 Welcome to this lecture. We continue to study the first order partial differential equations but now we move on to general non-linear equations from quasi-linear equations that we have been considering so far. So in general non-linear equations we are going to have 4 lectures in which we will establish the existence and uniqueness of solutions to Cauchy problems for general non-linear equations. In this lecture we are going to discuss about search for a characteristic direction. So we will be making regular comparisons with the quasi-linear case from time to time. So first we set up the notation for the Cauchy problem for general non-linear equations and the hypothesis. Then in the search of characteristic direction we will be led to the study of theory of envelopes and since we are anyway doing the theory of envelopes I would like to point out a few misconceptions which are there in the form of language sentences. So to define a general non-linear equation first order we need to set up some notations. The equation is going to feature the independent variables x and y. So we are going to deal still with the equations with the 2 independent variables. So x and y are 2 in number then the u, ux and uy there are 5 quantities which are appearing in the equation. Therefore we consider an open set in r5 called omega 5. Recall omega d we would like to use it for rd. Then let f be any arbitrary function denote f by f of x, y, z, p, q that means whenever we are going to differentiate f of something of a function of x, y then we need to do chain rule then we should know what are the variables we are using for the function f they are namely x, y, z, p, q and we need to assume that fp and fq both cannot be 0 at the same point. That means at every point in the domain at least one of the fp or fq is not 0 that is expressed by writing this equation for every x, y, z, p, q in omega 5 fp square plus fq square is not equal to 0 which means that f has to be differentiable. I said arbitrary function here but this already suggests it should be a differentiable function. Precise hypothesis we will also see when we are going to prove theorems till then this is fine. So such an f defines the most general form of a first order PDE of course first order PDE is defined for any arbitrary function which is going to come now. This always makes sense even if f is not differentiable. But we want to say that at least one of the first order derivative appears in f that is made sure by asking fp square plus fq square is nonzero. This is a differential way of expressing that ux and uy one of them features in the equation all the time. So we refer to this GE. So we used L for linear, SL for semi-linear, QL for quasi-linear, GE we use it to denote the most general form of a first order PDE. So sometimes we call it fully nonlinear equation. Sometimes we call it general nonlinear equation. So you should not be confused because fully nonlinear is a very standard usage for this kind of equation and I am calling it general nonlinear equation as mild you know step down from the word fully okay. Even though we know that L, SL, QL are all representable in the form of GE we know that. But still in this form whenever we are dealing we are doing a theory for this we do not know whether it is going to be linear or semi-linear. So that is why we call it general nonlinear equation. Now this general nonlinear equation reduces to quasi-linear equation which is here when the function f is of what type? f of x, y, z, p, q should be equal to so not x, y, z, p, q stand for x, y, z, p, q in the place of z u comes in the place of p ux comes in the place of q uy comes. Therefore f of x, y, z, p, q in the case of quasi-linear equation is a of x, y, z into p plus b of x, y, z into q minus c of x, y, z for some functions a, b, c okay. So therefore QL is a subclass of GE. So quasi-linear equations are contained in general nonlinear equations. Now aim is to extend the method of characteristics to the general equation general nonlinear equation. Of course questions come first that why find a method for quasi-linear first and then try to extend to GE. Why not directly find a method for GE? Of course there are answers. Before trying to solve a problem it is a good idea to explore simpler cases and this might help in designing a strategy for solving the most general form that is the original problem GE okay. Now coming back to quasi-linear and general nonlinear equation QL is not only a special case yes it is a special case not only a special case but also has a good geometry associated to it. We devoted a lecture for geometry of quasi-linear equations. So we use a geometry of QL to design a method to solve Cauchy problems. The geometry of GE is not as apparent as that of QL. Strategy is analysis for GE draws inspiration from that of their QL quasi-linear case. So frequent comparisons to quasi-linear case will be made throughout the presentation which helps in understanding the new difficulties and how to deal with them the process of generalizing the ideas from the quasi-linear case. The key steps in the analysis of general nonlinear equation is organized into the following steps. The key steps are this step one obtaining a system of ODs for what is called a characteristic strip they will be defined soon but I am just writing down the main steps involved so that we can keep our progress we can keep tabs on the progress where are we at each step. Step 2 is finding an initial strip. Step 3 is defining a candidate solution this was also there in QL case right this was I think step 2 there in QL defining a candidate solution and checking that candidate solution is indeed a solution to the Cauchy problem there is the last step. So for quasi-linear equations step 1 was carried out very easily in fact there was no characteristic strip there characteristic curve it was owing to the explicit geometric meaning of QL there was no step 2 and then step 3 and step 4 were called step 2 and step 3 in QL case. Now hypothesis on the function f f is assumed to satisfy C1 function on the domain omega phi where it is defined and assuming it is C1 omega phi is same as saying all the partial derivatives on the function are continuous functions on omega phi this is equivalent to that. And fp and fq satisfy this condition we already discussed about this this make sure that f of xy u ux uy equal to 0 is a first order PDE at every point. The projection of omega phi to xy plane is called omega 2 and to xyz space is denoted by omega 3. Now Cauchy data we take an interval i in R and we take 3 functions which are C1 functions such that f prime square plus g prime square is not equal to 0 for all is in i. Conscious space curve gamma described parametrically by x equal to fs y equal to gs z equal to hs. Recall this condition meant that projection of gamma to omega 2 which was called gamma 2 is a regular curve for a given space curve we already define what is the meaning of Cauchy problem. Cauchy problem for GE is consists of finding a solution that is a function which satisfies the differential equation and also such that it satisfies this condition u of fs gs equal to hs. S in i dash we do not require S to be in i as I already reminded you many times this is a notion of a kind of local solution. So i prime is a sub interval of i. So Cauchy problem for GE consists of finding a function u with a domain D which is contained in omega 2 such that for every x y in D this 5 to pull belongs to omega 5 and therefore I can apply f to it and f of that should be 0. And for each S in i dash u of fs gs equal to hs for some i dash sub interval of i. So geometrically speaking Cauchy problem consists of finding an integral surface on which a part of the datum curve lies. Since quasi linear equation is special case of GE you do not expect anything better than what was possible for quasi linear equations. Because any result that you show for a general GE equation continues to hold for QL therefore what you do not have in QL you do not expect for GE. So on the other hand you can ask questions what is true for QL is it still true for GE that is the question one can ask. So for example we neither expect we can neither expect a solution to be defined on whole of omega 2 nor the corresponding integral surface contains entire datum curve gamma that is we cannot expect solutions which are global with respect to domain. Similarly we cannot even expect solution which are global with respect to the datum curve because that is not true for QL so same thing holds here. So we can only expect the existence of a local solution with respect to datum curve this is what we have proved for the quasi linear equation and the corresponding Cauchy problem. Therefore we can expect this for Cauchy problem for GE as well. Now search for a characteristic direction because that is what we had the starting point for QL. So we would like to use our experience and expertise in quasi linear equations and try to extend these ideas to general equation. What is the idea in quasi linear case construct integral surface using characteristic curves construct integral surface using characteristic curves this was the idea of course there were difficulties in implementing this idea but we could successfully overcome all these difficulties and actually implement this idea. So therefore we would like to have something similar for GE. So here we had characteristic curves which are defined by characteristic direction. Now we ask the question what is the characteristic direction for GE how do we get it? Recall that for quasi linear equations the equation is AUX plus BUY equal to C we observed that at a point on an integral surface the normal direction will be UX, UY minus 1 and the equation tells us that ABC dot product with UX, UY minus 1 equal to 0. It means the direction ABC is in the tangent plane it is a direction in the tangent plane at P. This observation led to the definition of a characteristic system of ODE. Once we understood that this is a direction in the tangent plane then we thought of a curve which lives on the surface and such that the tangential direction is ABC that led us to the definition of characteristic ODE and solutions trace was characteristic curves and their union gave us integral surface for the case of quasi linear equations. Now remark that there is no automatic choice of characteristic direction suggested by the general equation the QL suggested very easily ABC is perpendicular to UX, UY minus 1. Therefore, ABC is a direction in the tangent plane but GE simply says F of X, Y, U, UX, UY equal to 0. So there is no directly suggested characteristic direction by the equation GE. Now how to get a tangential direction to a possible integral surface? Okay answer to this question is yes maybe there is no automatic choice very visible choice from the equation for a characteristic direction but still it gives something more actually it does not give one direction in the tangent plane maybe it gives the entire tangent plane that is a possible tangent plane we will see that. So GE puts a constraint on possible tangent planes that is all it says. Now we have to figure out one direction which is going to be in the tangent plane and in a consistent way as well consistent as we change the point we have to see that. Okay using the constraint on possible tangent planes we are going to see what is this constraint we follow a geometric argument to choose a direction at each point of omega 3 so that it belongs to the tangent space to a possible integral surface containing the point. It plays the same role that characteristic direction played for QL. So how are we going to get somebody who looks like a characteristic direction for GE we look at what is that constraint on the possible tangent planes and follow a geometric argument and then we will be able to choose one direction at each point which serves the similar purpose as characteristic direction did for QL that is the summary here. Now let us ask how do you find first assume that such a thing is there that is there is an integral surface somebody gave you integral surface understand that and then pretend that you do not know the integral surface and try to see how much of this information will be useful in getting what you want. So given an integral surface for the GE where U is a C1 function and P naught is a point on the integral surface in other words Z naught is equal to U of X naught Y naught that is a meaning right S is defined like that the third coordinate is U of first two coordinates. So equation of the tangent plane to S at the point P naught is of this form it looks like this it is a equation of the plane passing through the point X 0 Y 0 Z 0 having normal P Q minus 1 what are P and Q? P is U X X naught Y naught Q is U Y at the point X naught Y naught okay P Q minus 1 that is U X U Y minus 1 is a normal to the surface and then it will be normal to the tangent plane. So and the tangent plane passes through the point X naught Y naught Z naught so this is an equation when we do not know the integral surface how should we understand a tangent plane at a point P on S? S itself is not known. So we will only talk about possible integral surfaces and possible integral possible tangent planes to that at points of that. So admitting that U is not known to us what is left of this equation it is still equation of a plane where I cannot write what P and Q are like this because U is not known. But if at all P and Q are going to be tangent planes to the integral surface P and Q are required to satisfy the condition F of X naught Y naught Z naught P Q equal to 0. So we will not use this this is the explicit information which is known only if you know U. Now we are saying oh I do not know you admit admitting that U is not known we want to find the U. So U is not known so this is the equation of a plane any time only thing is P Q minus 1 is supposed to be the normal to a possible integral surface and that is required to satisfy this constraint F of X naught Y naught Z naught P Q equal to 0 because if at all you knew your integral surface this equation will be satisfied X naught Y naught Z naught U X at X naught Y naught U Y at X naught Y naught. Now I do not know this U therefore I simply put P and Q and this constraint. So this is the equation of a tangent plane passing through a point X naught Y naught Z naught and this is the constraint on the normals. For this formulation we need not we do not need to know U. So that is why the word possible keeps on coming here. Family of possible tangent planes through a point take a point in omega 3 done. We get lots of possible tangent planes to possible integral surfaces given by this relation. As many as solutions of T2 are for every solution of T2 you can associate a plane given by T1. We do not know how many solutions will be there for T2 it is a non-linear equation. So for each solution P Q of T2 in fact what is to be expected is T2 is an equation involving two parameters P and Q and one constraint therefore what we expect here is one family of solutions will be there. Corresponding to that you will have one family of planes. So for each solution of T2 P Q of T2 we get a plane that is true. T1 T2 users only the information which can be extracted from the equation. For example T1 it uses nothing it just equation of plane we always write only unknown quantities are P and Q they are supposed to satisfy this T2 which is coming from GE. So I am not making use of any explicit knowledge of a known integral surface here I do not know integral surfaces. I am proposing this T1 planes in fact going to be a family of planes which are one parameter family because two parameters are tied by one equation essentially means only one of them is free therefore this will be one parameter family of possible tangent planes. Now T2 let us see what it have what it means for the quasi linear equation case. So we can explicitly write down T2 for quasi linear equations this is one where P naught is the point X naught by naught Z naught and this is the equation right f of XYZ P Q equal to 0 is this it reads as this from here one of them is going to be nonzero either A or B let us assume B is nonzero then I can divide by that therefore I can solve for Q in terms of P I have solved. So you see Q is a function of P now I take this and substitute in T1 therefore T1 looks like this. So this is the equation of a possible tangent plane where P is free. So it is a one parameter of one parameter family of possible tangent planes the only property is that they pass through the point X0 Y0 Z0. So you have a point X0 Y0 Z0 now you are looking at a plane like that another plane the plane is infinite these are planes in 3D and another plane like that and so on. The only common thing is this point P0 they all pass through this point. So it is a one parameter family of possible tangent planes index by this parameter P in R it is because why how did we get this explicitly this P here because we could solve from the equation we express Q in terms of P explicit expression and then we substituted and got now for GE also we would like to do the same thing. Okay let us continue for QL this is the equation it represents a one parameter family of possible tangent planes index by parameter P. So for GE exactly this right this is T1, T2 as written earlier. So this represents a one parameter family of possible tangent planes at P0. So maybe we may have to analytically impose some conditions which say that Q can be solved in terms of P at least locally for around a fixed value of P let us say P equal to P0 small P0 you can express Q as a function of P or vice versa for a fixed value of Q equal to Q0 you can express P as a function of Q and then go and substitute here you will get a one parameter family of possible tangent planes. In the case of QL these two equations reduce to just this that is because we could express from here Q as a function of P went back and substituted T1 and we got this it is a family of tangent planes indexed by the parameter P in R. In the case of nonlinear equations we do not expect that we can have solutions global solutions therefore what typically happens is Q is a function of P in a small neighborhood of some fixed value of P. We will see the rigors of the details later, rigorous details. Story so far we were motivated by the analysis of Quasinary equations we were searching for a tangential direction that belongs to a possible integral surface but we ended up with a one parameter family of possible tangent planes. Now we need to pick up one direction from this how do you pick up that that is a problem let us see how that is overcome. However for Quasinary equations we have one parameter family of possible tangent planes as above in the previous slide and also the characteristic direction we also had a characteristic direction and the fact the geometric idea of envelope connects these two. The envelope of the one parameter family of tangent plane possible tangent planes that we described earlier envelope of that turns out to be a cone and the characteristic direction turns out to be a generator of that cone so therefore there is a connection. So we pursue this idea of envelopes for general equations. Now why the idea of envelope should succeed the one parameter family of planes described by T1, T2 and its envelope share a common tangent plane that is a property of envelope. If you have a one parameter family of planes and you take its envelope whenever this envelope intersects any member of the family it intersects it touches that means they share a common tangent plane. Therefore if you can choose a tangential direction from the envelope that will also be a tangential direction in the for the surface. It will also be a direction in the family of planes that we are considering which is what we want. So we hope that envelope is a good thing where it is easier to pick up that particular direction which was the case for quasi-linear. So it may be easier to pick a tangent vector from the tangent plane of the envelope we would get a characteristic direction that we are looking for. The envelope of planes is a cone I have put it in quotes because I am going to explain later since all the planes pass through a point fixed point P0. So a generator of a cone may be chosen as the characteristic direction that was the idea. Now let us have a brief excursion into the theory of envelopes. Envelopes of families of surfaces. So we take the family of surface of this special type actually graphs of functions z equal to g of x y lambda s lambda is given by this lambda is a parameter anything you want to do you must assume that functions are differentiable otherwise there is very little one can do. So lambda is a parameter g is a differential function. So I have got a family of surfaces. Now differentiate that with respect to lambda so you get this side it is 0 this it is just g lambda of x y lambda. Now for each fixed lambda let c lambda denote the curve of intersection of s lambda and 1 means set of all x y which are common to this and this. This is what we expect whether it will be curve etc one has to see okay this is a surface this is another surface intersection two surfaces we think it is a curve okay that is the background behind calling this. It all depends on how the lambda appears okay but what is the envelope first you look at the family differentiate you get some other equation look at the common points fixed lambda look at the common points set of all x y z such that this 3 tuple is there here as well as here that is what we have written the two equations. The envelope of the one parameter family of surfaces is defined as the union of c lambda s. Now let us go back and see what is the ql we had a family of tangent planes right possible tangent planes and what is its envelope let us see that. So this is the family remember it depends on p so instead of lambda we have a p here so we need to differentiate this with respect to p that means that this will be 0 equal to this quantity because p appears only here small p so this is 0 as well as this equation is satisfied what does that mean this is not there it is only this it means it says some relation between z minus z naught and y minus y naught please do this computation slowly so this is what we end up with cp p is the parameter note that cp is the same straight line for each p it does not depend on small p it is just straight line it all depends is abc at p naught cp is a line passing through p naught having this direction abc what is abc it is a characteristic direction at the point p naught analytic expression for envelope so we assume that we can write lambda as a function of x y from this equation and then go back and substitute in the equation for the family of surfaces which will give us z equal to capital G of x y instead of lambda I have expressed lambda as a function of x y so I write g x y that is assumption okay if you can do this then this is an expression one single equation for envelope let s lambda be a one parameter family of surfaces and let lambda represent solutions of this so that I am going to go back and substitute in this therefore e denotes the envelope we have defined as a union now it is simply this g of x y g x y let c lambda be the curve of intersection of z equal to g as well as g lambda equal to 0 conclusions assume that c lambda is non empty for every lambda in fact for some surfaces c lambda could be empty for some values of lambda in that case I do not want to talk about that therefore I assume that c lambda is not empty for every lambda then the envelope intersects every member of the family s lambda because I assume c lambda is not empty therefore envelope intersects every member of s lambda along c lambda that is my definition right important thing is the second assertion the envelope and s lambda touch each other that means at every point which is common to them which is nothing but c lambda by definition the envelope have envelope and s lambda have a common tangent plane proof of one so take a point on e definition of the envelope says that there is a lambda because the union of the lambda right of c lambda over lambda so there is a lambda such that x y z belongs to c lambda but c lambda means it is there in s lambda also because that is a part of the condition of the definition of c lambda therefore one follows that is simple proof 2 we prove that the envelope and s lambda have the same normal direction at points which are common to both of them that will establish that the tangent plane is same so we are going to enquire into the normal directions for e as well as s lambda so take a point in c lambda normal direction to s lambda at that point x y z is given by g x g y minus 1 this is a general fact that we have discussed many times whenever you have a function phi of x y z equal to 0 the normal direction is phi x comma phi y comma phi z in this case the envelope was this right z equal to g x g x phi okay here I have written typeset in small font because it was not filling so normal to the envelope is exactly this you need to differentiate this with respect to x x dependence comes in x as well as in the lambda that is why x derivative and lambda derivative once you take lambda derivative you have g their x dependence so g x similarly y derivative and minus 1 which on simplification it is this because this is 0 g lambda of x y g x y 0 therefore it is simply g x g y minus 1 that is it so it showed both are same so therefore they share a tangent plane that means they touch each other now let us look at this PDE this is a non-linear PDE f of x y z pq is equal to p square plus q square minus 1 now let us write the family of possible tangent planes z equal to px plus q y that is coming from equation of a plane and pq are constrained to satisfy this equation therefore p square plus q square minus 1 equal to 0 from here luckily like q like in ql we could solve for q in terms of p which is root 1 minus p square of course this makes sense only when mod p is less than 1 of course you have 2 solutions for q thus we have the one parameter family of tangent planes I go back and substitute for q the p now I will find envelope of this we need to differentiate this with respect to p that will give us this relation and express p as a function of x and y we get this now go back and substitute in this equation that will be the envelope envelope is z square equal to s square plus y square mod z bigger than 1 so envelope lies on the double sheeted cone with vertex at the origin the figure is there in the next picture and a few members of the family of possible tangent planes is also there okay these are various planes okay their envelope is this blue color the cone now a few misconceptions that are prevalent envelope of a one parameter family of planes is a cone this is not correct okay look at this example this is an exercise g of x y lambda equal to ax plus by plus c z plus d plus lambda times a1x to v1 plus vnz plus d1 equal to 0 it is a family of planes so find the envelope it turns out to be the intersection of these two planes so message envelope depends on the manner in which the parameter appears in the definitions of the families of surfaces or curves okay so we cannot have a blanket statements like this one parameter family of planes envelope will be a cone no such things are not true so in the literature there are at least three ways of defining the notion of envelope their interrelations may be found in a beautiful article written by coq note that for our purposes it does not matter what is the correct envelope that we have to consider correct definition etc what all matters is does it serve a purpose whichever you follow so the notion that we define is good enough as it is used only in guessing a characteristic direction and success of that we will see in the next lecture this is a reference for coq's article it is called envelopes notion and definiteness contribution to algebra and geometry he has written this article in 2007 it is a beautiful article let us summarize what we did note that the quasi linear equation defined a characteristic direction right away g e gives only possible tangent planes that is all fact is that geometric idea of envelope connects the one parameter family of possible tangent planes for q l and the characteristic direction we have just seen that we have proved this a brief theory of envelopes was presented in the next lecture we are going to work further the next steps in fact we have not yet found characteristic direction we will do that in the next lecture and continue the analysis of Cauchy problem for general non-linear equations thank you