Collective dynamics of quantized vortices in superfluidity and superconductivity
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Bao, Weizhu
Quantized vortices have been experimentally observed in type-II superconductors, superfluids, nonlinear optics, etc. In this talk, I will review different mathematical equations for modeling quantized vortices insuperfluidity and superconductivity, including the nonlinear Schrodinger/Gross-Pitaevskii equation, Ginzburg-Landau equation, nonlinear wave equation,etc. Asymptotic approximations on single quantized vortex state and the reduced dynamic laws for quantized vortex interaction are reviewed and solved approximately in several cases. Colective dynamics of quantized vortex interaction based on the reduced dynamic laws are presented. Extension to bounded domains with different boundary conditions are discussed.
 Okay, thank you for the introduction and thanks to the organizers for putting everything together. Wonderful opportunity for everyone to meet, to meet each other during this tough time. So this is John with Changdu and I think, okay, and Yan Zhizhang and Chen Lintang of Sichuan University and some current PhD students, doing this one. Brief outline. I will brief motivate you what is the quantized what is in super-calendar fluidity. The mass mass of models, there was introduced a center water state like a solid ton in exchange of solid in 2D. And there there's a cal size. If cal size go to zero, so there's many water states, then there's a reduced dynamic law kind of set of OD system, which is governing the water center dynamics. So that kind of let what you that for like a particle model for the water center. Well business with dynamic law for get the lambda equation, we can talk a few sense there. Of course, I will talk about some this kind of sense for shooting an equation with 90 as well as a bounded domain something like that. We know a quantity of what is kind of a particle or topical defect. So they are so you are other product complex other parameter or counter skill field. So side. So at the water center, the other parameter has to be zero. So in your fluid language at the water center is a vacuum. And near the water center, so it was x zero, water center near the water center side can be locally written as square root of rho, it's when I fee. So rho is a density, fee is a face. So if any gamma is a closed curve around x zero, the face, the integral along gamma, your face has a two pi n, where n is called as circulation, index or winding number. So this kind of quantum what it says is kind of key signature to support so fluidity. So suppose they support the dissipation is flow. So it's going to flow when they flow, there are no friction. So this is kind of a cartoon to show like a quantum what it says in the three, you know, there temperature is like a macro Kelvin some, some of that this is cartoon. I've got here this is real experimental result like a quantum what is in type two, so conductivity conductivity, so conductive conductor. So each what thought is a quantum what is in what it says. So here also are experiment result for the kind of in quantum versus in both transition you there because the top one is so many particles in the harmonic trap, you know, they're here and there's no rotations of ground like a bump, they're Gaussian type. Well, if this system rotation frame, you can see that little holes, each hole is a quantum what it says. So basically the quantum what it says exists in the superconductor model by quantum lambda equation, degree of chemistry, they could by two for the model couple the system equation like you have a VN and the row and the velocity density for a number of component and row as VS for superfluid for superfluid component. So there's a couple together. So they're also using the sort of if you just got a gross place equation to model as a phylogic model to model this one. Also the both kind of session is kind of not a shredding equation or gross place equation to model you also in nonlinear optics or propagation of these are being there you're not in a shredding equation or nonlinear wave equation, even nonlinear categorical equation to model this kind of problem. So here is some kind of typical machine model, you know, like here psi is the other parameter complex other parameter or will function. Like you get the model equation for superconductivity, nonlinear heat flow, complex flow, you know, here psi t is time, psi will function the other parameter. You have to say I intentionally write all the equations, right hand is the same. Right hand is the Laplacian psi plus one of f square, y minus f square psi could be on whole space in two dimension or a boundary domain omega with different boundary conditions. So this is a general equation of complex general equation or nonlinear shredding equation or nonlinear wave type equation or even nonlinear kind of equation. So here psi is a complex value to function or other parameter. I have a parameter epsilon is the definition of parameter which is sort of cos size of the problem. You can see that's the problem. The nonlinear part is a key, epsilon small. So basically in space, the typical scale is other epsilon. And in time, for the equation or shredding equation, the typical wavelength is other epsilon square. Of course, in wave type equation, the typical wavelength is other epsilon. Okay, of course, if you introduce so free energy, you have functional, you can see since there's four equations, the right hand side is same. So if you introduce the free energy or your functional, you can see the shredding equation is a dispersive system. So that the energy or the density of the modified density, they are all conserved. So this kind of dispersive system, they automate so that the solutions, particular like solutions, like any 1D, like a solution in 2D, canx or what is this, what I show you here. If you can find out equation, this is dissipative system, they're added to dimension system. So the single or central what is this we call red tail. So if I consider this, because all we get is whole space in R2, you can see, take an ansatz, psi equal, psi of x equal phi n of x equal fn of r, kind of this is i n theta. Here r is the polyconnates, n is the winding number. So I plug this ansatz into a system, I end up with this fn for this kind of two-point value problem and f r equal 0 is 0, r equal 1. So this is, so top here is fn r for different n, you can see when, so r equal 0 is 0, r equal 0 is 1. And formally, so the middle part is like the absolute value of phi n, you can see the inside that the origin is 0 of the density and 1 outside. So for fn r, asymptotically you can see where r between 0 and other epsilon, which is fn r is constant times r over epsilon to power epsilon by the n. Well if r much bigger than epsilon is 1 minus n square over times epsilon square over 2r square, basically near the core the fn is like a growth in term of r polynomially. And at far field the leading order is 1, next order the fn r decay quadratically in term of 1 over r square. So this is the phase, you can see when n equal 1, it's only when you go one round, two-point jump. So this next one is n equal 5, if you go one round, let phi, see that going one round, there are four, five times two-point jump. So of course, purely interested, all this kind of particle type, particle like a solution are the dynamics stable. Of course, if you set all the parameters, rho is density, v grad s is called velocity and g is current. So of course, there are some analysis for Kinsbler-Landau type, but for the Schrodinger type always some kind of assumption. So of course, when I study in America, I do the time, I don't want to show you this movie, I just give you the result. So basically, if I just have one, this is what I state, I do some perturbation for Kinsbler-Landau type, I can do perturb at this data. If you do little perturbation, where a equal plus minus 1, that dynamically stable. So if you initially have one, one, one, one, one in number, n equal to one minus one, that after some, some perturbation in this data, then then, a long time later, the steel one in number one. Well, if n bigger than one, you know, the dynamic answer, basically, the pattern is the split to n, what is it? When t much bigger than, much bigger than one. This is for this, this is for Schrodinger-Wilth type equation. However, for Schrodinger-Schrodinger equation or DPE, so this time, if you perturb an initial data, that dynamically stable for n and n, because the algorithmic citation is conserved. However, if you perturb the, it's not potential, for example, then n equal to one minus one, the dynamical stable, if n bigger than one, after n bigger than one, dynamically unstable, something like that. Okay, of course, since n equal to minus one is dynamically unstable, you are interested in studying what is the interaction. Then initially, I take my size zero of x equal a product of j from one to n, m phi of mj x minus xj zero. So xj zero basically is the location of the jth vertices, and mj equal to plus minus one. And phi m of x is the station of what is state. And also I can multiply or fix the independent initial data. So basically initially, I have capital N vertices, and suppose they are well separated, then it's basically xj zero and xm zero, the distance is much bigger than epsilon. And also, yeah, okay, this one. And then you just suppose, okay, suppose, so suppose, then you plug this, suppose solution size of xt could be something like a product of size j equal to one to n phi mj x plus xjt, where xjt is the water center of the jth vertices at time t. So you plug, plus a order term. So you plug this and that into the energy, you do some analysis, you will find that as not as you know, the distance, initial distance, they are much much bigger than epsilon, or the distance equal to infinity. In other words, so because the distance is much much bigger than epsilon, the left epsilon is zero. Or the initial distance is fixed, that's epsilon zero. And xjt will follow ODE and then later can prove rigorously this, this, you just dynamic law. So, so you can see this here, give you. Let me capture x is x1, x2 to xn, which is the set of points in R to N, which xj of t is water center at time t. So then I have introduced a functional table of x, which is like this one here. So remember this here, mj mk could be one, could be a minus one. So for Kinsman-Landau equation, kind of dissipated PDE, my so-called my capital Xj will follow this second order, that is the second order ODE system, nonlinear. So basically, so Vj is x dot, sorry, sorry, sorry, following the first ROD. So Vj is xj dot. So the velocity of the jth particle will follow this one. Okay. And for Schrodinger equation, this, this business from joining you, of many people are countries since this included Wynne, and the Fang-Hua-Lin and his group, and also for Schrodinger type equation, you have a j is a rotation, 0 minus 1, 1, 0, j times xj dot Vj t of Vj equals j dot. Follow this one, same. And for another equation, which is a second order ODE, like xj double dot, this one. So here, again, here, this one is almost the same, but the left side here is Vj here, but the rotation matches capital J, and for Schrodinger equation, this xj second order ODE system, this one is similar as many of you have studied this, for this correct dynamics for some set. But remember here, mj and all can be taken in here, plus 1 and minus 1. Of course, they're like how to, they can study from PDE to ODE system, like for another equation, they show for African ODE, you can start to study the qualitatively, qualitatively behavior. For example, if only have a pair of vertices, then capital N equal 2, and m1 equal m2 equal 1, the vertices will undergo the repulsive interaction. If the vertex dipole, the n equal 2, there is plus, that one minus, and go attract the repulsive interaction. And for example, the pinning effect of the other, so from this ODE system tell us. So recently we started this ODE system more intensively, what we can see more from this ODE system. Then for Schrodinger equation, what is it, we have the pair of vertices, like at the ideal field, and the PDE, like a higher order, like a radiation, something like that. Of course, for this ODE system, for this ODE system from Gizmo-Langda type equation, of course, if you have a plus minus, the finite time, the two-quartets center will go together, that's like a blow up or some kind of a merging of this one. Well, for Schrodinger equation, you can see at least for n equal 2, 3, or 4, one can show that no matter you initially have all the same sun, all you have different winding number, like some plus, minus what is that, what is this? You can always show that this ODE system is global well-posed. So from our democracy initiatives, this ODE system has not initially, what is location, they are distinct. The system, they are never two, what is it, go together. Okay, so now we try to, from this ODE system, what can we see more. So first we consider this ODE system from the Riddio-Tangai law from Gizmo-Langda equation, so this is the ODE system. So of course here, xj dot, because this one. Again, I want to remind you that the mgml could be one, could be minus one. So at the front, so the mass center, so this x bar t is the average of all the location. You can show that the x bar t is the conserved quantity. And also the energy, the w, the w of the x, this is decreasing. Of course there are some, like that, problems for example, if you initial, for all initial data, you shift one point, the solution shifts one point. And also to a scaling, you can do also scaling for the solution, also do a rotation, so initial data with one, like a q-theta is a rotation matrices, the solution also rotation this one. And also for some, if I have n vertices, if initially they span the uniform on a circle with radius a, you can solve this ODE system analytically. For example, if a equal to, a equal to three, a equal to four, for n and n, basically they are, they are, they are moved outwards along the line, connected initial location and the origin. And if n could equal to three, and if I, in fact, I have one, what is it at the origin, the rest n minus one, they are uniformly along the circle. Again the, the single one doesn't move and the rest n minus one, they move outwards. This, this is easy to understand, just plug the, use the symmetry. If initial have two, what opposite what is, what is dipole? Then you can use to show that here plus minus initially, this from ODE. Of PDE, the, the quality is the same. And of course you can solve analytically, you can see the solution when basically at a t equal a square over two, the two will be moved at the origin. Next case is, if n vertices n bigger than three, but opposite, I have one origin, I have minus one, one in what is this, and the rest n minus one with plus one, what is this? Then you can solve, again you solve, origin one stays there, but the rest will move into here, you can see my a equal four is a, so-called as a critical number. Because if a equal three, they still move, the two plus go to the motor center, move with the n minus one, the a equal minus one, and after more, there are only one plus left over. But a equal three, which, sorry, a equal four, which is a, the, the, the, the, the, the, the, the equilibrium, but this is unstable. The a equal five and, and the upper and, and, and, and the more, then they move outwards. Because the a equal four, they don't, they don't, there's this equilibrium, and less than four, they merge, and equal, and critical, critical five never merge. Then also we can start using the school that, not autonomous first integrate. So of course here we, suppose initial data, we have a capital N vertices vertices, so from an OD system, I suppose among them, I suppose I have N plus, number of plus one vertices, and N minus negative, minus one vertices. Of course the sum together equal N, and I didn't have M zero for this one. That defines this H one, X t, H two, X t, and H three, X t, this one. They need to show that if X t, X t, X t, X t for this ODE, then H one, H two, and H three, they are three, nine autonomous first integral of this OD system. So with this one, we can show that if I have N vertices, they have same one-day number, then there are no finite time collision, then there's a, that globe exist in the OD system. And the, and the addition, so at least the two vertices will move together at the particular time, in fact, one can do us, we can prove us stronger, so result at the, for any bounded domain, when t much, much bigger than one, there are the most one vertices in this bounded domain, when t very large. And also we can show that the so-called, if I define the distance between two, any two vertices, d t l equal s j minus x l, the absolute value. If N greater than equal to, less equal to four, and the n vertices have same one-day number, then we can show that this is the minimum distance, and the, between any two, and the, the, yeah, these two are, they are monotonic in collision. However, if N bigger than greater than five, this no longer, there are some cases that are no longer true. So this d minimum is, I think, is the minimum distance between the, between the, any two vertices, and the capital D minimum is the, the, the, the distance square. And okay, we also have some cases, if n equals three, we can see more, long times of behavior. If n equals three, and the same one-day number, if initially the three vertices are sort of located along a line, then when t equals infinity, we can see that two vertices will go to infinity. The, the, the, the, the, the one in between will move the, to the, the center of the initial, of the three vertices as the interlocation. And if they are initially not, not, not collinear case, then the three vertices are on any triangle. They are at t, at t large enough, then so-called the three vertices will, will, will approach to a red triangle when t large. Initially any, the, the three vertices is, look at any, any triangle. Then when t large, the three vertices will, will, will approach to a, of course, a red triangle, of course, the, the, the, the, the, the radius of the, the distance to the origin is become, go to infinity, something, something like that. And then one can have similar, have this kind of, kind of, so for n equals three, we have something like a similar, like a orbital stability of a self-similar solution for number. Initially we can define some so-called, so-called, so, so, so, so this is something like an x-tool of t is so-called with, you can say, the, the, the, the distance, the radius, the, the distance to origin is like a grow, like at t to power one half. The x-tool of zero is some, initially is a red triangle. Therefore any, any data, so, so, so, so this, this one will, so, so, so, so, so, so, so, the distance of x-t minus some, some, some, a, a, a center value they will go to, to, to the, the, to this, this solution up to a rotation. Some of that. So we have this, this kind of results. So similar of this, this, this, this, this for, so, for, everybody's always the same one in number. For, for, for, for any, we have some result for n equals three, you have some long-time behavior. Of course, if the, the opposite one in number, that initially have some plus, some minus, that, that, that, we can show that the, so, so, the, the, n plus, minus, n minus less than n. So we, remember what we defined as m zero, which is m zero is here. So m zero equal n plus, minus, n zero square minus n to power divided by two. So if m zero less than zero, you're always the, the, the, the, the, you can always show that when tick approach team acts, then, so at least two, they are, they go to a same location. So if m zero equal zero, the solution of, of solution is bounded. There's also sure we, we, we go to the, the, the, the, so often m zero equals zero, they're very super set up. And if m zero bigger than zero, there's no fine time collision, then the team, that group existence of solution, then we have also some, some other, this I can skip here for, for, for, for, for simplicity. Again, also for, we also give some, next condition for collision. Let me see. If, if there's, if for inverses and for, so, if they form a collision cluster, then the m zero less than zero and h zero, h one zero equal n h two zero and h three zero equal n minus two h two zero. So then the three, so the autonomous, autonomous, so first integral have a first policy relation. And we also give a second condition for equilibrium. Then if the, if the, if the, if a, if form a clear equilibrium, then my n plus must be equal one over n plus minus square root of n. This one less than n. Some of that, some of that. So of course, if n equal four, so you say if n equal four, n plus equal three, n minus equal one, this service condition. Then we also have some collision pattern if n equals three. So here I have, suppose I have m one equal m three equal one and m two equal minus one. So we should, if initially, if initially I have three vertices, if the, if the, if the initially they are, so called the, see a red triangle, sorry, a triangle with two, two segments with same length. Then, so they will, so the, so the three vertices will, will merge, will collide at the, so called the, at the x zero, at the center of the triangle, something like that. And the, the collision, collision time is equal, h one zero over twelve. Of course, if initially they are, so called the, so called the triangle, one is long, one, one is large, the other is smaller, there's a two one with, with, with short distance, they will, they will merge to each other and some of that. So, so, so that's why. And then we also have some for, for Schrodinger type of equation. So this is the OD system. So remember here I have a G, G equals the rotation matrix here. Then we define a sun, the mass center x two dot t equal one over n m g times x, x g t. So this one, remember, of course, if m g is the same one number, so this one is the sun mass center as same as the previous one. But of course, if m g has plus minus sun, this is different. And for Schrodinger type equation, this sun mass center is conserved. And also the energy is also the capital W two dot equals this one, this one is also conserved from this OD system. And again, we can show that if initial of n vertices, the, the initial, the initial d-baron circle and the, they will rotate each other because this is the, the, the, the, the circle, but they are counterclockwise with the frequency n minus one over a square. And if I have capital N, grid equals three, initial one at an origin, the n minus one unit for the circle. Again the order of the center will stay there to symmetry. The rest m minus one will rotate counterclockwise with m minus two over a square, this one, frequency. And if initial of two vertices, you can use to solve, they are moved parallel. So this is a solution. And you can solve, you can use to show this solution set as the OD system. And again, initial, initial I have n grid equals three, what are the opposite of what it says? One at this origin, which is a minus one, one at the number. And the rest m minus one at the unit for the order of the circle with plus one, one at the number. So you just again, origin one is, stays there to the symmetry. However, because not again, n equal four is critical. And you call four, n equal four is a critical equilibrium. And n equal three, they rotate clockwise. And any critical five, they rotate counterclockwise. Again, n equal four is some kind of critical number. So here of course, so of course, so here, of course, for this, for this equation, you can use it for n very large to our understanding, there are no collision, but I cannot prove it because n equal four, we can do it. And you figure the four, of course, some is for this OD system, if initially you have n vertices, they are distinct, then I did too, they are not the same. So since they are, they stay on each other globally. No, no, whether, of course, if the same one number is approved, if the opposite one number seems we cannot do it at the moment. And also if I'm here, if you initially they are, if I have four, two are closed, the other two are closed, but this two clusters, then short time, the two clusters will move like a leading order, they are moving like a similarly. So here, similarly, if you have n equal two, if two, one number is the same, what is pair, they are loaded at each other, if what is dipole, they move parallel to infinity. Of course, if you are, so this is from OD, you tell us, for PDE, then of course, for Schrodinger, another type of equation, you can solve PDE, then basically the qualitatively the dynamic from PDE is similar as OD. However, for Schrodinger type equation, when the cos size, the distance, if it's distance is less than some cos times cos size, so if you have one number, the two one number, they are opposite. What is dipole, you can merge to each other. So OD tell us they go parallel, but PDE, they tell them more. If the cos distance between what is, what is it, is smaller or at the order of cos size, the OD basically gives the wrong dynamics qualitatively. Then people also study on Bonnet domain. But on Bonnet domain, Schrodinger equation is the skin is same, but if for principle unknown equation, you need to multiply the riskier, look at the longer term, you need to multiply PDE at lambda epsilon, lambda over log epsilon something like that, so that you see dynamics. So here, so if I give it initial data, of course, you have a different, but kind of like a typical Neumann or here there are three scaling. One is the size of domain of omega, the other is the smallest distance between two and two vertices. And the third one is the one in the cos size of the vertices. So of course, for fixed n number vertices, if epsilon is zero, there are a lot of results. Of course, if you fix like a left epsilon zero and also n equal to infinite, you know, the sub-It have wonderful results, five years ago on this, for Schrodinger, for this study. So of course, for fixed n, there are lots of results. Let's add epsilon equal to zero, so the host was a limit. Of course, if you let the epsilon equal to zero and n equal to infinite, they are very challenging to study. So for bounded domain, the OD system, like of course, you can say you do need the host space OD system plus, okay, here I have an R, so due to boundary effects, the R is something like a, so like a laplace X, the capital X is the location of the vertices and the X is a domain. So basically, like a equation with the particular or particular function condition, this is the OD system. Basically, it's kind of like the OD with a non-local R, which depends on the location of the OD, of the, what is location? For Github-Lang, for Github-Lang, for gross potential equation, this is some OD. So basically, the capital R for a different boundary condition is something like, I don't know, but here. And for, if for, Github-Lang equation for the noise boundary condition, this is the OD, this is the dynamic law, for Github-GP equation, this is dynamic law. I think my time is almost gone, maybe I don't know, I just show some results. So you can see this is what is paired for, for, for, for with data validation. The left one is for Github-Lang, the gross potential dynamics, so the right one is for Github-Lang. So this is the gross potential dynamics, the right one is for dissipating. You can see how they go. You can see for the gross potential equation, the Roto-Rouichi order, for Github-Lang equation, they move repossibly. This is for what is type, what is type, what is error. So here is for what is dipole, so then what is our antipodacity. So one is when the number one minus one. Again, the left one side is for, for, for Github-Lang type, for gross potential type equation, right side for Github-Lang type equation, you can see how to say, you can see for, so gross potential equation, initially they move parallel, they put a multiballary, they move. For Github-Lang equation, they move very quickly to each other. And okay, I don't know if this is a skip. And I also show some, like here I show some, what is the pair for gross potential equation with dipole notation, but with dipole notation. So you can see this is for no initial, so called the impending, here is the impending, like a h equal to x plus y. So this impending will affect the dynamics. And I'm just going to skip this one. Okay. So, and also maybe, okay, I'll skip this one. I just show you, so for what is, for three, if for, for starting a type of equation, if for example, you have three, what is it, initially you can see this one is, what is it, with three random numbers, and this one with two plus one minus one center, you can see how do they move. For Bonnet domain, you can see, so for the left hand side, the same random number, they rotate clockwise, and up some time, due to Bonnet effect, they, you know, they move very into dynamics, and the left hand side, you know, again, they rotate clockwise, but up some, they move. So that is never much. I think I'll finish this one with the last, so this one is also show, so because the, a shooting type dynamic, a GP dynamic, shooting that dynamics, this is what dynamics, and a rectangle domain with pure foundation, because initially I have three, what is dipole? So plus minus, plus minus, plus minus, the, the bottom one is the density. So then the blue is zero, the top one is the face, you can see, the each has a, what, when a number, so I have three pair. So I have three dipole, you can see, when time evolve, they show some kind of lip frog effects. So you can see, the one with the largest moves smaller, you can see, this is just like a lip frog, the different pair, they move like a lip frog, I show you again, you can see, they just show like a lip frog. Like this is also observed in three dimensions for what is written in the dispersive dynamics of what is written, they show this also, here, we can show numerically here, for unit 2D. Okay, I think I finished here, and my time is finished, I think I stop here. Do you have any simulation for n quite large? We do like an n equal 50 something, but the dipoles are epsilon, if you have them smaller, your n cannot be too large. Okay, sure. Because you have two scales, one is epsilon, the other is n, the other is the domain size. Basically, my question is how different is compared to, let's say, simulation of fluid dynamics. My size should be at the outer of the epsilon, the cos size. So my point should be the so-called the size of the domain over epsilon to power dimension. If 2D, basically, dimension size over epsilon to square, this number of points is the minimum. And my time step should be taken as like this epsilon square or something like that. I think that barely there that we should leave to solve this.
 Okay, thank you for the introduction and thanks to the organizers for putting everything together.  Wonderful opportunity for everyone to meet, to meet each other during this tough time.  So this is John with Changdu and I think, okay, and Yan Zhizhang and Chen Lintang of  Sichuan University and some current PhD students, doing this one.  Brief outline.  I will brief motivate you what is the quantized what is in super-calendar fluidity.  The mass mass of models, there was introduced a center water state like a solid ton in exchange  of solid in 2D.  And there there's a cal size.  If cal size go to zero, so there's many water states, then there's a reduced dynamic law  kind of set of OD system, which is governing the water center dynamics.  So that kind of let what you that for like a particle model for the water center.  Well business with dynamic law for get the lambda equation, we can talk a few sense there.  Of course, I will talk about some this kind of sense for shooting an equation with 90  as well as a bounded domain something like that.  We know a quantity of what is kind of a particle or topical defect.  So they are so you are other product complex other parameter or counter skill field.  So side.  So at the water center, the other parameter has to be zero.  So in your fluid language at the water center is a vacuum.  And near the water center, so it was x zero, water center near the water center side can  be locally written as square root of rho, it's when I fee.  So rho is a density, fee is a face.  So if any gamma is a closed curve around x zero, the face, the integral along gamma,  your face has a two pi n, where n is called as circulation, index or winding number.  So this kind of quantum what it says is kind of key signature to support so fluidity.  So suppose they support the dissipation is flow.  So it's going to flow when they flow, there are no friction.  So this is kind of a cartoon to show like a quantum what it says in the three, you know,  there temperature is like a macro Kelvin some, some of that this is cartoon.  I've got here this is real experimental result like a quantum what is in type two, so conductivity  conductivity, so conductive conductor.  So each what thought is a quantum what is in what it says.  So here also are experiment result for the kind of in quantum versus in both transition  you there because the top one is so many particles in the harmonic trap, you know, they're here  and there's no rotations of ground like a bump, they're Gaussian type.  Well, if this system rotation frame, you can see that little holes, each hole is a quantum  what it says.  So basically the quantum what it says exists in the superconductor model by quantum lambda  equation, degree of chemistry, they could by two for the model couple the system equation  like you have a VN and the row and the velocity density for a number of component and row  as VS for superfluid for superfluid component.  So there's a couple together.  So they're also using the sort of if you just got a gross place equation to model as a phylogic  model to model this one.  Also the both kind of session is kind of not a shredding equation or gross place equation  to model you also in nonlinear optics or propagation of these are being there you're not in a shredding  equation or nonlinear wave equation, even nonlinear categorical equation to model this  kind of problem.  So here is some kind of typical machine model, you know, like here psi is the other parameter  complex other parameter or will function.  Like you get the model equation for superconductivity, nonlinear heat flow, complex flow, you know,  here psi t is time, psi will function the other parameter.  You have to say I intentionally write all the equations, right hand is the same.  Right hand is the Laplacian psi plus one of f square, y minus f square psi could be  on whole space in two dimension or a boundary domain omega with different boundary conditions.  So this is a general equation of complex general equation or nonlinear shredding equation or  nonlinear wave type equation or even nonlinear kind of equation.  So here psi is a complex value to function or other parameter.  I have a parameter epsilon is the definition of parameter which is sort of cos size of the  problem.  You can see that's the problem.  The nonlinear part is a key, epsilon small.  So basically in space, the typical scale is other epsilon.  And in time, for the equation or shredding equation, the typical wavelength is other  epsilon square.  Of course, in wave type equation, the typical wavelength is other epsilon.  Okay, of course, if you introduce so free energy, you have functional, you can see since  there's four equations, the right hand side is same.  So if you introduce the free energy or your functional, you can see the shredding equation  is a dispersive system.  So that the energy or the density of the modified density, they are all conserved.  So this kind of dispersive system, they automate so that the solutions, particular like solutions,  like any 1D, like a solution in 2D, canx or what is this, what I show you here.  If you can find out equation, this is dissipative system, they're added to dimension system.  So the single or central what is this we call red tail.  So if I consider this, because all we get is whole space in R2, you can see, take an  ansatz, psi equal, psi of x equal phi n of x equal fn of r, kind of this is i n theta.  Here r is the polyconnates, n is the winding number.  So I plug this ansatz into a system, I end up with this fn for this kind of two-point  value problem and f r equal 0 is 0, r equal 1.  So this is, so top here is fn r for different n, you can see when, so r equal 0 is 0, r  equal 0 is 1.  And formally, so the middle part is like the absolute value of phi n, you can see the inside  that the origin is 0 of the density and 1 outside.  So for fn r, asymptotically you can see where r between 0 and other epsilon, which is fn  r is constant times r over epsilon to power epsilon by the n.  Well if r much bigger than epsilon is 1 minus n square over times epsilon square over 2r  square, basically near the core the fn is like a growth in term of r polynomially.  And at far field the leading order is 1, next order the fn r decay quadratically in term  of 1 over r square.  So this is the phase, you can see when n equal 1, it's only when you go one round, two-point  jump.  So this next one is n equal 5, if you go one round, let phi, see that going one round,  there are four, five times two-point jump.  So of course, purely interested, all this kind of particle type, particle like a solution  are the dynamics stable.  Of course, if you set all the parameters, rho is density, v grad s is called velocity  and g is current.  So of course, there are some analysis for Kinsbler-Landau type, but for the Schrodinger  type always some kind of assumption.  So of course, when I study in America, I do the time, I don't want to show you this movie,  I just give you the result.  So basically, if I just have one, this is what I state, I do some perturbation for Kinsbler-Landau  type, I can do perturb at this data.  If you do little perturbation, where a equal plus minus 1, that dynamically stable.  So if you initially have one, one, one, one, one in number, n equal to one minus one,  that after some, some perturbation in this data, then then, a long time later, the steel  one in number one.  Well, if n bigger than one, you know, the dynamic answer, basically, the pattern is  the split to n, what is it?  When t much bigger than, much bigger than one.  This is for this, this is for Schrodinger-Wilth type equation.  However, for Schrodinger-Schrodinger equation or DPE, so this time, if you perturb an initial  data, that dynamically stable for n and n, because the algorithmic citation is conserved.  However, if you perturb the, it's not potential, for example, then n equal to one minus one,  the dynamical stable, if n bigger than one, after n bigger than one, dynamically unstable,  something like that.  Okay, of course, since n equal to minus one is dynamically unstable, you are interested  in studying what is the interaction.  Then initially, I take my size zero of x equal a product of j from one to n, m phi of mj  x minus xj zero.  So xj zero basically is the location of the jth vertices, and mj equal to plus minus one.  And phi m of x is the station of what is state.  And also I can multiply or fix the independent initial data.  So basically initially, I have capital N vertices, and suppose they are well separated, then  it's basically xj zero and xm zero, the distance is much bigger than epsilon.  And also, yeah, okay, this one.  And then you just suppose, okay, suppose, so suppose, then you plug this, suppose solution  size of xt could be something like a product of size j equal to one to n phi mj x plus xjt,  where xjt is the water center of the jth vertices at time t.  So you plug, plus a order term.  So you plug this and that into the energy, you do some analysis, you will find that as  not as you know, the distance, initial distance, they are much much bigger than epsilon, or  the distance equal to infinity.  In other words, so because the distance is much much bigger than epsilon, the left epsilon  is zero.  Or the initial distance is fixed, that's epsilon zero.  And xjt will follow ODE and then later can prove rigorously this, this, you just dynamic  law.  So, so you can see this here, give you.  Let me capture x is x1, x2 to xn, which is the set of points in R to N, which xj of t  is water center at time t.  So then I have introduced a functional table of x, which is like this one here.  So remember this here, mj mk could be one, could be a minus one.  So for Kinsman-Landau equation, kind of dissipated PDE, my so-called my capital Xj will follow  this second order, that is the second order ODE system, nonlinear.  So basically, so Vj is x dot, sorry, sorry, sorry, following the first ROD.  So Vj is xj dot.  So the velocity of the jth particle will follow this one.  Okay.  And for Schrodinger equation, this, this business from joining you, of many people are countries  since this included Wynne, and the Fang-Hua-Lin and his group, and also for Schrodinger type  equation, you have a j is a rotation, 0 minus 1, 1, 0, j times xj dot Vj t of Vj equals  j dot.  Follow this one, same.  And for another equation, which is a second order ODE, like xj double dot, this one.  So here, again, here, this one is almost the same, but the left side here is Vj here,  but the rotation matches capital J, and for Schrodinger equation, this xj second order  ODE system, this one is similar as many of you have studied this, for this correct dynamics  for some set.  But remember here, mj and all can be taken in here, plus 1 and minus 1.  Of course, they're like how to, they can study from PDE to ODE system, like for another  equation, they show for African ODE, you can start to study the qualitatively, qualitatively  behavior.  For example, if only have a pair of vertices, then capital N equal 2, and m1 equal m2 equal  1, the vertices will undergo the repulsive interaction.  If the vertex dipole, the n equal 2, there is plus, that one minus, and go attract the  repulsive interaction.  And for example, the pinning effect of the other, so from this ODE system tell us.  So recently we started this ODE system more intensively, what we can see more from this  ODE system.  Then for Schrodinger equation, what is it, we have the pair of vertices, like at the ideal  field, and the PDE, like a higher order, like a radiation, something like that.  Of course, for this ODE system, for this ODE system from Gizmo-Langda type equation, of  course, if you have a plus minus, the finite time, the two-quartets center will go together,  that's like a blow up or some kind of a merging of this one.  Well, for Schrodinger equation, you can see at least for n equal 2, 3, or 4, one can show  that no matter you initially have all the same sun, all you have different winding  number, like some plus, minus what is that, what is this?  You can always show that this ODE system is global well-posed.  So from our democracy initiatives, this ODE system has not initially, what is location,  they are distinct.  The system, they are never two, what is it, go together.  Okay, so now we try to, from this ODE system, what can we see more.  So first we consider this ODE system from the Riddio-Tangai law from Gizmo-Langda equation,  so this is the ODE system.  So of course here, xj dot, because this one.  Again, I want to remind you that the mgml could be one, could be minus one.  So at the front, so the mass center, so this x bar t is the average of all the location.  You can show that the x bar t is the conserved quantity.  And also the energy, the w, the w of the x, this is decreasing.  Of course there are some, like that, problems for example, if you initial, for all initial  data, you shift one point, the solution shifts one point.  And also to a scaling, you can do also scaling for the solution, also do a rotation, so  initial data with one, like a q-theta is a rotation matrices, the solution also rotation  this one.  And also for some, if I have n vertices, if initially they span the uniform on a circle  with radius a, you can solve this ODE system analytically.  For example, if a equal to, a equal to three, a equal to four, for n and n, basically they  are, they are, they are moved outwards along the line, connected initial location and the  origin.  And if n could equal to three, and if I, in fact, I have one, what is it at the origin,  the rest n minus one, they are uniformly along the circle.  Again the, the single one doesn't move and the rest n minus one, they move outwards.  This, this is easy to understand, just plug the, use the symmetry.  If initial have two, what opposite what is, what is dipole?  Then you can use to show that here plus minus initially, this from ODE.  Of PDE, the, the quality is the same.  And of course you can solve analytically, you can see the solution when basically at  a t equal a square over two, the two will be moved at the origin.  Next case is, if n vertices n bigger than three, but opposite, I have one origin, I  have minus one, one in what is this, and the rest n minus one with plus one, what is this?  Then you can solve, again you solve, origin one stays there, but the rest will move into  here, you can see my a equal four is a, so-called as a critical number.  Because if a equal three, they still move, the two plus go to the motor center, move  with the n minus one, the a equal minus one, and after more, there are only one plus left  over.  But a equal three, which, sorry, a equal four, which is a, the, the, the, the, the, the,  the, the equilibrium, but this is unstable.  The a equal five and, and the upper and, and, and, and the more, then they move outwards.  Because the a equal four, they don't, they don't, there's this equilibrium, and less  than four, they merge, and equal, and critical, critical five never merge.  Then also we can start using the school that, not autonomous first integrate.  So of course here we, suppose initial data, we have a capital N vertices vertices, so from  an OD system, I suppose among them, I suppose I have N plus, number of plus one vertices,  and N minus negative, minus one vertices.  Of course the sum together equal N, and I didn't have M zero for this one.  That defines this H one, X t, H two, X t, and H three, X t, this one.  They need to show that if X t, X t, X t, X t for this ODE, then H one, H two, and H three,  they are three, nine autonomous first integral of this OD system.  So with this one, we can show that if I have N vertices, they have same one-day number,  then there are no finite time collision, then there's a, that globe exist in the OD system.  And the, and the addition, so at least the two vertices will move together at the particular  time, in fact, one can do us, we can prove us stronger, so result at the, for any bounded  domain, when t much, much bigger than one, there are the most one vertices in this bounded  domain, when t very large.  And also we can show that the so-called, if I define the distance between two, any two  vertices, d t l equal s j minus x l, the absolute value.  If N greater than equal to, less equal to four, and the n vertices have same one-day  number, then we can show that this is the minimum distance, and the, between any two,  and the, the, yeah, these two are, they are monotonic in collision.  However, if N bigger than greater than five, this no longer, there are some cases that  are no longer true.  So this d minimum is, I think, is the minimum distance between the, between the, any two  vertices, and the capital D minimum is the, the, the, the distance square.  And okay, we also have some cases, if n equals three, we can see more, long times of behavior.  If n equals three, and the same one-day number, if initially the three vertices are sort of  located along a line, then when t equals infinity, we can see that two vertices will go to infinity.  The, the, the, the, the, the one in between will move the, to the, the center of the initial,  of the three vertices as the interlocation.  And if they are initially not, not, not collinear case, then the three vertices are on any triangle.  They are at t, at t large enough, then so-called the three vertices will, will, will approach  to a red triangle when t large.  Initially any, the, the three vertices is, look at any, any triangle.  Then when t large, the three vertices will, will, will approach to a, of course, a red  triangle, of course, the, the, the, the, the, the radius of the, the distance to the origin  is become, go to infinity, something, something like that.  And then one can have similar, have this kind of, kind of, so for n equals three, we have  something like a similar, like a orbital stability of a self-similar solution for number.  Initially we can define some so-called, so-called, so, so, so, so this is something like an  x-tool of t is so-called with, you can say, the, the, the, the distance, the radius, the,  the distance to origin is like a grow, like at t to power one half.  The x-tool of zero is some, initially is a red triangle.  Therefore any, any data, so, so, so, so this, this one will, so, so, so, so, so, so, so,  the distance of x-t minus some, some, some, a, a, a center value they will go to, to,  to the, the, to this, this solution up to a rotation.  Some of that.  So we have this, this kind of results.  So similar of this, this, this, this, this for, so, for, everybody's always the same  one in number.  For, for, for, for any, we have some result for n equals three, you have some long-time  behavior.  Of course, if the, the opposite one in number, that initially have some plus, some minus,  that, that, that, we can show that the, so, so, the, the, n plus, minus, n minus less  than n.  So we, remember what we defined as m zero, which is m zero is here.  So m zero equal n plus, minus, n zero square minus n to power divided by two.  So if m zero less than zero, you're always the, the, the, the, the, you can always show  that when tick approach team acts, then, so at least two, they are, they go to a same  location.  So if m zero equal zero, the solution of, of solution is bounded.  There's also sure we, we, we go to the, the, the, the, so often m zero equals zero, they're  very super set up.  And if m zero bigger than zero, there's no fine time collision, then the team, that  group existence of solution, then we have also some, some other, this I can skip here  for, for, for, for, for simplicity.  Again, also for, we also give some, next condition for collision.  Let me see.  If, if there's, if for inverses and for, so, if they form a collision cluster, then the  m zero less than zero and h zero, h one zero equal n h two zero and h three zero equal  n minus two h two zero.  So then the three, so the autonomous, autonomous, so first integral have a first policy relation.  And we also give a second condition for equilibrium.  Then if the, if the, if the, if a, if form a clear equilibrium, then my n plus must be  equal one over n plus minus square root of n.  This one less than n.  Some of that, some of that.  So of course, if n equal four, so you say if n equal four, n plus equal three, n minus  equal one, this service condition.  Then we also have some collision pattern if n equals three.  So here I have, suppose I have m one equal m three equal one and m two equal minus one.  So we should, if initially, if initially I have three vertices, if the, if the, if the  initially they are, so called the, see a red triangle, sorry, a triangle with two, two  segments with same length.  Then, so they will, so the, so the three vertices will, will merge, will collide at the, so  called the, at the x zero, at the center of the triangle, something like that.  And the, the collision, collision time is equal, h one zero over twelve.  Of course, if initially they are, so called the, so called the triangle, one is long,  one, one is large, the other is smaller, there's a two one with, with, with short distance,  they will, they will merge to each other and some of that.  So, so, so that's why.  And then we also have some for, for Schrodinger type of equation.  So this is the OD system.  So remember here I have a G, G equals the rotation matrix here.  Then we define a sun, the mass center x two dot t equal one over n m g times x, x g t.  So this one, remember, of course, if m g is the same one number, so this one is the sun  mass center as same as the previous one.  But of course, if m g has plus minus sun, this is different.  And for Schrodinger type equation, this sun mass center is conserved.  And also the energy is also the capital W two dot equals this one, this one is also conserved  from this OD system.  And again, we can show that if initial of n vertices, the, the initial, the initial  d-baron circle and the, they will rotate each other because this is the, the, the, the,  the circle, but they are counterclockwise with the frequency n minus one over a square.  And if I have capital N, grid equals three, initial one at an origin, the n minus one  unit for the circle.  Again the order of the center will stay there to symmetry.  The rest m minus one will rotate counterclockwise with m minus two over a square, this one,  frequency.  And if initial of two vertices, you can use to solve, they are moved parallel.  So this is a solution.  And you can solve, you can use to show this solution set as the OD system.  And again, initial, initial I have n grid equals three, what are the opposite of what  it says?  One at this origin, which is a minus one, one at the number.  And the rest m minus one at the unit for the order of the circle with plus one, one at  the number.  So you just again, origin one is, stays there to the symmetry.  However, because not again, n equal four is critical.  And you call four, n equal four is a critical equilibrium.  And n equal three, they rotate clockwise.  And any critical five, they rotate counterclockwise.  Again, n equal four is some kind of critical number.  So here of course, so of course, so here, of course, for this, for this equation, you  can use it for n very large to our understanding, there are no collision, but I cannot prove  it because n equal four, we can do it.  And you figure the four, of course, some is for this OD system, if initially you have  n vertices, they are distinct, then I did too, they are not the same.  So since they are, they stay on each other globally.  No, no, whether, of course, if the same one number is approved, if the opposite one number  seems we cannot do it at the moment.  And also if I'm here, if you initially they are, if I have four, two are closed, the other  two are closed, but this two clusters, then short time, the two clusters will move like  a leading order, they are moving like a similarly.  So here, similarly, if you have n equal two, if two, one number is the same, what is pair,  they are loaded at each other, if what is dipole, they move parallel to infinity.  Of course, if you are, so this is from OD, you tell us, for PDE, then of course, for  Schrodinger, another type of equation, you can solve PDE, then basically the qualitatively  the dynamic from PDE is similar as OD.  However, for Schrodinger type equation, when the cos size, the distance, if it's distance  is less than some cos times cos size, so if you have one number, the two one number, they  are opposite.  What is dipole, you can merge to each other.  So OD tell us they go parallel, but PDE, they tell them more.  If the cos distance between what is, what is it, is smaller or at the order of cos size,  the OD basically gives the wrong dynamics qualitatively.  Then people also study on Bonnet domain.  But on Bonnet domain, Schrodinger equation is the skin is same, but if for principle  unknown equation, you need to multiply the riskier, look at the longer term, you need  to multiply PDE at lambda epsilon, lambda over log epsilon something like that, so that  you see dynamics.  So here, so if I give it initial data, of course, you have a different, but kind of  like a typical Neumann or here there are three scaling.  One is the size of domain of omega, the other is the smallest distance between two and two  vertices.  And the third one is the one in the cos size of the vertices.  So of course, for fixed n number vertices, if epsilon is zero, there are a lot of results.  Of course, if you fix like a left epsilon zero and also n equal to infinite, you know,  the sub-It have wonderful results, five years ago on this, for Schrodinger, for this study.  So of course, for fixed n, there are lots of results.  Let's add epsilon equal to zero, so the host was a limit.  Of course, if you let the epsilon equal to zero and n equal to infinite, they are very  challenging to study.  So for bounded domain, the OD system, like of course, you can say you do need the host  space OD system plus, okay, here I have an R, so due to boundary effects, the R is something  like a, so like a laplace X, the capital X is the location of the vertices and the X is  a domain.  So basically, like a equation with the particular or particular function condition, this is  the OD system.  Basically, it's kind of like the OD with a non-local R, which depends on the location  of the OD, of the, what is location?  For Github-Lang, for Github-Lang, for gross potential equation, this is some OD.  So basically, the capital R for a different boundary condition is something like, I don't  know, but here.  And for, if for, Github-Lang equation for the noise boundary condition, this is the OD,  this is the dynamic law, for Github-GP equation, this is dynamic law.  I think my time is almost gone, maybe I don't know, I just show some results.  So you can see this is what is paired for, for, for, for with data validation.  The left one is for Github-Lang, the gross potential dynamics, so the right one is for  Github-Lang.  So this is the gross potential dynamics, the right one is for dissipating.  You can see how they go.  You can see for the gross potential equation, the Roto-Rouichi order, for Github-Lang equation,  they move repossibly.  This is for what is type, what is type, what is error.  So here is for what is dipole, so then what is our antipodacity.  So one is when the number one minus one.  Again, the left one side is for, for, for Github-Lang type, for gross potential type  equation, right side for Github-Lang type equation, you can see how to say, you can see  for, so gross potential equation, initially they move parallel, they put a multiballary,  they move.  For Github-Lang equation, they move very quickly to each other.  And okay, I don't know if this is a skip.  And I also show some, like here I show some, what is the pair for gross potential equation  with dipole notation, but with dipole notation.  So you can see this is for no initial, so called the impending, here is the impending,  like a h equal to x plus y.  So this impending will affect the dynamics.  And I'm just going to skip this one.  Okay.  So, and also maybe, okay, I'll skip this one.  I just show you, so for what is, for three, if for, for starting a type of equation, if  for example, you have three, what is it, initially you can see this one is, what is it, with  three random numbers, and this one with two plus one minus one center, you can see how  do they move.  For Bonnet domain, you can see, so for the left hand side, the same random number, they  rotate clockwise, and up some time, due to Bonnet effect, they, you know, they move very  into dynamics, and the left hand side, you know, again, they rotate clockwise, but up  some, they move.  So that is never much.  I think I'll finish this one with the last, so this one is also show, so because the,  a shooting type dynamic, a GP dynamic, shooting that dynamics, this is what dynamics, and  a rectangle domain with pure foundation, because initially I have three, what is dipole?  So plus minus, plus minus, plus minus, the, the bottom one is the density.  So then the blue is zero, the top one is the face, you can see, the each has a, what, when  a number, so I have three pair.  So I have three dipole, you can see, when time evolve, they show some kind of lip frog  effects.  So you can see, the one with the largest moves smaller, you can see, this is just like a  lip frog, the different pair, they move like a lip frog, I show you again, you can see,  they just show like a lip frog.  Like this is also observed in three dimensions for what is written in the dispersive dynamics  of what is written, they show this also, here, we can show numerically here, for unit  2D.  Okay, I think I finished here, and my time is finished, I think I stop here.  Do you have any simulation for n quite large?  We do like an n equal 50 something, but the dipoles are epsilon, if you have them smaller,  your n cannot be too large.  Okay, sure.  Because you have two scales, one is epsilon, the other is n, the other is the domain size.  Basically, my question is how different is compared to, let's say, simulation of fluid  dynamics.  My size should be at the outer of the epsilon, the cos size.  So my point should be the so-called the size of the domain over epsilon to power dimension.  If 2D, basically, dimension size over epsilon to square, this number of points is the minimum.  And my time step should be taken as like this epsilon square or something like that.  I think that barely there that we should leave to solve this.
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