Quasi - periodic standing wave solutions of gravity-capillary water waves

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Quasi - periodic standing wave solutions of gravity-capillary water waves
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We prove the existence of Cantor families of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x ) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. In addition we prove that these solutions are linearly stable. Joint work with Riccardo Montalto
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concerning quasiperiodic solutions the previous works which trying to understand what is the fact that of the letters and then only denominated and the former vector of the warm weather but perturbation conveys many that this was done by quarks enforcement and company officials I have been here 1 day this is of course still listening migration because that is just 1 that imitating the information concerned with respect to kids and then informed the last such fall the playing Gordon equation there was a work by Bush again for periodic solutions of this is that some sensible mythical too because it is less dispensing equation and then there is the more difficult and then with the Kadeer's permitting approaches that will be given a generic example of quasiperiodic solutions for the Steelers playing God equations with singing preparations that emissions so we've got word of the of depends on 1 day but different than when they in space and this 1 has to assume stock shot which is measured section which is relevant that which is analogous immediately this is annoyed the sublimely funny structure of this equation but it is natural to expect DB structure the rest of us actually uses a structure which is compatible with the presence of the quasiperiodic of solutions to control the growth of stubble of norms and there is so also for present and equations effect and has a nation before the visitors answer will stay with around applicable about the concerned that given the blast and 1 other enemies daunting political division which has a city that is so it is with a so which is generated by no means dawn in where the density depends on a gaining on 1 space there and then you want when where and what writing down the vector field in the medical field 1 rates but it is now has the shape so it is advising at so now let's go back
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constant coefficients and for his basic and made is in this way but then
of course we see the initiative that we mentioned before namely the is the development of these medicines was once and here there is a need to John J is any integer Saugus numbers always reluctantly to the Zito when 1 cannonball was the foremost of vendors of frequencies and again by discussed the lower bound of on the site but so that 2 billion as the peripheries inedible but it's the best operator because of the presence of Faxon was there and so this is the usual to modernize because of that public resonances the investigators it and the continuation of these audits it is not based on compaction
and then a way to solve this problem is through nationalism implicit function the amendment which uses has his own knowledge is made that on a new fundamental the horror of looking at this year's profit function given the and proximate solution you when 1 look for a better approximation you enclosed land in such a way that this is something divided the intersection with the condensate and this is the it that the scheme but the fact that ended the advantage of this Kemakeza has it is well known this aquatic scheme for the distance between 2 successive approximations listed on the square although they have recently Canterbury was 1 these are so fanatic who this but expanded of convergence enabled to compensate that is more divisive difficulties but the difficulty is that in order to write for example the skin and we needed to know the denies the operator after a function which is not only is it up like that you different things and so we are in fact accounted for in this project problems to show that the building owners operators using vegetable for more survivors of the bombing and that it's the satisfies estimates that the namely that the the Napa Sullivan on there can fixable and it was meeting with the respect to the right .period you where we and
this is the end of the shift is better than the 1 bride-to-be denies the operator for the what waves said he it has had bad the shape at is that what this is the 4th the way here due because the additional Predator B and the functions from the ingredients of the velocity potential after the free surfaces OK and so it is at the point is OK because it is exactly the operator here and that we would like to convert to prove that this is the foremost of of bonhomie doesn't approve estimates for last and so the proof is quite there is composed of main arguments that was not the to discuss was all alone but the real capabilities and nationals of proof that the accordingly has a here all had put the conjugation of amendment in order to to take away the vision with the parameters them that I will discuss a bit this past year namely the analysis of the denies that because of course of from this is an essential to the past OK about but only because the gender justice state 2 stinks of course has reset the body is all of this phenomenon has a view or not season on time really depends on a very complicated way on their campus at the time he doesn't affect practice which appear that I really complicated accountant and expand their soul really it's a very sensitive to know or not I hope to fulfill 25 non-residents conditions on the front and but in the effective the 1st accord will governors out their work before weirdest that is stronger than on generous a condition that for example was not satisfied In a celestial mechanics for the for the commander for the solar system and that has come a lot of news and then factories for example was a motivation although we can indigenous to conditions in political life and that and that is that have the right figure also of the sources that we will not satisfy the weakening business conditions here I want to discuss in here which was a and use the surface tension but I them off easily from a physical point of view which is also equivalent we can fix cap but and the considerable has abandoned the wavelength in here I put the tube by about 2 by land that is the end the parameter which enters into the waves so I fix the equation this solutions with existed for more survivors of the wasteland of the over there with and in that sense the same or 1 Thinker also to for the index to the depths of the ocean except the OK in this case is that the ingredients which I have and what we used which are important solidified his we on the generous conditions and the fact that the visiting the frequencies are indeed the analytic functions of the surface tension and there at least suitable as in politics at Infinity group of Jakarta 5 3 house and then there is a suitable number Janessa conditions and promote the condition that they explain which is fact the school the Gaussian condition which I think it goes back to the wall was precisely going up there saying that a but also problems under Russian people and that it is just to say is that there and if 1 thinks of the fragrances 1 looks at the founded the state we don't want to end then that image is not contained in any heightened so it is a storefront conditions there's so that the curve Omega trapped isn't which is an date is not identical contained the 8th when I saw analytical for any vector in and not leave the function of city Scott products Omega account but is not identical so it is not containing hyperbole that is called the abortion condition ended the April in this case in fact it's a competition that 1 they get the frequency of actor and the 1 shows that it is not contained in and I think and 1 shows that these vectors are indeed more attainable by the Finnish issuance and indeed that limited act the from here comes out Monday among them them on the continent and then 1 check that there is this :colon montages conditions .period then
that's OK and this is what they say about the the body and then I started to make the about the pity but that has
said that concern the analysis the spectral analysis of these time-dependent time and space dependent on which is the I think in this way and I repeat that the goal is to find their conjugation suitable change revivals fight which contribute to the sense when opening in a very simple British Al-Qaeda here I will be that putting the operator arrested at the border directions which I normally do they have tangential dynamics still OK and is that the quota is to compute the existing public expansion of and there always in the same spirit of the book market is abandoned in a conceptually different the steps the 1st step has the goal not reduce the sides all the good news is a constant revision but that sometimes but that the 1st step will be noted too worries about the acting thinks and to make it an atom square and so on but the 1st to reduce it to constant coefficients half-dozen without them which is in to EU relevant because when I hear that I was effects on high-frequency said for J 3 house is extremely high and that it is important that they should control at that time but it is a very old biggest contribution to the dynamics so the 1st step is to portable constant coefficients hazardous here the highest of landed 3 300 constant and then also will disappear the order liner again by the signatory of the city and the than the older than the X 1 half of which is always an unbounded contributions 1 hospital because of the fish when 1 of which is that all that is your people care about this a sufficiently while the effect that we have known to Union's situation at this point it is natural to stop a hair-raising that bonds which are excellent and an end to all square at some 3 and saw no need to performances and not 1 4 times this is removing so a little steps 1st to break the news articles and coefficients because seemed was and then they have bid to to reduce the size of the to jeopardize the effect of these so we needed to have his presentation for that for the United States and that's OK this is the case it is news will be attended by conjugation changes of mind was banned abandoned changes of light if we have a system which is a them in this case quest began thinking time we conjugate with the change of mind which is in this case was a chaotic in time then it transforms into another is the form of the new operator is written here that is the conjugation of the space operator in jest Western attitudes Fernandez 1 8 feet and then there is a better which canceled the contribution of the gives this conclusion OK and then the goal activities is to find their 7 at-bats formation aid which it in fact that we live in meaning that look for suitable any such a that abuse me does not depend on space on bank and that it is very that it is what that we would like to resolve was a of the use obtained by compositions of several transformations change the latter was over there to defend the nation this is the 1st
year it is that I started from
assistant I make a change of animal and I think users this agenda but it with something that this is a very judges lot that is the rules on the whole changes part of time and the part of space and then I will chose a in order to no it won't be like that I would like to do that if I don't win 1 step but in all steps that meant that you came to the the idea says Sorry sorry I said I wanted I choose fee which is a valuable known also as I have the status of course I check nutrition that the system with the vector field a transforming consistent with the defendant is adjusted transformation nothing and now we have to choose and prepare with the with the naked with thinks here there local experts say that in this context that we appears they wouldn't neck in the following this is the unionized ability effectively this stresses that the analysis the same has another about it then it is really operator 18 there with these a valuable there were norm which needs the conjugating that system with just his Maddox amid the multiplication port functional but the system called the contributing another problem we this 1 which is a much nationalist semantic and has also by John here so they it's that eigenvalues are in this context a purely imaginative because of an audit by the by I so the agony message thanks to this with knows that invented so we can spend much of this is the 1st change of which OK has visa change of valuable 48 is a Howie we said that in the beginning there that the Belgian right property of the system are important here we try to preserve old ways that condition because it is important it is necessities obsessively that and finally that the recipient in order to really the existence of this kind of conditions also we want always to preserve his property that this transformation president of the city so that wouldn't normally reserved a Sunni actually so the net amount from President Barroso listened consistently that measures so that we could as but nevertheless there for rest of it is sufficient and that because I've always listened but the character of the transformation is the 1 could be that much and then again has another there are some changes of valuables to which I contributed the from office while the dollar was on the rebound position of time such a bad that we are able after this change of animals that do not have to constantly fission the behest of the facts the action buying explaining what 1 has obtained the Golan Heights tolerance and then OK 1
has lived still some jobs to do this effect is not an equation with this system in Asia "quotation mark and the then and also the next goal is to seem many of these up to smoothing over the last fall .period inundated the off they had another dams which acts on the Cole born of each to pollution to support the proposed already negative order so standardization activities moving the operators so that the new system will be constant coefficients still viable proficiency on the which I act on each they have another once but then and all the argument initial becomes extremely small will say
something very good and then the next goal we want to do uh we had Vernell with the affected of at the 1st daughter is still viable coefficients so in order to do the same we conjugated with the floor the produce the rights of like this 1 and the the floor to the pedia like is 1 since we'd aid which is a valuable the reason for the Boston between each incident which France is nice sat at this very much and then there the reconciliation that did the conjugated banker we wanted to to analyze it when it satisfies the duties and has an equation is 1 session and then is equation has a solution that can be sold into the casing symbols thanks to the fact that the gloomy data infect the gains 1 of the Act and that because of this 1 can solve this equation and find that there a contributor the data decreasing seemed lost the new denim in front of the accident Is that is modified and then 1 chooses the function aerial acts in order to put the user to cost as it turns out that 1 can put it OK and
the same significant the additional How much Primary One is finished OK and then as well that is of the Tibetan again but I want maybe I see the Sundance and I think and then
just 1 water so and after this conjugation because we wanted to because the sides at this point we are adding a singing out situation and then the transformation the
analog of the normal form as this 1 we want to now to find the transformations which in the committees that the preparation from sites excellent besides at square so what conjugate there with the floor Over a function W to be found the new operator can we analyze the gain as before but then we expand the really empowers of absent and then we look at the new Of the new excellent and then we want to choose W so that the Vista appears in doing
this the was sold it and the
1 that is able to delete that III for this function of you and everyone how to obtain his winter from the bitter mission 1 has to divide this through the finances of ligand some on the face of it in medicine and so appears so that it is a necessity to boys and this the analysis and proper approach progress but there is liberalization process that 1 has to impose verified these conditions which in the original has the 2nd of the monarch of Morrison's conditions if 1 is able to is 1 firing up there is that thank you and