 Thank you Mr. Chairman and I want to thank the organizers for the invitation my first time here, it's a great place. This is joint work with my former student Dimash Spiegelman and let me begin with the physics. So the physics that we have here is a laser beam that propagates in the z direction through some transparent medium, air, glass, water, whatever. Direction of propagation is z, it enters the non-linear medium here is it starts at z equals zero and the equation of propagation to leading order is a two-dimensional cubic and a less. So here we have the diffraction terms which tries to make the beam wider with propagation but we have the non-linearity here, the Kerr non-linearity which tries to make the beam narrower with propagation. So we have this competition between focusing non-linearity and diffraction and we impose the initial beam profile at z equals zero so this is our initial condition and the propagation is in z so this is an initial value problem in z but since this is a mathematical audience most of the talk called this t to make it life easier but there are some z's that were left behind in the transparency so wherever you see z it's t. Okay so you have this competition between the focusing non-linearity and diffraction and actually this field started from experiments which are done in the early 60s by that time you had powerful lasers so one of the experiment people shown these powerful lasers into glass and what they show is that after some propagation in the glass there be actually the laser beam became narrower so much that it damaged the glass there was really damaging glass so they call this optical collapse or catastrophic collapse and the first explanation was given in 65 by Kelly who used the 2d cubic and less and he showed numerically and and wavingly that a solution of this equation can become finite in a can blow up in finite time which of course physically is finite distance so there's a lot to say about the blow up of solution and there are many experts in this audience and whatever I had to say about it I said in this book but today I'm not going to talk about blow up I'm going to talk about solitary waves uh so what do we mean by solitary wave we look for physically this our solution that propagate in the z direction without changing the radial profile so this is the well known equation for the profile of these solutions and the dependence on this new parameter is only through the dilation so it's really doesn't play an important role here and uh we are always in the 2d cubic case in this talk so we're only talking about l2 critical and therefore the power or the l2 norm of r mu is independent of this parameter and I call it the power not l2 norm or mass because in optic this quantity corresponds to the power of the laser beam how many watts or megawatts it has okay so here is this r equation it has been studied extensively there is an infinite number of solutions radial solution the one we usually care about is the ground states which physicists would call the town's profile uh it's positive monotonic decreasing and it has exactly the critical power for collapse and if you plot it it has a peak this is r so it has a single peak at the origin and I use here a notation which is slightly different usually people denote this solution by r0 I call it r1 because in this talk the superscript would always denote the number of peaks of the solution and this solution has one peak so that's why I call it r1 so what do we know about this solution uh this solitary wave if you take an initial condition this initial condition multiplied by c if c is larger than 1 then your solution blows up in finite time whereas if c is less than 1 then you have scattering as t goes to infinity and therefore this solitary wave is has sort of a dual instability if you perturb it to either direction it won't stay and so this is uh maybe trivial mathematically but physically this is a very bad news because what you want physically is to be able to take a laser like this and shine it let's say to several kilometers into the atmosphere and have it in a stable way how would you have it stable when I just proved that the solitary wave is unstable so this is a very important problem how can we stabilize laser beams when they propagate in bulk medium it's extremely hot extremely important problem a lot because of this atmospheric propagation but not just because of that so one very intriguing idea was proposed in 9p8 and that was to use a necklace beam so what is a necklace beam you take an initial condition and this uh so it's a setch a shifted setch in the r direction multiplied by cosine aim theta so here is the plot of this solution it has two m uh roughly beams which are called pearls to correspond with this necklace a seguv always has very illustrative names and uh they are at equal distance from each other they are all on the radius of r max and adjacent pearls have opposite signs or opposite phases and why use this configuration because if you look on this raise between each two pearls psi not is zero on this uh raise and by uniqueness psi remains zero along this raise and therefore if you look on just one pearl as far as this pearl seen its dynamics it only see these two Dirichlet boundary conditions on this raise so it doesn't feel at all the other pearls because uh a Dirichlet boundary condition is a reflecting boundary condition everything that comes to the boundary gets reflected and so the necklace structure would be preserved at all times and the way a physicist would tell you he would not tell you this uh using a Dirichlet boundary condition what he would tell you is that if you have two beams of opposite phase the repellage are there and that's why they don't interact so here is again our necklace beam so what was a seguv's idea let's take each pearl and give it power l to norm slightly below the critical power for collapse if it's slightly below it means that the non-linearity is slightly weaker than diffraction so it cannot collapse by itself so it wants to expand slowly but as it tries to expand it starts feeling the repulsion by the nearby pearls and so this would slow down the expansion of each pearl so just to show this numerically here we did a simulation we have a pearl with just a ring with four pearls here you cannot see that they have opposite phases because they plot the intensity but they have and this is the dynamics as z which is t goes on you see it expands it keeps the necklace structure but if you just would take just one of these pearls it would expand much faster at the same time so indeed this slows down the expansion so its expansion is much slower than for a single beam and sometimes this may be enough for your application but ultimately this structure either again would scatter or collapse in finite time and this is an experiment which came eight years later this was done at Cornell by Gaeta's group and here what you see you see an input beam this is a necklace they have opposite phases and this is the input beam and this is after propagation of 30 centimeters in glass you see the necklace structure is preserved now the question whether this is stable well it depends if you are a mathematician or a physicist for a physicist this is stability for mathematician i'm not sure but this necklace structure it's actually even been used as a mean not just as the end for example in this recent paper they use they set up a necklace beam structure which you'd see here in order to set up a thermal waveguide in air so sometimes you don't care by the necklace itself but you use it to do other purposes and let me point out that it may look like this is a vortex beam but it's not because a vortex beam is something like this times e to the i am theta and when you have a vortex beam and you plot the amplitude it's radial the necklace beam is non-radial so it's different from vortex beam okay so can we have necklace solitary wave in free space in two dimension the answer is no if you take the 2d cubic and ls it does not admit necklace solitary waves and by this what i mean is if you take this r equation and you look for a solution of this form there's no such solution now i put it in formal proof because we have sort of a arguments that probably could be made into rigorous proof but it's not rigorous but i believe it's correct uh actually they have been a proof of the existence of sort of a multi-peak science changing solutions of the r equation in these two papers but there they have a very complex construction they have two rings one inside of the other it it's not of this form that we have here so they are multi-peak science changing solutions but not of this form and anyway whatever you get here would be strongly unstable because any solution any solitary wave of the 2d critical would be strongly unstable and now there have been observation of necklace solitary waves with a different non-linearity this is in a photorefractive material where you have this type of non-linearity here and you have here some potential which you induce with another laser beam so in this kind of type of nls you can show that there exists necklace solitary waves and that there are even in some parameter range they are stable and this was shown theoretically numerically experimentally there's only one problem uh the medium that we usually take are not photorefractive we want air we want a glass we want water it's care it's not photorefractive so what can we do our goal is to stabilize a necklace beam in a curl medium so one idea which have been used since i think like 20 30 years has been to confine the laser beam to a hollow core fiber what is a hollow core fiber you have a fiber it's empty here in the inside and you put some a non-linear material which can be air or it can be some a noble gas so it has a cubic non-linearity that's what we care about and to leading order when you propagate here the walls are completely reflecting everything that hits the wall comes back in and therefore if you write the equation for the propagation it's the 2d cubic nls but now it's not in free space it's confined to some region the cross section of the fiber which typically it would be the unique disc although there are other shapes of hollow core fibers available in the experimentally so this brings us to the issue of the nls on a boundary domain so here is our equation we put this directly boundary condition so reflecting boundaries you can again look for solitary waves they satisfy the r equation on a boundary domain and uh this equation uh was studied by frank and myself in 2001 and later there have been some other studies of this equation these studies were mostly on radial solutions on b1 what did i'll summarize this studies into one slide uh so the important thing to note here is that we don't have the dilation symmetry anymore because of the Dirichlet boundary condition and this changes everything so for example if you look on the power now it it does depend on u it's not independent of this u um u starts from u linear which is the eigenvalue of the linear r equation the smallest eigenvalue uh and then uh from here the same the power increases until it reach the critical power as mu goes to infinity and let's know that the power is increasing as a function of mu what about stability in general so if you want to study the stability of solitary waves the easiest type of stability is linear stability so you perturb this r mu with a perturbation which has a real and imaginary part and some eigenvalue omega you put it into the equation you write the linearized equation you get this well known system with l plus and l minus which are defined here and what you want to know for stability is whether you have eigenvalues uh with positive real part because if you do then you have unstable modes and what uh was shown i believe first by vachitov and kolokolov that actually instead of studying this eigenvalue problem you can replace it with a much simpler thing so if you you look on a positive solution then uh you have linear stability if the power is monotonic decreasing in mu so this is if you look in the physics literature this would be called the vachitov kolokolov or the fika condition for stability and actually this condition is not only sufficient for linear stability it's also sufficient for orbital stability uh now on the unit disc we see numerically that this v condition vk condition is indeed satisfied the power is increasing as a function of mu and this leads to the following theorem that if you look on the ground that solitary states on the unit disc there are uh orbitally stable and when we talk orbitally stable on the boundary domain we mean stability up till uh phase changes of the solution and this is very different from free space in free space these solitary waves are unstable in the 2d qb case so the boundary the reflecting boundary is stabilizing the solitary waves and this is a general result Dirichlet boundary condition has a stabilizing effect on solitary waves so that's the good news the bad news is that if you look on the power of these solitary waves it goes between zero and p critical so the most we can propagate in a stable way would be slightly below critical slightly below because this is an experiment you don't have exactly the profile there's always some noise let's say 80 90 percent to be a critical power we want to be able to propagate much more than that so how would you do that so one we were trying to think of different ways to do it and one idea that came to mind is to use a necklace solitary wave because well if each one of these would have slightly below critical power together wow you'll get much more so that that was the motivation for studying this the physical motivation for studying this necklace solitary waves but what do we mean by a necklace solitary wave so we have this equation on a bounded domain and we can of course take a circular fiber and then the necklace solitary wave would look something like this with a plus minus plus minus but we can start a warm mathematician we can play with the domain so we can take a rectangle maybe and having a necklace solitary wave of this form or maybe even an annulus and have something of this form and with fabrication techniques today any shape basically can be fabricated if there is enough motivation as far as i know this solitary waves were not did not appear before on the literature and they are new so let's start with the rectangular one because they are easiest for the analysis so we want to compute them numerically how would we do this we would just compute one pearl first numerically on a square so this would be a positive or a ground state solution on the pearl and then we would extend it to a necklace by putting them one next to the other so this brings us to the question of how do we compute a single pearl a single positive solution ground state solution of the r equation on a square this is not very challenging numerically if you know which method to use which is this renormalization method you just need to use some known spectral version if i have time at the end i'll talk about it and if not it's all in the paper but here are the results so these are the solitaire way for different values of mu this is the power as a function of mu so you start from mu linear as before the linear the eigenvalue of the linear problem linear equation and as you bifurcate from this mu linear you are in the weakly non-linear regime so here what mainly supports the solitaire wave is the boundary but as you increase mu you go to the strongly non-linear regime where you are almost like in free space you hardly feel the boundary so you are almost like this and since we'll care about stability let's know that the vk condition is satisfied okay so we know how to compute one pearl and now we want to make a necklace out of it so we just put them one next to the other plus minus plus minus and this indeed gives you an honest solution of the r equation with a necklace structure so the only so on these interfaces r is zero and the only thing that you have to check is that indeed the equation is also satisfied with this interface but it does so it's fine it's an honest necklace solution of the r equation but you can start getting crazy for example you can put them in this form and this would also be a solution a necklace solution of the r equation so you can play with the shape as you like okay what about circle on necklaces so we take the r equation now on a circle Dirichlet boundary condition so again we need to first compute one pearl so for example let's have it on the quarter circle so you do the same technique same numerical method and here is a very similar picture this is the weakly non-linear regime low and wide the strongly non-linear regime narrow and high here you are close to free space this is very linear and the vk condition is satisfied and once you have one pearl you put them next to the other and you get your necklace i have a question if i'm allowed which is on one sector is it possible to get a sign changing condition there's one up and one down so my definition is that i would take a sector that i would have only one my numerical method can only compute positive solutions so i would always look for one pearl and then extend it all right uh you mean numerically or uh just in one's mind well so call all this a sector and then you'll get uh two signs right well i was thinking of a thin sector and have them differing radially ah different radially uh then you it so it's not so symmetric you put together a more something with two pearl rings for example i don't know i need the this track i need for for my numerical method i need to be able to impose this Dirichlet boundary condition so i need to know where they are and if you put this there are two there it's not clear to me where you would have the if you tell me where is the boundary then i'll be able to compute it okay and about the annulus so again we do the same procedure we compute a a single pearl a positive solution on the sector of the annulus and again the same behavior the weakly non-linear regime the strongly non-linear regime the vk condition is satisfied and once we we know them we just put them one next to the other and we get necklace solid wave so this brings us to the issue of stability uh so when we talk about stability here that we first should start with stability of a single pearl and only then move to stability of a necklace so when we talk about the single pearl uh so here for example you have the square the quarter circle and the quarter annulus and we take the initial condition is we take this and we multiply it by 1.05 and here is what you have after t at equal 10 it looks almost the same and in all the numerics we have done we always see that this uh positive solution on boundary domains are stable and it it agrees with the fact that they all satisfy the vk condition so this again it's not a regular statement but it's a numerical observation but what about the necklace stability well it depends what type of perturbation you put if you would multiply everything by 1.05 for example like we did previously then uh actually psi would remain zero on these interfaces because you don't change it in the initial condition and the more important thing is that you you maintain the anti-symmetries between these pearls and therefore psi would remain zero on these interfaces and so the pearls would not speak with each other so the stability of the whole necklace is the same as the stability of the single pearl so if you have any chance of destabilizing a necklace is to use a perturbation that would break this anti-symmetry of the necklace so there are various ways that you can break this anti-symmetry with respect to these interfaces one possibility is you just add noise another one which we usually prefer is to check just one of the pearls and multiply it by 1.05 and the reason i mainly prefer this to this is because this is a deterministic perturbation you can trust the numeric much more when you do a deterministic perturbation than when you do a random one okay so uh what about the vk condition well since each one of the pearls satisfy the the vk condition and the power of the necklace is n times the power of the number of pearls times the power of each pearl then the vk condition would also be satisfied for a necklace but this doesn't mean that the necklace is stable because the vk condition is good when you have a positive solution now this is not a positive solution anymore a necklace so therefore you need to go back to the original derivation of this vk condition which is let's do just the linear stability so you go back again to this system with the l plus l minus this linearized equation for the perturbation of the ground of the solitary waves and you want to check whether there are some eigenvalues with positive real part so let me show you the result of dyscomputation for the case of a circular necklace with four pearls so what we plot here this is the value of mu as we change it and we plot here real of omega so if there is any eigenvalue with positive real part we plot it if there's nothing it means that there's no such eigenvalue so what you see here there is a small region between mu linear and some larger value mu critical just here for which there's no eigenvalue with positive real part meaning you have linear stability but for all other mu's you always have eigenvalues with positive part here's just just one double eigenvalue and from here there's start a second eigenvalue so you have stability until here and now the million dollar question is what is the power here when you become unstable did you exceed the critical power or not so the answer is no you get this instability when the necklace power is one quarter of critical power so you can say oh that's not bad so if each one of them is one quarter you get a but no together they are one quarter of critical power so each one of them is like 0.08 so the good news is that this instability has nothing to do with collapse because you are well below the critical power for collapse but the bad news is that well we started up to be better than the radial solitaire waves and we ended up being much worse because with radial solitaire wave you can get up to critical power okay let's look on the dynamics of this uh so here we take this uh solitaire wave we protect with just this one we multiply by 1.05 and we let z go zone or t this is from you slightly below critical and this is slightly above critical and you see indeed that here it reserves the shape until z equal 10 whereas here already z equals 3 it's unstable so indeed you see this change of stability around mu critical it's not a blow up instability but it's an instability in under rise so now if you want we want to still not to give up on this necklace solitaire waves they look too promising so we said okay maybe if you understand the source of this instability we can be able somehow to fix it so in order to do this we need to understand what is the instability dynamics and in order to understand so if you look on the eigenvalue you just get a yes no answer stable unstable if you want to understand the instability dynamics then you need to look on the eigenfunctions of these unstable modes this tells you what is the instability dynamics okay so here are the eigenfunctions so for example this is the eigenfunction the of this line so it correspond to this the first eigenvalue okay you can look at it but what do you see when you look at it so the thing to see here and that's why I plotted independently the real and imaginary values are not just absolute ever so the things to see here is that this eigenvalue is anti-symmetric with respect to y like the eigenfun like the solitaire wave but is symmetric with respect to x so what does it mean it means that you start with this eigenfunction which is anti-symmetric but you add to it a symmetric perturbation so the both the result of this is that you see power shifts from one pearl to the other so what this eigenfunction tells you that this eigenfunction correspond to power flowing the x erection there is a second eigenfunction that correspond to the same eigenvalue this one has the symmetries of its anti-symmetric with respect to x but it's symmetric with respect to y so therefore here you would have power flowing the y direction and the third eigenfunction the one that you start only here has a it's symmetric with respect to x and y so it means that power flow between all these four pearls okay so uh that was the point of a mild uh yeah there may be well they would oscillate somehow because it's confined uh whether it's periodic in time the full frame which full family it's a good point actually maybe yes yeah we've seen sometimes that it's periodic in time but yeah sometimes yes sometimes no i guess it depends on the perturbation okay i can't help but say what there's several modes that interact why isn't it quasi periodic starting with we saw three unstable modes so why can't it have a three maybe four basic periods which interact with each other some of the resonance the y power stripped and the x power shift have to be the the same but beautiful one not necessarily so that would be beautiful but starting to get hard well okay so here it is that for stability we have uh this not as we expected so we wanted to probably get more power in a stable way so one very intuitive idea would say well let's just use more pearls why use four let's use six eight twenty or as long as it's an even number that's fine well it doesn't get you anywhere actually the power of the whole necklace goes down as you increase the number of pearls so a second idea would be to say okay so let's use another domain so for example if you use uh let's use a square with four pearls so actually this would get you something if you use a square domain not with six but as here but with four the threshold power for stability jumps up by a factor of two which was very surprising to us i didn't i thought it would be almost the same as circle and a square so i have no idea why is this factor of two but still this is well below what you have with radials so literary waves so then we said okay but we know the source of instability the soft instability is the power flow between the different pearls so we need somehow to be able to uh not to allow them to talk with each other so the idea that we had is to use a annular domain since if you put a hole inside well they can still speak with each other but much less so okay so this was our motivation to study the necklace solitary wave on an annular domain so here you see now the same plot that we had before but now it's for a four-person annular necklace and again we plot the real value of omega as a function of mu and it's very similar to before you have this stability region up to mu critical and surprisingly we put here a hole here the ratio of that small to large radius was one to two and it had no effect on the threshold power the hole completely didn't help to stabilize this necklace wave this was when the student was really depressed this was the bottom of the and there were many points of depression but this was the worst i was also depressed but then we looked again and we said wow what do you see here here again you have a stability regime here again uh at this large mu you have a regime where you again have stability and if you translate this second stability regime you ask what is the power here well it's above three critical powers so here the hole tremendously helped you finally get a solitary wave which are stable and just to show this numerically so we take these two cases so one is a power but in the unstable regime it's this one and this one is in the regime where it's predicted to be stable and we add some noise here so here you see this one it starts here and you see it's unstable and this one oops you see here with the higher power is actually stable so numerically we do observe now we have a stability regime who is much much more than one critical power and then the student started to smile and actually the annulus or maybe more generally a non convex domain or not a simply connected domain is very interesting uh if you look on this r equation and you look for positive solution so now not lackless now i look for radial for just positive solution on the annulus so i saw this r equation i impose a positivity and uh i want to compute the ground state solutions or the positive solution so it turns out that if you start at a low power then you have radial solutions these are the ground state solutions the positive one but there is some threshold value where you have still have this radial solution but they are not ground state anymore you have a bifurcation here here you have a second family of non radial solution that try to get localized more and more as you increase mu so we have here a symmetry breaking uh above a certain threshold and this is not very surprising if you believe the variation or characterization of the ground state that it tried to minimize the Hamiltonian clearly at a certain point it's easier to minimize it like this than using a radial form but nevertheless as far as i know and uh this is the first case where we have a two different positive solutions of this r equation or a positive solution which is not a ground state uh frank do you know of any example that okay you tell me okay so actually this phenomena have been observed for the annulus with a potential for the inhomogeneous nls but not uh in this paper but not as far as i know for the pure nls without the potential so what about the stability here so if you take this uh this radial solution on the necklace the radial solitary wave then below this threshold it's stable this is how it started this is how it behaves at z equal to 5 if you cross this threshold then the radial solution is unstable you see it tries to get low it tries to get localized but above this threshold the actually the non radial solution is stable it doesn't change with distance okay what about uh how do you compute these solutions so uh you want to solve this r equation in one dimension it's easy in higher dimension it's more challenging because it's a non-linear elliptic problem uh and uh worse we cannot even use any radial symmetry or something like this because there's no symmetry here and even worse we look for a non positive solution with a necklace structure so how do we do this so uh uh because we have this symmetry between the pearls the goal is you just compute a single pearl which is positive and then extend it into a necklace but how uh would we compute this positive solution on a bounded domain so actually when uh when you are in a free space there is a very simple method which basically is based on taking the Fourier transform of this of this pde doing fixed point iteration you do a simple rescaling and it converges beautifully very simple 10 lines of code all in the book but you cannot do a Fourier transform here because it's a bounded domain so it turns out actually that the Fourier is not the important part of this method there is the normalization is the important part so we just do this fixed point iteration but with a non spectral version so what do we do here is the equation we write the linear part here and then we can write it as fixed point iterations if you just do this fixed point iteration they always diverge either to zero or to infinity so all you need is at each stage to do this simple rescaling uh by some constant here in order to prevent this uh divergence and the way that you determine this constant is by requiring your solution to satisfy some integral identity which uh r should satisfy so for example you can take this multiple by r on both sides and integrate it doesn't have to be this this is just an example and from this identity you fix the value of ck at each iteration and this is enough in order to compute all this solution it's really a an amazingly powerful method that works for positive solutions and let me finish by saying that actually there's a very uh whole analogy of this in one dimension so and this was starting by a for kyozome, haj-sal and kikuchi so they're looking one dimension here for so a multi-pick solution sign-changing which are maybe the equivalent of these uh necklace solutions actually they are this one-dimensional solution they're both the analog of the radial ones which were studied previously and the ones that we have now and the theory in 1d and 2d is there are a lot of similarities okay so in summary we have a new type of solitaire waves of a 2d and a lesson bound of domains uh the threshold power for the necklace instability is a lot of time below the critical power of collapse so this is instability which is not related to collapse and for the case of the annulus we see that miraculously there is a second stability regime where we can propagate solitaire waves with power well above the critical power uh we see that the annulus and the lesson the annulus has many interesting properties there was a symmetry breaking here there and we saw the numerical method and everything is in this paper and thank you very much for your attention Merci, il y'a ti di question. Are there any questions? Yes please. So how do the vortex waves I mean in a disk vortex waves in disk practical purposes how do they compete with the necklace? vortex are unstable yes vortex are always remotely unstable we tried initially the vortex because the critical power of the vortex is four times the power of the solitaire waves so it would have been great but they are unstable yes I'm just wondering about a simple thing you see uh it's a bit related to the equation but you don't take the grand state but the the next one the first excited state the radiolumine yeah first excite that is so big vision sign okay so he has so it's a mix between uh the vortex and the when you do the things with four let's say but for uh have you tried to do something like that so for each one is the next either yeah for each one this one are stable yes yes so you have the same problem I want stability although the excited states are stable on bounded domain at low power but we want high power high power I had a question about the restabilization where you have a very tiny interval of red for low mu and then this big deviation and then it restabilizes when in the restabilizing regime the uh your solitaire waves it's the same of course and sectorial decomposition but they're very much more localized and so could you say that their stabilization is related to their separation the the the annulus domain walls are keeping them very well separated uh I thought a lot about this and I don't have a good answer uh because they cannot be too localized because if they are too localized then it's like like free space so they're unstable by themselves each one on the other hand when they are too uh non localized then it's unstable as well so there is this regime where they are uh mildly localized but why I try to we try to see whether we can say something more intelligent than that and we couldn't I mean to somehow get some better intuition why you get it there and I'm sure there is but we didn't find it for example you have small region of red but then instability then a large region of red that's very nice and then instability there's no more region of red higher than that is there not that we know not that we know if we checked I mean I can never say but we didn't spot it and then how about if you have very high many sectoral sectors like a really a necklace because the second region of red exists and does it is a nice big big one or how does it go if I wanted to get a necklace of pearls through a lot of atmosphere I'm sure to make very many sectors you succeed and so let let me tell you what's the problem the student graduated so there were many there are many good questions that I want to answer but yeah but actually the key thing would be to get an experimentalist to do this and trying to so it's not easy and the main problem so we have simulation which are very convincing but the problem is that when you do the experiment you have a limited control over initial conditions so you cannot make it exactly like this R it has some shape so then the question how can you find an initial condition that using experimental techniques how do you induce the necklace that's actually easy because you can put the phase place which would make the plus minus plus minus but then how do you make initial condition which is sufficiently close so that you will get a necklace at the high power so at the low power they were able to observe these נקלuses numerically but at high power you are always if you are unstable you like collapse and so I would say if the experimentalist would come then the student would also come to do this so I know this is very way there are tons of questions that I want to ask whether you can play even in the rectangular way whether you can have some sort of weird shapes that would give you more than more power I don't know it's open your questions thank you again very little