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# Kink dynamics in the \phi^4 model...

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#### Automatisierte Medienanalyse

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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 one-dimensional 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
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 well-being 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 well-being 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
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
proffer this and and that and the
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
results for this type of effort for this type of problem
OK say that basically we maybe you can improve OK
maybe you can improve this indignity but not too much I think you serve the end it was
difficult to prove that to you as tool Norman meantime abounded
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 three-dimensional 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 plant-eating 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 three-dimensional 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
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 Fachgebiet Mathematik 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/}), 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 Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general nonlinear Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.