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Necklace solitary waves on bounded domains

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Necklace solitary waves on bounded domains
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The critical power for collapse appears to place an upper bound on the amount of power that can be propagated by intense laser beams. In various applications, however, it is desirable exceed this limit and deliver more power. In this talk I will present new solitary waves of the two-dimensional nonlinear Schrodinger equation on bounded domains, which have a «necklace» structure. I will consider their structure, stability, and how to compute them. In particular, I will show that these solitary waves can stably propagate more than the critical pohe critical power for collapse.
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the term and to me 2 and thank you Mister Chairman there and a lot of things organizer of the invitation my 1st time here the please this is joint "quotation mark was my of student and the must be Monday and I let me begin with the physics so this is it that they would have in the laser beam the propagates in the the election some transparent media In the last Walter who will take over Our direction of propagation is the Intelsat the normally don't meet you here Is it started sequence and the equation of propagation and actually eating all the that put dimension Hubie In a letter I'm also here we have the defection terms which tries to make the BY that was propagation them but would have been only nowadays Gilda Carle lonely nality the which tries to make the being no was propagation so we have this competition between focusing on Monday nite in the fractional and and there at this week impose initial being Crawford's equals the also the our initial conditions and the propagation even this so this is an initial value problems but since this is a mathematical audience most of the token called this the hard to make it life is there but there are some that will left behind in their transparency so whether this is the it's the OK so you have this competition between the focusing on me now it diffraction and actually this fields farther from experiments which have done in the early sixties by that time you had a powerful laser also 1 of the experiment people shoulder this fall for laser simple glass and what the show Is that after some propagation ended last did the actually the laser beam became well so much that could damage the clock the glass those really damaging class so the qualities optical catastrophic collapse and their service explanation will be given in 65 but clearly we use that pulled the CU became the last and he showed numerically the that a solution of this equation can become final in Cambria lopping finite time which of course physically fine and so there's a lot to say about the block lotion endowment expert in this audience and whatever I had to say about it said in this book but to them not going to
talk about a lot I'm going to talk about some of the ways so what do we mean by some at the way we look for physically solos from the propagate and there's in action without changing their regular Paul so this is very well-known equation for the profit of the and I did dependence on this new there was only so that they relations so it's really doesn't play an important role in here and then we are always in the pool D cu Case in these talks will only go to about 2 critical and don't fall their power according to normal farm you is independent of this problem may call to Powell not until normal mass because in all optics this quantity corresponds the power of the laser beam out in what some analysts OK so here's this equation because studied extensively that of an infinite number of solutions regular solution the 1 we use of a chat about is the drums states which physicists would call the Times profile but spotted monotonic you'd be crazy anti-terrorism exactly the critical powerful collapsed and if you plot because their at this is altitude has a single ticket colleges and I use location which is slightly different usually people you know solution by Alan according to our 1 because in this stock the superscript would always know the number of speaks of this emotional and dissolution everyone speaks of this white-collar so what do we know about the solo show about this subject in a way that it should take an initial condition definition canoe multiple by the if poses larger than the 1 the roast and rolls up and find the time whether this is less than 1 then you have scatterbrained as he goes to infinity and there all this summit Delaware is an insult to the dwelling stability if you put them replied direction it wants to and so this
is ah be treatable mathematically but physically this is very bad news because of what you want physically is to be to take to laser light in these and shiny let's say 2 7 km into the atmosphere and headed to the stable away I would you have stable when I just wanted to afford collectors and so this is a very important problem how can we stabilized laser beams when their propagating bulk medium it's extremely important seemed important problem I know a lot because of this atmospheric propagation but not just because of so wonderfully intriguing maybe I was proposing 9 PM and that will put you over head and neck being so what is a necklace being taken initial conditions and if there so exist such a stood searching the other actions multiplied by co-author named so that the plot of this solution because the way I am there after he deems which are called terrorists vocalist bond with this that necklace Sega always has the restricting names and I doubt that it would distance from each other on the wall on the register file marks and adjacent paracetamol several opposite signs openly trees and why use this configuration because if you look at this race between each took those signed up is available long the race and by uniqueness side remains available along this race and also if you look at just 1 per photo as part of this process saying that the name makes it on the city's abundant conditions on this race so that feels like all the other girls because are bundle condition is reflecting back on the condition of this thing that comes the abundantly flipped and so the necklace stocks still would be preserved at all times In the way physicists would tell you he would not tell you this using additional bundle condition what he will tell you is that if you have to be encephalopathy trace the repairs which are there and that's why didn't don't so he entered the game dollar-negative necklace being so what was said to cited the let's take each step and give it power and belongs slightly below the critical powerful corrupt it's slightly below it means that the lonely narrative is slightly weaker than the friction so it cannot collapse by itself so it wants to expand slowly but as it tries to expand its outfitting the politician by day and nearby girls and so this would slow down the expansion of speech but for just to show the thematically here within the simulation will have a pair we've just ring was full pearls it cannot seem to have opposite faces because brought the intensity of the and this is the dynamics of the which is the goes on you see expands it keeps the necklace structure but if you just would take just 1 of the spread of it would expand much faster than at the same time so did this slows down the expansion so it's that expensive amounts lowered then for a single being sometime this may be enough for you consume but ultimately this structure I know the game and skittle collapsing
and this is an experiment which came 8 years later ,comma this will dominate the call by to the group now and hear what that you see input being this is a necklace that they have opposite the 1st and they said they enabled would be mandated after propagation this is also the sentiment of thing last Asiad the necklace touches is preserved now the the question whether this is fable of Wali depends if your mathematician of physicists policy this is the 5th ability for mathematician and not but
this necklace Genentech living being used as a means not just at the end for example in the and paper the use of the 5th set up a necklace being structure which was the other in order to set up a waded sometimes you don't kill but the nexus itself but use it to and let me point out that it may look like pieces of Baltic being but it's not because of all that's being is something like this policy to the I'm fate when I have evolved its name in April the amplitude it's ready the necklace babies from such it's different form but OK so can would have necklace solely to the way in free space include mention the answer was no I could take it to the CU and unless does not admit necklace a little weight and babies what they mean is if you take this out equation in Europe for the solution of the form there is no such solutions I put in informal process because we have sort of a arguments probably could be made into Lagos proves that this looks like but I believe that's correct that should there have been a proof of the existence of thought of our warranty peaks and changing solutions also doubt equation in this paper but that they have a very complex construction and they have to wait to rings 1 inside the and out it's not of the form that we have here so that I would be excited changing solutions but not of this fall then and it anyway whatever you get here will be strong and said that because missiles from the summit to the critical strongly insisted and now the head the observation of the necklace so that the way it with a different nominee out of the disease in the photo refractive billions will you have this type of money of the and you have some potential which you use was an neither later when another laser beam so in these kind of fare type of unless you can show that the disease so necklace only 2 waves and that doubt even in some the ranch stable and this was shown so radically numerically mentally there's only 1 problem I the medium that we usually take an old photo refractive we want and we want glass we want Walter its tail it's not sort threat
so what can we do Our goal is to stabilize the necklace Dominican so what they did which had been used since I think when 30 years it has been to confine the laser beam tool for local fiber what is the whole local fiber you 5 edits and the hearing the inside and you pulled some hair loneliness enough material which can be fairly can be something nobody has so it has a cubic nonlinear that's what we can and 2 leading all when you propagate here the walls are completely reflecting everything that hits the wall can come from and also if you ride the equation for the propagation of it's that put the QB m but now it's not free-spirited confined to some region the course section all the 5 Bell which did the killing it would be the this fall the other shapes of a local fight both abandoned experiment
so this brings us to the initial offer the end last on the bond domain so I really is our equation we put this village rebounded conditions hold reflecting pondering you can again look for selective waves of this city's the equation the bones of the living and the dead this equation than was studied by his friend and myself input pollen 2001 and later lived in some models of these Of this equation the studies will mostly on radio solutions on the 1 while the House summarized
studies into 1 slide so the important thing to note here is that we don't have the nation symmetry anymore because of the the Labonte condition and this change of everything so for example if you look at the power it is in doubt because of the it's not independent polls this year on used starts from you now out which is the idea value of building up an equation of the smallest rise in value and then there are from here the is day the pollen criticism .period Pritchett critical follows renewables to infinity analysts noted that the is increasing at a function what about
stability In journalists if you want to study this stability so it's a little ways that is this type of stability is stability so you gotta love asylum you always their cultivation which is real and imagined it's hot in the summer of heightened but only got to put it into the equation arrived in the Netherlands equation and that this well-known system with the plastered and wonderful child and is and what you want to lawful stability is whether you are buying and values I was positive in part because if you don't then you have unstable and what they want strong I believe both by the feet of the political that Exley instead of standing beside environment problem you can replace it with a much simpler thing so if you will you look on the positive solutions then I knew heavily stability is that policy is more not only can decreasing so this is the 1st at the physics literature this would be called the verdict of corner the speaker conditions .period and
actually this condition is not only sufficient for enough stability itself was sufficient for all the tells stability Our not on the unit Dennis Kuwait City numerically the disobey conditions make a condition is indeed satisfied the powers increment as a function of the and this leads the following ceremony that if you're open the grounded solely to the states on the unit these days all with Tiffany's stable and when we talk all the stable along the bundle domain would mean standing at the opulent phase change of undistinguished and this is very different from forests influenced by dishonored the words of unstable and it put the Cuba case so that by this reflected their their reflecting bundling is stabilizing the following :colon weights and this is journalism in this land boundary condition has a stabilizing effect some so that's the good news the bad news is that if you look on the power of the summit Delaware as it goes with wins it won't be critical so the most we can propagate and stabilize would be therefore slightly below critical slightly below because this is an experiment you can do you don't have exactly the process and the with some lawyers that 80 90 per cent of the pretty we want to be able to propagate much more than that so how would you do so while Iowa were trying to think of different ways to do it and 1 idea that came to mind is to use a necklace a little way to go away if each 1 of these would have slightly below critical power altogether while get much more so that that was the motivation for studying the use of physical motivation for starting this microphone but what do we mean by necklace on the way so we have decided equation on the bounded domain and we can of course taken a simplified and then the the necklaces away would look something like this with the plus-minus plus-minus but we can start the well-muscled edition would comply with the play with the demand so we can take a stand on and having a necklace so little levels the formal maybe even an analyst and had something this format With fabrication techniques to that shaped basically can be fabricated and that is enough motivation as far as I know this photo was will not the people did not appear in the form the letter turned on so let's stepped in with the
rectangular ones because the from numbers so we want to thank you Americans how would we do with this I thought we would just compute 1 pair of systematically on the square for this would be a positive on grounds that solution program and then we would extend the coordinator by putting the 1 next to the US so this
brings us to the question of how the we compute single parent single pulls it peaceful motion grounds solution of among the square but this is not very challenging the numerically funeral which method to use which is the spirit of a normalization method you just need to have some instead those and if I have time Indian and talk about it and if not its the but the
other results so this out there thought it away for different values of your this is the power of the function of new so you start from New Orleans not as before the leader of the eigenvalues you'll although the the program analysis of equation and as bifurcated from this mural in all in their weekly normally analogy so what's mainly supports the summit Delaware of is bound to but as they you increase annual you go to the struggling on another gene well you almost like InfoSpace you hardly feel the bond all almost like since when
catapults preventative at some of the disease conditions set for OK so we know how
to compute 1 per and now we want to make a necklace out of it so we just put them 1 next for the tough-minded plus-minus and I this event gives you an honest solution of that equation wasn't chip so only on these interfaces with 0 and the only thing that you have to to check if that indeed the equation is also fit his this interface but it does so it's fun to enormous necklace solution of the "quotation mark
but I didn't start getting crazy for them but you can't put them in this form and this would also be a solution and Nicklaus solution "quotation mark who can play with the shape as
OK what about silicone lessons so we take their art equation the said coordination of bundling conditions so again we need to 1st compute 1 but also for example lets of it on the quarter field goal so you do the same techniques and American method and here is there but a similar picture of This is the weekly normally analogy the and wide the struggling on analogy I don't know when I to free space this is limited and the VK conditions satisfied
and once you have 1 parent put them next to the other and then did you say have the on what researchers possible you outside Jing more was up my definition is that I will take a sector that would have only 1 my numerical method can only compute positive solution so would always look for 1 and then extended In human numerically equal to what some call all this effect under new media to perform closing since you have them different memories are different regularly and this not method so it was to I don't know I need them start I need therefore for minor Michael method they need to be able to impose distillates live under conditions I need to know where they are and if you put these days have told their it's not clear to me well the deal and there is a tendency was about to read them and the on the computer the no what about the Angelo Sodano would build a simple situation we compute the single parent of positive solutions on the sector lost animals and again the same behavior of the weekly analogy strongly known another gene vacate condition is satisfied and once when we know that we just put the 1 the other and media necklace on way so this brings us to the initial stability are so when we talk about stability here that would 1st should stop with stability of a single person and only then moved stability on the Net so when we talk about the single parent Sohail for example of the square of the Quantel silicone and their "quotation mark analysts and we take the initial condition as we take this and multiple it by 1 . 0 5 and here's what you have after the people that it looks almost the same and in all the demands we have done we always say that these a positive solution among the demands of all stable and it is the greatest was the fact that they all satisfied that conditions of this end in itself political statement but it's the American observation but what
about the necklace that well it depends what type of probation pool His you would multiply everything by 1 Posey will find for example like with the previously then I actually signed with the remains on this interface because you don't change it in the initial conditions I knew was the multiple is you you maintain the and defamatory is between despair and our folks I would remains the only thing the faces some the post would not speak with each other so the stability of the whole necklace is the same as the stability of the Singapore so if you have any chance of destabilizing the necklace is use a perturbation that would break and defamatory also their so Delvalle dissuaded you can break this and defamatory with respect to this interface so 1 possibility is that just annoying another 1 which musically savvy is to check just 1 of the Boston multiplied by 1 of of the and there is a prefer this pervasive because this isn't a Communist cultivation you can trust the balletic much more when you do that ,comma newspaper automation and legibility surrender OK so I was about the the condition was each 1 of the Pearl satisfied them their victory condition and the power of the necklaces and times the all all this is the novel's prose times the power of the expression then the vacate condition would also be satisfied necklace but the government maintains that the necklaces stable because a fake a condition is good when you have a positive solution now this is not a positive solution and the next day so therefore you need to go back to the origin of the prevention of this the the completion which is their analysts just the lead us to believe that the single goal back again to the system that was the end classes reminded us this is a leading arises equations for their perturbation of the example of the wages and you want to check whether there are some eigenvalues was positively so let me show
you the result of this computation for the case of a circle necklace with full so what we brought here this is the venue for new ways which changing a with broccoli all former got so if there was any item revenue was positive real public property could do not think it means that there's is no such thing so what you see here that is small religion between you Romania In some clouds over New New critical justice here for which there is no eigenvalues positive impart meaning it only noticeable but for all of them you will lose you always said reminding readers with positive popular don't just 1 that I can valuing fulfill the start the 2nd item so you have stability in the here and now the million-dollar question is what is the policy of 1 you become unstable the new exceed the critical power on so the answer is no did you get this instability when the necklace power 1 quarter of cryptic 2nd sound that's not bad so if each 1 of them is one-quarter but no together that one-quarter of critical policy of each 1 of them is like on so the was is that this instability has nothing to do with collapsed because you are well below the critical powerful collapse but the big news is that when we started out to be better than the readiness the way and we ended up being much with us because we sediments on the legislative where can get up to critical path OK let's look on the dynamics of these Soviet we take these suppose authority away thought there would just this 1 we multiplied by 1 . 0 5 and Willets eagles on the this is fun you'll slightly below critical this slightly above critical of and stated that he it was reserves the shape and this is equal to 10 words here already had closely it did so unstable so did you see this change of stability and new critical it's sort of blocking instability and a 5th and instability in London so now if you want we want a not to give up on this solid allege that look too promising so I said OK maybe fly understand the source of this instability I we wouldn't be able somehow to fix Seoul In order to do this we need to understand what is the instability dynamics and in all on the surface of funeral opened there I good value you just get a yes no sustainable unstable if you want to understand the instability dynamic then you need to rule on that item functions all this unstable this tells you what is the instability that India OK so that I can function so for them but this is the function but there that all of these lines that correspond to this the firefighting OK you can look at but what you see when you look at it so this enclosed stadium and that's why ProTour independently the real and imagined innovative vendors and not just absolutely so the seems to see here is that this kind of money or is anti-Semitic respect a wine like that item in front the like of the solely way but the scene clicked with spectral so what does it mean it may answer to start with design and function which antisymmetric but you had to adjust metric perturbations of the boat the result of these is that power shifts from 1 part of the island so what is I don't function tells said the cyclist function correspond to Powell's flow index erection the the 2nd item function that correspond to the same item value In this 1 has the symmetries of it's seemed anti-Semitic respect leaks but it seemed that it was better to wine so therefore here you would have power from the wider action and the 3rd I function the 1 that you start from here as a it's asymmetric was picked to X and Y sentiment that cultural between the wall and useful but
so far that was the point of fair-minded this year and you have to think about the future the plan pledged to the destruction of the Have you said that there may be pretty well you would also see late somehow because it's confined or whether that's pretty kind he should be here because of what the full effect which confirming sanctions because of of you need to know the it's a good point I actually do maybe the here yet here what we've seen sometimes at it's pretty the compiler you sometimes good sometimes but I guess it depends on the perturbation I can say there's certainly knows that "quotation mark throughout the list this Austrian stable looms what have you for this period this summer of of the wine should from this the problem is not that it would be beautiful story about look at so here it is at full stability we have this not as we expected so we wanted to probably more power industry so what less intuitive maybe I would say that while I just use mobiles wireless phone exceeds 6 and 21 as long as it's an even-numbered that's fine well doesn't get you anywhere actually the power of the whole necklace goes down this increases the number of so a 2nd maybe I would be to sell that's about another domain so for example refuses lets users square was faltering so actually this will get you something you quiet Domaine not was 6 but here but was for you contributed to this special powerful stability jumps by a factor of 2 which was surprising to us and in Paris sold it will be almost the same Silverman the school so I have no idea where the of the but still this is well below what was ready and unfolded way so then was OK but the the source of instability in the Soviet ability is the power between the different so we need some help to be able to to walk not to allow them to talk with each other so the idea was that we had is to use the angular domain since if you put the home side where they can still speak with each other but much less so OK so this was all
motivation to study the necklace so that there was only 1 of the main cellular signal the same prompted we had before but now it's far away for both on and and that left and again would probably invade former as a function of the New World and it's not dissimilar to befall you have this stability Legion of Boom you critical and surprisingly we would put together a whole field the ratio of the small to larger deals was 1 2 and it had no effect on the special poll the whole completely help to stabilize this summer with this was when the student was really depressed conditions the bottom of the handle many points of depression but this was the work of art I was also the surprised but then we looked again and so while others see him here I didn't have a stability converging here again in this large new you have original use again stability and if you translate this 2nd stability region we ask what is the policy of it's about critical Paul so here the whole tremendously happy to finally get so that the way which all stable and just to show this rightly so we take this school cases so 1 is that Powell but in the end insist unstable legitimate this 1 and this 1 is in the Aegean Willits it's predictably stable and we had some noise here are so here in the city this 1 it starts here in the States and stable and this 1 a book you see here was the highest power it's actually stable so numerically with dogs and now we have the stability gene is much much more than 1 critical and then the students started flying
initially the handles all maybe more generally and complex the meeting hall Malta simply connected with this very interesting if you look at the start equation and you look for positive solutions will now not likely not look for regular sold for just positive solution understanding 1st so this out equation 9 poses positivity and I want to the downstate solution all the positives so
it turns out that if you start at low power then you have regular solution this underground state solution the positive ones but there is some threshold value while you I still had this regular solution by Donald ground state anymore you have a bifurcation nearly every the 2nd film of non-random solution that tried to get localize more and more is increasing you so have the same approved above a certain threshold and talks that surprising if you believe there are variations on collectivization of the grounds that to try to minimize and Antonio clearly at a certain point it's easier to minimize it like to believe using them for but
nevertheless as far as I know this is the 1st case where we have a good different positive solutions of missile equation all of positive solution which is not advanced Frank development agreement with the pocket and noted so actually at this phenomenon has been observed for the analysts with the potential for the homogeneous and last but not star in this paper but not farther north along the pure and analysts without the so what about the
stability here so if you take these this is a regular solution on the necklace made of solid away than below this threshold stable this is how it started how would be added the proposed if you cross the celestial than the regular solution is unstable it tries to get lonely tries to get localized but about this threshold that actually did not write them solution is stable doesn't change this distance OK what
about their how do you computers solutions so you want to solve this out equation in 1 dimension it sees the new Hyundai dimension is more challenging because it's normally not only the problem I and Wells who cannot even annuals and Redding of funding that is because of losses and even while so will look for positive solution with the necklace structure so how do we do with it so because we have this symmetry between the top the goal you just computer single pole which is positive and then extended condolences but I do we compute this positive solution on the bounded only so actually win when you are lying there for space those of us simple method which basically is based on taking the full transform of this whole deal this could be came during sex .period iteration of both the simplest scaling and it converges you but a simple compliance of coal on the but but you cannot do it so it transform him because it's about the so it does not actually but added that the food is not important part of the method devised the normalization is simple part so we just built this year a six-point intervention but was announced by convulsions so what do we do all the other is the equation know about him and then we can write to fixed-point iterations if you just below the 6 . NIH that always leverage tied up was the old contributing so all you need is at its vegetable the simplest scaling by some constantly in order to prevent these they virgins and the way that to determine this constantly is about acquiring you'll solution to satisfying some table identity which should set aside for for example you can take days multiple buyout on both sides and integrate I would have to be this is just an example and a from this identity you fix the value of the correction and this is enough in order to conclude on solution this really is an amazing powerful method of for positives which and let me finish by saying that actually there was a valley all our knowledge of this and one-dimensional so and this was standing by you for Use Only Hearts silently "quotation mark so they're looking one-dimensional here for forum would big solutions signed changing which maybe the equivalent of the necklace solutions actually there are indeed this one-dimensional solution that both the analog of the regular ones are which will previously and the ones that we have now and that's still in 1 day and told the I know a lot of CIA similar OK
so here's somebody where new type of solid wastes of the told the analysts bungled remains Our the of power for the necklace instability is a lot of time below the critical part of collapse of this is instability which is not related to lot and wild 4 the case of the analyst with citic and recklessly that was the 2nd stability urging will we can propagate solitary waves was worried about the critical power are we say that the analysts and the lessened and the loss of many interesting property that the thing deliberating there are we sold in the local Lessard and everything is in this paper the problem is that you use your home you have to what worries me and it was this problem telling people vault at some unstable a yes vote Exxon always on the mood in the unstable we tried initially in the Baltic Sea because of the critical part of the Baltic Sea full times the power of the Fullerton waves of meat would've been great but the report from the conditional on the way in which he she was issued during the meeting which him you don't think nothing the yield on the 1st day of the regular room the traffic signs and that he should be change from the focus yes the solution to the problem 2 developments on along with news of the week for some sort of political tensions and questions so forward into 1 there the next season this year that's right he was asked the the heroes and wants to be although the excited said to have a stable among the domain at low power but we want type but when of questions about the regionalization where you have a very funny the overall rest of you the deviation the reason for the winners in recent quarters in the soldiers sustained growth and the center of the composition of influence and so the status their civilization is related to the separation with the emotionally involved he very well so have I suddenly thought about I of have good times a because there cannot be too localized because it's down to localized and it's like bad like free space Ford unstable bind themselves which on the other hand tool and not a localized than the unstable as well so there is this room bridging where there mildly localized but while I tried to we tried to will see whether we can say something more intelligent than there and we couldn't undo somehow will get some better intuition while you get to them I'm sure that is a button for example you have small you realize that this is really the whole lot of visionary courage and instability in the region requires that not that I know of none of you know who checked Munich analysts said weapons With wouldn't export however he have a very high and basis for the really and that was the 2nd Reading Room existence this is the most B we have to do what you so that approvals from a lot last year grew during the various rooms used to see solar and edited what's the problem the student attitude so had they were met them any good questions that they want to answer me about you said here about their bid exited the key thing would be to get an experiment in the do In trying to sort here it's not easy find the main problem so we have a single nation with letter convincing but the problem is that when you the the experiment and you have a limited comparable initially be condition so you cannot make it exactly like these are because some states so then the question How can you find in the initial conditions that using their experimental techniques were hard to induce the necklace that's actually easy because you can put the faithful Eastwood who make the plus-minus plus-minus but then how do make initial condition which is sufficiently close so that you will get the necklace and a power so the mobile of the well able to observe this necklace but the Taipower he always is unstable wouldn't close collapsed and so on I would say if the experiment in this would come then the student would also come to the floor I know this is that that also question that I want to ask when you can play even in their extended away when you can have Funchal we'll change that would give you more than Moapa told he fumed