A continuous family of energies for the Korteweg-de Vries equation...

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A continuous family of energies for the Korteweg-de Vries equation...
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A continuous family of energies for the Korteweg - de Vries equation and the cubic nonlinear Schrödinger equation The KdV equation and the cubic NLS equation are known to have an infinite number of conserved energies. I will explain the construction of a continuous family of conserved energies interpolating the classical energies
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and if you can get don't me have to do 2 well thank and could forcibly here and I want to thank the organisers for this nice in the Western Conference and I'm going to talk about consult energies the government is a staunch folk could entities fall analysts indicated the end the KGB and assist trying to talk with Daniels told before I state what exactly what to do about to give the equations so there's the analysts equation now sigh one-dimensional identity you plus you would expect is equal to plus or minus you square aren't you so classes and the miners cases focusing them solid on some the glass case it's deep focusing and there is no solid tones then there is the M modified gaining new creation on the Jesuit he last Nowitzki exists and it was the the 1st thing I didn't want attention to plus minus 6 but it divided you score out of this tool used to X is equal to steal focus now I want to do that in a plus minus and then it's the focusing on the focusing 1 wondered why the the absent value here not to allow all that complex solutions than the contents and Katie otherwise it's to be a long and then there's the KGB creations such as utility plants you know it's 6 minus 6 units units sealed In all these equations uppity connected to their all In the couple and spend a couple the 2 shows in the existence of luxe pale the no wonder divided in the following sections of the system sidekicks scientists 106 derivative of map to see to is equal to I know it when you lighted for the divorcing case last case I see you combine miners last time sea seemed so this is the last equation on to a complex complex Almeida I and the amazing factors that he complimented the immigration for the timed overtook please include to higher now I have to look at other times account remember these things plans to square when you use ground to see you the Monsey works -minus through see you follow my skin products ruled sees quelled plans use clout at sea cell and is said to and frost order equations and they're interesting the very interesting structures that bountiful want to solve them simultaneously 4 given daytime at 1 point 10 of them this is only possible if a compatibility condition is satisfied the compatibility conditions exactly the focusing analysts here on on the site so the and the connection to a and KDB is the following so for and has the same structure with the same operator heal its immigration you and a different equation on the site so that different different time the incredible the title to and then the same structure is solvable for Channel 4 forgiven data .period it's he was able that executes your ticket still far general data if and only if the then you will satisfies the MKT the creation and vote the for the for the focus cases but there are some sign changes which I don't want to go into it into a survey the on case is always sent use a minus sign and yield is more complex but this equation doesn't play at the colorful for what I'm going to tell now what is the effect of that too well suppose we take seat consumers they mention part of largely 4 to steal now they might look for your solutions music to better system so there a two-dimensional space of solutions and I want to normalize them on the left so if I want to help the city behaves like you 2 the 9 I see it see on the left for the case to the left To minus infinity down come on the right but suppose that she was contacted supported and then undivided decomposes in linear combination of tools 1 the in the fundamental to someone delight then on divided compositions that see them as being slighted To the minus 1 I want see you go to the airline has come to X and now here we have to give it more careful as Cannes 2 plus Infinity is seen in so that mediators linear combination of the 2 solutions which should have over fundamental system which would have if you would do would be equal to and I use this to define teetered the miners want which works well even if they mention part of Silas positive and this reflection coefficient is called the transmission coefficient of friction coefficient so tedious the transmission coefficient at now the fact that L the creation of a panel as a creationist down there ,comma turbidity condition for solving these equations allows us to solve this equation that class infinity and check what happens with this of infinity minds infinity and check what happens with the function the said that minus infinity they don't see the effect of you so what see element constant coefficient differential operator because tended to vote in terms of see you when all you so we can check this and if you take this on the left and that some time evolution to do it on the light that sometimes evolution and but is unanimous that the policy independent time if you have a solution we have a solution to the analysts extension so whatever we look for quantities which are for analysts or and KDB located the missing have to look out the transmission coefficient of deceiving can define it and this is a conserved quantities at each point in the integrated into whatever we want that and discuss the concept quantity so the this is some sort of old penny old it goes back to and in this context toward October it's taken as this is the but the New World and the call Sonia and so they did for the monitoring stations have down there is an
expansion "quotation mark to USC which is equal to or them minus 1 goal was to try 5 times this summer with his former didn't want this this is a formal summit and then there are the energies of the cheery seat to the miners to minus 1 well so this is I have seen tending to i infinity or depending on what
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at infinity and ban Yvonne is equal to the Indian coal use crack the eggs and each to institute between the call you acts and so will you too was equal to 5 times you'll you know want it's a mental and clean and some used you would want to have this year 1 and 2 people to in hope you exclude class but 2 times you to the Final Four suitable for the next witnesses it's just who he is so
bad and then you can continue and classical too expands the things and it because as follows and but I want to do is it wants to predicted in the trunk of his Daniel these sort of interpolated between these quantities in order to get a continuous family of conserved quantities Dow that's not exactly what we did to some modest more precisely the interpolated between the even bonds and not even that the interpolated between linear combinations of bonds so here's what the what they get Have love the things that we define so on this 1st at the focusing case finalist named Katie many years can be defined as an end the Cold War was over it is for shots functions before potentials as medical within the alliance there look with that of the local within which across the transmission coefficient is delegated act that Alexis over that he needs indicating some also functions and have would have to study the decay but you don't do that but that suppose that the locals must you L is a Schwartz functions will save it more about when these things are defined later so that 1 way defining it as to indicate over the legal access and could await which assumes something like that before we rate for Sobolev spaces and accede to the 2 West for this for a slot the minds of Hall and that's another way of expressing Batistuta in Conde to move to control indication as to what the line from 2 items infinity and that is the 2nd and this the 2nd former law that is equal to 4 signed pie as in the Vermont infinity so nobody product is an appeal fashion tossed minus 1 to the Paul as planned this vocalist and validated that I to sea of could had 1 which would be expected to delegated CSI over to on the left-hand side so we evaluated I told what to do and then they have to do some collections because the weight accede to the calls at infinity pool ammonia leak so we have to collect that in order to be able to do Oakland for indication so we subtract the 1 the most active we want to call these guys age so 1 page to the rear the a collective that we have to correct the behavior at Infinity a little bit thank you I'm also not so that's the only place where fault itself so we have to collect their behavior at infinity in order to get to that which we able to welcome the control indications of the control and if you do it in this fashion and this doesn't depend on how many times to collect if you have a liberty and there so we get this former you what you see here is that if is an integer and assigned pious managers and here you get the linear combination of the classical concert quantities from that sense the interpolate between the classical conserved quantity well then can do the same thing for the focusing on a less if you do the same thing for the focusing less than we have to the just for eigenvalues of occasion as operator so that they can operator has eigenvalues in the other half claims and we have to there just for them and also function as a way of doing that ability so you have to add some more with the Iden values with a multiplicity and she was armed with some countries sigh and nothing changes on the right side suffocated that the same structure the 1st stiff as a sequel to the larger than minus-1 1 than dozens of generalization of the classical formal there's a different function CSI just for the eigenvalues for the idea that as in this case on no 1 can change its and on the line for my plight to infinity there's a similar expression as for the analysts equation and again we have to adjust for the
classical energy something special happens at a sequel to minus 1 so the left-hand side makes sense for minus-1 and for the right hand side there is this part of the in the cold which condom makes has effective converted to that of function combined with assigned pie s so the outcome is down 8 times the transmission coefficient validated half minus but in the car you want you don't have to do some normalization which I don't want to take it much about what about so that these are conserved quantities for analysts and you they go back to Aiken as all 250 of them discrimination between the different wording in the calls down and these other objects they want to Brokers now defining these objects for Schwartz functions is to you because then on the scandal and transformed becomes something nice to look over the move to years faults function on the deal access and then it's an exercise in pundits analysis maybe not entirely easy 1 too get these form so much the day off since to use this independent of time and we have a solution or whatever you do with the it's going to give you a conserved quantities so the question is how to relate them to the things we welcome with an analysis onto the lake and supplement basis and Montague themselves 1st and then explain that the developers course thank you use you the tied it before but it's always different when I did it on the back porch want know 2 it OK because you still can have an impression of the form of nicotine don't all of them so all the human continue to talk so what we do as being a mail we look at the right hand side the defined the quantities by the right-hand side and then consulted it it makes all the sense in the left hand side so the 1st thing is if you know is a major test them both sides of all post was sizable defined but basically you want to look at the mourned the right hand side and so the map to yes His continues into you and he Olson as can ever you with sufficient difficult and it is
analytic but if I were to this is not my invention my account expected to be like in value if this so is and focusing case because they dysfunction scientists not analytic at 1 so this has to be excluded but otherwise it is analytic and it's jointly analytic and innocent you wouldn't the popular sense of whenever you want is age signal then it's analytic in S for S and many years analytic annual and sick not the new investment as less than 6 months I wonder why it but in their pockets entities has all the properties and would have 1 to have it L and the nice thing is they can play with both sides in order to get potable is more convenient said that part of it ,comma pale from this so this was test but it's analytic so whenever he was in and this is defined but the question is does this come told the International I felt so we can look at he has few minus Q emerged as well which is basically the same else the left hand side but for the fully transform evaluated at didn't collect all the Dexia to receive them so this is the 4th similar to the pinnacle of the left-hand side and this is political time the constant times gloom in major squelched and no I want to use a strange known here and I want to explain it later on but if you will In failed to the futile it's less than Delta wasn't built in so this tells us that the correct tragic apart Of analytic we can do an analytic expansion and you at civil and cathartic part of that is except the age of snow so for small if this commentary is small and synchronizing more about that in a moment of this quantity is smaller than this quantity in the international market closed you have to go and also what happens in that case is this pungent side Of the eigenvalues the chain is larger equal to a constant and comfort shade to the polls to work with the real problems of copper change occupants points to the port to Willis times dimensional part of the China so
all of this is small all quantities on the left-hand side of the former so there's a similar statement for creating the disease unless it is because they remind and those of similar theorem for KGB about now is the same sort of statement for this larger damn minus 1 the same statement and if this fall to minus 1 down at the same statement but image months-long due to of the potential for them the the minus 1 . 4 4 was not value .period shooting on which makes sense because then we want to validate the city single function the struggle with an apt no 1 house and that the candidates locals and one-liners and born in Tijuana and went on now some qualities I felt
struggling cope only if few years this scaling down and we get to that if you see what is in Manchester and unspecified pot which ultimately this strange bound the and that sort of clarification and the United has less than infinity and could be made more precise for the solution for analysts and Katie whatsoever the 2nd call areas well done under the vinaigrette electrons Solectron squabble calls on Titan values at Prevost and anything else what you want to define terms of the spectrum and the stables and protests for all as the French so this is a question of scaling you might scale so you look at the former lots it left-hand side and the left hand side of the smallest this month condition is satisfied and evidence positive and if you look at the 2nd line then this a little missed the 1 belt localism lofty vanishes on the Alexis and there is 1 constitution corresponding to the Icahn value so this is the lowest Melua years can take when you have been aware of summit on so this year 11 of functional and to minimize the persona and similarly at the people which is also characterized incomes of OK so there is 1 thing on the planet and explain and I don't want to go delve into it please look too much but I want to get some information I have is the EU to solve 1 pursuant to this something like age Klein of the heart but I think it's just how coming to work so ancient half the space profitable and it has doesn't have any good structure because there is no betting into continuous functions and 1 doesn't invention half so the space doesn't have parameters with this this space is not good for most things to do an analysis this you too was a replacement offered and it has all the good properties from which to half of many of the good properties from fish to have fun spaces offer derivative well into it's still on the same scale hits on and scandals in the sense that if you look at the best of space and the 1 house on Long and that's interview to advance into 1 half of 1 Infinity's also developed for space to general and the EU to these other formidable tips now think this inch models the the same sort of embedding and end come similar spaces for the same his need to and DVD to and that's a good duality between 2 things or 1 would would would have the haven't Britons into bounded functions of limited affinity for these functions you have the embedding itself in born into a the you to interview and so on so all the things which I
don't work for which don't look for it to half minds and how many of those things well for you 22 sold down and particularly was for them and tears of you can look at this sale to lead you to is defined by known to taken function you and then multiplied by smooth cuddle function X minus change for the Senate which is the palm-size you take this thing until you to 160 pieces and undertake an LP summation of the consummation or that this critics more pieces to something of 2 1 0 PM and now this L 2 if you look at this
altitude did you too long and this is less of equal to a constant times you and arrange that this for all of us not just in the minors so whenever you could possibly hope to get a good aren't like that and this quantity is going to be bounded into smaller than its component and of all Sobolev spaces that so could replace it by any chance he'll each man's house as epsilon the same statement who was met with him for years has been used in between 101 miners that the Infiniti so you have a lot of things you don't want to there since the EU yet in the chest neck during Rangers this lesson 1 that lasted less than mine yeah she then you have 1st Peruvian leader has the second-worst in Romania not so heated that going to be the assumption is that of the minus-1 helpers for both parts of ours so nothing was bound to make send them mature about their son's faults some sides of the equation and nothing makes sense if you that was the end of the year "quotation mark use you have to act so that should have given that sold for the KGB part you have to the place simply replace held to years to buy a chance cannot contaminants 1 to gage the booking them but what I percentiles connected for a lot of literature and account to go there and all of that I mean that in scattering part for the sake and the system is due to the top of its call Neville and Siegel and for the year from many also marks indicate that there because there's no point in competing this happens to doing in the scuttling and I'm sure that there other people in the audience and with it much better than I could do with them for their appeal we estimated should say there is a sort of a history which is different that is mentioned appeared on Friday to about the only estimates that fiscally and then tell for the nonlinear shrewd immigration I did some really estimates not you need not uniform in time but can toward wall all time responding to Tower to reach minus 4 and for the moment shredding and creation that there this is not a flock on the KTB equation at its invoking the KDD creation H minus 1 that this Buckmaster some uniform and time of way estimated age minus 1 which used to picked off in those Catalan but not much structural then there is the rope which was done simultaneously by killing fish John L. killing vision China and 4 1 gave a toll on it last week on the KDD equations known for S I think that the talk was for a stint in minus 1 it's conceivable that the office an extension beyond only 1 there wherever it is 1 possible and you will at 1st be 1 day so you worked with this told you go and talk to our act 2 2 at the effect is important
so that erupted this tournament biscuits is lost and to Sean equivalents of this single told you with age as small indicated notions of that which lead to publicly to something similar and strong on the right-hand side however the cinema single collection at Cheney equal to seal which is still 2 months away when you the same formal as this 1
and this that and the range of it so I want to have to you the heart of it of it well welcome too 2 the strategy is to balance tilted to control that in the "quotation mark plant on the right-hand side is the most essential part once they have control of the Indigo part on the right hand side it also gives the control From my limits by supplements but all of the whole inside and down if you look at the the focusing part of this involves 2 on the 1st look at the transmission coefficient not in the whole half plane but in the the stoppers looking at them some known as conditions of we controlled the game replied when told of transmission coefficient to you all the way from the Alexis and the smallest condition ensures that the can kill things so that they will get it somewhere we also get it it's a 1 health if you want to and now and is Mullally the happen harmonic function and the Republican of seasonal 1 expansion which virginity folks and case has appealed part no less than Singapore Seoul and then we are able to interpret this as an the settlement function the house ablaze and then they feel part of that is going to be a mission under the law also defined the left hand side so if you stand that in people was part that USC has also had to want the the localism House Report of the radicalism of and then we can take the place of the apart from that it's an interpretation of the performance on the left hand side for the full presentation also has to use in beckoned and form but it don't want to go into that would have wanted was want to explain how we can prove Loconte .period holding the courts well so we go back to the Aiken system so there can assist a mosque the 1 client is equal to remind us I've seen 1 plus you will see 2 see to plant is equal to the C C 2 plus anyway he among the wonders of on the solutions so that we won't see his life had to miners died c CEO and text input but among the things OK so that they can try to solve this equation because new to get a poet expansion in terms of you so in the 1st step the this equation and then began to seem to want this fall To that end the call from minus infinity to act times you through I have seen the effects of ticket epitomize but my -minus new won't of X P To the miners see the text of and and then we take you at the so this stands for the duration C 1 2 and they get something similar and then there's the head of the house while right so this is 1 of those farms were so and down to get the 1st began to heat is equal to 1 class new commitment as next tends to infinity no I see no 1 wanted what next at the last minute to this situation and called the the sky T 2 C R E 2 the I since then you can also produce and begin to fall in these terms have denies structure so if you do with them now and this team to trade policy is equal to in the halt and you 1 understand why 1 of the debate could think of to chain yeah we get them please to the 2 0 0 5 season there is some level likeliest whereby please minus the sum would the X K but times you'll all life wanted you the life change you pile of snow on the mobile phone exchange cleaned so To the donation and then you get to scrutinize the presentation allegedly part of CBS this positive so this is negative and I will this thing is not in this morning and we have otherwise on the right-hand side so that leads to exponential decay of how and when and the other half look so you do without the right to control the things like isn't funny but all the time so I have to decide what I want to focus on now so down the coast there are estimated to have estimated that this is the dispute would change of scene is bounded by and that don't want to Gold you go into that and constant times the you he had to -minus I Yield context and thanks you In a tool and put it in the June hot you receive room did you want to add to the Polish so this is esthetic qualities of the spaces still to come in nicely you have to take to put them also tore apart into your and then there the defending naturally this estimate here his imaginary part of scene caused bonds to the localization in space to intervals of size and Megen part of the command center this exactly the scale on which this exponential decay but this this is contemplation I want to say that about that but I guess I'm going to skip that so this is the 1st estimating that walls to get convulsions Paul blighted provided this quantity is small and this is basically the other conditions the field comes in OK so far also them we don't want to get estimates of in terms of this thing but we want to get estimates in terms of age as norms so how do you get a call from you treat as norms that held began the war inventing something to get some villages and and betting sealed and that the scaling Almeida and get Palace of the National apart and this is going to allow it to get some of the calls but it only goes up to a certain point and then you have to look at the look of after the government look at the closing of the sink and hold on to 2 this In to go to the local from them that they will actually the way they didn't think that cannot have signed home because of and so we get you want to take a look at the front of that of Peter the minus so this is the key to the minors 1 is somewhat in the courts which I applied in this fashion so this is the turbulent occult video is still fills in the cold which he got here this is the 2nd time cult of unilateral but it's visible so this codifies the information that X 1 is less than by 1 the next to last and by 2 and ended with the intercom and then you have missed the 3 things that 4th and then if you take a look at whether we get a lot of common topics you can spit the indication domains into it the sense that the order of the variables and this is message this is the screening so I did it for the 1st time I got the 2nd part to explain what did in the cold stands for in but I think I computed by hand these these things you ought to hear and then I was convinced that this Hutchison will persists but the and a Iowa stock is down your stock is that didn't know what to do but it was important for us to get this structure and explain what this country is so this goes so for it was offer just going out and put an X for the aperture going down and put wife and the negative you an orderly had it's wonderful as the next 1 is the next it's the best environment this is going up the text for Hispanics for less than life we missed in life the industry's don't matter but this codifies this in the call him then I think you is that we all of have the last wife on the side of every X so this year has a much better to carry on the domain of indication as the original and the call because it decays exponentially than ever 2 of the valuables are 5 so where's this this information you leads to decay Of all the basically said that it was initially part of seat to the mind of change because this is the local as the size of the corresponding from the simplex he'll like the image part of seat the 20 minus the minds to plus 1 because the last line is on the right path so the last so you can stitch these things into presented most part of it but no matter what do was but this decays exponentially been able the put things part of the ubiquitous in into calls could help the situation that's why 1 is close to 1 and White was close to extort but buttocks to from by 1 and then you wouldn't get decay although the the company I paid for that as a comment on that so it and as I was the 14th situation that there in this book and that was the people from probabilities and I showed this question to Martin Martin I had told me that I should look at 2 of budget process and that it would be deleted 2 things for women budget passed so my colleagues in Bonn still the a minute told them about but about the button in the colleagues notify the excited you can't have a hit the well but then I tried to get it into the context of the amendment that didn't work so the basic thing is it's not much of a big effort to show that this is a whole project ,comma because it's very close to a standard apply that it can be assumed a difficult eschatological and the landlord the event that give the conclusion is that would be a core commodities of budget puppet a commodity funds so that sounds almost the same but it is not that 1 but down L still infant dies to do things by Ford basically none at all In the calls are and that's the structures and at the end so well and now we have this estimate younger tedious which controls to the expansion of the book above lofty so do get to the estimates we need the never change this not compelled to as so could tired I order polynomial and things are easy but since we get that we have to look at the lower order moments case-by-case and if you look at the conserved quantities and it's clear that something has to has to happen because this is less than 1 day less than stable then there is no Consul quantity which which holds that is equal to you get built 1 or the other to handle the this is easy but this is easy because with the all too long found he check T 2 projected net if 2 then this this the to you In page script so that requires some simple calculation for the 2 Thomas OK no yeah but now at the level of the of the next 2 on at the T 2 on that corresponds to the next the it added that at the level of the T 2 to 4 times you expect their held for 1 on the power forward to show up at the level is equal to 1 so this has to be taken care of no this this connected to touch show you get up to this point and the idea to go beyond that is pretty simple that this is the 1 indication by parts in the indication suspect 1 you you do an indication that hearts delivered to the falls on this and so it could be an indication that pops the suspected this well is exponentially so you can effective 1 overseeing and the derivative this and get the same form of affordable to all you get to the ball under the terms and many have and in the course of one-dimensional less and then you continue to do that and if you end up with no wind a one-dimensional in the call in the end we get the conserved quantities and otherwise dissimilar estimates explained before so different than what happens implement and you wouldn't buy this is the so this is a very long way of idea freight all know amendment pull it has something to do with that on that on the quality of the casino that better than I do but United the so only lament the very people who say that this is exactly over amid the exactly operational costs from the series yet they always series of review like the little said the rules made the information out of them instead as it is a regional phenomenon in which he's afraid because the those integrity of the season of misfortune police found a great deal the end of the year use the next few years this is a nice because that is exactly what makes purses alone my friend we always used the vapors is my only goal the become yeah somewhat from nearly there was that with these things about me but what I think is true is that there's a lot of common at all behind which continues expressed in the various ways and it should be it should be connected on 1 thing and that is the hope that it would deletion invite modern in the whole budget lies I think different it's closer to what what I presented to you and none of the fat determined but L the thing that just as barristers COM it's not difficult to get and it's not difficult to use stage in order to compute the things to any order but those people I asked the results of stock with explaining what the coefficient of this expansion should look like but the coefficients should look like so this is full of technical it gets connected in the calls which is what they need but they don't have any access to the size of the coefficients immediate effect on the town and to handle all to get that way waste to there's a lot of comatose behind wouldn't understand there is a general idea of what it is standing in Europe ,comma exactly as it appears as if you will review the Jerry funky generated with coal a number of decided lower the material that is exactly the jury found that the number and a lot of work songs talking but I would think the Netherlands I could put those having to get to get that but it's clearly violated many things but there was this summer estimates have been seized in front of you and so it was stipulated in the road the dissolution of the Euro-Mediterranean the provisional is all of you and you are the views of the world history is the sole source of the machine which the world is are hold more furniture political relations In an few and