Interaction of small mass solitons for the half wave equation
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Title 
Interaction of small mass solitons for the half wave equation

Title of Series  
Part Number 
1

Number of Parts 
23

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

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Abstract 
I will discuss the construction of solutions for the one dimensional focusing cubic half wave equation based on the interaction of two small mass concentrated solitons, displaying growth of Sobolev norms of high regularity. I will focus on the study of the modulation system and on the energy estimates. This is a jointwork with Enno Lenzmann, Oana Pocovnicu and Pierre Raphael.

00:00
Dispersionsrelation
Group representation
Multiplication
Presentation of a group
Lecture/Conference
Wave equation
Characteristic polynomial
Equation
Mathematical analysis
Circle
Absolute value
Group velocity
03:38
Wave equation
Resonance
Lace
Theory
Subset
Number
Power (physics)
Sign (mathematics)
Lecture/Conference
Term (mathematics)
Moving average
Energy level
Conservation law
Liquid
Uniform space
Descriptive statistics
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Area
Series (mathematics)
Conservation of energy
Closed set
Mathematical analysis
Peg solitaire
Local Group
Turbulenztheorie
Convex set
Equation
Cubic graph
Object (grammar)
Resultant
09:44
Point (geometry)
Maxima and minima
Existence
Beat (acoustics)
Momentum
Observational study
State of matter
Wave equation
Translation (relic)
Solid geometry
Mass
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Food energy
Rule of inference
Scattering
Group velocity
Causality
Lecture/Conference
Velocity
Natural number
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Devolution (biology)
Hydraulic jump
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Product (category theory)
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Concentric
Expression
Functional (mathematics)
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Principle of relativity
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18:30
Maxima and minima
Group action
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40 (number)
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Propositional formula
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Frequency
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Natural number
Operator (mathematics)
Square number
Special functions
Friction
Social class
Process (computing)
Spacetime
Coalition
Duplex (telecommunications)
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Infinity
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27:01
State of matter
Wave equation
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Equaliser (mathematics)
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Sign (mathematics)
Square number
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Soliton
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Elliptische Funktion
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35:55
Point (geometry)
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Presentation of a group
Multiplication sign
Wave equation
1 (number)
Resonance
Parameter (computer programming)
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Integer
Subtraction
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Spacetime
Forcing (mathematics)
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Separation axiom
40:59
Logical constant
Dynamical system
Collision
INTEGRAL
Multiplication sign
Direction (geometry)
Correspondence (mathematics)
1 (number)
Insertion loss
Parameter (computer programming)
Mereology
Food energy
Square number
Series (mathematics)
Descriptive statistics
Physical system
Area
Closed set
Structural load
Moment (mathematics)
Infinity
Thermal expansion
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Radical (chemistry)
Category of being
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Hausdorff dimension
Order (biology)
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Equation
Right angle
Resultant
Point (geometry)
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Rule of inference
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Wellformed formula
Operator (mathematics)
Energy level
Stochastic kernel estimation
Subtraction
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Hydraulic jump
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Scaling (geometry)
Conservation of energy
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Physical law
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Family
Limit of a function
Separation axiom
00:06
and in and around the world to do that and the mood on the way out on massacres in the Shakespeare uh misuse organized at his office said it given Yasuo on an ongoing budget community over the use of its 2 could look at the issues I'm very pleased to to open this workshop in Aichi uh it's pleasure to be here and it's a pleasure to talk about analysis you and so the the world can going to talk about it is issued from Long joint work with and collaboration with and the woman's man on Apple ,comma who and clear Fed and slightly that some of you already he said 1 of us talking about it In the last few years during the last few words it does all that we change the but a little bit of the presentation not we made in the paper faltered and then but I would like to apologize to for those of you who listen to me recently and particularly to at them because he sent twice twice that to the store but this 1 is slightly different I would focus on and some different features I never discussed so far so so that is the main idea is to talk about some some very simple onedimensional equation which is called the half wave equation so and so we are working on there are some versions of on the on the circle but I will not discuss it here and the and when introducing the physical location in the whole of 1 over on the DAX and so my my multiplier is my amendment applies in this story is Monday which corresponds in the 48 representation as modification by absolute value of so the half wave equation is the following onedimensional but also you use a complex valued sanction of red able to and the following equation ITT you minus Monday you 1st what you square you can cost so 1st of all you should you should show that this is a wave equation in the sense that if you look at the characteristics of all the dispersion relation is moxie so you have to and to cities group velocities which are plus or minus 1 Wilkinson so
03:39
at least from the I'm level for the linear Paul this is really the wave equation in fact it's a half wave question because your allotment from the classical there I mentioned is just I did the minus 1 did I'd like to Crestmont again so so this is the reason of half of not not the 1 is 1st with the nonlinear review which is due which is suddenly in this context the most
04:09
natural nonlinear you could have and secondly you have here a plus sign which means we see this is a little bit later on that very shortly but this breast size means we are in the forecast thinking the fastest focusing case and if you know well the workers of nonlinear wave equation particularly and you know that focusing case is known to admits more money in your objects in there in the language of pot for instance and we will we will use this very much even if the number with the focusing case could be very interesting to Kingman but probably more complicated for what we're going to do OK so why looking at these equations in fact the main innovation the about about this equation this is to look at what we call sometimes with turbulence but with turbulence is very complicated and not that well identified area so I prefer to talk about growth of some which is crucial the strict subset of of wave turbulence in the sense that it's it measures infants and the the appearance of smallscale so while looking at this equation in this context when you can you can for instance look at papers which shows a series of papers by my Michael Fleeman back but also Starting in there in the late 90 s and and also was 100 the US new yen while many and the people I'm sorry for all the wrong to follow them which tried to give us some some general description of what onedimensional wave turbulence and and study that kind of equation here with an the case and the notice that 4 of 5 bigger than 1 all 4 fortified you love the nominee affording a question cubic nonlinear for lingering question in this equation is known to be integral with plenty of very strong conservation laws and the end it's it's a classical result the coming back to suffer a French about that in fact all the all these Sublett moans or controlled by the conservation if you know what that if you lower the powers of you would see that's OK if you make some kind of resonant analysis that will not go into details yet but the resonance analysis would tell you that they resonant novel nontrivial resonant terms the firefighters bigger than 1 and you can find them if other 5 small 1 for some convexity reasons an awful lot fight was 1 your precisely in the 1st case where you have you have many recent aren't in fact for fight was 1 you have a big presence which make which make I mean the it make the appearance of with children's favorite a certainly the most favorable situation for the appearance of which OK so this is 1 of the reasons there is another 1 and there's a reason why we look at this equation is that this equation in some regime is very close tool but none integral the question which is called the signal a question that I started with said something in the air the during the last 5 or 6 years now and in that in this context was studied in her Beijing this is by far 1 of the group so on the fact that we are close to an system will help us to study uh more more closely the liquidation and solution in particular but will goalie with the regime of modulation of all of traveling with Seoul in fact what we are going to do what we trying to do is to study the small last solitaire all this equation sold In a end and as a 2nd plot which is the main want their interaction and we will see that this interaction will lead to some girls of summer so let me Commonwealth specific bearing in mind the new Of all the theories so maybe the coaches theory could not quite classical call for the equation was completely
09:45
understood by you Clinton and have failed In the United States .period 2013 and
10:00
the point is that you have essentially so 1st of all conservation rules but which will help who we free conservation was the first one is the caustic energy corresponding to the meeting and of the Senate and insistence so this is half of what do you use for this the In a product the rear part of the U 2 in a product of the sinking minus 1 full the final 4 you have the momentum of the which is you you can finally have mass with his viewing and to squeeze him so these 3 guys all conserved and they're using the first one and the 2nd 1 exactly as we did as well as was done by a by Michael Weinstein around the VAT is you can't prove the 1st at the global existence theorem and what as at least a half which sponsored to the energy space here global existence for the initial data all formats if you prefer smaller that the mass of some prostate QC rule so choose 0 is a state solution Of these equations what the Cubans 0 the cause future world where queues so this is really the same idea as the world by staying for 4 and 2 critical analysts observe that this is a critical by the way if you solution you have TX and change the scale connect with them that the 1 of you than that the there you get still at solution so as this nonstate solution was proved to be unique In the nice paper front and lends money but to enforce a translation that and and so but this this is the 1st call the 2nd part is about existence all of critical must know what solution so there exists in fact minimall must know what solutions which is obtained by some kind of modulation of stews Europe was something a small and and this is of course very close to the study but polls done by by fog and then he and his collaborators except that the the the the the theorem by fault about characterization of such minimal must solution is is not available it's an open question but because you can find exist examples OK and and finally the 3rd results which appears in the road these beautiful paper by the gallons a month and a half is that there exists in fact and this is very different from from nonlinear fretting over there exists small last Fullerton In fact for every epsilon small you can find the Sultan of mass exodus and before every accident I wasn't a very exists for months solid of must clean particularly you don't have any kind of scattering for small mosque scattering fuming and this is of course very much good connected to the fact that your equation is not disperse because of that the devolution of the group velocity the governor sitting here remind you of what title in the very beginning the Grobbelaar cities plus or minus 1 so the velocity is essentially not moving except that's a point at which equals 0 wouldn't it hasn't jump but otherwise you don't have any kind of ahead of knowledge over it's complicated so now let me tell you put it a bit more of all these are the small amounts and this will be the 2nd part of my told so the idea of a lawful choosier remember accuse you role is obtained by minimizing the following fashion which is clear but do you view you want to square wear divided by you with find that this is the classical got the elderly and functional so you minimize it and you get objects which are essentially choose wrote to translation in and went to petition and deletions and duplications by the no it does all that here you can modify this sanction without making it a I'm in a room with keeping an equivalent juicy role by introducing for every beat of the minus 1 the function urging the Battelle view was minus the miners the dead the clue who went square divided by you before flying even the gulf between minus 1 wall strictly and that of course the term which is which shed here still equivalent to more than you given the market so I can as well and minimize J. because on the integers based nature Hall OK so what happens if I mean my JB the nature half again just argue a little bit with the concentration compactness and I can find minimizes that is minimized if they're not if you rights the guy the only article equation of these minimizes you will find after restating the following expression so let me divided by matters be I can always do it this is better for me but I'm going to say I'm sorry expressed should be Hey the the what should be desk where so this is the the only on aggression and they said you but if you look at the function you reach out she acts which is equally continues to be that of X minus the that is provided by 1 his because this function Is a traveling wave solution is a solution to your request Of course I could I could avoid this 1 might have been begun here and we avoid this it would be absolutely the same why did I put it just because in fact so this is a problem with solutions With the velocity the speed of speed that uh which is at the top and the guy remember is smaller than 1 which is perfectly In in accordance with through relativity principles OK we so so because of the speed of light here what can but not now I'm in fact interested in Cook continuing with the with the metaphor of with the with the withstood guerrilla activity and an interesting in the Falkland so what happened when the tide goes to war because when the government to
18:31
walk while in fact you can prove that when everything goes to war your chili off itself is bonded in receives bonded in every solid so it's a very nice friction and so that's the most of you because most of you because is oneliners because transport 1 which is a massive tribute and so I just might change of and so I think that goes to wander the man's goes to 0 can you if I can prove that you be Gulf States bonded in nature so why is is June doubledigit nature because homes as the utmost to 1 you would see exactly what happened in this equation so the line numbing of what you have missed his all food operator Monday miners believed that the will what is this operator in fact you can you can see it exactly if project reviews the single project which of the project also on the positive or negative on the variables because so few project for instance on the positive on see positive Monday is the but then the stock is just the oneliners be BWV writing by 1 of the be divergent views so you get the Q metropolis list you without plus the cost by class of cubicles square computer between knowledge you have had your project ,comma she negative you will get that is a very different equation which is 1 Busby top provided by 100 Jewett minus Stewart minus cleanliness of Cu does question so in fact you control using them that of course would surely die minuses banalise as being the most 1 and with it's essentially was goal to 0 while the Chili Peppers will have a limit and the limits and this is in fact the 1st results that would prove is that has become goes to 1 end up to a subsequent so we have that if I take such a minimizes solution of this Saudi Oregon equation I get and after some translation that these goes to simply to he I go along convention duplex objects and children so that his solution with Lisa question which is precisely the question I get bypassing delineated in the projected the 1st projected equation so this due breasts will be supported in the forties based in St. Paul so it does not matter who gave good so well and so well what do you know about these equations and terms of that we know everything because because because of 1 of the the job during her visits and on Apple could called in the visit hints of every nice paper in 2015 she proved that Duplessis is precisely game that due to some translations 1 over experts on explicit this 1 experts either connected by it will force myself to face factor by some translation but you you can in fact characterized all the solution not only the minimizing all the solutions to the equation in the In Asia half and the others and that's a very nice here is extremely useful because even if we don't know about you beat up itself we know its limit as beaten with 1 after OK so and so what I I would like to do to say moreover is that in fact so this is the 1st public opposition while the 2nd part of Proposition 1 there's you that if you impose moreover that Cherie town is also about to find Q blasts and include GX chip grass Claudelle these guys these guys the span of IQ blasts and yes cupolas Ontario as a real space is precisely the chairman of the leading rise parading along Cuba's for the equation a role for Cuba this thing a writer parades plays a very important role in all that kind of construction of modulation the rain we know how to identify the cup OK so if you demand that you meet all is all too little to these guys essentially for being that close to what what you have characterized you because then you'll dice characterized that you read that tends to blessed in that case exactly has been done since to 1 for the dialog and the best start dancing and in fact the the mad because used to be ,comma please is small matter is moved fact ponder the enjoyable but the star 1 but you've been effect and in fact we know more about the stupid we know more about this history recall that right to demand for you for all Cureton president expansion for the banks and small Utah security personal but the big like you brought as AIG's goes to infinity while over X times somber modulation factor which is very interesting which is therefore 1 of oneliners because divided 1 1 plus the tactics this can't on overall I by excuse and this function as is a special function told is the integral from 0 to infinity of of thought in the mind of undivided by come from minus the iPhone which was essentially going forwards you roll on the store small close to 0 . 2 1 and if don't goes to infinity it is like I to so what does that mean it well then you will see some kind of guy multiscale competition a competition between the Bendix and the inverse of the small parameter oneliners beat with and so for instance this will be like 1 of Is there is a long line of the cupcakes school that 1 but on the other hand it will be a fine divided by 1 man has been that it's clear if oneliners because I don't think you and so on and of course you will have some some some some intermediate regimes and daunting .period region around Mexico 1 of the 1 9 so this is very important cost in the coalition where the August so this is the the 1st search results and now what we are going to do is to study the 2 interaction so for this holiday interaction I chose 2 of them in fact both for use on comparative some comparatively studies at
27:05
the so between the centreright coalition which is the question we write it down but that this is a question of all of which hubris is depriving wave versus the halfway yeah you this is the only thing that you have to be here yes if you assume if you demand this the result of an entity conditions what can and should be desolation of the equation I wrote to the government equation then its unique formula that close enough to walk this is implicit function during isn't Of course you could you could do Janjaweed amid this conditions but if it should be close to the heart in order to apply the implicit function here and then Jue know that you that Institute plus it's a small branch off running away from the game the frightening waves have small knots and the mass is going to 0 as the buttons "quotation mark OK so not so what is it the same year Gration What is the halfway halfway we rolled altered down already is that you write it again for you cigarettes that right it's like this plus minus the blast was supplied bless you bless clear to cluster was so I put depressed just to remind you that you as before you transform supported it's the nominated so what kind of information and I discussed the 1st kind of information on conservation soap again fault for the halfway and I already told you we had 3 many conservation interview momentum and last here we also have energy but the infighting that is the energy is essentially the along the fault 21 fall but what about the momentum and the mustpass something infatuated something very big because they this a question as I told you was integral but life I'm interested in 2002 interactions in fact I need just 1 more To have a closed system over conservation laws invitation in order to get the system which is integral in the sense of duty and we see so the are consolation for 1 more year 2nd .period what is there to soliton interactions here so what good is it to Fullerton solution while the other 2 solid on solutions something extremely simple it's solution like this some from Jake was wanted to pay off Cuba obsolete for the scaling factors the new right at the home of G of the cute loss of X minus pages of the divided by the cabbages of and what is office which is really quite exceptional in such a
30:51
situation is that we have exact solution of a mole remained so if you start from the date on which is of that kind that equality role where all 4 jails are complex numbers cabbages all positive numbers and injuries are reimbursed many of you keep such a solution for all time this is of course an expression of this thing to gravity properties and the fact that you have this fall conservation along with a little you I don't know To solve completely in a complete explicit way the audio system all these on these dates parameters a threeyear parameters here you have to really parameters 1 1 4 times to a so you have a it's real parameters so it's a simpler to many full dimensions 80 flights and you have 4 you have for conservation using information to get read you've eaten into gravity and you can solve it expressed historians don't know the interact but in in such a way that they don't produce something new and the remainder to but they interact because you would see that the 48 will have some special behavior for a long time so was especially heavy of the audience rule to system of awarding wouldn't it be better cancel you don't need any kind of elliptic function workers will get its console in the Niger breakway it's completely explicit OK but was long 1 by 1 or 2 and and then and then what happens as the a goes to infinity a single infinity and on some kind of resonance conditions so you have a 790 full of these sympathetic manifold dimension Bates a manifold dimension to opened fire of good I mentioned to you which is given by the by this condition of long divided by X 1 minus X 2 minor spike about 1 brisk opportune but which was up at 2 divided by takes 1 minus 2 plus all I can 1 this capital 2 so this is a money for reducing violence by the would system by the wrong with you can imagine there is 1 and all this summer and fall you know that the March the meeting on all these factors cabbages of when Jake was once controlled most of the raw like 1 of the 2 square as he goes to infinity what does that mean what it is that if you have a stickler to infinity managing Frist and this is the 2nd 1 capital than this bubble will be extremely concentrated and if you compute if you computer high Seibel of small source of Heisterboro known here it means a chance for as bigger than the half typically 1 for instance the pitch 1 moment will be delighted to square the 1 on square will be like to square looking them computers so full of mechanism of this is quite interesting here the mechanism is that have 2 provides X 1 X 2 and this is the 1st the 1st soliton as the goes to infinity will be the fastest and this will be the most concentrated you have that kind of thing and then if you go back and then go to the past after goes to minus infinity then they will exchange their there will exchange their their constant with In some sense and that this 1 will be the most concentrated and withdrew faster to minus infinity the list of yeah for the in there are several of the regime but the the 1 I'm going to describe this 1 at some point when their distance is minimal there with optical OK at the it's perfect symmetry now what happens full halfway so full of where you just have 3 conservation laws and and the fear and I'm going to states more precisely in in 1 minutes is is that while you can find soliton solutions all that kind of course in that case we will have remain determined write it down immediately and and then what I'm going to change a little bit that the description here because what is important here is the moderation of the student down so I have these stupid idea but we thought we turned on top with depend on time which is quite nice because I know but the ranch small indeed and I can't even computed derivative and it with respect to beat a not of what was
35:58
again X minus X J of the provided by not 1 because you to for these this is exactly the UB doll which is here except that I need some face factor In the idea and then I need some of those some of the parameter which corresponds to to the modulus of what Europe by jury an act which I call them the J. Paul and digitally 1 cookie I just applied skating to this like other veggies such solutions at police near of course the the veggies equals 1 so in fact I'm going to introduce some small parameter itself and end the sprout small parameter he does each of such move parameter find families you off of that In the the main difference OK suggested that a force at sea will be will be small for instance at infinity excellent English ones will be what would we go to as the goes on in this is just 1 side the mystery that the went to press infinity and so what happens now that maybe this as the spirit that more in detail and we continue with this comparison here so here I choose 1 of the bubbles To be faster than the other 1 again it's possible to do that so let's say that this is the 2nd of all so so in the case of all of sixacre equation of course if you look at the each 1 allsquare as he goes to to infinity in the case of on the resonant conditions here and the resonant conditions present condition is equally or which you respond that have something like this will constitute the square a parable now for half wave equation what will happen it will happen that so assault from both overtime given time whatever it could be 0 if you like but what is interesting here is that you will have some some growth of find time for the square at time where you have some turning point in the turning point is strongly connected to what I am not a race which is this competition between the 2 skaters uh aches and won over oneliners began and after after the turning point you have some saturation so here you have some friends and jubilant evolution and after some sometimes the depend on life parameter we face the parameters of that essentially Siamese that is 1 of the it's all here you have separation which means that the H 1 is essentially a for a long time OK so this season this is the the comparative situation of course you don't which means that we don't get a solution which is embodied in each 1 small solution which is symbolic in each 1 but we get something which is very excited for every for every time we get something so that for instance the new initialed in each 1 is abundant In the year initial after some time the the you of of the at the sometime in 2012 will be a very big like a something like 1 over reductions in to support some power In fact we have even better we win even have of all this fall as between 1 another something like 200 so this is important because for for all those of you know this beautiful paper by going on under key that the coca and the end of what was done little so by the by people like you like you and it was a day of it could be issued on Monday near fronting India but you get you always get results In the same integers space so for instance if you start from something which is bound small in each 1 of you will get growth in each 1 but not in a chance for as more than 1 here you have some some room because you have a description which is much more precise we can compute the solutions just by reading a graduate of the at the age of not just by reading all these so it's
40:59
not enough for me to to state the result more precisely essentially this is the swamp except that I'm going to not through to introduce some Contra fighters so if you quantify for precisely this year and let's say OK at the date is a little difficult and what it says something like this have to be absolutely honest I think this is laughter the at this way out of consideration also area in which we portrait so this you that tho there exists among some universal small mold needs to fix small prime ministers that for every of between 0 and you just saw that is the solution you solution Paul can be halfway immigration so be halfway focusing cubic conflagration on the time enjoyable 1 over the death of the infinity so that I small parameter but fixed and it is a small parameter of misery small parameter so this guy is big but cannot but be kept said so as not so what did so you have this expression I already gave you so this is this expression With epsilon goes to 0 goes to infinity and waivers Nikon can explain exactly what happens for the 2 brothers so we're getting the front bubbled with this century I mean quiet so the better ones all the land edges for forceful language will be essentially like 1 they move but not that much I need them of course In my in my system but they they will not have a complicated and singular behavior be that 1 1 1 9 has become 1 will be like OK so eager Allegheny the small parameters so the fact that the Taiwanese close to 1 of 2 weeks on
43:48
that we are close to the Sega regime what but not the nice thing the nice thing is of for the AJC because you are essentially like the badges with the Tyminski Ontario and so they follow the same kind of overall behavior like instant that the behavior of this all and and they also follows these this behavior here will be that to accept that because dependent you know and what is really important is 1 oneliners bigotry 1 line has been up to we have to something like this so often seems smaller than 1 over in dire need of fact the small parameter also here Delta where he got this 1 been that too is reunited it is really a key square it really a T square itself so that you know it's faults like like uh load like you take it just made it up to be 1 prostitute to choose didn't offer lawful initial but then after starting point unilaterally these roles for In the transient gentlemen situation and after this time you have something like that essentially you and then throws saturated Of course there are constants yearend this constant depend on the small promise you done that's why we need In order to to close our big sister so of course the men ideas here of course is the the main point is the main singular dynamics is here and this corresponds to this rule of law yes you Will you solution is found only in 1 of the years while I can define it before but I cannot describe its 14 going to minus infinity this means that the story is that your problems a the latest 1 inch buys a small but fixed parameter look like insulation that is the belief of dissident USsponsored that for every look at the you know you can use this 1 no need in general since I put it out very very small this guy is smaller than this 1 yes it was I can face after role I can fix it up in terms of the deal I can fix that they don't believe that the cautioned OK but they're not given that all if adults small enough this is amended to read from the book for years all stories holidays this then you agree of course of course otherwise I understand your question was it up to the Delta costs well that's the problem with with with 2 chalked but that it has the advantage particularly in these situations are so so this is a situation again Sumner I still have something like 5 minutes more several panel all of this is true troops start in history so I guess I can tell you exactly in 2 minutes what is the details of this proof of course it was as if In many other results in particular in the very nice hindering work by my dad and sold worldwide must there and so were we would have fired their several I think the first one was in so they are essentially 2 blocked which are so these where for for KTV useful for summer also fall from analysts In the 1st it is construct an approximate solutions so constructed that approximate a solution of course you get brag did you promise expression that I write to gain here but I'm going to write it slightly differently this so it to the ideology of divided by land that you 50 to the pond of capital region of X minus X shifted divided by language of keep oneliners with that you've 10 depending on some parameters which are 1 of that to be there won't be that too and some of them at 2 miners gonna wanna know also it's to minus 6 OK so you promise into the equation and you extract from the heart the system all equation of villages and V1 and V2 and the system stuff like this what the miners be hedging the divided by 1 man as the budget plus 1 digit minus regions where the dream of so that this is the starting point this is really the this equation except that not of course the remainder to which is very big OK so you have a very important remained determined which is a linear combination essentially although of us of objects which are made with 2 pledges and and some remain neutral but will give the me me Muneca operated on this and of course you initialize so region would be something like Julie budget but some remember terms Chinju want Fiji to accept and in fact these guys will be really remained jump in the sense that they they decay like 1 over all where are these these parameters for this is the scale of the scale stance between the 2 but those so you have these 2 centers here is getting with oneliners 1 which would be top finally and essentially this is like 1 overall obviously was like 1 overall square and you make some kind of geometric optics expansion for doing this expansion you need to know very well where the structure of the count of the legalized equation here because as soon as you put this in the 1st here you will get equation but you know we precisely you know it's in the guys close 1 because it is still not where you can compute because you know that this is the limit case the kernel that you propose so you know that the Journal these guys is the span all right you with the jury and the next with you this is extremely important so you have a good dimension to range so applying for those in the fury you are going to solve linear rise equation that too introducing tools to directions which will help you To also belong to through to the terminal will to be committed to the meeting in the major sending dealing with the cipher joint operators OK so and so the studio erections will be introduced here and the coefficients of these 2 directions will be parameter of which are extremely important and would you will see in the systems on all these parameters and end in these guys you have in particular you haven't particularly expression of Schubert also you have something like all all where are is this parameter here which raises a very important role of so this is the main thing is of course what part which is extremely important in this construction is to study the the system he not just 1 minute for GAM I'm already late the 2nd part is of course the interviews to to construct an exact solution Close to the approximate solution to this awful that there is not some kind of roadmap and would stay in the lung a so that there Ugandan you doing to you have your your approximate solution here and you pick a sequence of times which to press infinity and about time all aggression With your city and it was you out of now what you're going to do it's too right you not too as you are but some remain determined but with some puzzle you up corresponding to other parameters here that's where pointed to the bus amended widened to include another along 1 course we expect that it's not about fall but because essentially I need some awful properties properties of these 3 manager related to regain the trailer all of my leading arise operator OK so the result of levity properties are extremely important to the loss then to study to introduce and then I will be extremely the very localized energy functional all this liberalizing their viewpoint was difficulty here the fact difficulties you have to those and they are strongly interacting why are they strongly interacting precisely because the outlook for to meet top do not the jail exponentially they became very very slowly OK so that even if they are far away from each other the still interact and so but the point is that the revised energy French should take care of the heart and what you do in fact is that you write everything in the frame which is connected to those those lower all these 2 brothers so the season with the but 1 here and then you you right the energy on these guys so this localizing energy function unfortunately is not completely costs you lose a little bit over the issue half known for plus pot of these guys and so you have to really to propagate these by using very precise it interviewed to for them and the rest would be the people hopefully before the summer thank you very much weaves into 1 I'm very concerned the new rules yes yes we tried and we are trying to fashion the problem is that the problem is is that we need to respect this the fact with that we are close 2 and to the resonant situation so here we assume that I didn't tell you when we have we assume that our parameters our initial parameters are close to this this this is initial that this resonant manifold and this is not that easy to do to manufacture them in more than 1 and this is there is another problem which is if you look at the 3 solid guns In the securities and it was of course you can make them control like the square but never more because of the fact that you had another solid done in the security and does not make you a droll to infinity faster so this means that there is some kind of long again maybe some kind of saturation this region to even bite by adding some so we still have the hope to to look at more than 2 skeletons and we get something which which goes for finally to infinity but we are not sure maybe there's something and that there are some some computations some simulations of the article done on some different kind of system that not that far from this 1 where generating you see rather some kind of separation so maybe that's the the generic thing minute that this is because of the Virgin Mary any role in this it plays a role because because of that because because of the fact that uh you know you need you need to know the description of the arise the parade and of course these theorized that operator clause In the dams old friend honesty dates of the leaner either Prater for cigarette and the rise writerdirector falls ago is completely studied because of the integrity of the onetenth the level of since the level of estimates it's not the it's it's not that I mean we are we are returned more I'm in free with respect to security in impractical in fact we need to be because of the phenomenon is different but but it's different after some time so that the 1st part of the phenomenon is really like cigarette but we don't need to use your right the the formula the explicit formula from the integrity Forbes again because In fact we we we did that we we manufacture the the system in such a way that it looks like hours ago equation and the singer question in this very particular case you can do it really In mentoring there is that this system of Sega in that case is rudimentary once you found the 4 conservation from now you can say that we are very happy we found the false consolation away or you could be more more Coniston said that in fact they are infinitely many consideration was and we dared to do that because we knew that it was integral then it's a matter of taste this is 1 of of the and so you mentioned in the history of the Over the years he has been told that Universal had to fire on the way to make this regulatory making the world is going on in fact we have already said they had the 1st thing you know for years and years and years essentially we we we are in this situation you here we avoided colliding of silk you To do it to forward OK so for 4 straight years we can study the collision case it's completely bring this intense uh but that the problem is this To keep the smallest and and of the smallest of oneliners because for all time so for the moment we don't know how to do that again that a sum but it's a very natural hold on what will happen to the solution 40 going to minus infinity I don't know the country of and maybe just this once more for a while this is a question about his criticisms square whatever seems to be accomplished however the use of as a that it that you would at that you divide and the square so if Delta gift is that over the top he said that you know it's not just answered your question of sorry I didn't but you know that I small but fixed parameter he have some some constant here depending on the time the 1st 1 of those 3 letters from his career noticed yes there the United might buy it at that well although there is it maybe there is no I'm sorry there until no you're right it's a tidy 1980's square sorry sorry sorry sorry it had divided by the square I mixed needs this and they each 1 sorry it's it's not divided by the square of yeah yeah it apparent that 1 of I'm main I'm in the same the same the same area that they do will every time know depending on the thickness of the things I always and but sorry and thank you for what the