Thanks very much, first of all, for the invitation at the workshop and also for being here

at least until 5 o'clock, which is a heavy job. OK, so in any case, this is a joint work with Ricardo Montalto, currently at Zurich University. And OK, so the system is the same that was described yesterday by Jean Marc. So the Euler equation for a fluid,

mid-dimensional fluid, and the periodic boundary conditions, a little bit y. So the fluid is below the graph y is equal eta of x.

It is ruled by the Euler equations. And in the case of any rotational velocity field, one velocities field has the gradient of phi, which is a velocity potential, if the fluid is assumed

to be incompressible, hence it is an harmonic function. And the action, so here we put infinite depth, could be done also with phi depth, but here it is. And then, OK, it is the fluid that

moves under the action of gravity, which is g. And then there are surface tension, so there is capillarity effects at the free surface. And so as this was discussed yesterday, the Euler equation boils down to this Bernoulli equation,

the continuity of the pressure at the free surface. And then one also assumes the last equation, which is a kinematic condition, which tells it the particles which originally were in the fluid on the surface, remains forever on the surface. So there are no breaks in the wave.

And of course, so the unknowns is to know how the fluids evolve. And so to know its velocity potential phi. And this is as it was reminded yesterday, so it's an infinite-dimensional Hamiltonian system.

The canonical variables are the graph, so the function eta of x, and the trace of the boundary of the velocity potential. And in fact, if one knows the eta, so the shape domain, the domain occupied by the fluid.

And then if one knows the value at the boundary of the velocity potential, one finds the harmonic, unique harmonic function in the domain, occupied by the fluid with these boundary conditions. And we shall follow the Zakharov-Krigsland formulation,

which is this one. So eta and psi evolves, in fact, it's the Hamiltonian formulation. Eta and psi, this is the evolved, has the Hamiltonian system. It's the L2 gradient of the Hamiltonian, which turns out to be expressed,

as it was in yesterday, by the Dirichlet-Neumann operator of the surface applied to C, which is a linear operator in psi. And then there is the second equation, which is, on the contrary, quite longer. And the Hamiltonian, it is what one should expect.

Namely, the Hamiltonian is the full energy of the system expressed only through these canonical variables, which are eta and psi. So it is the kinetic energy of the fluid that, when expressed in terms of the Dirichlet-Neumann operator, takes this form.

Then there is the gravitational energy, and then there is the energy due to the capillary forces. Okay, so in fact, this system has, it's an Hamiltonian system, but it has also other symmetries. One, it is that it's a reversible system.

It's a reversible system from the point of view of the Hamiltonian. Meaning simply means that the Hamiltonian is even in the potential psi. This is the symmetry, extremely simple but quite important, as we have also seen yesterday. And so the Hamiltonian H is invariant,

left invariant by the action of this symmetry. Which is this evolution as a square simply changes the sign of psi. And then it is natural to look to solutions which are reversible.

Namely, that are in time are even. And that in space, sorry, in the first component which is the space are even in time and in the velocity potential in time is an odd function. So particularly at the time t is equal to zero, the velocity potential is zero, as it was yesterday.

Then another property is that the water waves vector field leaves invariant the subspace of functions which are even in x. So if this subspace is left invariant, and we shall confine to this case.

So these functions, so for us, always we shall take functions which are even in x. In this case, also the velocity potential phi is even in x. And hence, the x component of the velocity field, namely so,

phi x is odd in x, and so it vanishes at x is equal to zero. And because of 2 pi periodicity, it also vanishes at x is equal to pi. And so the situation is that physically the fluid has the velocity field, whose component, the x component when x is equal to zero always vanishes.

So it is like this, there is no flux of energy of fluid here. The same happens because of oddness and 2 pi periodicity at x is equal to pi. And then the problem that we are studying physically,

it means that really there is no flux of fluid at the walls. x is equal to zero, x is equal to pi. And so really we are studying the standing waves. So it's the evolution of a fluid, we confine it between two walls. And so, okay, these are.

Then as we said yesterday, the mass is a prime integral. Then in the case of infinite fluid, there is also another peculiarity. In any case, so we shall always confine it to solutions. Which have also we shall, so this is preserved and for simplicity,

we put that the average of it is equal to zero, as well as the average of the velocity potential. Okay, so we look for quasi-periodic solutions. We want to say that it is a very typical phenomenon that we should expect in this case to have most of these kind of solutions.

I remind what is a quasi-periodic solution. A quasi-periodic solution is a solution defined for all times, which has a specific form. So it is the composition of, I call the n frequencies and basic independent frequencies.

So it's a composition of a function u, which depends on new angles. So these are new angles, which belongs to Tn. So the function u is a 2 pi periodic in phi one,

2 pi periodic in phi n, composed with the linear flows, namely so each of these variables have a rotation with frequency omega one, omega two, and omega n. Of course, so when n is equal to one, there is only one frequency. These are periodic functions with frequency omega, period 2 pi over omega.

And so these are the interactions of more superpositions of more frequencies, of circular frequencies with frequency omega one, omega n. So if one has, okay, the frequency omega in order to be really quasi-periodic has to be independent.

So we assume that the vector omega has no integer relations. So omega dot k is different for all integer k in Zn, which are of course not zero.

So solutions of this kind, solutions global, global in time, defined for all times. Then it means that in the phase space, there exists a manifold parameterized by n angles.

So in a certain phase space, dimensional phase space, if the solution, the initial datum starts here, then the solution will remain forever on this manifold. And actually the flow applied to this initial datum will be nothing but, it remains that it is nothing but just the rotation of for the linear flow omega t.

So the dynamics are restricted on these, the dynamics of the initial data restricted on these manifolds is just the linear flow. There's the translations.

And okay, we look for solutions of small amplitude. And so of course it is relevant to know what is the dynamics of the linearized equation at flat ocean eta is equal to zero with zero velocity, psi is equal to zero. We linearize the Euler equation and we get this system,

which is this constant coefficient system, where g of z, the Dirichlet normal operator of the flat surface is just the free multiplier models of the x. And then one look for its solutions. Actually, so these are all its linear,

are all the standing wave solutions, which are even reversible for this linear equation. So there are even in x. So we have expanding the basis of cosine of jx, avoiding the average in space, so j greater than equal to one. The first component in time is even,

the velocity potential in time is odd, psi j are parameters. So the linear equation actually has this structure. All the solutions of the linear equation have this structure.

Out of this form here, even with an infinite dimensional torus, here with infinitely many frequencies, because here I put the series for j greater equal to one. And this kind of non-resonance conditions on the contrary, of course, it will depends on the linear frequencies on the values of the parameters, gravity and surface tension.

According to the values of g and k, there are different non-resonance conditions. And the question that we pose is to know, actually whether these solutions of the linear equation can be continued holds for the solutions

of the full non-linear equation for small amplitude. So the problem is quite involved. And so, in fact, we shall continue solutions which are supported in Fourier space on finitely many modes. Namely, we take finitely many indices,

integers, j1 bar, jn bar. And then we look at solutions which are supported in Fourier on these sites. The other modes are at rest, zero. And then rephrasing in this case, the linear equation has invariant tori,

which are explicitly this one, parameterized by the angles phi 1, phi 2, phi n. Which support the frequencies which are now, in fact, the vector omega of the linear frequencies, which is this one.

Now, it is intuitive. It's also well known from mechanics and since 100 years, even more, that are relevant. Well, no, less because the Kolmogorov theorem is after the second war. But what would be important to know whether these solutions will persist and their perturbation is to know

whether the linear frequencies of oscillations satisfies suitable strong inner resonance conditions. And in particular, if the frequency vector satisfies the conditions for which omega dot and integer l is not only different from zero

but from a quantitative point of view, it has a lower bound of this type here, which is so-called the Dafontan conditions. And the point here is that the linear frequencies that I recall here are... So the linear frequencies depends on g and k and r.

I wrote in the previous slides, so r, j. The linear frequencies... Indeed, one can show that for most values,

for example, of fixing j, for most values of the surface tension parameter, this vector, this n-dimensional vector, actually fulfills these conditions. So that along the curve, omega kappa, the vector indeed encounters a lot of points

where these conditions are fulfilled. So at least the initial... starting point in order to hope for the conductive these solutions will persist also in the non-linear system is satisfied.

So here we are in a situation which we have the linear system which is like an integrable system and the non-linearity for small amplitude solutions is seen as a perturbation. So this kind of problem has been studied considerably in the last, say, 30 years under the name of K-M theory,

which has been quite developed for semi-linear equations by many people. But in fact, we started starting since out of years, motivated especially from water waves, the issue of quasi-linear PD because in fact most of the equations are of this type.

And the first results that obtained with Pietro Baldi and Riccardo Montanto four of K-M type results concerning in fact the K-DV equations plus a quasi-linear small perturbations. And then in fact the further development of this strategy

it is what I will describe in part for the water waves. Okay, so for water waves there are what are already known as some results of periodic solutions. Starting from a series of remarkable works

by Toland, Plotnikov, and Jos Plotnikov Toland in different cases. So when first where the first work was in the case always for standing waves, finite ocean, and in the case of a finite fluid,

the first results proved and without capillarity the first results proved the existence of periodic solutions, finite depth only under the action of gravity. And then Jos Plotnikov Toland in the case of more difficult case of infinite depth, in this case it is more difficult because it is,

okay, there is a degeneracy in the kernel, and then still with only gravity. The case with the capillarity in fact, which is exactly this case, has been solved by Alazar and Baldi, proving exactly for this system periodic solutions in time.

And then there are also some related results of traveling wave solutions, which by Craig Nichols and Jos Plotnikov, which are in higher space dimension. I think that the traveling waves in dimension two, I think is periodic is levi-civit, if I'm not wrong.

But okay, it is not a small divisor problem. And so, okay, the natural question is exactly continuing the result by Alazar and Baldi to look whether quasi-periodic solutions exist. They proved periodic and we won't like to see that.

Okay, so for concerning quasi-periodic solutions, previous works which try to understand what is the effect of derivatives in the non-linearity under four vector fields, where the perturbation contains derivatives, was done by Cookson first and Kappeler-Pechel,

KdV adding one derivative. This is of course still a semi-linear equation because there is just one derivative in the perturbation concern with respect to KdV. And then for NLS, for the Klein-Gordon equation, there was a work by Bourguin for periodic solutions. This is in some sense more difficult

because it is a less dispersive equation and then, it's a bit more difficult, and then with Lukaviansk and Michi-Laprochese, we have given a general result of quasi-periodic solutions for still Klein-Gordon equations with semi-linear perturbations.

Perturbations where the vector fields depends on one derivative in time and one derivative in space. And one has to assume a structure which is algebraic structure, which is relevant, which is reversibility. I have no idea about the possible Hamiltonian structure of this equation, but it is natural to expect a reversible structure.

Reversible structure, it is a structure which is compatible with the presence of quasi-periodic solutions to control the growth of Sobolev norms and as we saw also yesterday. For quasi-linear equations,

in fact, as I mentioned before, the first results we obtained with Pietro Baldi and Riccardo Montalto concerned KdV, plus then one add an Hamiltonian perturbation which has three derivatives, so it is quasi-linear, so which is generated by an Hamiltonian where the density depends again on one space derivative. And then, when writing down the vector field,

the Hamiltonian vector field, one reads that it has the shape, so it is a quasi-linear system. So now let's go back to water waves and I wanted to precisely give the result of quasi-periodic solutions for this system.

So these are again the equations and it's an autonomous system, so the frequencies of the quasi-periodic solutions is not fixed a priori. It will change according to the non-linearity when and according to the amplitude. So if the solutions will tend to zero,

the non-linearity will determine we have effect on the frequencies. So we look for quasi-periodic solutions with frequency omega tilde, which are unknown, which are to be found. And then the theorem is this one,

namely the statement is the usual result that one could expect. Namely, there exist quasi-periodic solutions. The quasi-periodic solutions of this system are done in the following way. As I said, take any arbitrary finite subset of indices,

s, j1, j2 bar, jn bar, then the quasi-periodic solutions are in Fourier space, mainly supported on these harmonics, cosine of jx. The amplitudes are mainly, the main part of the solution

is described by the linear equation. In fact, these ones. But the frequencies are shifted. In fact, there will be a shift in the frequencies, omega j tilde. The frequencies omega j tilde will tend to the linear frequencies of oscillations

when the amplitude xi tends to zero. So when xi tends to zero, the frequencies will converge to these ones. The frequencies omega j tilde, as I said before, has to satisfy strong, suitable, non-resonance conditions. As a consequence, the solutions do not exist

for all the values of the parameters, but they exist for most values of these parameters. And we say like that, that in fact, so for most values of surface tension parameter, for example, considering the gravity g is equal to one, the solutions in fact exist.

So our quasi-periodic solutions and an extremely important property for the proof, not only because it is interesting from a dynamical point of view, but also for the proof, is that we know that these solutions are linearly stable.

I want to explain now what I mean. It is linearly stable in the meaning of dynamical systems. I have slides on it. So this is the typical results that one can expect. One can, of course, also rephrase the results,

saying one can take, for most values of the surface tension parameter, fixed, one fixed equation, for most values of xi, there exist solutions of that fixed equation. Now, some comments, very natural.

Well, the first is that the fact that there are these restrictions is not technical. There are these restrictions on the parameters, is not technical, because otherwise one should expect completely different phenomena. This is the typical situation that one should encounter, say, in Hamiltonian systems in here.

Namely, outside, there are resonance phenomena that destroy the solutions. And there are completely different solutions, kind of solutions. And now, the system that has been described before,

in fact, even probably a formal proof is lacking, but it is not integrable almost surely. I don't know, Walter will comment on it. And for example, also if one looks at the dynamics of the third order system, approximated system,

there can be sources of instabilities and there can be a resonant dynamics called the Wilton ripples. And so, the choice of these values of the parameters, g and kappa, of course, avoids these effects,

is the choice for which one does not see these instability phenomena. And similarly, in fact, as we heard in the talk of Jean-Marc yesterday, for most values of g and kappa,

solutions with an initial zero potential exist for a long time. We don't know if solutions exist for all times. And so, in some sense, in fact, in the previous theorem,

one in fact is selecting initial conditions for which actually the solution avoids forever all the resonances and so is a survivor, in some sense, for all times. Maybe in between, there can be regions where, on the contrary, can be blow up, actually probably, and would be these other kinds of dynamics.

And so, this is the picture, very natural picture that one could expect of the complicated dynamics between stability, Km, and then hyperbolic orbits, horseshoes, and very complicated thing. Okay, so I said that the quasi-periodic solutions

that we found are linearly stable. They are linearly stable really in the sense of dynamical systems, really in the sense that in the following sense. Okay, in a suitable set of coordinates,

they linearize the system that unfortunately I have not written, so I write it. Okay, so in the sense of dynamical systems, I mean that we have an Hamiltonian system, UT gradient of U, and then we have a solution of time. In this case, it is a quasi-periodic solution.

And then we linearize the vector field. We linearize at these solutions, at this solution, the system, at this solution. And then we have an equation which is, of course, linear with time-dependent quasi-periodic coefficients.

So this is a linear operator, which depends on time here in a quasi-periodic way. And the linear stability is that, of course, the dynamics of an object like this

can be very complicated. And the whole point, the main point of all the proof is that we are able to control completely this operator. Actually, we diagonalize it, so we are able to compute the synthetic expansion

of all its eigenvalues, of all its floc exponents, and this is, in a sense, the core of the matter. So because if of the linearized equation like this, I know I am able to control x spectrum. Of course, I'm able to control also the non-resonance that can appear there. And in fact, so the precise statement is that

there exists a suitable set of coordinates, good coordinates, which integrate the system. In a suitable set of coordinates, these very complicated PDE, linear PDE, becomes this type, this constant coefficient system, namely, okay, phi and y are finite dimensional coordinates.

They live on a finite dimensional torus, and they describe the tangential and the normal dynamics across the torus. I call it phi, phi one, these are the angles. But then, let me see particularly

the infinite dimensional part. This is a PDE, so in fact, this must be a PDE. And it means that that PDE, now forgetting the action angle dynamics, is conjugated to a constant coefficient operator,

decomposing V, I mean, in Fourier series, as we also saw this morning. It means that these really are completely diagonalized in these variables, and they are simply harmonic oscillators

with lambda j, I call it mu j, which are, in this case, real. And for which, in the next slide, we shall give also an asymptotic expansion of this.

So, because of this, if this complicated PDE, the linearized equation of the system, has been conjugated to this set, to this constant coefficient PDE, so this set of infinitely many harmonic oscillators,

decoupled, well then, of course, with mu j, which is real valued, then, of course, so for example, one has the stability, the Sobolev norm of this system does not increase. Zero. And I mu ij are what are called in dynamical system, the flaky exponents of the solutions.

The flaky exponents of the solutions, so of course, if I know everything, if I know them, then I completely control everything about the dynamics nearby. And what, in fact, the method provides, an asymptotic expansion of these flaky exponents.

These eigenvalues have the following form. They have the form mu j,

the ampere-turbed frequencies, where I put g is equal to one, so exactly this one, where the gravity is equal to one. Which have to be corrected. They're corrected by a constant, which is close to one,

and when the solution stands to zero, these constants will tend to one. It's correct. These frequencies must converge to the linear frequencies of oscillations. Plus, lower order corrections, there will be another correction, lambda one.

There is, by cancellations, due to the symmetries of the equation, there is not the correction of order j. The next correction is at order one-half. Again, this is another real, small correction. Plus, lower order corrections, real, small.

If one want, the method would provide, at any order, the asymptotic expansion of this. So, again, the Fourier multiplier are a perturbation, of course, as it should be, of the linear and perturbable frequencies.

And that is, of course, a fundamental information, because it allows really to control, under perturbation, the nearby dynamics. The second information concerns the change of variables. So I said that one can conjugate this linear PDE to this constant coefficient, to this system.

Through a change of variables, I have to control this. And the point is that this change of variables actually maps Sobolev spaces, HS, into themselves for any S, however large it is. Why this is a relevant information? Because, so it means that essentially, if I want the information of high Sobolev norms,

well, I can achieve these conditions here, which is simple. And then, by the inverse change of variable, I will have, of course, the informations about the evolution of the initial datum in Sobolev norms, also for this one. So these are the two fundamental informations

that exist. There is a very good spectral analysis of the linearized operator. So the linearized operator is conjugated through changes of variables, which maps Sobolev spacing to themselves for any high Sobolev norm into something, into another system,

which is the diagonalized. And we control in a constructive way, in a very precise way, all the asymptotic expansion of the inverse. It's clear that these informations enable us to control and to overcome the small divisor difficulties.

Okay, this is an easy explanation why this is a small divisor difficulty. Why this is a small divisor problem, I rephrase it. Suppose that we look for zeros of this non-linear function, those of this non-linear operator,

where one looks for an embedding of the solutions, replacing dt with the omega dot d phi. So dt becomes, in quasi-periodic section, omega dot d phi.

So one needs to look for zeros of this operator. When u is equal to zero, because we want small amplitude solutions, we start from the flat ocean with the zero velocity potential. And then for the implicit function theorem, we want to find eta, and the phi has a function of the other parameters,

kappa, omega. And then one is interested in linearizing the operator, which is essentially what we did before. And this operator is this one. Constant coefficients in Fourier space can be diagonalized in this way.

But then of course, we see the difficulty that we mentioned before, namely the determinant of these matrices are these ones. And here, l is an integer, j is any integer, so these numbers always will accumulate to zero. And one can impose, for most values of frequencies omega and parameters kappa,

lower bound on this type, so that the linearized operator is invertible, but its inverse operator, because of the resonance effects, loses derivatives. And so this is the usual small divisor problem. Because of the resonances,

the continuation of these orbits, it is not based on contraction. And then a way to solve this problem is through Nash-Moser implicit function theorem, which is, as it is well known, it is based on Newton's method

for looking at the zeros of a function, given an approximate solution un, one look for a better approximation, un plus one, in such a way that is obtained by the intersection with the tangency. And this is the iterative scheme.

But in fact, and the advantage of the scheme is, as it is well known, it is a quadratic scheme. So the distance between two successive approximation is less or equal than the square of the distance between two previous ones. This quadratic speed of convergence enabled to compensate the small divisor difficulties,

but the difficulty is that in order to write, for example, this scheme, we need to know the linearized operator at a function u, which is not only zero, but u different from zero. And so we are, in fact, encountering for here in this problem here. The problem is to show that the linearized operator

is invertible for most values of the parameters, and that its inverse satisfies estimates, say, taim, namely that they map a Sobolev-Nore, there can be a fixed loss, and it must be taim with respect to the point u, where we linearize. And this is, the difficulty is that when one writes the linearized operator

for the water waves, it has a bad shape, at least, well, this is the form, where, so here, geovit is the division operator, b and v are functions, share the gradient of the velocity potential at the free surface.

Okay, and so it is, the point is, okay, this is exactly this operator here, and we would like to invert and to prove that for most values of parameters and to prove estimates for the inverse. And so the proof is quite, is composed of many arguments.

I will not discuss, of course, all, but okay, there is a Nash-Moser proof that we formulate as a theorem of hypothetic conjugation of Hermann in order to take with the parameters. But then I will discuss a bit more

this part here, namely, the analysis of the linearized PDE, because, of course, this is an essential PDE part. Okay, about parameters, just to say two things. Of course, as we said, all these phenomena, as we all know, since a long time,

really depends on a very complicated way on a counter set of parameters. There are fractals which appear, there are very complicated chaotic dynamics. And so really, it's very sensitive to know how to fulfill, to verify non-resonance conditions

of the Fahrenheit type. In fact, the first Kolmogorov result, they worked with a very strong non-degeneracy condition that, for example, was not satisfied in celestial mechanics for the solar system.

And then, in fact, this, for example, was the motivation of the weaker non-degeneracy conditions, particularly by Hermann, and then there is the result by Feijoz about the solar system, where are satisfied in

weaker non-resonance conditions. Here, I want to discuss, here we can use the surface tension parameter, or physically, from a physical point of view, which is also equivalent, we can fix kappa and consider, as a parameter, the wavelength. Maybe here I put 2 pi,

but 2 pi lambda is another parameter which enters in a non-trivial way. So I fix the equation, the solutions will exist for most values of the wavelength. And it is essentially the same. Or one could think also

to the depth of the ocean, for example. Okay, in this case, the ingredients which we use, which are important to verify these weak non-degeneracy conditions, are the fact that these linear frequencies

are indeed the analytic functions of surface tension. And they satisfy suitable asymptotics at infinity, square root of kappa J three-half. And then there is a suitable non-degeneracy conditions, a non-triviality condition that I explained now,

which is in fact called the torsion condition, which I think it goes back to, I don't know precisely, appears in a Roman, but also probably in other Russian people. And it just says that if one takes the frequencies,

one looks at the map from the, let's say, R into Rn, then the image is not contained

in any hyperplane. So it is a torsion conditions tells that the curve omega kappa, which is analytic, is not identically contained into an hyperplane. So analytically, for any vector C in Rn,

not zero, the function C scalar product omega kappa is not identically zero. So it is not contained in any hyperplane, that is called the torsion condition. And the proof in this case, in fact, it's a computation. One take the frequency vector,

and the one shows that it is not contained in a hyperplane. And one shows that these vectors are indeed obtained by differentiations, are indeed linearly independent. From here, it comes out a Vandermonde determinant, and then one check that this non-degeneracy condition is fulfilled.

Then, okay, this is what I say about parameters. And then I start to make a bit of the PD part. The PD part, as I said, concern the spectral analysis of this linear time and space dependent operator,

which is obtained in this way. And I repeat, the goal is to find the conjugation, suitable change of variables phi, which conjugate this to an operator in a very simple operator here. Okay, here,

I have already put the operator restricted to the directions which are normal to the tangential dynamics. And I said that the core is to compute this asymptotic expansion of the eigenvalues. And in the same spirit of the talk of Jean Marc,

this is done in two conceptually different steps. The first step has the goal not to reduce the sides of, because this is a constant coefficient plus epsilon terms. But the first step will be not to

worry about the epsilon things and to make them epsilon square and so on, but first to reduce to constant coefficients up to smoother terms. Which is irrelevant, because when here there are quasilinear effects on high frequencies,

J three half is extremely large. And that is important. If you don't control it, that it is the real biggest contribution to the dynamics. So the first step is to put two constant coefficients as it is here, the highest order terms. Lambda three at order three half constant.

And then also will disappear the order one, here again by the symmetry of reversibility. And then the order dx one half, which is also an unbounded contribution, one has to put two constant coefficients. When one reaches the order zero, okay, that is sufficiently mild.

In fact, we have down to a semilinear situation. At this point, it is natural to start erasing terms which are epsilon, and then epsilon square, epsilon three and so on. Namely to perform say a normal form analysis, removing sides. So these are the two steps. First to reduce to constant coefficients

in decreasing symbols. And then at the point to reduce the sides of the perturbation to diagonalize. The effect of these two things will lead to this representation for these eigenvalues.

And okay, this will be obtained by conjugations, changes of variables, time dependent changes of variable. If we have a linear system, which is a linear system in this case, quasi-periodic in time, we conjugate with a change of variable, which is in this case quasi-periodic in time,

then it transforms into another linear system. The form of the new operator is written here. There is the conjugation of the space operator a, just by similarity, phi minus one a phi. And then there is a term which comes from the conjugation of the dt part. dt gives this contribution here.

Okay, and then the goal is to find a transformation a, which in fact will eliminate, look for a suitable a such that this b does not depend on space, on time. And it is diagonal, that is what we would like to do.

And so of course, a of t is obtained by compositions of several transformations, changes of variables of very different nature. The first, it is, I start from a system a, I make a change of variable, and I obtain a new system.

This is a change of variable, a very general change of variable. That is the rule of how changes the part of time and the part of space. And then I will choose a in order to. So now you want to make b like constant? I would like to do that, yes. I don't do in one step, but in a lot of steps.

I think you meant to say you choose phi to make a- Yes, yes, sorry, sorry. I said, I want to say, I choose phi, the change of variable. No, no, sorry, sorry. Sorry, so of course, I checked, looked at a certain variable so that the system with vector field a transform into a system with vector field b. This is just the transformation rule, nothing else.

And now we have to choose a proper way to make good things. Here, there are a lot of experts. In this context, we appear as the good and non-linear, in the following way. This is the linearized operator. In fact, this first step of the analysis is the same as in the other body.

Then this is the linearized operator. And with this variable, the good and non, which means conjugating that system with just this matrix, b, the multiplication for the function b. Well, the system obtained conjugating this linearized problem becomes this one, which is much nicer, symmetric.

And as was said by Jean-Marc yesterday, its eigenvalues are in this context purely imaginary because they're not divided by i. So the eigenvalues, thanks to this good and non, the eigenvalues becomes purely imaginary. So this is the first change of variable.

Which, okay, has this change of variable, okay. Have we said at the beginning that the algebraic property of the system are important? Here, we try to preserve always the reversibility condition because it is important.

It is necessary in some sense, either the Hamiltonian, either the reversibility in order to verify the existence of this kind of condition. So we want always to preserve this property. That this transformation preserves reversibility. So the good and non preserves reversibility. Actually, the linear version preserves also the simpletic nature.

So we could also, but nevertheless, reversibility is sufficient. And to preserve always the symplectic character of the transformation is a bit more complicated. Could be done, but. And then again, has inelizaribaldi. There are some changes of variables, which are diffeomorphism of the Taurus

and the reparametrization of time, such that are able after this change of variable to put it to constant coefficient at the highest order term. I have no time now to explain it, but one has obtained the goal at highest order. And then, okay, one has still some job to do.

This, in fact, is not an equation, but it's a system in H bar. And then the next goal is to symmetrize, name up to these two smoothing operators. So to eliminate the off-diagonal terms,

which act on the component H bar to push to very negative order. So symmetrization up to very smoothing operator, so that the new system will be constant coefficient, still variable coefficients on the term, which act on H, diagonal ones. But then the off-diagonal terms in H bar

becomes extremely smoothing. I will say something later. And then next goal, we have now the effect of, at the first order, very still variable coefficients. So in order to do this,

we conjugate with the flow produced by epsilon PDE like this one. And the flow of epsilon PDE like this one, since with A, which is a real value, is well-posed between HS into HS is nice. It is tame. And then by conjugation, the conjugated operator,

we wanted to analyze it. It satisfies this Heisenberg equation. This is essentially the Lee method. And then this equation has a solution, can be solved into decreasing symbols, thanks to the fact that the commutator,

in fact, gains one derivatives. And because of this, one can solve this equation and find the conjugated operator in decreasing symbols. The new term in front of the x is modified, and then one chooses the function a of x in order to put these two constants.

It turns out that one can put it to zero. Okay, and the same for the conjugation of time. How much time I have? One minute's finished. Okay, and then, well, that is a bit technical, but maybe I say, no, I don't say nothing.

And then just one word. So after this conjugation to constant, we wanted to decrease the sides. At this point, we are in a semi-linear situation. And then the transformation, the analog of the normal form is this one. We want now to find transformations which decrease the perturbation from sides epsilon

to sides epsilon square. So one conjugate with the flow of a function w to be found. The new operator can be analyzed again as before, but then we expand really in powers of epsilon, and then we look at the new epsilon term.

And then we want to choose w so that this term disappears. In doing this, one solves it, and the one is able to do it. If this function w and w1 are obtained this way from the perturbation, one has to divide through differences of the sum and differences of the again values.

And so appears that it is necessary to impose in this diagonalization process, one has to impose, verify these conditions, which in the literature has the second order meaning of non-resonance conditions. If one is able to do this, one finally will obtain this diagonalization.

And then finished.