes and I will... Yes? Of course, the second situation, your main theorem, you have type 2 blue wash necessary or... No, we don't know. But we suspect it's type 1, but it hasn't been proved. Okay? This is a very good problem. Why is it below the long slope? It's a no. There's an antecedent of this result, for example, that is due to Levine, that if the energy is negative, then there's finite time blow up. And it's not known if it's type 2 or type 1. Okay? So even going other questions? Please don't hesitate, okay? Well, of course, if you overdo it, I'll stop you. Okay. So the next thing I want to discuss are the traveling wave solutions. After all, if we want to prove solid and resolution, we have to identify who the traveling waves are. Okay? So let's take a solution to the elliptic equation and let's take a direction, L, which is less than 1. Now we will use another invariance of the equation, which is the invariance under Lorentz transformations. This is a well-known fact that the nonlinear wave equation is invariant under Lorentz transformations. So let me write that down. So I'll write Q sub L of X, T, and I will write it in this peculiar way. So this notation shows explicitly that these objects are traveling waves, right? They're just traveling in one direction, the direction given by L. But what does Q sub L of X, 0? And this is a horrible mess. This is Q, I'm missing a little bit. L is a vector. X is a vector. This is a number. So this gives me a vector. Yes, the whole thing multiplies out. You want to put another? I can put another. If that makes Frank happy, that's OK. So this formula is a horrible mess. What you should do is choose L to be in the direction of X1, let's say, and make it a scalar L, little L times E1, and write down this definition, and you will see what this is. Yes, is there more? Q, I'll write it, sir. Oh, I am sorry. I'll write it right the first time. Thank you. So now this is a solution of NLW, because this Lorentz transformation maps solutions of NLW into solutions of NLW. And this is explicitly a traveling wave solution. And then I have a theorem here, which is what I will conclude with. These are all the traveling waves. So if you have a traveling wave in a direction L, necessarily L has to have length less than 1, and the traveling wave then is the Lorentz transformation of a solution to the elliptic equation. So that's what I mean by this theorem. So at least now we know how to formulate the Soliton Resolution Conjecture, because we know what the traveling waves are. Okay, and this is where we stop. Thank you. I have a comment. Yes, yes. For the Soliton Resolution Conjecture for the cubic NLS, 1D focusing, I think it's not even that case, no, in general. No, no, no. In the 1D case it is... It will have, for example, infinitely many solitons. Oh yes, oh yes, oh yes. So you don't know... No, you know it generically. Generic data. I can give you the reference. Carlos, increase the reference. Okay, I don't know. I think it's not 100 percent. Well, maybe it is not 100 percent true, but it is certainly suggested. What? Some Soviet author. Okay. For KDB it's proved. Now for NLS I don't know. The level of precision that you want in the proofs. And of course, you know, for Modify KDB you have to include breathers. I mean there's all sorts of things that you have to do, but I didn't want to get into that.