 Thank you, organizer, SFN about nice conference and opportunity to give this talk. So I will be talking about ideal to dimensional hydrodynamics and also discuss how it could relate to… There will be efficient tool to describe two dimensional turbulence. So let me first mention collaborators of this talk. Это Сергей Деченко, моя форма, кредитная студентка. О, кажется, что у меня есть батарейка. Продолжаем. Я не думаю, что это работает. Я не вижу ничего. А, вот это работает. А, я поставил центральный кнопок. Так что я должен использовать. Да, окей. Итак, первое, коллаборатор Сергей Деченко, мой форма, кредитная студентка. Он сейчас постдокат-браун. Мой департмент коллег Александр Крадкович. И сейчас я моим форма, кредитная студент. Денис Силантьев. Он просто защищает свою диссертацию в последнюю неделю, и в следующей неделе он двигается к постдоктару позиции в Курант институте. И Владимир Захаров. О, я уверен. Все говорили об этом. Владимир Захаров. He has many affiliations, Landau Institute, and University of Risona, and many others. Окей. So, I will be talking, as I mentioned, ideal hydrodynamics. Let me quickly start from 3D when reduced to 2D, when it's really necessary. So, we're talking about earlier equation and gravitational field. And actually, we're looking for incompressible flow. So, density, we just set to 1. So, we have divergence of v0. In this case, we can introduce potential flow with velocity potential phi. So, our incompressibility just immediately results in Laplace equation. So, inside fluid, we have just Laplace equation. And first earlier equation actually comes important only for boundaries. So, here we can factor out nablin with equation and arrive to this form after we use this nablin phi, and we are here in square brackets we have just Bernoulli equation. So, all devil here comes in boundary conditions. Boundary conditions, I can see the situation with dynamics of free surface. So, typical situation it's water waves in ocean for velocities of wind which small enough, you know, but when we can in many situations for long waves we can completely neglect wind. So, we consider this situation. So, we have some free surface by this red line. Generally, we have surface tension, gravity, and we have y equals eta shape of free surface. So, in many situations in ocean it's sufficient for long wave to consider two-dimensional situation. So, a typical situation with ocean swell when we have long waves they nearly one-dimensional. So, they propagate like this. And in this transverse direction dynamics is very slow and often practically it's not important for long waves and in this case just two-dimensional hydrodynamics it's completely natural to consider this is why I already restricted to just single horizontal coordinate x and vertical coordinate y. Now, as I already quickly mentioned all devil here comes from boundary conditions and there are two boundary conditions again I just remind everybody absolutely standard stuff at this point. We have kinematic boundary condition which basically says that our free surface moves with fluid which comes for example in this form for our surface elevation eta and we have dynamic boundary condition which just states that above fluid. So, again we have now two-dimensional fluid so we forget this third dimension this is our fluid surface here above fluid we neglect any air so here pressure is zero and then we just enter fluid on a boundary when pressure if there is no surface tension pressure will be always zero. If there is a surface tension it will be jump in pressure just determined by this exact form with sigma surface tension coefficient and this pressure p we just plug into Bernoulli equation after that we don't have pressure anymore it's completely excluded from equation everything is expressed in terms of velocity potential phi and our form of free surface again here boundary Bernoulli equation we need only on boundary every way deep into fluid it will be satisfied automatically and this is what is called dynamic boundary condition together we all form closed equation system of equation so we need to solve Laplace equation with dynamic and kinematic boundary conditions and we need to do it at every moment of time so this is basically our form of mathematical formulation then by Vladimir Zakharov in 68 he derived he showed that this form of equation is equivalent to the following canonical Hamiltonian system so here psi is a value of our velocity potential phi at boundary it means that this is nothing more than Dirichlet boundary condition and it is our surface elevation and h is Hamiltonian and h has very simple physical meaning of total energy with fluid h is consist of t which is a kinetic energy so I just take square of velocity over 2 and integrate from the depth here minus h if we include case of finite depth what typically in this talk I will talk about infinite depth so h is just infinity up to top of the fluid and then integrate over horizontal coordinate and then we have potential energy it's gravitational term and surface energy term and this is already expressed in surface variables but here I would like to pay attention this is still integration over volume while our Hamiltonian equations from Vladimir Zakharov were written completely in surface terms so it means we need to do something and basically we need to do in principle very simple integration by parts in our words use Green's theorem to convert this integral this terms which I will show next slide this is surface form so here this kinetic energy is transformed because Laplace inside fluid is 0 so from Green's formula only surface terms survive and this is the normal component of velocity naturally defined with unit normal vector defined here so it's everything looks nice what is the trouble that in this Hamiltonian Vn is not known because for Hamiltonian formulation we need to write everything in terms of canonical variables so it means that we need to express Vn through psi and eta through our canonical Hamiltonian variables if we can do that then of course we can solve everything but of course we understand that for hydrodynamics life is not so simple we cannot solve everything what we can do first of all nice mathematical formulation because as I already mentioned psi is nothing more than Dirichlet boundary condition for phi and Vn is nothing more than Neumann boundary condition just normal derivative it means that in mathematical terms what we have to do at each moment of time we need to find Dirichlet Neumann operator so we need to solve Laplace equation to find relation between Dirichlet and Neumann boundary data this is written here in symbolic form but of course generally for some complicated domain generally we cannot solve Laplace equation analytically we have to do something and straightforward way to do that which was originally by Vladimir Zakharov just do perturbation in power of small parameter which is just small elevation of surface but of course this is working for reasonably smooth waves not for really large situation situation of large waves but this talk I would like to address situation when we have really big overturning waves and when perturbation theory is not completely not efficient but in two dimensions we really can use power of complex analysis so we can just introduce parameterization of not parameterization we can introduce new complex variable plus IY horizontal vertical component of velocity we know from undergraded books on complex analysis that for stationary flows we can use complex analysis so basically what we can do for time dependent case it's more complicated but it's exactly in this style so and to go to find explicit solution of Dirichlet-Neumann operator if we look if we work with complex functions with homomorphic function then we can do of course conformal map and if we able to find conformal map from simple domain showing here in auxiliary variable W which is a low complex half plane and map this domain into our domain occupied by fluid so everything below blue curve and this real line is mapped into this blue curve then of course for this simple domain we can find Dirichlet-Neumann operator find explicit solution of Laplace equation for each moment of time so this transformation just will be given in terms of this new variable W which includes real part U imaginary part V as analytic function Z of W so this will be our canonical transformation time enters as a parameter so it will be time dependent conformal transformation now I remind you again basic facts from undergraduate books of complex analysis that we can introduce for two-dimensional fluid we can introduce stream function theta such that is just defined here so we y v x means just component of velocity and this new function automatically satisfying compressibility condition if we just plug in this stuff into divergence of V we get identical zero but if we now define complex potential phi our velocity potential plus this stream function then our condition for definition of theta will be just nothing more than conditions for analyticity of this function phi capital so phi will be analytic function of Z but if we now do conformal map for Z for this simple domain of course in new variables W phi will be also analytic function inside fluid so this is idea for example we can immediately by differentiation introduce complex velocity do all this machinery now time dependent hydrodynamics again this stuff already not in a degraded books complex analysis then it was shown in this paper looks like yeah okay so that we can derive the following equation which equivalent to ideal hydrodynamics with free surface we just look now on real line so we have this physical plane and after conformal map I remind you that we have low complex half plane here so we just let's stage us in a real line so our W will be just equals to U because imaginary part V is 0 in such a case we in this work in this work we obtain exact equations and here equations written on real line so here subscript means derivatives partial derivatives now our free surface is represented parametrically as X of U and Y of U and it means that also that we can describe situation like this with over turning without any troubles while in this old form Y equals eta of XT of course such configuration would not be described so it means that we can just complicate solutions I apologize I think eventually I have to touch better and better ok so we have equation for Y you see there is time derivative we have equation for psi and equation for X defined here just because X and Y are imaginary part of analytic function so actually if you know why we immediately recover X by this integral transform and I remind you that Hilbert transform in Fourier space is just multiplication of sine of K and imaginary unit so it's really convenient for example for numerics so again this equation are exact so such type of equation first derived in different but equivalent form by Avsyanikov around 1975 when it was reproduced by several authors including Salik Tanvir who is here and gave excellent talk yesterday all about different topic but I use here exactly this formulation which is convenient for me of these authors ok now kind of idea we use power complex analysis but what we can solve initial hope quite a while 20 years ago that check for integrability because it was some initial signs of integrability in sense of inverse scattering transform but it was shown in this work of the Chandler of Zahara that at least in terms of inverse scattering transform with time independent parameter of inverse scattering which we usually call lambda it's not integrable it's kind of there are some initial signs but fifth order matrix element is not zero but kind of problem but now my group and my collaborators we try to explore is to focus on dynamics of complex singularities why complex singularities of course inside fluid everything analytic but if we looking for entire complex plane here above fluid we allow to have singularities and typically what we see branch cuts it could be different situation they could have different forms sometimes it could be finite length and what Professor Tanvir showed in this early works in 1991-1993 that some of them can can show up in infinite small time but in our for example in our simulation we typically do not see much formation of branch cuts in infinite small time because in principle of course if you can introduce infinite small noise you can introduce you can put some branch cuts but we will be in some sense weak because jump across branch cut is very small so typically we can see at least initial condition which we can see so far is to look situation dynamics of widths branch cut and ultimate goal to describe two dimensional turbulence in ocean through dynamics of this branch cuts because of course it would be much easy to restrict because basically if we know jump on each branch cut we know everything we immediately get solution everywhere inside fluid and if we have tool to efficiently describe this branch cut then we will have everything including for example efficient way to describe turbulence through statistics of this branch cuts but this is kind of general program with only at the beginning of this program and I will show you what we got so far so this is for example let me start from numerical examples this is what you see here it's time dependent simulation so we have time dependent conformal map which start from pairs of poles here above free surface and these poles actually now after some time here in the center they move down while what we see in this simulation that there is a formation of branch cut of the following form so there are two poles moving down and there is a branch cut which forms in a small time and they all extend down here with finite velocity but what we see there is a slow down so we expect that touching of free surface to happens only on infinite time while so this is again this is complex plane while here this is form of this solution in a physical plane so it's xy so you see it's kind of bubble which try to pinch off and we can study this very peculiar property that for this type of solution we showed that it's possible to have poles but these poles they always produce branch cut somewhere else so this branch cut is completely unavoidable yeah by way in this simulation which already stop this branch cut is represented through so called pade approximation set of poles so when we do numerics we really have to scale our numerical precision until very high limit to show that increasing precision we have more and more poles and to show that we have continuous limit of continuous jump on a branch cut which in some simulation to demonstrate that we really have to go to super high precision up to 200 digits sometimes what we need is of course challenging simulation but still possible our group doing this type of simulations to again numerically show that we have robust convergence with discrete poles which just numerical results towards continuous branch cut another example which I show you it's another solution when we start just from single pole and previously it was pole also for z here poles for complex potential so in this case if we have start from single pole and in a small time it produces branch cut and two branch points they extend with time in opposite directions which is shown on result of these simulations and there is also analytical solution which shown in early work but this solution for weekly non-linear case so it's applicable only for finite time according to this weekly non-linear solution there is a touch and branch point here what you can guess from this solution square root branch point pair of square root branch point one of them touches real line in a finite time so basically here if time is zero when this square root is disappear we have just pole if t is arbitrary small we see this extending branch cut but for example if we look for full non-linear solution compared to this weekly non-linear analytical result when we see that there is a slow down of approach of this low branch point to real line and this is shown by this green dots so it looks like this approach happen in infinite time again weekly non-linear solution would describe finite time touching of singularity by the surface which would mean that we have some non-smoothness of here I think the solution in second derivative of surface here also blue line it's a result when we regularize some way breaking by hyperviscosities surprisingly it's of course not physical but it's important if you really want to describe oceanic turbulence we need to do something with way breakings an idea to kind of introduce some hyperviscosity which allows to save main statistical properties of turbulence now what happen if we add gravity previous results were quickly described was without gravity if we add gravity what happens in this plot when this branch cut when basically in physical plane this solution correspond to jet which going up and up at the point when it stop there is something like this happening on the top and at this point we have branching of our branch cut into two branches like here so this is what we see here so it's kind of demonstration with simulation that in many initial condition it's really reasonable to have reasonably simple structure of branch cuts ok now this is what I mentioned already with jet profiles of jet at different times and for example time dependence of this profile it has really nice form so this is rescaling to psi with L-simile variable what we see is this distance this position of low branch point and it really after this rescaling collapse we single curve so we have square root branch point all is definitely not end of the story yeah by way an approach what we obtain so far approaches approximately with Gaussian approach of this lower singularity to real line but definitely it's not end of the story real full linear solution is not simple analytical solution which I showed you previously from 1994 paper this is our jump along branch cut and this branch this jump along branch cut so basically this part is our square root and our square root here but here in the center we have instead one fourth law and basically from our results on different topics it looks like but I think I still have a lot of time I thought you already pushing me out yeah so that if we have some power different than one half what our experience with analytical solution teach us that we will have infinite number of sheets of rim on surface if we try to cross branch cuts this is typically what happens here and I will demonstrate with you in a few minutes an example of Stokes wave very quickly so just now talking about Stokes wave just give you some idea without real technical details so this is also analytical part of our work Stokes wave is just wave moving with constant velocity in conformal variables we have this form so until they just move on into free surface so here we just can solve this equation for single wipes I can be excluded it's advantage of Stokes wave I remind you Stokes wave it's a non-linear wave propagating with constant velocity in gravity case so what happens for Stokes wave Stokes wave has our count limiting Stokes wave solution which has singularity famous 120 degree solution found by Stokes long time ago and in addition Grand found next order correction which has transcendential power which again indication that solution here not simple at all from point of view of sheets of Riemann surface but there is a paradox here that let me quickly mention we can ask question why Stokes wave is important this is some example if you perturb Stokes wave it's immediately start produce over turning on capillary waves when we will see this formation of craper solution for capillary waves which you know it's really high precision simulation which we can do and we see this is typical mechanism of one of mechanism of formation of the destruction of big waves through emission of capillary waves and also there are some results with overturning but let me skip it okay so basically we have two approaches which we use together one approach is a numerical approach here I already mentioned we have high precision but just brute force high precision often not sufficient here what we do here if we have some area near singularity so it means that some branch cut very close or touch real line for Stokes wave for limiting Stokes wave then this area we use extra conformal map basically to zoom into this area and for example just give you quick idea one conformal map which we use is like this so U is old variable again it's now parameterization of real line and Q is new variable and L is small parameter so what happens if we use in Q we use uniform grid 4M methods completely straight forward in U it will be strongly non uniform grid and beauty of this transform what we showed that this transformation it goes through all equation for hydrodynamics general conformal map will not do that because basically it will mix what will happen on minus infinity here so just yeah I think I skip this little slowly but just idea of this conformal map what we can do we can take this original so this is our original W plane this is our singularity if we apply this conformal map so in Q plane we push this singularity up by this small parameter you see denominator but limitation that there are extra singularity of conformal transformation itself which moves down and there is optimal height optimal value of L and corresponding optimal height of this singularity new plane and this basically gain square root of Vc so our conformal map for your spectrum now decays much faster and gain which we got so far in simulation let me skip this yeah yeah by way using this for example we reproduce classical results of longer higgins without infinite number of oscillations of parameters for Stokes wave I would like to pay attention this is high C basically position of singularity it changes here by 12 of magnitudes so this is only possible with this extra conformal map and maximum gain which we got by this method so far speed up simulation 10 to the power 8 times so we have generalization of this stuff to multiple singularities so it's more tricky but it's also working so we have this tool now and for example this picture which I already showed only possible can be possible through this tool to get this high precision now Stokes wave returning to Stokes wave very quickly again we have limiting Stokes wave but if our linearity a little smaller it means our Stokes wave is not limiting so we don't have singularity so this is example this green is Stokes wave and blue is non-limiting Stokes wave but very different near to the tip so I remind you with Stokes solution again Stokes solution returns in terms of our conformal map and deviation here near center near to the tip but analytically it's completely different solution question why answer is the following let me maybe skip this sorry again I returned to work of professor Tanvir from 1991 that if we have non-limiting Stokes wave so this singularity in arbitrary small but not 0 distance from real line we only allow singularity as square roots while Stokes wave have power 1 third it's completely different solution there is no simple transformation between them and what we did we basically explore our sheets of Riemann surface and just give you idea what happened this is first sheet so this is fluid and we are not allowed to have singularities here but if we cross branch cut in a second sheet we have pair singularity here non-physical sheet we are allowed to have another square roots and what we proved that we are all singularities as square roots and starting from first sheet we have also diagonal singularities now how always complicate picture to come together it come together in a single analytical solution which let me show you it looks big it's a basically infinite product of increasingly nested square roots so this is one square root next product we have three nested square roots here fifth five nested square roots etc but if you look for this high sea with distance goes to zero we just reproduce Stokes wave because there is geometric series so we start one half one over two to the power of three one over two to the power of five we have this product gives Stokes wave and it's possible to show again this stuff looks like I really go too slow today comes together because this solution reproduce with limits but solution is really complicate and now this constant alpha they are completely determined by this positions of this what I call diagonal singularities so they not some fitting parameters they really defined so all they come together to form this really nice looking solution and just to conclude here some technique how we explore how we explore this singularity different sheets but I skip it so when we compare with numerics we have really excellent convergence between numerics and analytics because here if we looking if we try for example to do Taylor series in half integer powers we have limitation two singularities in all sheets except first one these limits of radius of convergence by this by this nested square roots we really goes many thousand times above this radius of convergence for small parameter and produce this nice solution so let me come to conclusion basically I kind of try to give you several snapshots and what we doing so again ultimate goal to look for two dimensional turbulence of ideal fluid and we have numerical methods different techniques to really resolve this analytical structure and at the same time we work on Stokes wave which really taught us that if we have some non-trivial power in solution except one half we immediately expect infinite number of sheets of Riemann surface as we really mastered how to do for Stokes wave ok, thank you