Kink dynamics in the \phi^4 model: asymptotic stability for odd perturbations in the energy space
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Kink dynamics in the \phi^4 model: asymptotic stability for odd perturbations in the energy space
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Kink dynamics in the \phi^4 model: asymptotic stability for odd perturbations in the energy space

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4

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23

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CC Attribution 3.0 Unported:
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2016

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English

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Abstract 
Kink dynamics in the \phi^4 model: asymptotic stability for odd perturbations in the energy space We consider a classical equation \[\phi \phi =\phi\phi^3,\quad (t,x)\in\RR\times\RR\] known as the \phi^4 model in one space dimension. The kink, defined by H(x)=\tanh(x/\sqrt}), is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virialtype estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Kortewegde Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general nonlinear KleinGordon equations with potential: the interactions of the socalled internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

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take the time to act and send it to and so 1st I would like to to thank the organisers for inviting me was it's a pleasure to be here I've been here for almost 2 months and fortunately dying almost over so I will talk about a joint project to work with moment that include new neurons the paper is already on archive and on Meet the Press that we can use the problem but the problem is solved problems said very simple onedimensional problem with which is known in the city's cultural this complex certified for Moldova and before a committee before a competing with this model and we make a little aggression with about half the generality of each which sent the similar similar properties all as far as the nonlinear structure so it can I am of the sole people considered to equations solicit political equation of public inquiries and ack ack ack ack and it can be considered in piping in bounded domain soaring in the whole space this problem from Spain sold in often this you know this equation is referred to to as the Allentown equations but or by stating that encases very easy to see is looking at the London wasn't called by stable the sleekest 3 zeros 100 gold this was legal matter OK because 3 0 Sunday from this is a little bit picture the point is that this area is equal to the area and also the good all this the other way around I didn't I invented it but when it's clear that this extends 0 spot as opposed to the state To date so this kind of thing questions have studied the law and that there a lot of free literature about and there are nice and in a sense that they have certain connection with geometry so the parabolic question this is very related to what this could mean curvature flow and there lived a key questions related to to the minimal surface .period and they mean started from the seventies said I think that Jordan was the 1 who introduced them to the support of mathematical Pt a monument to keeping the public in the community great souls so I like to see this certified for is a part of a bigger family and there's some might think affinity between between those theories also OK so it is good thing and sometimes I wasn't Austin to write to the model is a system and then I will abuse their locations so for me 5 sometimes is a scholar and sometimes selective going this already game and so it's obvious that there this this this equation has stalled conserved quantities that are important the energy and mental landed both play a role in what I was saying In quake and this is for the alleging I just pulled Hugh W. OK in which is so aquatic polynomial and so this is where the main 5 4 model comes from because the full force of inquiries and there In fact I don't come from this culture so I inevitable this problem from a from a by paper written by which OK and in which he mentions among other models this model and argues that the the meeting but that is the PRI's why this said this potential W should be this fall this system is set for order and the equation that reports of a polynomial and not something that's going to this if it the physical physical reasons for 4 OK in other companies that Windwards this model appears to Lotus if you open a book ,comma on quantum field theory that typically 1st chapter is devoted to the fight for more than entangled ways so the basic object people study and this this is substances ,comma also from the island ,comma equation is the king In fact in the fight and in the quantum field theory also everyone started at the beginning the kink so it's something very simple in this case as its explicit solution just hyperbole pungent answers and solutions will be that of course say Well it's not unique if unless I say that should be increasing and all
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other ways it's not because I can translate or is a symmetry the whole problem symmetric also met with soul but if I look at the at the way the question the king generates twodimensional invariant fault a piece of invariant mindful duties there just by Lawrence transformation of dash became vacant changes to for it every wireless upholstered and sees the speed between minus 1 and 1 ahead solution 2 To approach we think and some basic question is OK what is what kind of behavior is expected around this type of solutions of the flow of what will happen so fast that think Dole told the panel that described little of it is on the 1st of the 18 so just make compact somehow notation to avoid remained too many letters there what I will that I will do all the kinks so OK still to be translated landscape plan and its derivatives so can think of H & H prime as a senior Salvator associated to the solutions in the the position and velocity so that I can look at but the product of the perturbation all this initial data and then it's not so difficult to see that that in fact taken without loss of generality taken calls the perturbations in such a way this speech corresponds to the initial momentum and the perturbation the 0 baby 0 is a vector so it's 1st compliment he spent 2 orthogonal blades price going and this is very easy to total supply stocked with any new Sunday X 1 day they can 1st took to see and then I kind of just why in order to satisfy your supplemented this sort is just a matter of choice when a convenient always so but maybe a little bit pretenses because because they attributed this to you to myself and Yvonne Cloud but in fact we have not found there we we not of the hearing before and this is simply a bit stability of the king but maybe it's a part of a fall the focus and decided so if you take a solution will be in use of the site described which is additionally small Indiana displays then the glove solution OK as I should say that global solutions for this problem existence it's it's not very difficult to show that case that that's that's known the governor solution will be such that it said it's better basin viii you yourself are not always to which Prime and the momentum use His fixed as it should be and then moral there it is so small in the Indian into space so this is simply the orbit of stability of of the king became sold while not talk about the problems of his sister rather easy just use the conservation of energy and momentum and the spectral property of all the king the fact that there was a previous as I should I should say that there was a previous pools of water because WET Canada Henry Patterson Rosinski inmate being 83 uh and they proved that kinky sorbet stable also Indiana displays assuming that sees equal to 0 so that profits maybe a little bit more to other article said that the they prove it for other money narratives as well but on the other hand the fund sees equal to seal signifies the beginning of the is almost there no I mentioned that part of parabolic problems so the parabolic problem he said is very diverse and this very well and stalled the and in some sense it's but it's not trivial but it's a it's well understood so if take at the parable advertisement consider king banking his use of its asymptotic exponential asymptotic is stable and he fuck OK fuel think offered that the initial data that change Simon signed what then this thing is so they will evolve to a king and then we can move things in some way and there was this example there from maybe 97 it's Paterson went work then there is a mess also a classical example of an intangible system that is it better fight for more than it's not entangled in way which is that somehow considered together with the Fife modified for modern is the same Gordon questions so equations and so some of you may not like the sign that in fact the development of the site here and there and this same question has said world solutions that are interesting sets of muscles and other interesting in particular has a keen that this explicit as a solution that's called breeding and something that's called wellbeing King so that means that so you hold this so a reader in fact is not a single solution said is is a family who carries a 1 parameter family in the same way they will be keen to want Parliament before so will be
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Kinkel case so he will take care of our own this is the direct character is the King of Siam and then I have snapshots of wellbeing came with certain terms of parameters greatness on the coast of the snapshots and go over that period the notice that 1st of all let the way they chose the site of the is such that the kink changes between 0 and 2 by and eat this if you take as there was access by minors the king will be all Korea at the same Woodall blinking CPI minus the wellbeing can will be king will be old and also notice that doesn't stop the asymptotic the fact that the while being king and assigned garden and the king for are indistinguishable as they did their asymptotic says the same amendment at exponentially going here have overwhelming and the reason for this are snapshots of a brigadier oscillated around 0 before signing of ocelots and 0 so 0 pedestal sort of stable state of signs great the agreement oscillates here and here we're
12:34
going so somehow you can think of who will be key is set as the combination result superposition convincing that the a kind of combination of of Wiki abbreviated endit and of course while the king said that Kincaid is not asymptotic is stable for signs order and then somehow people believe that the breeder causes it in some way so if you have an Armenia claimed that immigration with in which affect breathe there may be a kinky for if there is a kink in the same questions will be state so and the Breeders by themselves out I think are very very delicate objects are better than on genetics and and that does that prove that they do not persist after an hour and then small generic perturbation of single factor it said it and they're kind of got a medicine while the bases so there is only 1 type of perturbation and then he claims also that even for those perturbations Breeden doesn't promises and there is a war it's not very difficult very easy a factor would exclude possibility of existence of readers in out of there is a result that I found out that time it's from the ladies as well there a view that more than 1 who shows that certain classes of craft also is a condition for seniority so that readers Olympics the UK and the US is because the figures are used to the whole world will see in through real time period ball I didn't say that but they batted let me go over the years they are absolutely periodic because of the signs the water leaking through yes absolutely absolutely yes insisted that account of the number of reasons freedom which the move the absolutely 1 absolutely so this on this earth .period gives I didn't say that sort this might be the fiscal on new to indeed deserve this Alpine periodic solutions focus will be on display yesterday was that the kids are all great there this I don't know what they're about to know it's 1 of the things that the new CD noted it that it is as if it was like is OK yes well again this is this isn't there so what I mean by this is of the power periodic nearzero small solutions prevailed in Indiana despite the great because his see there's not enough space here right if I want to make this trust fund has decided to use the kind of shortcut :colon it's a very simple in fact it is as follows from that our method that I will explain later on foreign asymptotic stymied his out is rather simple and say even observation in you for a very special kind of playing on the questions that that preserves all space so immunity must be called it off a small readers so this means starting the energy space and there is no additional sources that are a small Indiana space and at the time .period so and not by the way Florida signed odometer Waterbury there is this is not obvious even of course this is not a contradiction it was somehow all readers what cannot exist for the sole defining wellbeing Kingsford defy formerly the fact it was peoples I think we're told it somehow and in the 80's and try to find sort of solutions that will look like wobbling Kingston and readers and there is that there is still some work of cedar and and cost considering but in particular was consider sold sort of show that there are no small breeders for the 4 5 for the fight for more than their bite about that by then back to pay for them small read enemies agreed and that all slaves of their own to what minds OK so as we talk a little bit about them the fight what is known about the 5 formal design got some results among many in the world but the which I found the relevant may in an interesting because they look at the broader from the angles so fast that there is that there is a couple of Paterson maybe more maybe more comfortably 2 worth 3 or 4 papers of commented cooperative about and they analyzed that money applying gardening questions that look like something like 5 formal there but they're not fight for modeling date because but that there is some of the smaller prophetic nonunion oddity that do not include the fact that the perturbations are instances not small because they severely protagonist seniority the but they have been they were able to solve the study the of the keen that in the Allstate solely only alterations to the context of our work well the entire dimensions day there is the work of alQaida from its maybe 2 thousand dead and he got so that a plumber from Seoul to you take the
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king and the cost of you do lifted 3 dimensions of great concentrating in the dimensions in space and then you look at us topics that BT of all this object that he sold as asymptotic study the and and it is actually part of the vagueness of the spaces 3 Anderson simplifies the lot because then you can use the special good in in complementary dimensions so in particular cases toward 1 of the more difficulties from the start of those I wanted to mention workers of at the bank the fault line a little bit the result there we which is not so easy maybe to explain that it is a kind of a lockout existence is out of paying higher dimensions musicals attraction was meant that the it's a construction loan of local solutions look on and time that look like taking the old represented the certain there meanwhile service Greenland space all the talk about about our results introduced and admitted to it so I will talk about all that keen topped only by all perturbations this is this is very important it the White is important because you know that David said that the kink you can you can boast advocate and let it come off but if you're follicles to the altered the bases there is no motion so see 0 so this is this is this way you take out 1 degree of freedom paid a 1 1 0 modulation and this simplifies I think analysis England so way OK so you look at the altar to races and then immediately from owning a problem you pass but that of the beauty of the previous fall you passed to a different kind of equation that this take the lead in your I spotted the millenium Park and so now the problem is not signed Gordon quite it's not claimed Gordon because the public has applied for the potential foes is different OK and then the problem is reduced asymptotic studied the old 0 in the old space so this is the problem I have with mister and this is something important that I will use a lot which is the fault of the nearest operator agreed is the nearest operators is classical and can say a lot if not all you know about specter and so the result is is the following scientific Baker perturbations in the old space and displays which are sufficiently small look at the global solutions that with this initial data goes to 0 the lockout In a lockout energy space program for the norm here is so here is isn't the whole space that his own all abounded we could solve this of course OK we don't even have a and and but on the other hand in some sense if you are mistaken if you restrict yourself to be in the space is said this is an optimal results because that having been said that he had that you can only take long time no can because he it was if you were able to show that congress 2 0 in the hole in the full knowledge that effective didn't show anything so in order to explain a little bit there is out and how we prove it I have to talk about the spectrum of the Unionist operators saw this as soon as I said it's explicit you have 0 it's obvious in value corresponds to translations and there's another Reagan value which is very important for all its engine functions very important for what I will talk about because if it has to be called and then you have a continuous spectrum and stories as announced that there is a lamp that I did about again functions even tho he doesn't play a role the entire cities are said to have the translations and then you have firm the 1st mold and if not 0 mold which is followed this is what is important so In order to understand the the dynamics of this convenient to do is to become polls the solution OK being told that the discrete mold all the molded by described the great multiplied by some function of the and the rest of organized have radiation this part of the school's internal oscillations sold notice that this area again function is look is localized pays localized around the thinking that's why the people call it the call the smoking journalists relations OK so there is furious that is the followers In London assumptions of the previous day or is the following you have the following estimates to use this the z y z tool to power for integrated Teams and again it's a kind of local energy awaited an engine or both the radiation abbreviation parked integrated team time is bounded by being sold so this even you can see is 2nd kind of weak form of asymptotic studied being factored the previous result follows from this place by an additional round argument which is taking a very difficult but this is this is basically the important is up now for the soul
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of this information that is given in the sphere it seems so it seems weak but is as I said in the United States that is difficult thing to do but it said something that could to another column in context the functionality ,comma context increase and that they had the basic approach to
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proffer this and and that and the
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previous 1 stronger result is based on all material estimates basically this said this was motivated by the work of Martin Marietta for creating the animal and are file unless especially this works because think else we'll see at the end of the governor somehow but there is a difference in the difference is said to take into account and analysts relations so this is this a new element of and then we OK I should say that the general
25:00
results for this type of effort for this type of problem
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OK say that basically we maybe you can improve OK
25:08
maybe you can improve this indignity but not too much I think you serve the end it was
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difficult to prove that to you as tool Norman meantime abounded
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some the sowing of the 2 schools say about what to because I want to explain that is that the probe but I also want to explain what is going on is the mechanism and so in order to Lewis I think it is fair to all they had to talk about the paper of sulfur Weinstein's was somehow started this business and that savings in some way the paper from 1990 be days they consider that this type of class million trying question with QB community with Appleton style of painting in threedimensional space for the potential of courses introduced us to avoid the translations of the defeats that the object they were interested in a in space and what they were interested in is a status similar to questions that star asymptotic study at the old 0 4 member the title my problem doesn't thought experiment using his various in and they make certain assumptions of them their and the nearest operators of the assumption is that if as well as the pagan value 1 K and there is a continuous spectrum without his analysis is the same abstractly also they make another assumption which is called defending golden rule will firmly golden rule I think was previously but it comes from a specter serial OK and when I don't really understand why it's called family golden rule because in the end few open book offer of reassignment and you look at this golden rule something completely different about me the people for the reasons to college like this I follow and I I will try to get the OK I just say that the family golden rule in pricing particularly that 3 times only got OK is bigger than answers have end here and have only gotten the result break relations between them and the fact it seems to me that this is a very important some who sold as the result of sulfur lifestyle carry something something like this they also the pay the the composer's solution to internal oscillations and the radiation in a similar way and then there so that the internal solutions because they I'm trying and feeding fact time with the rate and the rate is also the case in a but they did right so was or what is that I think that when you when you don't understand the strategy of the profiles understand what is going on so many explained the strategy of the of the Pearl but the strategy of the approved I think it's best to to get rid of men in the Union test and just stick to to the core of the problem and the core of the problem is the following year the coupled equations of for the radiation and for the internal oscillations so this is a nonunion PDE will pay and this and this is simply an OBE is that OK this just mean oscillate and so their strategies is is something like this to 1st solve the man you solve this equation for beat up you planteating blew the the 2nd question then you have just the ammonia oscillator you for millenia of for normal form of this later and then you find in the cave in the and then the decay of the tough follows from that and sold by the way if they see that you have take you here and and this ECU which is very important at this 3 almost got bigger than it has to do with this skew gave it was formed in the former governor of the the amendments OK and it's important in this strategy I think that the state's history that is said threedimensional so you can heal department aligned to specify estimates and close estimates of gracefully in 1 day mentions that this strategy is not to think of a is not so easy to to follow especially if you if you work
29:33
in the United States so let me explain what is this strategy that we will be proposed OK case of fast OK suddenly recalled the problem so this is the problem of claimed that this kind of Union operators and that was like the composer's I said good according to address to the spectral this operator so this is steadily described and then the project because this problem into 1 1 and 2 ,comma complement Citadel Entertainment promptly questions a festive have been a finite dimensional cradle nonunion oscillator and then have been in the business of said and what I always say if it systematically and only its 2nd contends the verifier order soldiers to be viewed their relief they can be controlled and what they need fees to use the corridors of the of the problem being on the cover of the problem is to understand how this story questions are coupled quadratic you go there coupled quadratic good and in Seoul that somehow they think OK well maybe I II and III of the coupled and some in some way but this is impossible as this is the I thing this is where the problem is you cannot decouple that you have to keep you have to work with them or her to understand the way the we the function and human worries explicit surmise that became functional group In sense of place the electoral so the idea of a city school somehow reviews in a way of a sold or you won't would like to find normal form the house that is not so easy that is all too well I think it's good complicated but you can find something like a normal form that is said so let me in he explained this so OK so what is the number of policy lines of for me normal form is a form of the problem in which you can understand how their models that evolves so you can kind of don't have to face so this is what I a to blow that they change variables to introduce new variable file which is simply OK this is the part of z squared if you look at disease is a complex number In the days the imaginary part going and make a change of variables in the states of variables and clean said it delivered the problem in such a way that you can but the infinite dimensional problem just in terms new function view that is basically like you but then he now thought OK which is the square and then you can write also said the problem for our final beta greater choice of this of this change of violence makes it possible and then it is seeded the problem is is not money animal that is I have certain nonunion actors that I can control and that I have at a lenient problem here and the your problem he says now that it says that at this moment that you can use .period identities could insult to use identities the result is the objective of the identities objective is to show that this object to carry this isn't about being time is bounded and it and it is important to me when were away when when the this analysis it's important to have more because that before OK because it allows in fact to control a lot of nonunion of tents so they the identity said with the 1st part is something that everyone knows a thing the 2nd part is maybe not so obvious but it's kind of natural OK because you exacting makes the way you would like to do do outside the pool and then OK although this identity is the point is that you have unknown functions in the G going there and we have 2 children in such a way so for the types of sigh that there is a natural choice this you won't site look like X solely controls a function that approximates X in some sense because if you just back from the subject may be no more looking better good but about the but for G there is no obvious choice then this is something you have to work out then weighed in if you plant this scene OK what if you defend seats or a few defenders status of Iowa noted that in many cases this store objects that this blueeyed function are bounded become because of the that small so now fantasy them and Emmywinning differentiated into buying our for on the 1 hand and some stands gain that involve of finding 1 some explicit function 8 and 10 which I defined the explicit and dysfunction gene is not explicit and so the point is not to chills the science Genie and used at all unless he steps away that this mania for is about whether scholars is this is the to whole thing In another tho it does not well but in order to justify actions GE it's where we use the family golden rules in order to well not so maybe it I don't extending details that kind of thing involved calculation but at some point you have to you see that in order to choose do you have to solve the equations and this a question can be solved in a way you want to do it only if you have mingled so it's heated itself but it appears that was not what they want to have this is not enough because he was still missing the tool and data but this is this is easy it you just the compute massacres of fun data you you find 2 other VTL identities and you computer the integrated so so this this is not so not to so now I will let me uh let's talk a little bit about the CBT office of his biting uniform group sold so as I said in order to just sorry when you have to choose some sort of approximation of X would go in and then it was not sure about the many choices but we had which was the site of the dysfunctions so it's related to the king going and and well I think the stresses is quite good some sense it works very well and I'll explain why so cases have just buying and that if you like it or they do the lid is the storm had to change a little bit because of this pride factor here so you have to change variables for the changes very much volume just going to have to take square what he would like his squirreled only 1 instead of spirit of outside Prime the 1 guy who was detained variables we could I would just say prices quadrupled Taiwan which is the 6 account hyperbole they change variables biologically attending functional and dysfunctional this is nice because of potential that this potentially 650 can compute what it is and then that's a basic fact that would prove is that under the plan all this assumption and also the malady In this biting uniform is choruses of income it's all over me well I I think this year so something we've just to explain a little bit how how we how we work out the course CVT sewing factories making our former said is seen as the desert is quite complicated so it could consist of tool parts 1 part of the price in gave the was not sold maybe it's kind of thing both but there that's not so important not so difficult that some kind of message but the 1 is very nice O'Grady 1 that is inviting our form that is the source of his dog you may think you sort of his square the 1 is a mania for associated with classical operators can and then when in fact you you can fight you open a book of devoted to this beach Marshall some classical book you find this kind of operator and the environment spectrum computed and encouraged to see that the His mania informational negative and in fact this is the operator you Justice Scalia and this is where I the at some point I mentioned that we use the approach of many landmark Montana and Maryland and offer and almost not only that we use the approach because this is OK use the approach but also at the end it's everything changes on on the same kind of classical but shedding the operator so I think there is a very interesting and kind of connection and structure that is behind this the 20 it's OK to be understood OK so but he said if the question questions you're missing from the signing of the what's when was removed old allows the reader to exist I don't know I think because of fears from that the U.S. spectral structure is not all know it's not similarities that OK OK OK the SES and no there is no internal before signing gardeners have 0 and and continue spectrum there is different and the heads of the continuous spectrum allows but it's all OK so this quite different yes so if you really want to do is to remove the onus is on the table so it was you the euro reaches a salute from all of the Lawrenceville alliances can promoters have followed you also get resonance at the end of the spectrum users were supposedly more serious or yes I think it's it's uh way more serious I don't think it's a technicality and so it's a serious Michigan they don't understand why they just to the right and that was registered with the problem will carry into the manufacturers allows which this looks like a cost that would project formerly owned this you see the thing oscillating going through as close as he want to the entered space was limited because I think it's a company it's complicated university you know which in the based on the theory goes through the the formula some of the workers choose from this kind of that's news to me ,comma you of OK that's eluded beyond my expertise to answer this question but also it's a very honest but did OK it has to do with that was the moment when Simpson's "quotation mark and if you look at him who formerly it with an ax and just works nicely "quotation mark versions In the few future and