 Thank you to the chairman and first of all, thank you very much to the organizers for the invitation to this really excellent program It's a pleasure to be here and second of all small apology for Changing the title of my talk from what was at least originally advertised. I Had an idea that maybe in this summer school setting It would be a good idea to present something a little bit more say elementary and and Probably less technical than what I'd originally planned to talk about and that's this the this topic of Stability of periodic waves so periodic waves by periodic waves. I mean spatially periodic Standing wave solutions of the nonlinear Schrodinger equation It's a topic which I am new to and So far don't have so much to say about the problem yet, but what I do have to say is part of joint work with Stefan Lacoste in Toulouse and Typing Psy from UBC Okay, so this is just the the program. So what I'll do is I'll describe these periodic wave solutions for a one-dimensional NLS and then Describe the various different variational characterizations of these periodic waves So anybody familiar with stability theory knows there's a close connection between variational characterizations and the stability After discussing variational characterizations and the related notion of the orbital stability of these waves I will move on to discuss a weaker notion of Stability, which is a linear or spectral notion of stability for periodic waves and then finish with some open questions So this is a business where I have many more questions than Answers so maybe you can think of this as an invitation if if interested to some problems Which I think are interesting and challenging problems. Okay, so the equation is Very simple. It's just the one-dimensional cubic non-linear Schrodinger equation in both the focusing and de-focusing forms I guess when B is plus one, it's focusing B is minus one. It's a de-focusing Of course as with any nonlinear Schrodinger equation, you're interested in its standing wave or solitary wave solutions Which are solutions of this form if we restrict ourselves to Standing waves which have real-valued profile function, which I'm going to do in this talk So from now on only the real-valued periodic waves will be discussed Then of course this profile function satisfies a very simple ODE which you can It's a nice little exercise to draw the classical mechanics pictures You can figure out what all the different possible periodic solutions look like But even more than that This equation is so special that you have special function descriptions of all these real-valued periodic waves. In particular, they're all At least the non-trivial ones are all Jacobi elliptic functions Depending on the sign of the non-linearity in the de-focusing case They're sonoidal waves and in the focusing case there are their conoidal or denoidal wave elliptic functions Of course, it should maybe not be so surprising that We have explicit special function representations of these these periodic waves because Famously, this is a so-called completely integrable PDE a completely integrable Hamiltonian system. It's one of the few Examples of this this phenomenon and so You see a lot of things like explicit formulas As a consequence of the complete integrability for example You can in principle write down infinitely many independent conserved quantities for this cubic NLS and so on many other properties So somehow my foot even though I'm considering the cubic here my philosophy is to avoid the integrability as much as possible with the idea that one wants to be able to say things for Potentially non-integrable equations if you change the non-linearity a little Nevertheless the integrability is going to crop up in various places Okay, so the the periodic waves are these Jacobi elliptic functions You haven't seen them before this is Some idea of what they look like They come with this parameter little k which ranges between 0 and 1 From which you can determine their period through this elliptic integral Well, how is it four times this integral is the period of the sonoidal and the cannoidal waves and the denoidal one has half that period For a relatively small value of the parameter k. In fact, you can see the Canoidal looks a lot like a cosine the sonoidal looks a lot like a sign And the denoidal just looks like a little bump away from one But if you take more extreme values of the parameter k You see more localization of these profiles become much more sharply peaked and in particular this this denoidal wave Starts to look more and more like the classical soliton on the real line which is Exponentially localized, but of course these are all periodic functions. They repeat endlessly in space Okay, and the main And obviously the big difference between the denoidal and the sonoidal and the cannoidal is that the denoid is positive solution and the cannoidal and sonoidal waves are sign-changing makes a big difference in the analysis of their stability All right, and that's the main question. I want to address here is the stability of these various different periodic wave solutions and When you ask about stability you can ask a bunch of different questions For example, you can take different classes of perturbations so maybe the simplest thing you can do is consider these solutions as spatially periodic solutions of the NLS with period Equal to the fundamental period of the wave itself in other words Take perturbations of that wave which have the same period That period will be preserved so you can essentially consider NLS on a circle With one of these waves sitting in the circle and make a small perturbation which preserves the periodicity So you stay on the circle Another thing you can do is you can consider perturbations of the wave which are some integer multiple of the basic period of the wave in other words you're taking several periods on one circle and then considering solutions with that periodicity And the other class of perturbations you might think about is spatially localized perturbations So you might think about the periodic wave on the whole real line and just make a little localized perturbation of that and Ask what happens? That's quite a different question than stability against the Period of multi periodic perturbations and so I'm not going to address that one here I'm going to stick to periodic and the multiply periodic perturbations Okay, and somehow the the motivation here is that For solitons, let's say of the Nonlinear Schrodinger equations on the whole space on the line or on the plane or whatever This is the stability analysis of these objects is a very classical business Has a long history and this has been developed into a more or less a fine art But the problem in a periodic setting is I think not nearly as well understood. So there are lots of open questions Okay, so let me be a little bit more precise about stability Okay, so let's take one of these periodic waves as people probably know in this in this room Natural notion of stability for standing waves is the orbital stability More precisely your equation has some symmetries. We have translation symmetry and we have a phase symmetry and so a standing wave really belongs to a larger symmetry orbit consisting of all the translates and phase rotations of this wave and It's really the stability of this orbit that you want to consider rather than the individual way Okay, so just a little notation let's consider functions which are locally in h1 function so in the energy space if you like and which have a given period capital T so by Orbital stability in this space PT. I just mean The usual statement of orbital stability, which is written here Where we are measuring of course, it's very important what norm you measure the perturbations in measure the perturbations in in the energy norm in h1 unless Otherwise specified So if you start close to one of the waves you will stay as measured in the same norm Close to the orbit of that wave for all time. That's orbital stability there's a much weaker notion of Stability which you can look at which is a purely linear notion So you linearize the equation around the standing wave you look at the linearized operator particularly you look at the spectrum of the linearized operator and The notion of spectral stability is just that this spectrum stays purely on the imaginary axis So at the level of the linearized equation you don't have any exponentially growing modes. So that's a spec spectral stability It shouldn't be too hard to convince yourself that Spectral is weaker than orbital in the sense that Orbital stability requires spectral stability If you had exponentially growing modes at the linear level then in reasonable under reasonable conditions You can bootstrap that to a nonlinear a contradiction to this statement of orbital stability So it's so much weaker notion okay, and the other little piece of notation I have to introduce is this Notation a 2k so a 2k stands for half anti periodic Okay, so to be more precise this Sonoidal and conoidal ways if you look at their picture oops They have an extra symmetry like the sign in the cosine So if you go out if you go halfway through the period You just pick up a minus sign so they are half anti periodic functions and moreover This family of half anti periodic functions because the the nonlinearity is then the The nonlinear potential in the equation is then periodic. That's a preserved class of functions So you can ask about the stability of these objects within this subspace of half anti periodic Half anti periodic functions as well Okay, so let me try to summarize in one kind of crowded slide Some things that we know about this stability question One wave at a time. So let's start with the sonoidal waves. So this is the de-focusing case So if you restrict as we as I just indicated if you restrict to these this family of half anti periodic functions Then there's a result which says a result of Galilei and Haragus, which says that The sonoidal waves are orbitaly state This is essentially an argument which which follows the classical lines for example of a Grilak is shot off Strauss type argument More somehow more more spectacularly and certainly much more integrably So if you're willing to really exploit the integrable structure there was a nice observation made by a botanic and de conic and Nevala and somehow Formalized by Galilei and Polanovsky That by exploiting the integral structure in particular exploiting one higher order conserved Quantity one higher order conserved functional One can obtain an orbital stability result Not just in For functions perturbations of the same period, but in fact perturbations of any multiple of the period of the sonoidal waves So I'll say a little bit more about how that goes Later one thing to notice is that because they're using a higher order functional which lives at the level of h2 This is an h2 level stability statement. So this is h2 orbital stability and doesn't say anything about the energy space Okay, so one thing that I'll present a little bit later on is Somehow much weaker results. So I'm gonna look at the spectral stability Certainly if you have orbital stability you have spectral stability so that it's spectral stability is a known result from this integrable integrability-based work of Galilei and Polanovsky But I'm going to try to give a proof of at least the spectral stability of The sonoidal wave in a way that does not really rely on the integrability at least not too much So that's the philosophy is to avoid the integrability as much as possible Okay, the conoidal wave. So that's the focusing case again, there was a result of a Galilean argues at least for some range of parameters about the orbital stability Against half anti-periodic functions of the conoidal wave Again, it's a grilakis-schattastraus type argument So I will I will show you a Let's say an alternate proof and one that applies to the full range of the of the parameters Roughly speaking They're looking at the local minimization problem, but in fact, I'll show you we have a global minimization property For the conoidal wave among half anti-periodic functions if you remove this Anti-periodicity assumption you go to the full period then we don't have any results about orbital stability as an open question But at least we can say something about spectral stability And in fact, it turns out that the spectral stability was established by Ivy and the fortune again somehow exploiting the integrable structure in it in a heavy way and explicitly computing the entire spectrum of the linearized operator So what I will indicate What I think of as a less integrable way of getting at least this spectral stability For the conoidal wave And then finally, okay, so at least at the spectral level at the linear level You can see that the conoidal wave is stable with respect to its own period perturbations It turns out that if you go to higher period higher multiples of that period Then the conoidal waves become unstable This is somehow known a long time ago in the 70s by some work of Rollins which we Which we somehow Formalized give a rigorous proof of the instability. So if time permits, I'll say a little bit about Where the instability comes from and just to complete the picture there's not much to say about the denoidal wave So the denoidal waves that's that's the positive one in some sense It's theory is very much like the theory of the usual ground state soliton the real line because it has this Constrainted minimization property and so it's automatically or both Orbitally stable with respect to its own period perturbations and it's very easy to show that it's unstable against higher multiple perturbations so the first thing to discuss a little bit about is the Orbital stability and related to that variational characterizations of periodic waves Of course, these are everybody's favorite conserved functionals for the nonlinear Schrodinger equation the energy the mass and sometimes the momentum and Deliberately leaving off this list all the infinitely many more you get from integrability because I want to avoid them as much as possible and It's Certainly the case that if you look at the equation for the solitary wave standing wave profile Then it's a critical point equation for some combination of the energy and the mass sometimes called the action and it's a sort of it's a classical result that going back to the 80s that if You can conclude that your standing wave is a local minimizer of This functional Subject to a constraint of the mass or maybe the momentum or maybe both then Then you have orbital stability, okay That's that's a loose statement, of course You need you need something about non-degeneracy or about compactness of minimizing sequences But basically if you're a constrained minimum and the constraints are conserved quantities Then you're orbitally stable so it makes sense to look at the various Constrained minimization problems in the periodic setting So if you try the obvious thing you constrain the mass and minimize the energy among functions of a given period Then that will give you your denoidal waves the positive ones at least for some range of parameters and in the focusing case But in the de-focusing case you won't get anything you just get a constant So natural thing to try if you want to find the sign changing waves the conoidal or the sonoidal natural thing to try is Maybe to impose this extra anti-symmetry half anti-symmetry condition Minimize among half anti-symmetric functions and indeed if you do this it will yield I'll show you on the next slide. It will yield the conoidal waves in the focusing case So you can say and in fact this is you can see this is not a local minimum This is a global minimum So the conoidal waves are global minimizer constrained minimizer In the de-focusing case, however, you don't get the sonoidal waves. In fact, you get a plane weight the complex value solution So I'll completely clear how you might In a useful way generate the sonoidal ways variationally one possibility would be to add one more constraint Which is zero momentum constrained, but I don't actually have a good I don't have a nice proof of this or indeed any proof of This this fact that if you constrain If you restrict to half anti periodic constrain both the mass and the momentum Then you'll finally get get these the sonoidal waves as you minimize this Okay, you just say a word about The variational characterization of the conoidal way So I said it's a it's a constrain minimizer of the energy we constrain the mass and you restrict to half anti periodic functions and here you can see Right away the difference Between dealing with the sign-changing solutions and dealing with usual positive ground state solutions So what you might be tempted to do when confronted with a minimization problem like this is you might immediately In your minimizing sequence you might replace you by its absolute value Which you know which you know does not increase the energy and preserves the constraint But of course the problem here is that destroys The anti periodicity, so that's an illegal operation So instead what you can do is you can use a kind of rearrangement on the Fourier side So if you write the function in terms of Fourier modes because of the half anti periodicity in fact You'll just get the odd modes and You make a rearrangement of this type on the Fourier side It's immediate that you preserve the L2 norm you preserve the kinetic energy and So the name of the game is to show that this procedure increases the L4 norm Which you can do with just a little bit of combinatorics and linear algebra Okay, so you get this global minimization problem for the cannoidal wave as a corollary You get an orbital stability result for the cannoidal wave within the half anti periodic functions And let me just emphasize again if you remove the half anti periodicity assumption As far as I know, this is an open question whether or not this is an orbitally stable wave for the sonoidal wave We didn't have the variation of the same kind of variational structure as for the cannoidal wave But as I mentioned at the beginning Both Mendeconiq and Nevala and Galilei and Pellinowski have used the exploited this higher order functional which comes from the integrability to say very strong things about the stability of the sonoidal wave So more precisely They take this functional here this Higher order functional lives at the level of h2 Together with some combination of the mass and the energy I didn't say it on this slide, but of course, this is a conserved functional for the cubic NLS That's consequence of the integrability And it turns out that the sonoidal wave is in fact a local minimizer of this functional without any constraints and with respect to perturbations or within the class of functions which are periodic of any multiple of the basic period of the sonoidal wave So this is some kind of integrable systems miracle, I guess and So you can then build you use this functional to control perturbations to get an orbital stability result so their conclusion is that the sonoidal wave is in fact orbitally stable with respect to any multiple perturbations perturbations which are periodic with any multiple of the basic period as Measured in h2 H2 because you need to use this function So as far as I know if you want to make a statement of orbital stability in the energy norm in the H1 norm then that's Still an open question Okay, so that's I think all we can say about orbital stability for now Let's move on to this weaker notion of spectral or linear stability Okay, so just Show you the linearized operator. So if you take one of these periodic wave solutions and you linearize Around it probably most of you have seen something like this before the linearized operator has this form here this Made up of these two Two self-adjoint shorting or or Sturm-Leoville type operators L plus and L minus But as anybody who's worked in this business knows that the Relationship between the spectrum of these self-adjoint operators and the spectrum of the truly nourished operator is a somewhat complicated one Anyway spectral stability when I say spectral stability I mean that the spectrum of this truly nourished operator lies entirely on the imaginary axis Certainly a necessary condition for any kind of any other kind of stability Okay, and the situation is is kind of like this in general In these stability for Schrodinger Standing wave problems you normally have a Certain amount of information about the spectrum of the self-adjoint operators the L plus and the L minus which make up the linearized operator But not so much about the the spectrum of the truly nourished operator And the name of the game is kind of to go from one to the other in the case of the cubic in one dimension In fact, we can be very explicit indeed about the spectrum of the The low at least the low energy spectrum of the operators L plus and L minus And it's essentially all summarized in this picture, which I'll try to explain so essentially what you're seeing here is is some kind of Another algebraic miracle, which is presumably connected in some way with Integrability and that miracle is that if you consider these the self-adjoint operators the L plus and the L minus For the three different families of elliptic functions They're all just translates of each other and so they all have the same eigenfunctions and so from this very simple but Algebraically Amazing fact you can pretty easily generate explicitly all the low the low end of the spectrum of these operators So you can start for example with the L minuses over here You can Look at the zero eigenvalues So the fact that you have SN as a zero eigenvalue of L minus for SN That's a consequence of the phase invariance So you get the zero here and the zero here and the zero here just by phase invariance And then just by translation these become eigenfunctions for the other operators as well And you can play the same game for the L pluses in this case It's the derivatives of the waves which give you zero eigenvalues and you can translate those to generate all the other low eigenvalues Except that you will be missing some and it's easy to see that you're missing some by Counting the zeros In other words by Sturm-Leavitt theory So if you look for example here at this derivative of SN up here You count the number of times it crosses zero as you go around the circle it crosses twice So it can't be the ground state the ground state has to be positive. So you know there's something below and With a little bit of diligence you can compute exactly what that is in fact So you can compute this zero like this ground state eigenfunction as well and then the last piece of information on this picture, I guess is the these subspaces up here, so Explain before what p2k and a2k are so p2k those are the half periodic Functions a2k is half anti periodic functions and you can make a further decomposition into odd and even Because the conoidal wave is even the sonoidal wave is odd So their squares are even so these are subspaces, which are also preserved by the linearized evolution Okay, and maybe the other the other thing you can see immediately from this picture is why the sign-changing periodic waves are different than the positive ground state that you might be more familiar with and It's just for a very simple reason that you have too many negative eigenvalues So look over here for example for the L minus SN you have two eigenvalues below zero And for L plus you have another one And if you think about the usual Conservation laws that you can exploit the mass the momentum maybe the momentum and the energy you just don't have enough You don't have enough conservation laws to control the negative directions. That's kind of that The essential that the extra difficulty here in the periodic setting okay, so let me just Sketch an argument which I hope relies as little as possible on the integrable structure for showing based on this picture that You have spectral stability for the sonoidal wave and the conoidal wave with respect to perturbations of the same period I Should point out that this this argument that I'm going to show actually breaks down at some at some range of the parameters for the conoidal wave Although in fact, we know that the conoidal wave is still stable beyond that Spectrally Yes, please Can I can ask a question? Yes So if I do this thing on the whole line, which probably means your little K goes to zero to one Then doesn't the L plus only if one negative eigenvalue. Why do you have three negative? Which you know eigenvalues probably from translation you're looking at L plus right exactly the Waves are those that become the oh the denote right, okay So this is a confusing picture for the denoidal wave because this is this is already twice the period for the denoidal wave it's not Because the normal way that's half the period of the sonoid and the conoid That's why you have the extra negative. I in fact in fact the DN In this setting in twice twice its period is actually unstable Okay, but if you would somehow draw it just with the its period then they would only be one Correct, you it would be the usual picture. You're familiar with from the ground state of the NLS You'd have zero mods. You have one negative eigenvalue. Yeah, okay, so let's see how we can extract at least spectral stability from this This picture Okay, so the first thing we do is we break the piece problem up into small pieces using the symmetry so I can split the the periodic functions up Into half anti periodic and half periodic and it's useful to make a further decomposition of the half periodic into odd and even and Then we can use our variational information. We know already from the lost my pointer From the half anti periodic setting so we know I showed you these variational Characterizations earlier for example of the conoidal wave which says that it's a minimizer constrained minimizer among half anti periodic functions And so this is enough in within this subspace the a2k to give you the spectral stability For the senoidal wave, we didn't have the variational characterization, but you can nevertheless By using Just just by direct computation more or less You can show that it's not necessarily a global minimizer But some kind of a local constrained minimizer and that's enough to give you the stability in a2k So the a2k part is Sort of the classical part. That's the that's the part that's related to variational characterization P2k minus if you go back to the previous picture, you'll see there are no negative eigenvalues in there So this there's no problem. It's automatically stable And so all the problems that you might have come in this subspace of half periodic even functions So here you need to do something which is not coming from variational information So here's what you can do You can use a very simple coercivity or Sometimes it's called crime signature argument to conclude the following If you can find Some eigenfunction in this subspace Which has negative energy in this sense here? So it's negative energy from the point of view of the self-adjoint operator then That eigenfunction will somehow use up the entire negative subspace That you have for the self-adjoint operator. And so you can't have anything else. It's unstable So this is some some kind of sim just a simple version of a Crine of a crime signature eigenvalue count due to for example capitula capricchitis and Sancti So if you can find just one eigen eigenfunction With the negative energy Then that kills the possibility of any unstable eigenvalues And here it is And this again is presumably Some integrability rearing its head once again is you can just write down some Some appropriate eigenfunction. So here it is in the case of SN down below in the case of CN You can compute directly that this is an eigenfunction with purely imaginary eigenvalue with negative energy So by this crime signature argument, it rules out any other unstable eigenvalues So in a sense, this is probably Probably coming from integrability in some sense But at least the problem is reduced to to a single eigenfunction in a single subspace Okay, so in just the last few minutes Let me talk about the instability of the conoidal wave So what we just showed is that the conoidal wave is stable at least in a spectral sense with respect to perturbations which preserve its period So now we go in the other direction we Consider perturbations which are very high multiple of its period or if you prefer we're putting a whole bunch of conoidal waves Or one one conoidal wave with many periods on the circle, and then we make a small perturbation of this guy Okay, and here's a theorem which says that if you do that if you take Sufficiently high multiple of the period perturbations and indeed you have unstable eigenvalues So this is a it's a kind of perturbation argument, which I'll explain a little about in a moment and somehow you do the perturbation theory and It boils down to some heavy heavy elliptic function computations And we did these computations and we're very happy to find this instability And then we learned that the these heavy integrables a heavy elliptic function computations were already done in 1974 by Roland Although his argument was more or less formal so We made the perturbation theory rigorous, but the computation was essentially done in the 70s And here's what's behind it. So what's behind it is is a floquet theory So floquet theory is is how you describe for example the spectrum of a linear operator With respect to localized or L2 perturbations on the entire line when you have periodic potentials It says that you your spectrum is made up of these bands which come from considering Reduced problems this L theta. So you take the linearized operator you conjugate it with a phase where it is here Conjugate it with this e to the i theta x Consider that eigenvalue problem with respect to periodic boundary conditions and then take the union of all Those eigenvalues that come up with respect to this parameter theta that generates the entire LT spectrum That's what it's okay theory tells you So in particular what you can do is you can use this parameter theta as a kind of perturbation parameter Perturbing from in our case. We're going to perturb from theta equals pi over t which exactly corresponds to the anti periodic the Anti-periodic subspace and the reason for the anti periodic is because that's where the Conoidal wave itself lies So if you take the the anti periodic operator Then it has a null space in fact it has a generalized null space made up of the conoidal wave appropriately phased And the derivative of the conoidal wave those lie in the kernel and as usual you have general generalized eigenfunctions corresponding to l plus and l minus as well and You can do perturbation of this generalized eigenspace at the origin Away from this Anti-period half and or sorry this anti periodic Value of theta use theta as a perturbation parameter and from this you do if you do this carefully you can you can construct An Eigenvalue which is genuinely complex non in fact non-zero real and imaginary parts boils down as I said to some Calculation involving elliptic functions, which turns out to have been done a long time ago Numerically, this is what the spectrum. This is the L2 spectrum the the full band spectrum for the conoidal wave And so you can see The perturbation coming from the origin in this direction into the complex plane Which is what this theorem is computing Okay, so that that's the L2 spectrum, but as a consequence you have some range of theta some range of this Floquet parameter for which you have unstable eigenvalues in particular you can conclude from there that for sufficiently high Multiples of the basic period you have unstable eigenvalues Okay, so let me finish with some open questions of which there are many in this business it seems start with some Relatively modest once I don't know how modest they are actually so for the sonoidal wave We can get the orbital stability Against half anti periodic functions perturbations, so that's a more or less it's related to variational more or less classical argument and by using this elliptic Or sorry by using this Integrable machinery we can get the stability against higher perturbations if we go to h2 to level of h2 So it'd be natural to ask whether you can descend to the energy space whether this property of orbital stability in the energy space holds for this anoidal wave Same question for the conoidal wave We know even less about the conoidal wave and in particular. We don't know orbital stability outside of half anti periodic perturbations We do know that for the conoidal wave we get instability for a high enough multiple of the period perturbations So there's some kind of transition there between stability and instability, so it'd be natural to ask the following question fix fix the multiple of the period For which values of this parameter k do you have a stable wave for which is it unstable? Okay, and then More more ambitiously maybe I don't know that anything is known about stability of periodic waves against localized perturbations should be a hard problem, I guess and something that maybe maybe I want to look at next which is To remove the integrability so to change the nonlinearity to something non integrable where you don't have these higher order functionals to exploit and see if we can See if we can get any kind of orbital stability or even spectral stability in that setting Okay, thanks for your attention Just a comment it seems that your fourth question even on a numerical level if you think of simulations It's already a hard question Yes, I'm sure it's a hard question. Yes But you okay, you can dream right Please I guess Presumably interesting object, you know, well My philosophy is to of course avoid the integrability as much as possible And so I'm sure if you're if you're very good at turning this crank then you can generate formulas for such things Of course, you can you can cheat you can Just play with the period right, but I guess but I guess there are there are formulas for more complicated I would guess there are formulas for more complicated solutions. I know nothing about it Absolutely you can yes, but we don't really have any answers Yet So it would be quite natural to ask about this Stability or instability then do Yes, you can go down you can go up Sure, I mean people people know how to do some things like this on the real line by using integrable transformations, for example, but in the periodic case, of course It's a different different story Certainly it's a good question So now you have a complete description of the phase space like this Changer variable Of course, it's not that explicit that it looks like I mean but but You know anyone who tried to attack for instance this question of stability h1 because stability h1 through this change of variable. I don't know of any such attempt But it's a good Yes