 So first I would like to thank the organizers for inviting me. It was a pleasure to be here. I've been here for almost two months, and unfortunately my time is almost over. So I will talk about a joint project of work with Ivan Martel and Claudio Munoz. In fact, the paper is already on archive. And well, let me, first let me tell you what's the problem, what the problem is. So the problem is a very simple one-dimensional problem which is known in, let's say, this culture or this context as a 5.4 model. And maybe before I continue with this model, let me make a little digression about general PDEs which share similar properties as far as the nonlinear structure. So people consider two equations. So there's an elliptic equation or parabolic equation. And can be considered in bounded domains or in the whole space. So let's think of these problems in the whole space. So often either of these equations is referred to as the Allen-Kahn equation or by stable. OK, so it's very easy to see looking at the nonlinearity where is it called by stable. So it has three zeros. How it goes this way, it goes like this. OK, it has three zeros. And there, in fact, this is a very bad picture. The point is that this area is equal to this area. And also, oh, this is the other way around, I inverted it. But well, anyway, it's clear that these extreme zeros are supposed to be stable. So these kind of equations are studied a lot. And there is a lot of literature about them. And they are nice in a sense that they have certain connection with geometry. So the parabolic equation is very related to what is called mean curvature flow. And the elliptic equation is related to the minimal surface theory. And they've been studied from the 70s. I think that George was the one who introduced them to this sort of mathematical PD, nonlinear elliptic PD and parabolic PD community. So I'd like to see this phi-4 model as a part of a bigger family. And there is, I think, a affinity between those theories on a certain level. So it is convenient sometimes I use it often to write the model as a system. And then I will abuse the notation. So for me, phi sometimes is a scalar and sometimes a vector. And this is all real and everything. And so it's obvious that this equation has two conserved quantities that are very important. It is the energy and the momentum and both play a role in what I will say in deep way. And as for the energy, I just put here w, which is a quartic polynomial. And so this is where the name phi-4 model comes from, because the fourth order equation. And in fact, I don't come from this culture. So I learned about this problem from a paper written by Witten, in which he mentions, among other models, this model and argues that, in fact, there is a deep reason why this potential w should be of this form. This is a fourth order equation, fourth order polynomial, and not something else. There are physical reasons for this. And another context in which this model appears a lot is if you open a book on quantum field theory, then typically first chapter is devoted to the phi-4 model and sign Gordon equation. So the basic object people study, and this, in some sense, is common also for the Allen-Kann equation, is the kink. In fact, in the quantum field theory, also everyone studies at the beginning the kink. So it's something very simple. In this case, it's an explicit solution. It's just hyperbolic tangent and a solution to the ODE. And of course, well, it's not unique unless I say that it should be increasing an odd, for instance. Otherwise, it's not, because I can translate it. Or there is a symmetry. The whole problem is symmetric, is odd symmetric. So if I look at the wave equation, then the kink generates two-dimensional invariant manifold, or a piece of an invariant manifold, at least, just by Lorentz transformation. So if I have a kink, I can change it. For every y, y is a constant and sees the speed between minus 1 and 1, I have a solution to the problem. And so basic question is, what kind of behavior you expect around this type of solutions, of the flow? What will happen? So first thing to describe a little bit is orbital stability. So just to make compact somehow the notation, not to avoid maybe too many letters, what I will denote the kink suitably translated and rescaled and its derivative. So we can think of h and h prime as initial data associated to the solution simply, initial position and velocity. And so then I can look at the perturbation of this initial data. And then it's not so difficult to see that in fact I can, without loss of generality, I can consider perturbations in such a way that the speed c corresponds to the initial momentum and the perturbation v0. v0 is a vector. So its first component is l2 orthogonal to h prime. And this is very easy to do, because if I start with any initial data near x, I can first choose c. And then I can adjust y in order to satisfy the orthogonality condition. So it's just a matter of choice. So that's a convenient choice. So OK, so maybe I'm a little bit pretentious because I attributed this theorem to myself and Ivan and Claudio. But in fact, we have not found this theorem before. And this is simply orbital stability of the kink. But maybe it's a part of the folklore, this I don't know. So if you take a solution with the initial data as I described, which is additionally small in the energy space, then the global solution, as I should say, the global solutions for this problem existence, it's not very difficult to show that that's known. And the global solution will be such that its perturbation v will be orthogonal always to h prime. And the momentum is fixed as it should be. And then, moreover, it is small in the energy space. So this is simply the orbital stability of the kink. So I will not talk about the proof. The proof is rather easy. It just uses the conservation of energy and momentum and the spectral property of the kink. So in fact, there was a previous, I should say that there was a previous proof of orbital stability by Henry, then Henry Peres and Brzezinski in maybe 83. And they prove that the kink is orbitally stable, also in the energy space, assuming that c is equal to 0. So that proof is maybe a little bit more general because they prove it for other non-linearities as well. But on the other hand, the fact that c is equal to 0 simplifies a little bit the argument. In fact, the argument is almost trivial with that. Now, I mentioned the parabolic problem. So the parabolic problem is very well understood. And in some sense, it's not trivial, but it's well understood. So if you take the parabolic version and you consider a kink, then kink is asymptotically, exponentially asymptotically stable. And in fact, if you think of, let's say, initial data that changed sign once, then this initial data will evolve to a kink and then move in some way. And these things are known from maybe the 90s already, 80s and 90s. Then as I mentioned also, a classical example of an integrable system that is, well, 5.4 model is not integrable, by the way, which is somehow considered together with the 5.4 model is the sign Gordon equation. So this is the equation. So some of you may not like the sign, but in fact, it doesn't matter the sign here. And this equation has solutions that are interesting, several solutions that are interesting. In particular, it has a kink that is explicit, has a solution that's called a breeder, and something that's called a wobbling kink. So let me show you how this. So a breeder, in fact, is not a single solution. It's a family, it's a one parameter family, and the same with the wobbling kink. It's a one parameter family. So a wobbling kink, OK, so here what I have are the red curve is the kink of sign Gordon. And then I have snapshots of a wobbling kink with certain choice of parameters. The other curves are the snapshots and go over the period. And notice that first of all, well, the way I choose the sign Gordon is such that the kink changes between 0 and 2 pi. And it is if you take pi as an axis, then pi minus the kink will be odd. And the same with the wobbling kink, pi minus the wobbling kink will be odd. And also notice that asymptotically, in fact, the wobbling kink and the sign Gordon and the kink for sign Gordon are indistinguishable. Their asymptotics is the same, at exponential rate. And here I have a wobbling breeder. So these are snapshots of a breeder. Breeder oscillates around 0 before sign Gordon oscillates around 0. So 0 happens to be the sort of stable state of sign Gordon. A breeder oscillates here and here. So somehow you can think of a wobbling kink as a combination. It's not quite superposition. You cannot say that. But it's a kind of combination of a kink and a breeder. And of course, a wobbling kink says that kink is not asymptotically stable for sign Gordon. And then somehow people believe that the breeder causes it in some way. So if you have a nonlinear Klein-Gordon equation in which you have a breeder, then maybe a kink, if there is a kink in this equation, should not be stable. So breeders by themselves, I think, are very, very delicate objects, very non-generic. And so Densler proved that they do not persist under small generic perturbations of sign Gordon. In fact, they are kind of co-dimension one perturbation. So there is only one type of perturbations. And then he claims also that even for those perturbations, breeder doesn't persist. And it's not very easy, in fact, to exclude the possibility of existence of breeders in general. But there is a result that I found, maybe it's from the 80s as well, of Willer Moe, who shows that certain class gives a condition for nonlinearity so that breeders do not exist. And with the yes. Just because I think it's obvious to you that a wobbly kink and a breeder are time periodic. Oh, I didn't say that. But let me go back. Yes, they are absolutely time periodic because of the sign. But wobbly kink too? Yes, absolutely. Absolutely. Yes, yes, yes, yes. That's a count of the number of degrees of freedom which they can move. Absolutely. Which is one. Absolutely. So these are periodic. Yes, I didn't say that. Sorry, maybe this is obvious to me. But indeed, these are time periodic solutions. So that's why they are. Can I ask another question? Yes, tell me. What is the definition of the result? Ah, OK. This I don't know. But it's one of these things that when you see it, you know that this is it. So what does your answer mean? OK, yes. Well, again, this is it. So what I mean by this is solutions that are periodic near zero. Small solutions periodic in the energy space, OK? Because you see, there's not enough space here if I want to make these transparencies. I had to use a kind of shortcut, OK? It's a very simple. In fact, this follows from our method that I will explain later on for asymptotic stability result. It's a rather simple, I would say, even observation. For a very special kind of Klein-Gordon equations that preserve odd space, so that nonlinearity must be odd, you don't have small odd breeder. So this means if you start in the energy space, there is no solutions that are small in the energy space and time periodic, OK? So now, by the way, for the Klein-Gordon, breeder is not odd. It's even. So of course, this is not a contradiction, OK? So somehow odd breeder's are objects that cannot exist, OK? So finding wobbling kinks for the 5.4 model, in fact, it was people, I think, worked on it somehow in the 80s and tried to find sort of solutions that would look like wobbling kinks and breeders. And there is some work of Cigar and Kruskal and Cigar. In particular, Kruskal and Cigar sort of showed that there are no small breeders for the 5.4 model, but for them, small breeder means a breeder that oscillates around 1 or minus 1, OK? So now let me talk a little bit about what is known about the 5.4 model. So I gathered some results among many, which I found relevant, maybe, and interesting because they look at the problem from various angles. So first, there is a couple of papers, or maybe more, maybe not couple, maybe two or three or four papers, of Komach and Kopyleva. And they analyzed nonlinear Klein-Gordon equations that look like 5.4 model, but they are not 5.4 model. I mean, they consider perturbed version of this model, perturbed nonlinearity, that do not include. In fact, the perturbations are, in some sense, not small because they severely perturbed nonlinearity, OK? But then they were able to show stability of the king in the odd space. So they consider only odd perturbations, which is the context of our work as well. Now, in higher dimension, OK, there is a work of Kukarnia from maybe 2010. And he considered the planar front. So what you do is you take the king and you consider it in, you lift it to three dimensions, OK, you consider it in three dimensions in space. And then you look at the asymptotic stability of this object. And he showed this asymptotic stability. And it is actually important that the dimension of the space is three, and it simplifies a lot because then you can use dispersion, OK, in complementary dimensions, let's say. So in particular case, two and one are more difficult, I think, from this point of view. And also I wanted to mention the work of Gerard that I kind of outlined a little bit the result, which is not so easy maybe to explain, but it is a kind of local existence result, OK, in higher dimensions. It's a construction, better say. It's a construction of local solutions, local in time, that look like a king around, OK, centered around certain minimal surface, OK, in Lorentz space. OK. So now let me talk about our results so just to introduce a little bit. So I will talk about King perturbed only by odd perturbations. This is very important. And why it is important? Because you know that, OK, I said that the King, you can boost it, OK, you can let it move. But if you only consider odd perturbations, there is no motion. So C is 0. So this is this way you take out one degree of freedom, OK, one modulation. And this simplifies, I think, analysis in a certain way. OK, so you look at the odd perturbations and immediately from nonlinear problem you pass of the previous form, you pass to a different kind of equation that you take a linearized part and the nonlinear part. And so now the problem is not Klein-Gordon equation because it has a potential. So it's different, OK. And then the problem is reduced to asymptotic stability of 0 in the odd space. So this is the problem I will study. And this is something important that I will use a lot, which is the form of the linearized operator, OK. This linearized operator is classical. And you can say a lot, if not all, you know about its spectrum. And so the result is the following. So if you take perturbations in the odd space, energy space, which are sufficiently small, OK, then the global solution with this initial data goes to 0 in local energy space, OK. So the norm here is the whole space, but here it's on a bounded interval, OK. So there is, of course, OK, we don't give any rate. But on the other hand, in some sense, if you restrict yourself to the energy space, in some sense, this is an optimal result because, I mean, in a sense that you can only take local norm, OK. Because if you were able to show that it converts to 0 in the whole, in the full norm, then, in fact, you didn't show anything, OK. So in order to explain a little bit the result and how we prove it, I have to talk about the spectrum of the linearized operator. So this is, as I said, it's explicit. You have 0 is the obvious eigenvalue that corresponds to translations. And there is another eigenvalue, which is very important for its eigenfunctions, very important for what I will talk about, because it happens to be odd. And then you have a continuous spectrum. And 2 is a resonance, but the resonant eigenfunction is even, so it doesn't play a role in this, OK. So as I said, you have the translations. And then you have first mode, non-zero mode, which is odd, OK. And this is important. So in order to understand the dynamics, what is convenient to do is to decompose the solution, OK, into a discrete mode, the odd mode that I described, OK, multiplied by some function of t, and the rest, OK. So we have radiation. And this part is called internal oscillations. So notice that this eigenfunction is localized, OK. It's localized around the kink. That's why people call this mode internal oscillations, OK. So there is the theorem is the following under the assumptions of the previous theorem. The following, you have the following estimate. So this z1, z2 to power 4 integrated in time. And again, it's a kind of local energy, a weighted energy norm of the radiation part integrated in time. It's bounded by the initial data, OK. So this even you can see as a kind of weak form of asymptotic stability. In fact, the previous result follows from this, OK, by an additional argument, which is, I think, not very difficult. This is basically the important result, OK. So this information that is given in this theorem, it seems weak. But as I said, in the energy space, it's rather difficult, I think, to get something better, OK. One would have to go to another context, a functionality context, OK. And the basic approach to prove this theorem and the previous one, the stronger result, is based on virial estimates, basically. And this was motivated by the work of Martin Merle for KDVE and Merle and Raphael. I mentioned especially these two works, because I think you'll see at the end that they are very relevant somehow, OK. But there is a difference, and the difference is to take into account internal oscillation. So this is a new element, OK. And then, OK, I should say that general results for this type of problem, OK, say that basically maybe you can improve this integrability, but not too much. I think it's rather difficult to prove that you have L2 norming time bounded, OK. So in order to say about what is, because, OK, I want to explain the proof, but I also want to explain what is going on. What is the mechanism, OK. And so in order to do this, I think it is fair to talk about the paper of Sofer and Weinstein, which somehow started this business, let's say, in a certain way. It's a paper from 1999. What they did is they considered this type of nonlinear Klein-Gordon equation with cubic nonlinearity with a potential, OK, in three-dimensional space. So the potential, of course, is introduced just to avoid translations, OK, to fix the object they were interested in, in space. And what they were interested in is similar questions, the asymptotic stability of 0. So remember that I reduced already my problem to asymptotic stability of 0. So it's very similar, OK. And they make certain assumptions of the linearized operator. So the assumption is that it has one discrete eigenvalue, and then there's a continuous spectrum without resonance. So you see it's the same, abstractly. Also they make another assumption, which is called the Fermi-Golden rule. So Fermi-Golden rule, I think, was previous. Well, it comes from spectral theory, OK. And I don't really understand why it's called Fermi-Golden rule, because in the spectrum, if you open a book of Reed and Simon and you look what this Fermi-Golden rule is something completely different. But these people, OK, they had reasons to call it like this. So I follow this. And I will not write it, OK. I just say that the Fermi-Golden rule implies, in particular, that 3 times omega is bigger than m. So you have m here, and you have omega, and there is algebraic relation between them. And in fact, it seems stupid, but this is very important, this algebraic relation. So what is the result of Sofer and Weinstein, OK, something like this? They also decompose the solution into internal oscillations and the radiation in a similar way. And then they show that the internal oscillations decay, OK, in infinity, in fact, in time, with the rate. And the radiation also decays in L8, but they give a rate. So what is the, I think when you understand the strategy of the proof, you also understand what is going on. Let me explain the strategy of the proof. So the strategy of the proof, I think it's best to get rid of many nonlinear terms and just stick to the core of the problem. And the core of the problem is the following. You have a coupled equation, so for the radiation and for the internal oscillations. So this is a nonlinear PDE, OK. And this is simply an ODE. It's just nonlinear oscillator. And so their strategy is something like this. You first solve, given A, you solve this equation for eta. You plug it into the second equation. Then you have just a nonlinear oscillator and you look for normal form of this nonlinear oscillator and then you find the decay of A and then the decay of eta follows from that. And so by the way, you see that you have A cubed here. And this A cubed is very important. This 3 omega bigger than m has to do with this cube. If it was 4, then you would need 4 omega bigger than m and so on. And it's important in this strategy, I think that the space is three-dimensional, so that you can use Duhamel formula and dispersive estimates and close estimates. So in one dimension, this strategy is not so easy to follow, especially if you work in the energy space. So now let me explain what is the strategy that we propose. So let me recall the problem. So this is the problem with this kind of linear operator. And well, I decompose, as I said, according to the spectrum of this operator. So this is the step I already described. And then you project this problem into Y1 and it's to complement. So if you do that, you get two equations. First, you have the finite dimensional nonlinear oscillator and then you have the infinite dimensional part. So again, what I do is I systematically omit certain terms that are of higher order. So you have to believe me that they can be controlled. And what I leave is the core of the problem. So the core of the problem is to understand how these two equations are coupled quadratically because they are coupled quadratically. And so somehow we think, OK, well, maybe I need to decouple them in some way. But this is impossible. I think this is where the problem is that you cannot decouple them. You have to work with them. Or you have to understand the way they're coupled. The function f here, by the way, is explicit. It's a nice decaying function. And this function plays a role later on. So the idea, basically, is to somehow reduce in a way. OK, so Y1 would like to find normal form, perhaps. But this is not so easy. Turns out I think it's quite complicated. But you can find something like a normal form that is a. So let me explain this. So OK, so what is a normal form? Because for me, normal form is a form of the problem in which you can understand how the modulus evolves. Also, you kind of don't have the face. So this is what I, so to do that, OK, is change variable. So introduce new variable alpha, which is simply, OK, this is the real part of z squared. If you look at z as a complex number, and beta is the imaginary part, OK, and you make a change of variables. And this change of variables cleans a little bit the problem in such a way that you can write the infinite dimensional problem, just in terms of new function v, that is basically like u. But then in alpha, OK, which is the square. And then you can write also a problem for alpha and beta, OK, the choice of this change of variables makes it possible. And then you see that the problem is not non-linear anymore. That is, I have certain non-linear terms that I can control. And then I have a linear problem here and a linear problem here. So now it's at this moment that you can use virial identities, OK? And so to use virial identities, so it is the objective of virial identities. The objective is to show that this object, OK, this integral in time is bounded. And it is important, OK, when we do all this analysis, it's important to have orbital stability before, OK? Because it allows, in fact, to control a lot of non-linear terms, OK? So the virial identities, the first part is something that everyone knows, I think. The second part is maybe not so obvious, but it's kind of natural, OK, because you exactly mix the way you would like to do, alpha and v2. And then, OK, in both of these identities, the point is that you have unknown functions, psi and g, OK, and you have to choose them in such a way. So for the choice of psi, there is a natural choice. Because you want psi to look like x. So you can choose a function that approximates x, in some sense, because if you choose x, this object may be not integrable, OK? But for g, there is no obvious choice, OK? And this is something you have to work out. But anyway, if you plug this in, OK, what? If you differentiate, sorry, if you differentiate this, oh, by the way, notice that in any case, these two objects, these two functionals are bounded, OK, because of the orbital stability. They're small, in some sense. So now you differentiate them, and then when you differentiate, you get a bilinear form on v1. And you get some mixed terms, OK, that involve alpha and v1, and some explicit function h and f, which I defined before, which is explicit, and this function g that is not explicit. And so the point is now to choose psi and g and use the oddness in such a way that this bilinear form is bound, well, it's coercive, this is the whole thing. And in order to, it turns out, well, that in order to choose the function g is where we use the Fermi-Golden rule. So in order to, it's not so maybe, I don't want to explain it in detail, it's a kind of involved calculation, but at some point you see that in order to choose g, you have to solve an equation. And this equation can be solved in a way you want to do it only if you have Fermi-Golden rule, OK? So it's hidden somehow, but it appears. And then, OK, once you have this, this is not enough because you're still missing v2 and beta, but this is easy. You just compute. Once you have alpha and beta, you find two other virial identities, and you compute, and integrate, and so on, and you get it. So this is not so difficult. So now I will let me talk a little bit about the coercivity of this bilinear form, OK? So as I said, in order to choose psi, you have to choose some sort of approximation of x, OK? And then it's natural, well, it's many choices, but we choose psi to be this function. So it's related to the king, OK? And, well, I think this choice is quite good. In some sense, it works very well, and I'll explain why. So OK, so you have this bilinear form. Notice that if you like, OK, this form had to change a little bit because of this psi prime factor here, so you have to change variables. So the change is very natural. You just take square root. What you would like is square root of v1 instead of square root of psi prime v1, OK? You have to change variables, OK? With this zeta is the square root of psi 1, which is the secant hyperbolic. When you change variables, well, you magically get a new functional. And this functional is nice. It has a potential. This potential is explicit. You can compute what it is. And then it's a basic fact that we prove is that under the oldness assumption and orthogonality, this bilinear form is coercive, OK? And so now let me, well, I think I finished earlier. So let me just explain a little bit how we work out the coercivity. So in fact, this bilinear form is, you see, this potential is quite complicated. So it consists of two parts, one part that I hit in B2, OK? B2 is not so maybe it's kind of involved, but it's not so important, not so difficult, but it's kind of messy. But B1 is very nice, OK? B1 is a bilinear form that is associated. Oh, there is a W missing here, sorry, W squared. B1 is a bilinear form associated to a classical operator, OK? And then, well, in fact, you can find, you open a book of, well, devoted to this, I don't know, TeachMars or some classical book. And you find this kind of operator. And you find it's spectrum computed. And you can see that this bilinear form is non-negative. And in fact, this is the operator, you just rescale it. And this is where I, at some point, I mentioned that we used the approach of Merle and Martel and Merle and Raphael. And so not only that we use the approach, because this is, OK, we use the approach, but also at the end, everything is on the same kind of classical Schrodinger operator. So I think there is a very interesting kind of connection and structure that is behind this that maybe it's still to be understood, OK? So I'll stop here. Thank you. Are there questions? Could you compare with this signed organ case? What's the one loss that really exists? Oh, what allows Breeder to exist? I don't know. I think it's because. What's the case with someone? Yes, spectral structure is no. No, it's not similar. It's OK, OK, OK, OK. Yes and no. There is no internal mode. So for signed Gordon, you have zero and then continuous spectrum. So it's different. And the edge of the continuous spectrum is a resonance, but it's odd. So it's quite different. Yes. So if you were to remove the oddness assumption of the initial data. So of course, you get the zero, which is a symmetry mode of the Lorentz transfer to find so this can be more or less handled. But you would also get the resonance at the edge of the spectrum. And this is supposedly more serious? Yes, I think it's way more serious. Yes. I don't think it's just technicality or something like this. I think it's a serious difficulty. You can understand why. You just write the problem, OK, and you have a resonance, which looks like a constant. We'll project formally on this. You see the things are oscillating in certain. And then they are as close as you want to the energy space. It's very difficult. I think it's complicated. Yes. The relation between the basic estimate as it goes to group 32 and positive commutator because sometimes it somehow looks like those choose functions. Sometimes something looks like a call out x in order to make a commutator positive. OK, it's a little bit beyond my expertise to answer this question, but also call it virial estimate. It's a virial estimate, yes. But OK, it has to do with the momentum in some sense. It's just this. And if you look at, you know, if you formally do it with x, it just works nicely. OK, thank you.