 First of all, I'd like to thank organizers for inviting me to this nice conference and give me an occasion to stay here for two months. Today I'm going to talk about Kp2 equation, which is actually this equation. This equation is very close to the Kdv equation. Here we have x derivative of Kdv equation and some other terms here, which takes into the slow variation of the motion of waves in the transversal direction. And here, sigma is a parameter which is related to the surface tension. If sigma is a positive number, it means the surface tension is weak. And we call it Kp2 equation. And that's what I'm going to talk today. And if sigma equals minus 1, it is called Kp1 equation. And in that case, surface tension is assumed to be strong. And let phi sub c be this function, written in this way. Then phi sub c of x minus 2 times c times t is actually one solution of this Kdv equation. And Kp equation is the model to explain transversal stability or instability of Kdv in soliton in the 2D setting. And for that purpose, equation was derived by Kadmosev and Petrovsky-Shivri in 1970. And they also did some formal analysis of stability of line solitons there. And actually, both Kdv and Kp equations are known to be integrable system. And this is a photograph I followed from the website of Professor Alphavitz, which is a picture of the ocean. And here, there are many waves which are uniform in transversal direction of the motion. And they are crossing. And actually, Kp equation I also used to describe such the intersection of line soliton waves. But today, I'm only talk about stability of one line soliton. So in my talk, there does not appear any intersection of solitons. And first, I'd like to introduce conservation laws of Kp2 equation. And in the conservation law, I use this no-station, which is defined by Fourier transform eta divided by c times u hat, the Fourier transform u. And inverse Fourier transform of this quantity is defined in this way. And apparently, as Kdv equation, autumn norm is conserved for localized solution. And moreover, it has a Hamiltonian, which is written in this form. And actually, the Kp equation I wrote in the previous slide is formally equivalent to the fresh derivative of this Hamiltonian multiplied by this q-adjoint operator d dx. And formally, it's easy to see that this quantity is preserved and this quantity is also preserved. And for Kp1 equation, actually, when sigma equals minus 1, the first two terms has the same time sign. And as a result, there are paper. In the next slide, I explained that there is a grand state, which was proved by Anduin and so. But for Kp2 equations, the first two terms has opposite. Although this term is 0th order, actually, in terms of scaling, y-derivative is equivalent to twice of x-derivative. So this term has the same scaling order with this one. So these two terms have the principal term of this Hamiltonian. And for Kp2 equation, both those two terms have opposite sign. For this reason, this Hamiltonian is not so useful to estimate time global quantities. On the other hand, this one is quite useful. And also for line soliton, it is quite useful. And actually, along line soliton, this quantity is almost preserved. And we can prove it if we prove that the modulation parameters, which I will introduce later, has a good decay estimate. And before I introduce non-results on Kp equation, I would like to introduce some famous results on stability of Kdv1 solitons. Actually, there are not many, but I think these three results are quite famous. And the first one is the definite stability of one soliton in the energy class. And the argument uses both two conservation law and also the Hamiltonian for the Kdv equation. Unfortunately, such an argument doesn't work because of the bad nature of Hamiltonian for Kp2 equation. So unfortunately, this argument doesn't work. The second one is to prove asymptotic stability of one soliton of Kdv equations in exponential white space. What Pegwan once did is to use some set of group of estimates in exponential white space. Usually in L2 space, because Kdv is a Hamiltonian system, we cannot expect decay estimate in L2 space. But in weighted L2 space, the weight function, if the weight function reflects the wave, the motion of waves, we can have some kind of decay estimate. I'm sorry. And later, by Martel and Mel, they proved asymptotic stability of Kdv soliton in H1. And the key in their paper was the virial identity, as well as the UVO theorem. In this talk, I don't use UVO theorem because I don't do classification argument. But the virial identity, the usage of virial identity by them was quite useful in the argument of Kp2 either. And for L2 stability of one soliton, it was proved by Mel and Vega in 2003. And what they did is basically, by using the mirror transformation, which is sort of backland transformation, they connect soliton solutions with kink solutions of de-focusing modified Kdv equations. And by using another mirror transformation, they map kink solutions or modified Kdv to null solutions. And actually, they gave a local isomorphism between these typical solutions. So as a result, it gives a local isomorphism in L2 sense between the neighborhood of 0 and the neighborhood of one soliton. So this is the idea. And this argument cannot be applied directly to the stability of line solitons in L2 R2. But the idea is still useful to investigate linear stability of line solitons. So the next rise is known results for Kp equations. And the linear stability of Kp2 equation, of course, there is a formal argument by Kdv and Kdv. But somehow, mathematical argument was done by Vazir first. What he did is by using inverse catering theory, he finds some resonant continuous eigenmodes, which is related to dynamics of moderating line solitons. Later, Alexander and Pego sucks proved that in L2 space, if the linear rise of later of Kp2 equations along line solitons do not have any eigenvalues or continuous spectrum, other than on imaginary axis. So they proved neutral stability. And such a result was also proved by Marianne Hargass in 2010 on this setting. So they used periodicity in X variable, and they considered Y variable in the whole line. And also there is a result by Avrilov and Avrovic who constructed a formula on solutions along line solitons by using inverse catering theory. And what I did with Zvezkov in 2012 is to use mirror transformation, as well as for the Kp2 equation. Idea is almost the same with Merlin and Becker. And in this case, the methods of Merlin and Becker worked very well because there is no resonance in the spectrum. And actually, in this case, the mirror transformation gives local isomorphism in L2 sense. And there is also several results on Kp1 equation. But since there are so many results, I will only introduce some of them. So the first one is done by Zaharov in 1975. What they did is to find an unstable eigenmorph for linearized operator, and he proved linear instability. And later, the Russians, Zvezkov, investigate the instability of planar solitons for Hamiltonian system in quite a systematic way. And they proved also non-linear instability. And why? So for Kp2 equation, it is expected that line soliton is stable. On the other hand, line soliton is expected to be unstable in L2 for Kp1 equation. Why is the situation so different? Because I think it's related to the difference of the sign of the Hamiltonian. Here we have different sign of the Hamiltonian, and it lays a lot. And actually, for Kp1 equation, it has a stable ground state. It was proved by Duvall and So, and as for the well-portedness, the first result on well-portedness was obtained by Uka in 1989. And later, by Brugan, using Fourier restriction theorem by himself, he proved well-portedness of Kp2 equation in this setting. And later, Morine, So, Takao, Zvezkov, and I forget to like many other people, but they improved Brugan's result in many boundaries. They proved similar results like this in different boundary conditions. And what I used there is a well-portedness result along line solitons. And it was proved by Morine and Sovans-Bezkov in this year. And what they prove is that if you add some L2 partial elevation to this line soliton at t equals 0, then the difference between solution and line soliton stay in L2 space. And this quantity is continuous in L2 norm with respect to time. And my talk is also related to normal results on heat equations. So I want to introduce some of them. And the typical one is a stability of one kink solution of this heat equation in Rn. If n equals 1, it has a kink solution, which moves with a constant speed. And actually, phi prime is an, since this equation is translation invariant, it has a traveling wave, which doesn't, and its special derivative belongs to the kernel or the linearized operator. So if you write the spectrum of the linearized operator for n equals 1 case, then the spectrum is like this. So there is a nice rated eigenvalue 0, 0. And all the other spectrum rotates in stable half-frame. And there is a spectral gap. But if you think of the transverse stability of this traveling wave, so it means that its uniform, we extend this kink solution of one dimensional, one dimensional kink solution to uniformly to the transverse direction and think of stability. And then what happens is that if you think of the spectrum, in this case, because thanks to the transverse direction, there appears some extra spectrum, which converges to 0 here. So here is continuous spectrum. And 0 is not a nice rated eigenvalue anymore. Of course, there should be some stable spectrum here, either. But anyway, if you think of the transverse of stability of kink solution for this equation, there appears some continuous spectrum like this. And if you want to investigate the dynamics of solution, we make use of these answers and substitute this formula into this equation. Then we get a solution of heat. Then we find that gamma satisfies a heat equation of n minus 1 dimensional. So the original problem was in n dimensional. And the motion of this one is described by n minus 1 dimensional heat equation. And it decays in this rate. And also, there is a rated results in Hamiltonian system, which was obtained by Kucanian. What he did is think of this wave equation and think of transverse of stability of one dimensional kink. Again, this translation value, this function, is described of PDE, which is one dimensional smaller, which locates in two dimensional in this case. Because the original problem was three dimensional. And this one satisfies a 2D wave equation. And because this is a solution of 2D wave, it is expected to converge. It gamma tends to 0 as t equals to infinity. Now, I want to return to KB2 equation. To analyze KB2 equation, I make use of answers of this form. Here, C is basically the amplitude of line soliton. But it depends on not only on time t, but it also depends on transverse of direction y. And also, the phase parameter here depend not only on t, but also on transverse of direction. So if you think the closed section view of line solitons using this formula, it means that the center of the cross section view is here, x of ty. And c of ty means the local amplitude of line solitons. And if you observe the crest of line solitons from the line soliton, then because x of ty depends also on transverse of parameter y, it is not necessarily a straight line. But it's close to a straight line. So if you think of the y derivative of x, this basically means this angle. And now, I want to introduce my result. First, I say u starts with a line soliton plus some small part operation, which is written as an x derivative of L2 function and whose norm are small in this sense. So each one norm is small. And also, the minus half derivative of v0 locates in L2. And this autonome is small. And with this assumption, there exists some modulating parameter c and x, such that u of tx, y is always close to this modulating line soliton all the time. And the difference of autonome between these two is over of autonome of v0. And also, if you think of the difference between this modulation parameter c, ty and the original wave, original amplitude of line soliton, its difference in hk norm in y is also of this order for all the time. And if you think of the angle of line soliton, it's x sub y. And if we square this quantity and integral it on the entire domain, we also have this quantity. So the x sub y norm, the hk norm of actually L2 is enough. But L2 norm of x sub y is this order for all the time. And this just says that x sub t is almost close to 2 times c. So basically, x sub t means the speed of the motion of line soliton. So it has this function. And also, if we think of y derivative of this one, or x sub yy of the norm of x sub yy, it tends to 0 as t goes to infinity. So this means that so combining these two results, we can say that in an infinity sense, this one converges to original speed as t goes to infinity. And also, we have some local convergence like this one. If we choose a collect phase shift here, then the difference between these two converges to 0, at least locally in space. So this just means that the last one means that this is x direction. And this is a line soliton. And it's moving. And if you see things in the moving coordinate, so this is x ty. This line is x ty minus r. So in this local frame here, we have asymptotic stability of lines solitons. And the first assumption I made here is almost the same with this assumption. That is the integral of v0 along the axis, along the line, which is parallel to the y-axis, is equal to 0 for all y. This is assumption I made in this theorem. And actually, we can replace this assumption by this assumption. So if the perturbation v0 decays polynomially in space, it's enough. So this is what I like. If instead of assuming some smallness of this norm, we can assume smallness of this norm. And if we have smallness of this norm, we have the same sort of result. And this is a remark, which I explained already. So I skip. It says that basically my theorem says that if you perturb line solitons, it converges locally to the original line soliton as t goes to infinity with some phase shift, which doesn't depend on y. But if we see things, here we limit ourselves to this domain. So this is on the at least for y, this is for compact set. And if we see things in entire line along y-axis, things will be different. This is another my result. And to obtain this result, I assume that perturbation is exponentially localized. And if perturbation is exponentially localized, and also it is L1 in some direction, the perturbation is in L1, what we have is that the difference between the modulating speed of line soliton and the original speed, or the angle of the crest of the line soliton, is a linear combination of the self-similar solution of the Vargas equation at the first order. This is the result. And here this one means, I mean these two notations by this function. And actually, this one is a self-similar solution of this Vargas equation. So if you think of dynamics of modulating line solitons, its dynamics is described by Vargas equation. That's what I want to say. And actually, this theorem can be proved by using theory of Kawashima on Vargas type equations. And this kind of observation was not for myself, but actually there was a work by physicist Fedasen who showed that similar result can be obtained for business systems. And since business systems and the KP2 equation are quite similar, I think that this type of result can be obtained for many other business equations in at least if it's considered, at least they describe the case where the surface tension of the water wave is small. And this is a heuristic picture of what I wrote in the previous slide. So if I integrate this formula in Y once, what we have is heuristically like this. At least around the center of the Y cross, around the compact region, in the compact region of Y cross 0, X of ty becomes flatter and flatter as t equals to infinity. But I don't have any estimates actually around here because this region is extending in linear order in t. But what the previous theorem says that this part can be written by an integral of the self and it has similar solutions of Vargas equation. So as a result, there should appear some gap here. So there should be a small difference of height here and here. And if we estimate the ultromb of difference of modulating line soliton and original line soliton here, then we should have some sort of instability in it too. So that's what we have here. If you just think of all line soliton, the set of all line soliton solution, then there should exist a solution of k52 equation whose difference between the set of all line soliton solution and solution itself becomes close at least in this rate. And this happens because if we disregard modulation equation, I said it's a Vargas system. But if we derive the modulation equation on x and t, c, and if we disregard deflection term, actually it's a kind of diffusion term, then it becomes a wave equation like this one. And it can be written. Then the solution can be written by using the Landwehr formula in this way. And because of this term, there appears some phase shift like this here. So this kind of result, this theorem is peculiar to the case when the modulation equation is described to the 1D wave equation. And now I'd like to explain the strategy of the proof. First, I want to recall the result by Pego and Weinstein for KDV equation. And what they did is to prove stability of family of line soliton waves in this weighted space. And if you linearize the KDV equation along one soliton solution, we obtain this with a spectrum of the linearized operator. It has, again, value 0 here. So this is for KDV in weighted space, the spectrum of KDV in weighted space. So anyway, it has a spectrum at the origin. And because of the weight, the continuous spectrum moves to the stable half thing. And the spectrum is a sum of this isolated eigenvalue and this stable one. This happens because any oscillation tails of KDV solution moves slower than the main soliton wave of KDV. And also, if the soliton wave is smaller, then the propagation speed is smaller than the main soliton wave. So what happens in the weighted space is that the weight function grows exponentially. And suppose this is a center here. And the main soliton moves with the same speed with the center of one soliton. So if there is a smaller soliton wave, the propagation speed is much smaller. And that means that when the center of this weight function moves, this wave is getting smaller and smaller. Yes, thank you. And if we add some extra time, something like this, actually it stabilizes soliton wave more. So if we linearize the KDV equation around u equals 0, then we have this dispersion relation. And if we compute the xc derivative of this omega, then we have the group velocity. And the first two times is exactly the same as KDV. And the latter one has negative sign. So this also means that if there are some oscillating tail for Kp2 equations, such oscillation, the propagation speed of oscillating tail should be slower than that of KDV, which means that those waves are also exponentially stable in weighted space. Actually, if you think of the solution of this equation, we have exponential stability in weighted space. But as with the same with heat equation, there should appear some effect of transversal direction. And actually, for the Kp2 equation, there appears some continuous spectrum of the shape of parabola, which goes through 0. And if we truncate this spectrum, then we have the same exponential stability as the solution of the linearized equation around 0. So that is the first step. And to work in exponentially weighted space, I need to interpret what the antiderivative means in this space. If the function u is a derivative of L2 function, then the antiderivative of u can be written in this way by using this integral. And since the continuous function with complex support, as I said in the section of continuous function of complex support, and the x derivative of L2 function is dense in this exponential weighted space. And also, this integral gives a bounded operator in this weighted space. We can extend this formula directly to the weighted space. So this is how we interpret the antiderivative in exponential weighted space. Then this is the formula. So for Kp2 equations, I wrote that this is Kp2. If you think of Kp2 equation, there appears some continuous spectrum here. If you cut off these continuous modes, then we have exponential stability. That is the result here. So there are continuous eigen modes like this one for a joint linearized equation, linearized operator. And if we cut off continuous modes which are in these places, then we have exponential stability like this one. And now I want to rely on moderation equations for the motion of moderating, to describe the motion of moderating line. So let us use these answers like this. This is a moderating line. And this one is a perturbation term. And this is an auxiliary function. And this function is needed because if we substitute something like this into Kp2 equation, what happens is that we have something like this one. So and if we compute this quantity, there appears something like this term. So this is the integral of function. And this function tends to 0 as x goes to infinity. But it tends to constant as x goes to minus infinity. So this one is problematic. If it remains as it is, we cannot say that v is remaining l2 for all the time. So I use this adjustment to ensure it's satisfied. And this function is a lot of freedom to choose. But anyway, I need to choose that the difference of these two is equal to the sum, the sum, mass of this function, the mass of phi type C. So with this adjustment, we can show that v stays in l2 for all the time. And for all the time. And to analyze stability in exponential weighted space, I need to truncate some small parts. And what I want to do is this one is not in exponential weighted space. This one doesn't belong to exponential weighted space. So I want to try and remove something further so that this one belongs to exponential weighted space. And the candidate of that one is a small solution of small solution of the Kp2 equation, whose initial data is exactly equal to the perturbation, which is added to line solitons. And if we decompose the remainder time v into a sum of v1 and v2 in this way, since the mass of v1 is very small, what is expected is that, so here is a line soliton. And v1 is very small, so its propagation speed is small. So what is expected is that this one locates far behind line solitons. And if we subtract this component, then the rest of the time belongs to exponential weighted space. Since v1 is a solution to the Kp2 equation, we can use any known result for small solutions. And for this one, for this v2, I use exponential stability results by exponential stability of the semi-group. And to use exponential stability of the semi-group, I need to cut off continuous IMOs, which belongs to this continuous spectrum. So I need to impose some assumptions like this one. So this is a joint of the linearized operator. So if v2 is also known to this one, if we presume that v2 is also known to this one, we can apply the semi-group estimate to v2. And to do this, we choose parameters c and x from this L2 space, the sub-space of L2, whose Fourier transform has a compact support. And actually, I take eta 0 to be very small. And if I take eta 0 to be very small, what happens here is that I assume this assumption for eta belongs to this region. And this one is very close to actually the generalized kernel of a joint of the linearized kdv equation operator. So actually, it can be written, actually, this is very close to these two functions, phi sub c, and the integral of the c phi. And so this assumption can be basically can be interactive, like v is also known to this mode and also to this mode. And if we compute the time derivative of this quantity and substitute this equation into the resulting equation, what we have is actually these two equations. And since L is explicitly written in the previous slide, in this way, we actually get a PDE of c and x. And it's not so clear what we get, but the diagonalizing of the modulation equation, we can get some PDE, which is much better. Of course, I need to differentiate x instead of x sub y instead of x, because otherwise, it's a wave equation. And it's not so easy to approximate. But then, if we think of the system of these two functions, we get the Vargas system with some remainder time. And diagonalizing this equation, we have this very good Vargas equation. And actually, this Vargas equation satisfies a monotonously formula like this one. So this is enough, actually enough to prove that the L2 norm of the remainder time v is almost conserved. And what we need to do is to something, the remainder time v remains in L2 in some L2 localized space. So this is what I need to do in the rest of the time. For v1, it's very easy, because it's a small solution of the KVT equation. So I just need to use very identity written in the paper by Boudou and Martel. And we have this type of estimate. So for v1 part, we have this information. And to do the rest of the analysis, I also need that L3 norm of v1 remains small. But this can be done by using the theory by Hadak-Harr and Koff. Actually, the U proves scattering for small solution to the KVT equation in the critical space. And using the Wustrak argument, we can easily get a bound of this norm. And what I need to do is to estimate. I have the estimate for this one. And to do this, I first truncate v2 in low frequencies y and apply the same group, for that part. And if I truncate v2 in low frequencies in y, then it's almost an equation of 1D. So we can estimate this part by using the same group of estimate exactly in the same way as Pegel-Weinstein's paper. So it's quite easy. But for high frequencies, just we use the same group of estimates, we face some sort of loss of derivatives. So to get rid of such a thing, you need to use Villar identity. So this is a Villar identity for v2. And there are some extra terms coming from potentials and some others. But for this reason, and since we already know this type of estimates in low frequencies y, what we need to do is to obtain the same sort of estimates for high frequencies y. And for high frequencies in y, if it uses a Fourier transform to rewrite this part, then the symbol becomes something like this. And using the Cauchy-Schwarz inequality, we have some estimate like that. And if we consider only high frequencies y in y, this means that this part is quite huge. So this part can be absorbed into this part. So in that way, we can prove that high frequencies in y also belongs to this function space. And in that sense, we can prove us in stability of line. So I think I want to stop talking here. So if I understand why you not only prove stability and asymptotic stability of initial line solitums, but also of initial distorted line solitums, you should take such initial data. I think so, yes. So what would be the assumption that you would make on distortion that would be allowed somehow in the y direction? I think this one is a bit small. And also, I haven't checked, but probably this. So I might be wrong, but I think this kind of assumption is enough. This should be very small. I find it curious that in a dispersive equation, you have some piece of the modeling using a dissipative burgers equation. Now, I know you derived that for us on slide number 22, I think it was. But that went by a little fast. So can you just describe it a little bit? Why do you get burgers? Maybe it wasn't 22, a little bit earlier. There you go, burgers. Ah, this one. No burgers. Oh, I see. Yes. So why would you find dissipation in a dispersive equation? I think there's an explanation in paper's part. I think this function, dissipation is important. So this is a schematic picture of line-seated wave, moderating line-seated wave. So this part becomes flatter, and flatter as t goes to infinity. And this part cannot move because of the well-posedness. Because otherwise, if part. And there should be some gap at least here. And what Miles did is that there is just a jump. And what Miles did is just to use wave equation to describe here. But the Peterson later says that there should be, if you derive moderation equation more precisely, there should be deflection. So this shouldn't be something like jump. But there should be some heat to moderate here. And this part is expanding in the order of this. I think this is an observation made by Peterson. And that's how high it appears. And also, from the spectrum, to derive the moderation equation, I use also, now, to these I moves. And this is not only strictly on imaginary axis, but it curves like a parabola. So this is another. It's a little bit the exponential weight. Yeah. Which is it left? No. Actually, those eigenmoles actually close exponentially as x goes to minus infinity. So if you think of those eigenmoles in a two-space, it can never be a continuous spectrum.