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Modified scattering for the GrossPitaevskii equation
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Title  Modified scattering for the GrossPitaevskii equation 
Title of Series  Trimestre Ondes Non Linéaires  May conference 
Part Number  4 
Number of Parts  21 
Author 
Thomann, Laurent

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  10.5446/20766 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2016 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We consider the cubic nonlinear Schrödinger equation with harmonic trapping. In the case when all but one directions are trapped, we prove modified scattering and construct modified wave operators for small initial and final data respectively. Actually, we prove that the asymptotic behavior of the solutions is given by a simpler equation, depending on the resonances of the GP equation, and we exhibit some particular solutions of it. 
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00:03
what we and on and on and on and on and on and on and picture which with the introduction of ink you for the invitation it's sufficient for me to do look to many here saw a tree and we talk about to modify country for analysts and the circumstances of the longtime dynamics of ammonia string equations so indeed they we present some joint walk was always been widely there and if he petulant are announced I also mentioned some some contributions which Kagermann From the individual curiosity and sigh honey from Georgia Tech so long as it and just beginning recent introduction so that they question we we conclude that he is due to begin in less than a week the room some modifications on its late on I begin with the analysts on a wave of the day many forests so it's crossed the I walk upon the tourists which uses identified with all of his arm on to 2 Phase III and the training immigration is the following so it's IGT you In who did so at the end of the year their radiation the viability in off and then you have the Laplacian unit forms of in 2 2 and conceded the crucial problem with an initial condition in some civil space age all of this and he so the viable designed to fully inform so it's up close and analysts to and we are interesting in their own kind in equation and I will begin we started with the results of highly put their sweats growth and the CPI but who have obtained them with the fights Katrina Ford is a Christian and a piece of this talk is to see the wall of the residencies and the induce 2nd toppled the talker we studied over the question is a result of Domo perturbation by a convolution potential and they would shoulder up to the reserves are different so the 1st to reminder the results of an EPA that should go and Geagea so to begin with I will give you the the resonances 10 which is associated to the music equation so this is the integration which on the concerns of the viability of Hawaii what we wanted to and that community but but but but but but but the story played here that the crucial problem .period so it's change its key so if ideas I identify a functional receipts for years Serena's J is the sum of the G T X and then the exponential progression of before this 1 and this J has a foolproof way transformed from X to CSI which is due hat and you have is the 48 just in the index by about viable which is in and this jury heart has to satisfy new system so as you see exciting here is only a parliament so even if the a question also depends on the viability and of course key tools the CSI here does not to play worried you can just see this as a Parliament so in some of this is reduced to dimension of contained exchange of money into 20 what we want to see always the link between between those equations and we will see that precise description of the systems will give you all the behavior of this 1 for a long time and the CIA this is the Cuban knowing I reading but to reduced to remain reason of interactions so those are the rules do the frequencies which player a the longtime dynamics on this and other things that can be canceled out by year integration by parts all all of the coffin of my former arguments is history the sticking to prove so this is they do and those are the many questions and the spaces that we use in the in the following argument by just 2 of the best known which user pages norm Clinton you for some large in the following pennies large say you're Larson density and you also need this summer localization in Texas because of disease to Buchanan and some of the things that you also need the to localization and then I need a stronger North to Tuesday's blasts 1 which has the same features French and EU blood a few more David Stevens and and also but the you ask that the functions of the small localized the finally ending at Beijing not because it is the senior operator and acknowledge the status of the the result of Of the total up so the 1st reason they obtained concerns the Long Valley and behavior of the flinging a position and and he'd say is that if you take a smaller initial conditions in the U.S. press and they if you think you the solution of defending immigration in recent years in jail conditions and then you is defined on air blasts or even and agent value them and the you can't find a planned initial condition the system so that it is Gina and the solution jail 12 thousand soldiers on Tuesday reason sequentially said that you can be approached by your genes for "quotation mark time in the following sentence so if you
09:25
consider that that's an initial conditions is a question and you do corresponding solution then he behaves like the kind of free will rule should apply to of piety Ellen where J. satisfies these equations which actually is simpler than this is a modified scattering results so if you if you will if this that they would not depend on times a year just was a constant demands that we would have a scattering result but here it is his duty to modification and later on we will see you where it appears there is also out of danger the Congress resort which can be seen some of the popular dynamics of the answering a question and this visit stated that if you consider any solution In the solution of this equation then there exists you which satisfies the known nationally accretion such that the source forceful and Bennigan's construct to connivance and support of the precise so if you consider gene which is in its press and smaller and managing the dissolution of the recent and then there exists a real solutions although analysts said that recently switched from and the size of a load them to construct a very interesting behaviors In particular the have growing new norms solutions so long as a Kodori they have obtained that there exists a new solution of configuration said steps in this award if Norman tends to infinity in the following sentence this so it's the millions of the generally by its so it's something which goes down and grows so I think there's a way to use a load of money it's so this has been criticized by your dad conversion so the idea is that such a solution can be found in the system and the system has been studied by you or by his side and um by you Europe and by European Ryder securities institutions whoever exhibited some of the moment creations and the later on that this has been precise value regarded Koshien with smaller than an ecosystem techniques standing here they have proved prescient the rating in growth so it seems to me that the disease through the 1st example of growth of someone norms for analysts type equations wizard None integrity so adjusted to the just just the it's just the thinking can face a hole to move to to go so a few of the studied the system itself without the parameter have to to do what he did 3 few to either side by a so that you of integral billions in X as you see you ask that a new solution is a localized and and remind banks under the new system is transparent to respect to the viable exciting so once you have to obtain the solution of these equations coated and 180 not so which depends only on the viability and saying so why is the violence which is new Gandhi find such a function it's success life it's just that here you use this carrying out of the question and fight is a present function 1st class because of urges knowing I Christian has a skating because you do not tell you there's no reluctance Stillman lenient on him so you will need to reduce scanning in time so you can you can escape to you can do is to respect to fight you want function and then issue us that this is a regular on the once smaller than you can build such a function solid media just you some references about growth of closer with Helms incident energies procedure for the specific question so so there were results by a young for the way the creation then as I mentioned you will go through contender he steady and they can go on trial who walked for analysts into the and they showed that too you you can have the inflation of norms and later I have walked on the on the systems and as I mentioned young and can oceans no I like to mention the results for the so these concerns and interests and concerning to the QB Christian walks by you Juhasz tend to go with you also From the when the book of the people who have some growing solutions solutions 1 by 1 the line and they walked on so the 2 total messages that I would like to do even the store is the 1st day the I'd like to see it so the resonances of the nonunion found dictate to the dynamics of the accretion want for long times and the 2nd to the messages that the Soviet government unlimited used by you and put it at its school from the studio for the survey question and to show that to dismiss an admitted he said were best and need a place to work and the as we would see so during its existence under the law the crucial of the presidencies of the equation and analysts produced mean 1 of the and that is to me Rashidiyeh which drew walks on the latch class so the equation so that's way ladies so it's all a product many so now what I'd like to state that although results we have obtained the reserve good bear Baturin and this concerns not only the into the equation and analysts but now I have to be wizard wizard become convolution potential and that we will see that such a phenomenon and does not appear when you anymore and so we've seen that the season have played a crucial role so let me just say usually Christensen allies concede the union operator which is going to be so previously was the Nets and now I would I replaced the ways you know it's only recently which is the United players the convolution value potentially under convolution is
19:18
only in the X valuable so the CIA
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DVDs please makes the following phrase that you keep the same Eigen functions with the convolution and if you choose a dutifully coefficients of the you can play with Eigen functions result changing the Eigen molds Is it because of that the wider yes only 20 years so did these changes that were previously by use of the castle where this and now it's a this grasp arrested for concessions of the afraid had put the the project both but then shut it was the BU more physical relevant but then I changed the structure of of arrogant since it took the well on the other so the the 3 In the frequency is on the wall the number of civilians and inattention and cleared the squares disgruntlement luckless operator pressed BP has dutifully could petition of of the and we made the following assumptions on Monday the piece on Monday so that this assumption is satisfied on Mussolini the following is a it's a small viewed as David Davis is reserve was story morning there a single you ask that the differences of the eigenvalues inspired 0 all larger than some rights to it is the following and he and the human condition and the following so when you come to their aid the the
22:08
subjecting your right but so the injuries the disk and vanish if the peace accords to queue for example some diapers to entice actually and I think he's the only depends on the quality of the motives of those pay war I can see it's made assumptions so this convention and if he doesn't you asked that it is larger than some people were of what both P a new treaty is the very large doses of the 3rd largest among the default settings the meeting this means that the if this is smaller and you ask that at end you and then at least 3 of them have to be large so this gives you a a control and such an essential and is the is the wellknown and studied in the context of 110 PG soulful people studying Torrens and difficult ,comma homes so this is an instance in which you can or you can prove that it is satisfied that almost stories UP :colon you don't need to have this is also the mom were the 1st cubicles or was it ideas through that assistance system fulfills its history or a sore arm injuries injuries down reason that's why the Richard no logic it's my fault so its reasons the reason it follows that this ceremony yes that's it it's serious and it's not result from its that's it it's key construction work and this continent so that such a
24:18
control can be can be assumed
24:25
so as at the Beaver again so this can vanish and if he does not vanish new us that at least 3 of them online In the previous cases it was ideas you on this 1 because we were playing with integers and then the system so due to go into part of the creation of on that for the newly Christian is the following so excited I did J this is any question that the no I I have only decent restrictions worries that so I didn't want to read because here you have do do isn't terms are the following that's it so if you want that to this exactly you cancers than they have to work the same so if there is an insistence measures how much fewer attendant and produced again because beautiful privacy and in its I'm dates but you have a condition with the squares but not all this is larger restrictions so they Sistani's much where much more in behaved them and we will see that the Serie does not to exhibit new rows of solutions so you have to use most of the residencies and as a consequence we can fool the following results so the reserves will obtain these and improve their opportunity and my sense is do the same conclusions just killing 1 and 2 2 we as a reason you know with plays that by yearend and as a corollary so if you consider that an initial condition which is a regular and smaller and then the corresponding solution of In analysts this was president Clinton and this stuff the solution but then the solution is isn't known leader in India so what if no 1 you even proved that the normal tends to a constant when goes to instantly so you show that any solution is bonded so on most order of perturbation potential decisions strong going to parties the existence of the growing old the untouchable accretion so you see who the resonances play here you were so I'll it if there are enough food exact user can prove the growing resistance of growing old beats that here we cease to began essentially you proved that any solution remains falling any known so I'd like to give you know so few ideas of pool the very beginning this gives you some just so some ideas of commuters you can ask to we used to obtain a longtime behavior of an interstate accretions so do so so I screw SoHo to get access 2 were long and behavior 12 analysts say so well at when when it and you can use is the use of difficult normally forms for example if you walk there so let's consider for example the designation culminating song 22 please so we're using the golfer Norman the ball was in and under the bed have proved that if you added to the care potential so like furiously you can prove that the 4 on most social order potentially lose any solution remains a bonedeep for lunchtime so say you should begin in
31:23
summer supported norms or it's also in lots of with no longer than it's remains a small fall 4 times which are at the are indeed putting new maritime units in so this is occurring in the spirit of the rule of difficult moment home another type of resorts are came reserves so EDS earned and Cooksey enough proved that there exists a came real solutions to secretion so again using the because centuries the therefore that they exist the designated solutions so those remaining more for all times but now only heroes some derided DVDs concerns all its it's more initial conditions and came reserves concerned some of the console but all types of assault this is because it failed solution but the case of the cost was 0 is so this is a picture in to the context of very ridiculous valuables but now if you consider the on the other hand the equation on on the venue expects to scattering results of with me just give you will and this victory this series who considered the question on Monday was the Austrian In the new executives concluding the users and the such results go back to work to need below and into and that he had the idea to consider our product many followers used to tools use situations in between and the study has been initiated by youths with governmentissued for foreigners always in the case so the the ideas of following in 1 of the you have the reduces the suspension is to make it so that in the infinity norm in X is less than 1 over to to acquiring follow 1 onehalf this is the disposition its unity should believe it will To provide that you will do stated in your solution and then if you consider the nonunion terms like like a potential terror then the staff should behave like the 1 over the Titans you and so I did do they mentioned in these higher Alderman ladies has a larger pool of then this becomes integral but and you can prove scattering reasons and here you are exactly into India limits case and actually do looked around which isn't In these results comes from more than 1 of you which appears here so it's nonsense to consider the issue of a stronger poet this becomes integral and that he had this listened to entreaty appears later constant and here you will see the discounting appeal but here you are just 2 fingers in the case so now and we try to give you an idea socalled weekend makeshift residency reasons appeared slim particles that affected underneath so we consider the profile of fuel so you you conjugate resident your operator then the function as some of the and the questioned whether the new about this being canceled so it looks like this so I don't know if this Jerry depends on dynamite until the season ends and and another to study its you can do work for you to position so boasts in the next by about 19 why you valuable sold to very very for years from X to society of the known this can be written as the researchers and then the weakened so the full year the project to fully of the product is a convolution so you have such talent appearing the commission so this is the sort you here you have the terms of this is the for the Topeka for Yukon call conditions of the moment the moment to here those are the Eigen venues of the senior operator and here is the convolution which appears scenes of the product and honored to be here to give you an idea of the longtime behavior of such terms so he would you can
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prove is that using integration by parts so Uganda divided by this fact also integration in time and the actually the is these factories lodged in the day we have a smaller contributions and in any case I can control them using 2 residents assumption by so this is really need immediate loss of their relatives but that more is so the 1st time assumption is that you can 2 of terms in which our key and huge congress must send a signal to the world and this implies that has the same length as queue up to presentations to see what I have read in the privacy of key and can lead us and maybe it's just 2 them we tend to to see it so here I had the control I said that I have to go to evade assert largest and favored here on the UC when you want to study months union towns uh you you you are not allowed to work to heat so do many of the frequencies you we won't so if you want to study say you're a term which means they could enjoy your from using the devil you know when you want to make summer and estimate of heat so as to whether to estimated read by duality if you have full terms so a you V and Avenue and you want to make a good decision made so you do with regularity so that you can do it you you're not allowed to hit ready when the 1st time because it's it's 1 of the Serbian it's at this function church so do the 1st 2 largest is not enough again and the 2nd largest so please is not that loans also because if you you want to do the same the same estimated that consumer richest norms then you would like to heat on you on the do you determine which hazardous instruments so the 2nd is also not an old but then Uganda heats here and here you you may lose to the derivative I you have the denoted by John and this shows that then you up people into work to obtain a good solutions .period argument in some gunmen this is a roughly speaking to the thing and didn't the his teammates is served largess was crucial and previously used by you walks in and by you can bet on collaborators so this was a restriction to the reasoning terms and the 2nd to restriction come summer phase of formula we especially face from solar here you you were trying to arrest the test in the face to the phases find isn't it to cap on and on so this faces questions tension on the eve of counterparties 0 and then so no no nearer it looks like a brother to restore the England died in the large enough so you have this factor P 1 nobody comes from the fastest in the phrase from Lyon deep so would do to mentioned here are often in the cap attack and then you still have to do contribution reserve a typist was 0 so it's a F Q but with side and flew off part of CSI and if this parts and and here we have the think of the reason systems so we see how well we comes from and the view of the value over to you can cancel out by a change of idea violence here you despite you have some questions it is single 5 remaining minutes and really just the government on some other mothers Yukon study research ideas so such arguments also walk on the moon for that money faults I cross tools so you take that their stand on his feet so this has been the mentioning of people and he couldn't answer confrontation here so this has been done for the Mexicans because you have to wave of character so the why you absolute value and it has been done by your height and should and in this context the reason and so the question which is associated is disfiguring Christians and saddled her to construct a growing old
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you can also study system so it is soul the couple and their Tennessee commissions and other diseases under investigation by you and the dog and the lesser of 2 and the reserves I honey and the gentleman we we have obtained results for the for the Gorski today equations which which is there and there is a question that freezer confining potential source you can refer to the retreat to to their so you have a potential which is as quick and is a question is that is studied in the context of globalization strangling the sites where wherever we wish to use to the Europe received cigarshaped condensates and in this context we can prove also that you have a modify scattering results and under underlying reason to question is also wellknown known he's related to the West Longo questions the lowest level in normally questions and this notice to construct some practical solutions to produce a question I think you he said he knew what was going on and on and on and on the parameters for the same Pereira explained that in the last line work at the meeting so you're equation for G previous 0 an increase for all involve 1 parameter yes yes I did he said the value of it's just the cost of this convolution of 3 functions to parameters Cienega of the but the stationary phase ideas CDC's worried about needed what students interest so that I have a question about the Our equation which is not 1 equation is in fact a continuum family relations each 1 needs a solution the Red Sea is a proud yes and for Oxy the solution yes the renewed resonance conditions which resists the French no I gave you and you walk on the question resolved the and then I can from too a candidate what sentiments studied the question with the parliament I I I do not have to compete description Eichinger practical solution on the question which does not depend on the perimeter and Nikon added to the point so I can do 1 parameter 160 years several C is not always for all to see yeah sentiments he claimed that the bureaucrats at the Ministry of discussion at the venue on also questioned even work on the questions of pop OK well so long as you hear in his 1st day in the future and my
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Musical ensemble
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09:21
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19:20
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24:18
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31:22
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