Lecture 20 Shocks and Patterns in Hyperbolic and Hyperbolic Parabolic Balance Laws

think I'm going to start then. Okay, this is the final pair of lectures for the the semester course and thank you all for being part of this class. It really enjoyable. And don't get You need to send me the address where to send the bill. think it's the chair's office Bob Gerard and it's Canada All right, you Toronto.ca Okay, so This is the first two slides are just repeat of of where we stopped in in the previous lecture which was to say Okay, periodic patterns are totally interesting. And we looked now at the reaction diffusion and we found ultimately the behavior as predicted by Whitham equation is approximately phase modulation by single parameter the phase evolving according to Burgers equation. however, when we add some conservation laws the the Whitham equations become more complicated. They're they're larger, they're system. We'll see in minute what they are. And just want to talk about the extensions to that setting. So that the first part was done by by Milkie, Schneider and others. It was rather complete theory and extended by Sandstede and Dorman. Yeah, know. But if you don't need the extra hour tonight to practice, then stay. If you do, then go. don't know. don't know. see. Somebody had it should be muted. think it's Christiana. Are you or or Yes, sorry. Yeah, okay. All right. Okay, so And we're motivated by thin film flow. Okay, so here's our setting. It's periodic traveling wave solution, but now have more general class of equations. So, convection, reaction, diffusion. But the diffusion and the reaction is partial. It So, leaving So, it has zero coordinates and the rest non-zero. Presumably full rank. That leaves dimensional constants of motion, basically. As we'll see in minute. Well, see in minute. And that leads to more solutions and more interesting things. gave you some examples previously. repeated here. Okay, so now let's go Let's talk about the existence problem. Okay, at this point, so we talked about should should have said here oops, I'm going back. Okay, so there are two extremes, right? The extreme cases are is zero. There are There are no conservation laws. We That we've already discussed. The other extreme is tilde is not there. So, it's complete set of conservation laws. And those two cases are kind of nice because they're easier to bookkeep. They're completely one way or the other. So, I'm now We've already done reaction diffusion. I'm now going to specialize to complete conservation laws. You can do all the other cases sort of interpolating between the two, but would have lot more indices to write, so I'm not going to do that. So, here here when we have conservation laws, we can integrate, as we did in the shock wave case, and get traveling wave ODE that is just first order. Okay, and that means the dimension the phase space for the traveling wave ODE is dimension the the original dimension capital of the full system. That's right. little Saying this right. Okay, so we have how many parameters? Well, there's constant of motion that's parameters. We have speed we have the frequency, the wave number And so altogether if we're looking for periodic solution, we take the initial value, we take and then we take the period, which is 1 over So that is plus plus 2 total parameters. And feel I've let's see where capital is 2 minus Okay, so here it's just 2 2 times little Why? that's the initial data and Okay, so and we have capital constraints. We must come back around to where we started. did say that right? Yes. Therefore, we're going to have Let's see. plus plus 2 parameter solutions. so said something wrong. There are little constraints. Yeah, there we go. Okay, it comes out to be yeah, little so it's it's Am saying this right? In the end, it's plus 2-dimensional manifold. And what what one nice way to parameterize this is by the the wave number. shift, cuz there's always translation invariance, and then is defined as the mean over one period of the part the part of the solution that has conservation laws. So, in in this case, everything. So, okay. So, in the reaction-diffusion case, of course, there was only There was no So, there are the the point is that there are lot more parameters in the solution space of just periodic waves. There are additional parameters, where is the number of conservation laws. Okay, so what are the spectral stability conditions in this case? Well, they look almost the same. They're going to be almost the same. We need negative real part when the block locate number is not is non-zero. away from zero, we have the real part bounded by diffusive rate, negative theta squared. and lambda equals zero is an eigenvalue. Okay, in the reaction-diffusion case, we said it's simple eigenvalue. But there, we expected two-parameter family of solutions up to sorry, one parameter up to translation. Here, it's plus one parameters up to translation. So, at at most, we're going to have multiplicity plus one. And actually, that's not so clear, right? Because could take variations in every direction of this plus two-dimensional manifold of solutions. And how come don't have plus two zero eigen functions? The reason is It's the same as in reaction-diffusion case. is one of the directions, but if move in that direction, change the period. So, that's not allowed. That's not not in the zero eigen space. Okay? So, there will at least be plus one, though, cuz there are plus one directions that don't change the period. And that means okay, so now we have to think more about what we want. Well, it turns out that just these assumptions alone imply that the eigen values that bifurcate from zero. So, there are plus one zero eigen functions and there are zero eigen values which which bifurcate smoothly. That's really surprise because even though didn't say it, generically, there will be Jordan block here in the in the eigen structure. If perturb Jordan block by by xi, can perturb it analytically all like, but it usually doesn't expand analytically. It usually expands like square root or or some higher root of xi. But, that's not the case here because of special structure. And interestingly, this is pointed out by Denise Serre in very important paper 2005. And it looked pretty bad for stability at that point. But, it turns out the same kind of structure that showed this was generically Jordan block shows that it actually perturbs in special way. So, maybe I'm just going to write this quickly. So, this is basically the problem. Okay, so have Jordan block. If If this entry is non-zero, I'm going to have perturbation like I'm going to have an expansion like square root of xi. But, it turns out this is zero. That's very special. And this means can do rescaling rescale by transformation. So, take times all of this times inverse. And the effect of that is well, it moves this one it multiplies it does nothing to diagonals, it multiplies upper off diagonals like and one over on the bottom. So, this one gets this one moves up to order squared. This one gets promoted to this place. these are demoted down. So, this one comes in here. this one would be horrible, right? But, it's it's not it's not it's not present. So, that So, we're still So, what we end up with is zero in the first entry plus times several things. This will be one. We don't know what the rest are. So, we get new matrix perturbation problem that is just let me take outside. And this is nice standard one. It's no longer Jordan block unless you're very unlucky. Generically, it should not be. And it turns out though you know for for most interesting cases, for us it's not. Whereas, the the original setup was always Jordan block. But, after the rescaling, we find it isn't. And now, this as long as as the eigenvalues of this matrix are distinct. So, first of all, just by this decomposition, know that it's it's perturbing smoothly in the sense that wrote. know the first the eigenvectors of the first matrix are the AJs. And then it's little of xi meaning just continuity of spectra. but and the worst it could be with in this as in this example would be of xi to the 1/2 would go here. sorry, order of xi to the 3/2 would be the the worst. But in fact, if the AJs are distinct, it would be analytic by standard matrix perturbation theory. Once they split, you're good. Okay, so that is that was one thing that actually now now that we understand it, it seems very very simple and okay, it's standard, but that really blocked us for while. Just mental block, how can we proceed further? And it wasn't because so for ODE, if you have Jordan block, you have time algebraic growth. You you it's not even neutral. Here that's not really what's going on because it's in the block flow case setting, you integrate in xi, it doesn't really work that way. But what you do need is that projectors be analytic and bounded. If they're if projectors are not analytic, they would be if if things split, you'll have an unbounded projector. And that would be bad. So it it it really was important to understand that. Okay, so the basic nonlinear stability result looks exactly like the non-local sorry, it for localized data says that there is certain phase shift psi and if we modulate by that phase shift, we decay like heat kernel. The the modulation error and the derivatives of the shift decay also like heat kernel. The unmodulated error is just bounded. That's still an orbital decay result, but let's see. Is that think it's right. That's still bounded stability result, but it's not asymptotic stability. So, it it's something like an infinite dimensional orbital stability result. Exactly as in the what these decay rates look like are exactly the ones for the non-localized data for reaction-diffusion. Interestingly. The modulation is responsible for the Sorry? The modulation is responsible for the asymptotic stability. Yeah, think it's needed. The the decay is is only for the modulation, yeah. Right. Is that what you were saying? Yeah. Yeah. Okay, so so that that resolved question that we opened in with Muñón in 2001. And we had some some decay estimates, but they were actually assuming that there was no Jordan block, and we just really didn't know what to do if there were Jordan block. Yeah, okay. It's weak decay compared to the localized case of reaction-diffusion. Everything decayed like the derivative of heat kernel. That's much faster, and so this is just more delicate analysis. All right. And there are some results in in multi-D as well. Okay, so now want to go little bit further and talk about the Whitham equation, which is really interesting in this case. So, here we do the same trick, the same Whitham trick. So, you rescale to capture the large time and space behavior, you rescale XT to epsilon epsilon and this converts the equation to something with singular diffusion. We had that last time, but we also had it looked little different. It was an epsilon squared here. it it was just different in the reaction-diffusion case. But, this is what you get here. And then, now you get modulation approximation, the same one. but, bar of MK, where this is the manifold of possible periodic solutions with phase psi of XT. Now, if rescale back, this would be like psi over epsilon. Okay, there's I'll show you in minute how that goes. And here's what the equations look like. There's the KT plus omega equals zero. That's the the usual one. That's the one familiar from from reaction-diffusion. But, then there's also MT, there's conservation law. MT plus FX equals zero. And here, is again the mean over one period of the the the part of the of the variable with conservation laws. And is the mean over one period of the flux. The the the conservation flux part. Okay. And as before, omega will be the speed times which is negative psi will be negative psi over psi That's implicit function theorem for wave front. And omega has some non-linear dispersion relation. It comes from the existence problem. For the reaction-diffusion case, we had equals zero, was not there, and it reduced to scalar conservation law, and the characteristics were the the group velocity omega prime of okay. Here, it's system of conservation laws, first-order system, and we don't necessarily know that it's hyperbolic in the first place. That's That's quite interesting. So, this is more like what Whitham derived for KdV. And consistency of this approximation requires that you well-posedness of the first-order system, which means hyperbolicity. So, it must have So, the characteristics are eigenvalues tilde of DF omega over DM of the full system. And we require that these be semi-simple and and real so that that system can be solvable and even make sense as descriptor of large-time behavior. So, that is that's definition in some places, like in in physics literature and and engineering literature, that would just be the definition of stability. definition of like back-of-the-envelope stability. If that system is hyperbolic, that's stability. Otherwise, it was clearly pretty clearly unstable intuitively. And this is there's fundamental theorem, and this was proved by Denis Serre in that 2005 paper, really fantastic paper that drove lot of this development, but it's anticipated by Frisch and Oreol in in 1976, little earlier, but in the specific case of Kuramoto-Sivashinsky, and they they made lot of deductions based on symmetry, but they but they also proved rigorously for that case that AJ is AJ tilde. What were the AJs? AJs were the spectral expan- the coefficients in the spectral expansion of the neutral modes. tilde these are the formal predictions of of propagation of behavior in the Whitham system. So, it's very beautiful that they agree. Well, not only is it beautiful, but it it rigorously justifies this this physical definition of stability. Hyperbolicity is necessary because suppose AJ tilde had complex part. Then, the first-order term in the expansion would be * AJ * And there would be real part because AJ tilde would then the real part of that would be negative complex part of AJ tilde. And by changing the sign of can also make always make an instability. So, that's no good. And you can continue you could continue the expansions and you it's much harder. You get second-order prediction. You get some diffusive part. And actually, we we need that for our for our result on actually disorder because otherwise we can't go to long time, right? We we want to go to time infinity and the left-hand side if just put zero on the right, it's not going to go to time infinity cuz shock could form, could break down. We don't know. Okay, so here's the theorem with Johnson, Noble, Rodriguez in 2014. So, now this is the analog of the the final theorem of the previous lecture with non-localized data, so allowing phase shift at initial time. So, if we have an initial phase shift whose derivative is small in L1 and for which the the difference between the So, the modulational difference between your perturbing solution and the the background solution shifted by if that is small both of these and assuming our stability hypotheses, these diffusive conditions then defining script and script to be solutions of the the Whitham the second-order Whitham system with data sub and something This is This is an interesting point. What you know you can do formal things, but then what's the data? That's that's something you kind of it turns out you need some analysis for that and and you can find what it should be, but it's not so simple actually. It's not not obvious. But you find it. And okay, and setting psi to be the antiderivative of of kappa then if the difference between tilde and the modulated both and kappa modulated solution with phase psi predicted from the Whitham equation decays faster than faster than heat kernel. And let's see. And the meanwhile this part decays like heat kernel. So, that's very interesting. For the reaction-diffusion case the localized data decayed like derivative of heat kernel. Non-localized data decayed like heat kernel. For the conservation law case, there was no difference between the decay rates in the localized and non-localized. It didn't help very much. That was little bit of mystery to us. but what you can see is if if the equation decouples, for which there's it just means that the change in speed of the in in the non-linear dispersion relation, the the the variation with respect to the the mean vanishes. So, there's no effect on the the speed. Then you get faster asymptotic stability. You get decay at the reaction-diffusion rate. And so, that tells exactly why this is happening. what what what else? And again, we mentioned this before, there are multiple group velocities. There are multiple characteristics A1 through AJ. So, it's not clear how one would ever do renormalization type analysis, and but it's well suited to our our conservation law tricks that were designed for such system. So, so we apply the non-linear iteration that we did before. So, the analysis goes exactly as in the previous lectures. However, finding the cancellation is little harder. There's more bookkeeping, as you can imagine. There are more terms. And and new new aspects, but the the basic scheme is the same. Okay, so in general when you have an that is present, so whenever is one or higher, you'll have system, and you'll have richer behavior. And now want to go back and mention once again the the paper of Frisch and Ore in 1976. couldn't believe it here. They completely anticipated. This paper was titled viscoelastic behavior of Kuramoto-Sivashinsky periodic Kuramoto-Sivashinsky equation, and what it really means is that Well, they they they show that the main behavior is that the phase should satisfy damped wave equation. Okay, so at the level of the Whitham equation, that would mean isentropic gas dynamics with viscosity. And here in Lagrangian coordinates, and there you have characteristics which are plus or minus the sound speed. So, they're equal and opposite, and so and also you can see that the the diffusive effects are also equal by symmetry. This is picture of troughs and valleys of periodic solution that's been perturbed. So, it's been perturbed in the initial data just right here by Gaussian perturbation. If it were not perturbed, all these lines would be vertical because it would be So, this is space space to the right and and time vertically. Okay, and so when you know what to look at, you can see very clearly here we've drawn the two characteristic velocities A1 A1 and A2, and here we've drawn the quadratic envelope that we expect for the the diffusion, right? For for the the diffusion expands roughly as square root of With certain parameter certain multiplied by the parameter. So, you when you compute these things and and plot them, it fits so well to the perturbation. It's quite beautiful. or hope this So, hope this will run. But, it's too fast anyway. This is So, there's kind of picture though in this paper by by Frisch and Oriel that the periodic cells act like little masses on springs. Damped springs, and they just bump against each other. And so, we also thought it would be fun to to do that. So, this these are pictures of So, the troughs and the valley This this is the actual solution, and we put the the same green and blue dots for Sorry. Sorry. Peaks and valleys, not troughs and And it may or may not run. Okay. It ran way too fast in any case. You're not missing much. It ran so fast, but you can If you look very quickly, you can see them rush out to infinity. So, had feeling this would happen. Okay. So, but maybe What if click okay? No, think that's bad idea. Okay, we'll just skip that one. All right. let's see. How are we doing on time? think we have enough time that want to go back. want to show you how this how this WKB expansion goes. And I'll go ahead and I'll put the epsilon in from the beginning. And we're going to look for multi-scale expansion. So, it's going to be of and psi of over epsilon. Remember, that's how it went. And then, we'll plug this in and expand to all orders. This is what you do. So, you substitute and expand. And what you get is Okay, what's the worst order you could get? It's epsilon order of epsilon inverse. That would be at that order you get what? So you So sub and I'm going to call as I've been doing before, this variable I'll call theta. So as as an argument of bar, that's theta. And it And then I'm going to use chain rule. So get So get theta prime. sorry. You use of theta derivative with theta times psi prime. And I'm pulling out the epsilon inverse factor. I'll get psi prime over epsilon. And I'll get DF of theta psi not psi. Psi psi and get Okay, so this is little funny. I've got an extra epsilon. That's what saved me. So here get theta theta psi squared. That's the only term that survives. And the others will have extra epsilons. Okay, so this is exactly This is exactly the traveling wave equation with omega If If psi were omega and psi were And so and and And this is supposed to be forgot to mention this should be one periodic in theta. So that's an assumption in this expansion. Okay, so this is going to be one periodic Well, it's got to solve the traveling wave ODE and this This is necessity, right? It must be omega and cuz there are no other periodic solutions nearby other than the ones that satisfy you know, periodic solutions can be phrased in this way with an omega and That that's what these have to be. So, that tells us by mixed equality of mixed partials, we get sub plus omega of sub equals zero, usual. Now, now where does the other one come from? That was the first that was the the final equation, right? That's the same as reaction-diffusion. But, now I'm claiming there's another conservation law and that comes at order order one. didn't plan well. Okay, I'll do it down here. At order of one, now some other things start to come in. So, have for example, have just sub have sub These are the slow variables. Okay, don't get don't get any contribution here from the slow variables because there's an epsilon. And then, I'm going to get lot of other things which has have theta derivative. All the other things that get, there are some fast derivatives, but they all can be shifted out like this because in this type of expansion, you treat these as independent variables. this is the yeah, DF. Okay. Now, Yeah, but now I'm going to integrate in theta. And maybe it's it's clearer if write this like like this, meaning knows understood there's no theta we're we're freezing there. Okay, if do that, then get the mean over one period. Lost the mean of the flux over one period. Just immediately. There's nothing to it. This is the beautiful argument that Sierra gave. That's great nice argument for the first-order Widom system. There are other ways to do it, but mean, like this way for obvious reasons. If you want to go to the second-order term, it's lot hard that's it's mess. Now now you have to do some real work. You don't just see it like this. But that's where it comes from. So, saw it is is really direct path to the Widom equation. Okay, so now we have at this stage we have really very complete non-linear theory assuming we know the spectral assumptions. So, how can you verify those spectral assumptions? have already hinted at this, but so, in general, you could do this is periodic. So, you can you can do sort of Galerkin method. You have discrete Fourier transform to work with indexed though by the block Floquet number So, that makes it little harder. But for each you can you can track the spectra. And that's actually very very nice way. There is If If want to play with that, there's Let's see. Spect can't remember. I'm not I'm not supposed to say Spectru, but it it starts with Spectra and then it has capital capital which is Washington. And that was done Yeah, that was developed there. think you're supposed to say Spectra Washington or something. But everyone says Spectru. But it's really handy. They they made it so easy to use. So we we we used that for while and then it worked so well, we we built our own so we could change the parameters. But but it it's fantastic. And and you can you can compute interesting really interesting problems just using their online menu. Very easily. Yeah. So, recommend that. or shooting methods similar to what hinted at for the the shock wave case. and we've done that. That's basically the only way we know, the only two ways we know for large amplitude waves. But there's also the interesting problem of small amplitude limits, like just at the onset of these patterns. And as mentioned last time, that is it's not like Turing bifurcation, but rather long wave bifurcation that's sort of generalized Kuramoto-Sivashinsky to KdV limit, it turns out. at least for the cases that we're interested in. So, here's the KdV and we're perturbing by Kuramoto-Sivashinsky type and diffusion multiplied by small parameter delta and delta goes to zero as as we approach the onset, the the bifurcation limit. And so, very nice very nice result of Ercolani-McLaughlin-Wrightner, '93, is that the profiles do approach conoidal periodic waves of KTV as delta goes to zero from above and that's not not simple. You know, that that requires proof. here's the next one the stability then Barker and Meerski in 95 showed by spectral perturbation analysis from the conoidal wave plus numerical study so you you do spectral perturbation analysis, you get an expression in terms of elliptic integrals for the first order corrector. Then you check that that is always negative in real part and you you lose 50 strange you oops, you you lose 50 digits of accuracy in the course of those computations. It's really get tough. Very ill numerically ill-conditioned thing. Okay, and that's the one where then Barker, my student Blake Barker in in 2014 converted that to numerically rigorous proof, which was quite job. using the unreasonable effectiveness of analytic interpolation basically that it's exponentially converging in the number of interpolation points that that supremely good conditioning can overcome the really bad conditioning of the other stuff and and win. Okay, and so there's they find there's limiting stability region. So the let's see the existence region is somewhere in here. So periods from 6.2 to 48. The stability region is from 8.4 to 26. Roughly. And and you have result. And so this will What's interesting, of course, is this applies this limiting behavior tells you what things are doing also when you're close to onset, not just at onset. So it describes all small amplitude behavior. And it tells you really where it's by un rescaling you can figure out what the what the period range is for any small delta. So think these are really interesting results. what was the intervening JNRZ 13 was just to show that that Saint-Venant also reduces to this case. Which is after after some rescaling. That's the contribution there. And so here's the picture. You're you have wedge here or ramp with slight inclination. You run waves down the ramp. And these are the roll waves that you see, the smooth roll waves. And here are the equations. The long wave delta would be square root of minus 2 where is Froude number. This is probably not important, you know, from far view, but okay. So we show that also for the rescaled Saint-Venant we get the same we get the same asymptotics. And there's nice picture here. You start Where did this come from? This comes from incompressible Navier-Stokes with free boundary. And that leads to Saint-Venant by very non-trivial modeling problem. And now what we have done is we've completed this leg here. So we show that that goes to this generalized Kuramoto-Sivashinsky to KDV limit and then that's treated in the previous slide. This leg is still open. It would be very nice to go straight down, but seems seems difficult, very interesting. And here's here's picture of our our heroes, these waves, these roll waves. And you can find these on any rainy day. You just have to find street that is that has some kind of gutter, and it has to be not concrete, but asphalt. It has to be rough. And you'll see this all over the place. My kids are experts. Took many pictures on their phones. And do have there's one more very interesting to thing to say about all this. So, this is picture of 4 Saint Venant. It's typical phase portrait for the existence problem. So, how this breaks up is it's saddle node kind of bifurcation where you get homoclinic that breaks up into not quite homoclinic. So, you limit cycle somewhere in here. You have from the outside things approach, from the inside things spiral out, and somewhere in between there's limit cycle. That's your periodic solution. Now, as you let parameters vary, the periodic wave can either shrink to the center, so that would be Hopf bifurcation in the middle, or it can go out to the home almost to the homoclinic. And then, what does it look like? Well, that's kind of where we are here, cuz look look how far these waves are from each other. They have really long flat spots. That's like coming very close That's that's almost homoclinic. So, that's near the homoclinic limit, we would say in in hind hindsight here. We can see it in the picture, but we you know, okay, in the beginning we didn't really see that, but And here's here's the really funny thing is for the actual Saint Venant, mentioned this Froude number, and that Froude number equals two is where bifurcations happen. And why? Because constant states become unstable. But the funny thing is all constant states become unstable, not just one fixed one. All of them. And so that means that any constant is also unstable. And it means that homoclinic solution is automatically unstable. Because it's end states have unstable they're unstable as constants. And remember that that leads to the by Henry's result that means the essential spectrum is unstable. So it's just going to be unstable. This was quite puzzle because people often see this it looks like this one looks rather periodic. But in some experiments it doesn't always look that periodic. And actually there was theory that these things aren't periodic but are just bunch of home homoclinics that would eventually combine into one. They would they would interact and and finally coarse and down to one if if allowed to go forever. And the idea was just well our ramp doesn't go forever so they didn't get to do that. But that turned out not to be true. So let's see. yeah and there's more. there's one more thing. so there's that and there's also stability cannot be not all of these periodic waves can be stable because that the limit in the center which was constant solution is unstable. Because constant solutions are unstable. And the limiting homoclinic it's also unstable. So both limits are unstable. It has to be something in the middle. If there is one at all it will be band or several bands of periods that could be stable. And the others would be unstable. So you know that from the beginning for Savart. So here's picture Okay, this is little confusing. This is family of all the orbits. It's not phase portrait. That would be Hamiltonian phase portrait. It's not Hamiltonian. Here we just put all the orbits together for different parameters and it happened that they were concentric, which is lucky for the diagram, but very confusing. It looks strange. Okay, and so what we the very first wave that we ever tested stability for was this green one. And green is for stable. So, we were really hoping we had all this analysis by this point. We have this one example that maybe might show that it's not vacuous. That there is physical example. So, we really wanted periodic solution. We And we were stable periodic. We were prepared to test all of these. We tested one, which at the time took quite long time, and then we all gathered and we went for coffee, came back. The green light came on and we knew that everything is good. But, we were pretty lucky because here are the stable ones. And so, it's unstable below the red. Everything below the red is unstable and everything above the blue is unstable. Well, you can't see the the blue really. it's Well, you can, but it's superimposed on the stability on the existence boundary, which is black. You can't even see it. So, it's really you get really close to homoclinics and you're still stable. Strangely. And so, that's kind of funny. And yet the homoclinics are stable. It seems strange. And so, that's probably the Let's see. Okay, and here's This is kind of fun. If Have you looked very much at spectra of of periodic waves? It's it's very interesting. So, we're here we're letting the we're varying the Floquet number, and and tracing out the spectral curves. And there's one that looks lot like the shock wave essential spectrum. And there's another one that looks like translational eigenvalue that got like like blown up little bit. And this is actually what they are. You can show that those those are the limits. And then there's some other stuff. Well, that doesn't matter. There's always some stray negative spectra even for the conservation law case. So, that's kind of what it looks like. And okay, so here's the big mystery. The homoclinics are unstable. That's kind of starting point if you look at the model. And yet they seem to dominate large time behavior. So, people had studied these from this from different points of view. The one one point of view was to show that they could still they could persist for very long time. That cuz they're slowly interacting. Okay, that's interesting. That was more or less this big paper Schneider-Ucker paper. But our resolution is that the homoclinic arrays, they can be stable if you extend them out to infinity and choose their spacing properly. And here is let's see. Okay, so maybe can don't have it there. I'm I'm sorry. Okay, let's see. So, Yeah, maybe I'll try quick one more picture. So, if we have homoclinic wave, homoclinic wave, homoclinic exact homoclinics and we just kind of superposed them with some distance Okay, what happens? If you send perturbation in and we've done this. It's very interesting to see. Okay, so let's take small perturbation. What does it do? After while, it looks like this which is which is classic wave packet of an an unstable wave packet and this is growing exponentially and it's in the shape of Gaussian envelope and it's oscillating rapidly. Okay, it's growing. That's bad. But it's also moving toward the homoclinic and when it gets here it interacts with the homoclinic and it comes out like this. It comes out smaller. There's some some deamplification or shrinking that happens and then off it goes. So if you do the thought experiment of varying if is too big you can't win. This will grow too much. It will grow more than this deamplification can correct. If it's too small, this picture is not right cuz this was assuming they don't interact very much and so there should be you know we tested this by by putting exact homoclinics together at different spacings and then running the experiment and we find exactly that that's another way to find the upper stability boundary near the homoclinic limit. Okay, so that kind of that answers the question. No, they can't they're not individually stable but they can stabilize each other. So it says little more than it's different statement. Yes, they interact slowly and so they could persist for long time but they can persist for all time by sort of you know mutual stabilization of the periodic pattern. find that really beautiful, surprising. Okay, here is more. So we're able to compute by this was done by these shooting methods by Blake Barker compute the stability boundaries. So here is here's this this KDV KS limit. So, we've got these stability boundaries. They're rigorous asymptotics for small amplitude. So, that means small Okay, let's see now. This is log of And so, Okay, we're down below one here and it's against log of of period. There's reason for that. But anyway, so this exactly matches the upper the blue and the black here. They exactly match what we expect. there's actually green line below. The black is the Sorry, the black is the experimental boundaries. As are the blue X's. The dashed lines, we'll see they're two different things. There's green line, that's the the small amplitude rigorous theory. But but in what we see up here, this is different. It's completely different behavior. This green line goes off. It goes somewhere else. It's nowhere connected to this. So, the small amplitude, the near near two behavior is gone at about what was it? 2. 2.5 or so. At as No, 2. 75 2.5 Anyway, pretty small And afterward, it's something else. And the interesting thing is in hydraulic engineering, like canals or or dam over dam overflow, is like 6 8 10. It's much bigger than two. So, for those applications, it's no good. The the small our small amplitude stuff stuff is out the window. Okay, but but on the other hand, huge surprise, it's very regular. It's even better. It's power law. That's what straight line is. And it took ages to compute this. We kept really bothering Blake because he did few and we saw wow that looks kind of like almost straight line. Looks pretty well-behaved. But maybe it is straight line. Could you do few more? 100 hours later another dot appears. that's really looking like line. How how about five more dots? So the whole thing took like month, you know, to compute this picture. And what you but if you look more closely at it it looks like it's kind of homoclinic if you look where this falls in the traveling wave order. It's near homoclinic and it's kind of near inviscid as well. So this really motivated us to look at the inviscid problem. And why was that surprise by the way? Cuz let's see why was that surprise? Because we had earlier looked at this. This is the the generalized Kuramoto-Sivashinsky with both small parameters included, epsilon and delta. And people had This was the gen- the generic thin film behavior which includes also vertical and and horizontal. Here's epsilon and here's the period. So fixing let's see what did we do? We fixed epsilon squared plus delta squared equals one and we we varied epsilon. And on this let's see on let's see so epsilon equals zero is where we are. That's That's salmon on. Over on this side it's it's Kuramoto-Sivashinsky. In between look what happens. It's chaotic. There are so many bifurcations. This is what we expected to see. We expected that there would be some small amplitude behavior predicted by the the small amplitude theory and then it would go crazy and we would say we so you really need this numerics to to study it. But instead we saw no there's formula in terms of power law that people could use for engineering that you could compute in second. Cuz even though Okay, you even though this took month, you compute it once and then you know the slopes. That's that's You could carry it around on little card and and compute, you know, canal design using that. Perhaps. Okay, but so it's not this. And okay, so this this one is was supposed to be an open problem at some point, but it's not anymore. We've We've done this. We We studied went back to the inviscid problem, which by the way is the one used in the industry. People who who design these things and and do numerical approximations, they never include the viscosity. And they still get good results. So and so we got much better result. We got actually oops. We got formula for the lower boundary at least in terms of cubic equation in and This was rigorous result. It says it's very surprising to me now. special feature of the equations we took advantage of and we were able to do this. And the upper stability boundary, all can say is it seems to be very close. So here they're not so close together, but in the inviscid limit they get much closer. So you so you don't even need to do the month-long calculation. You just solve cubic and then you hope that the other one's pretty close. And that tells you roughly, well, what do you want to know? You want to know the amplitude and period of of the waves that survive. Cuz the roll waves that survive, though these are actually bad for canal. They could splash out water. Maybe they even depending on the application, maybe they damage the the structure. So you really need to know for your design how big are they going to be. And they're quite lot bigger than the background flow, the laminar flow, just the quiescent flow. Okay, so but this tells you essentially the height can be computed from the bottom curve cuz they're pretty close together and the bottom curve can be solved explicitly. So this was don't know. That's the think that's the best applied math result that I've I've gotten where we say we don't know the formal asymptotics seem sometimes contradictory. There are lot of conjectures, but no one knows. So let's throw out that stuff and do an rigorous spectral perturbation theory. And at the end of that we got something simpler than all the all the formal stuff and it's backed up by analysis. So that that's You have to have some luck there. And you have to have good equation. So also want to promote Saint-Venant up with KTV. think it's you know, KTV is so ubiquitous, everywhere. And it's wonderful equation, but Saint-Venant it's not it's not completely integrable. It doesn't model quantum things, but it does model shallow shallow water flow and it models not only the small amplitude, but also the large amplitudes and almost everything in this model can be solved explicitly. The the profile equations are polynomial scalar ODE. You can you can find formula for the background solutions up to arbitrary amplitudes. You can compute exact explicitly the stability boundary, all kinds of things. It's very friendly equation, so you can really learn lot from it. But in the large amplitude case. Which differential equation do you have to Okay, so go back Let's see. Let me just make sure I'm in the right spot here. Yeah, that's Okay, so that's Where did it go? Back at the beginning. Okay, yeah. This is the This is the you know, it's shallow water or thin film on an on an incline. It's It's both. And then But then there's also this is the other one. This is the small amplitude limit that encompasses so small amplitude waves even on vertical, which would be Kuramoto-Sivashinsky. purely horizontal would be KDV and it interpolates between those two those two extremes as epsilon and delta vary. Asymptotes of nonlinear diffusion. see. Yes, okay. Right, the wetting boundary and all that. That's This is has no dry boundary. This is assuming you the water goes out forever. So so that Yeah. So that you don't see this degenerate diffusion of the vacuum or the or the dry the dry edge. All right. Yeah. And am done with this part except to mention Okay, so the main open problem. So Barker did numerical proof for the small amplitude waves. Why not large amplitude waves? Well, it's different problem, right? But but that's actually the same problem as as for shock waves more or less. And think that people have the tools, Blake in particular, to to maybe do it now, but it hasn't been done. And seems like the time. okay, and then the Hamiltonian case is always mention that. and then Okay, and then small amplitude asymptotics for the full Navier-Stokes free boundary problem. That would be fantastic. Okay, so I'll stop there and we'll take break. Come back. See what can do this. Okay, well, let's get it started again. think everyone's here. Okay, so this last lecture, I'm just telling more some further direction that one can go on and I'm not giving the analysis, but just telling about it. And this is again, modulation of periodic wave trains, but slightly more difficult problem and in particular stability so stability of defects, in particular source defects. So I'll explain what those are. Okay, so we're we're still going to look at background periodic solution. We know that the behavior should be approximated by some modulation or we think it is governed by Whitham's equation. Second order. And where we have we have the non-linear dispersion relation given by the existence theory that gives the links omega and and then we have initial data given by This this this this is the measure This is the derivative is the derivative of the phase. And so it should be the derivative of some initial phase and that should be in L1. The derivative of the initial shift should be in L1. Okay, when we look at that, that is like that's like perturbation of constant solution. It's exactly what it is. Single wave. But now you could ask what what if we go further? Well, I'm sorry. First, we we just talked about the generalization to systems, but there's another generalization. Now, let's go back to the reaction-diffusion, and instead of adding conservation laws, we'll say, "Let's take still less localized." Sorry, sorry, not So, originally, we took psi, the antiderivative of to be localized. Those were the first results. Then, we talked about non-localized, meaning that was actually localized, but not psi. What if we take just asymptotically constant? Why not? Especially because, when you run numerics on these things, you see that doesn't seem to hurt anything. you basically almost any any modulation, well-spaced, seems to be pretty stable. okay. So, in that case, what what do we expect? If it's asymptotically constant, it turns out, it's not obvious, but well, yeah, yeah, actually, would say it is obvious, that if we believe the Whitham equation, then asymptotically cons- constant solutions, those are shock waves, scalar shock waves. So, now we're back to the beginning of the course, through this nice, this link of of Whitham to from periodic to to hyperbolic systems. Okay. All right. And now, now I'm going to be I'm I'm talking about work really of of lot of Sonestedt and Scheel in the early in the late '90s, early 2000s, and physicists before that, but okay, it turns out there exist what are called defect patterns, which consist of two periodic wave trains at plus and minus infinity. So, you basically got one wave train here, another one here, and they meet. They somehow interact across an interface. And these could have certain speed. So, okay, if it were lax shock, we'd expect the speeds to come inward. That's called sink type defect. There are also transverse defects. It turns out that pass through. And there are source defects. And source defect emerges from the interface. So, that means And and this this interface is called defect cuz you have nice periodic patterns and then there's defect, just like in crystallography, right? But in the source case, the defect is really running the show. Everything emerges from that source. And that there are relations to these Nozaki-Bekki holes in complex Ginzburg-Landau that come up in in physical discussions. Also in spiral waves, this the the source the center of the spiral is source. This this is that this is the heuristic of Sonin and Shiel that say the the 1D source is somewhat like slice of of of spiral wave where you see the center emitting all the controlling things. Okay. But here's the beautiful thing. This this already really catches the eye. What if you had such pattern? Then and and what if you're looking for wave that remains Okay, what kind of wave could you have? Well, it's got to be at best time periodic, right? It can't be stationary cuz you have these two things impinging and interacting, but it could be time periodic. So, that's the idea. So, if you look for time periodic one, now you have strong constraint because out near infinity, that's just traveling wave and it's traveling with certain speed. And okay, so it will be time periodic with time period depending on its speed and its period. So, that is it Okay, so here CP is the Okay, is the background velocity of the of the exact wave train, so the phase velocity. CP is the group velocity. And so, the relative velocity is the difference between the two. That divided by the the period is the time frequency. And this must be equal on both sides. Well, dividing that by the period is the same as multiplying by the wave number. And if multiply the wave numbers together, get in this case, omega of and in this case, CK. So, it just the the the Okay, whatever it is, the the phase velocity times And these must be equal at plus and minus infinity. That means if compare them, if take the jump from plus infinity to minus infinity, get jump in omega equals jump in That's the Rankine-Hugoniot condition for scalar conservation law. Beautiful. So, now we have tion also at this level. But something maybe should bother you little bit about what I've said. I've said that very important solutions are not Lax type. So, maybe the analogy doesn't fit that well. Well, kind of left off some some really interesting detail. We focused on the Lax case cuz that's the main one, but actually there are cases of non-classical waves, under-compressive waves, so more waves going out than than Lax type, over-compressive with more coming in, and they are physically relevant. It's just that the the details of the inner structure were not taken into account in the inviscid theory. So, you can treat those. In fact, that was my entry into the field, and spent probably my first you know, 10 years of work on viscous shock stability was really about understanding under-compressive and over-compressive shocks and how that could fit. And you really need to include the the viscous problem to understand it. Okay, so here Well, that's you know, even more structure than viscous problem. So, it's believable that it still could be stable even though it the inviscid theory kind of says doesn't make sense. And it turns out they are. And the the sink type and the transverse type, those were treated by Sandstede, Scheel, and others previously, and I'm just going to tell little bit about so, the source defect needed seemed to need, and and we use some techniques coming from the shock wave course, but much much extended. So, here's just picture. Well, okay, I've drawn it on the board. This is what what it looks like. There's core. There are two wave trains there emerging from the core. And this is what it looks like out in the far field, say. You have group velocity, you have phase that's different from the phase velocity. So, waves can move around like scalar conservation law. That's all I'm saying. And this this motion is the characteristic speed for the Whitham equation. this one very unlikely to work even though it works on on my computer. Nope. Okay, too bad cuz this is kind of good one. There's Well, you you can imagine. You just see the waves coming out exactly. Yeah. It's animated version of the what's to the right. Okay, so All right. So, existence. There are There's beautiful series of papers by Sandstede and Scheel in the early 2000s where they they took these isolated kind of physical observations and they put it all in framework and classified all these possible waves and showed how they can emerge through bifurcation. And highly recommend those. yes. Okay, so then you could talk about That's existence, but of course physically persistence or stability is also important and that was resolved by them also Yeah, al- al- also using similar techniques as in the in the existence problem for the sinks and the transverse waves. And basically the method that they're using in both cases is what's called spatial dynamics. This is where since you're looking for periodic solution in time you look on the space of periodic solutions with certain period. And now you view as your as sort of your your evolution parameter. Which is very much the same as looking at the eigenvalue problem in way in in the in the previous cases, but the issue is this is infinite dimensional and you'll pose as Cauchy problem. So, you have to be that's the goes back to Kurganov and this, spatial dynamics idea says you can still make sense of these problems even though the dynamics in strictly speaking, is ill-posed. And yet, the boundary value problem from plus infinity to minus infinity can be well-posed. The Cauchy problem is ill-posed. It it's not contradiction. Okay, so Yeah, so the question is two question, well one version that kind of believe is that you could just take almost general modulation and let it run and it's still going to be staying within the family of modulations. From what see in numerics and, what we've been able to prove that seems pretty likely in at least in some cases, but have no idea how to prove that cuz now you're far away from any solution. But, something that you could show along the way to that that's sort of like that but weaker is just say, "What about these source defects which are specific modulations that interpolate between the two end points? Could those be stable?" And then that would be evidence for one and, well, independently interesting, would say. Okay, so that's, what we're analyzing and, okay, and by the way, should say, "Oops." So, that this is work done with Margaret Beck, Björn Sandstede, and Tuan Nguyen, who was student at IU, originally before many other things, but now he's at Penn State now for long time. Okay, so here just gave some pictures of shock wave problems. We're just, we exactly what we spent the last semester talking about. So, I'm going to basically skip that and go okay. straight to here. yeah, okay. So, this was the end result by using some pointwise estimates we were able to get algebra time algebraic convergence. Green function balance that looked like this. and okay, this last thing w- in this course only did LP LQ to LP versions of stability. Nonlinear stability arguments and at least up to the time of this work which was around 2000 and 13, think. the only treatment of undercompressive shocks had been using pointwise estimates which we didn't do here and then it's more complicated. You in your bootstrap argument you keep track of the shape of the wave basically. that's actually no longer true. just just last year Zhao Yang and introduced an LP LQ to LP undercompressive treatment. So, wonder if there could be it's not clear that could be used in this case though. This case is different. But at least for usual shocks you can do it and it has an interesting feature. It involves Strichartz estimates. Like in in this version got So, that's the trick. Okay. All right, so basically so how how do you get at this? Well, we're going to again do linearized equation, but now the very the coefficients are both are periodic in space and time. You're going to have linear solution operator defined well, in the usual way. How are you going to find it though? You're going to find it by Floquet inverse Laplace transform representation. And here, okay, this is the tricky part. The Floquet resolvent kernel is it's it's periodic in in and lambda is not what you think it is. So, lambda is the Floquet exponent and the is the actual time derivative here. Okay, then we we invert on on finite domain. Well, this is much This is the Floquet inverse. All right. And we estimate that by spatial dynamics. And Okay, so here's what you get. This is even the the the end result without talking about the the analysis, think it's interesting. It's kind of do turns out to be dual to the shock wave, the the Lax case where everything goes in. This under compressive case Okay, let's define few things first. There's of it's difference in two error functions. There's no here. okay, it just the error functions. There's theta, that's Gaussian. It's Gaussian going out on the right and out on the left. Again, no just And here is here are the estimates on the on the green curl. It's basically the translate you So, you have translation invariance in and in now. You have two different translation modes. And what you get is sub times Okay, times something. This is like the little SP. And UT times E2, that's like little That's like little SP, but now for time. And these depend on and But how do they depend? They depend like this. Big is little of and minus times beta of Or and plus something smaller. Okay, beta is exponentially decaying. This is actually just the opposite of the green kernel of the principal part for shock wave. If you exchange and they would be the same. So, that's kind of weird. All right. And then what else? Yeah, okay. So, that's the main part. And then Yeah, then there's some Where did it go? okay. So, so all right. And there's some some GJ is These are Gaussians moving out to plus and minus infinity. And they get turned on. We we've artificially cut it off. We we say it nothing happens at time zero and at time one and later these things show up. That's technical bookkeeping. And then the non-linear theorem is Well, if the difference between Okay, if the difference between the initial perturbation and the background wave the is much smaller than Gaussian so to the minus squared over naught for some naught big then we can do modulational error estimate. So, tilde of plus psi of plus phi of so we have to modulate in both time and space and subtract from that the background wave. This will be bounded by constant times the theta. Theta was the the pair of outgoing waves. So, this is very strong localization. This is pointwise argument. It's more It's not just L1. Now, we're saying it has to fit under Gaussian with double exponential decay. And phi plus psi remains small and their derivatives are bounded by Gaussian times epsilon. Epsilon was the size of the initial perturbation. So, you just get yeah. And but maybe maybe the best idea of the behavior already comes from the green kernel. It tells you what the response is to delta function. And you get these plateaus, these these big E's that are very very strange coming out. Okay, now should mention this the the this one is quite general. The second one is so far done only for complex Ginzburg-Landau equation with cubic quintic nonlinearity which is important. The main thing is that because it's complex case, we land out as separation of phase and by using appropriate gauge function, you can decouple the phase in way to to verify this and the cubic quintic is to make it non-degenerate so you can get existence and and and know of the existence problem. But believe that this this should be true in for general oscillations. Or the for roughly the class considered by Sunstein and Ahluwalia. All right. That's it. And that is surprising. So think it's because skipped the part to