 We start the discussion of method of characteristics for quasi-linear equations from this lecture onwards. The outline for today's lecture is, first we recall the assumptions and notations which are used in the context of quasi-linear equations and we emphasize the importance of mathematical precision. We see couple of statements which could wrongly imply wrong things which are written in language and that can be read in many ways. So therefore, this highlights the importance of writing or importance of understanding mathematically what we write in language. And then we take first steps into method of characteristics for quasi-linear equations. First we present the inspiration behind this method and we carry out the step one out of the three steps. So in the next lectures, step 2 and step 3 will be discussed. So assumptions and notations used in the context of quasi-linear equations recall from lectures 2.1 and 2.5, quasi-linear equation we denote by QL and that is an equation AXYUUX plus BXYUUY equal to CXYU, where ABC are assumed to be C1 functions defined on omega 3. Needless to say omega 3 is an open subset of R3 and connected we already discussed whether we should have the connectedness or not. This is the most important thing. We do not want the coefficients of the partial derivatives UX and UI namely A and B vanish simultaneously at the same point. So therefore, we require that at least one of the A and B must be non-zero at each and every point in omega 3. And the projection of omega 3 to XY plane is denoted by omega 2, omega 2 is those elements of R2 such that XYZ belongs to omega 3 for some Z in R. Now Cauchy problem, given a space curve gamma described parametrically by gamma X equal to FS, Y equal to GS and Z equal to HS, S belongs to I, where I is an interval and FGH are C1 functions defined on the interval I and such that F dash S square plus G dash S square is not equal to 0 for all S in I. This we discussed that this corresponds to the projection gamma 2 X equal to FS and Y equal to GS. This is a curve in plane and this curve is what is called regular. This is this assumption is something to do with the smoothness of the curve gamma 2. This means the tangent is well defined and each and every point of gamma 2 for example, this is gamma 2 and you take a point, the point looks like FS GS and at this point the tangent line, this is not a good picture, we will just write a good place, this point, this has the direction of F premise G premise. Find a solution to the quasi-linear equation such that U of FS GS equal to HS. That means on the curve gamma 2, FS GS describes a typical point on gamma 2, when you take U of that then you will get into 3 dimensions Z equal to that, Z is going to be H of S. That means it is lying on the surface Z equal to U of XY and this we require for S belonging to a sub interval of 5. So it means that a part of the curve gamma lies on the surface. In other words, we are looking to construct a surface, construct an integral surface which contains a part of the curve gamma. So what we are going to see in the future lectures is that if you are given this curve gamma, this is in R3 and given any point on that, that looks like FS0 GS0 HS0 under some conditions we are going to show that, let us call this point P0. Some conditions we are going to show that there is a surf integral surface S Z equal to UXY such that it contains this point P0 on some curve nearby that, some gamma, the initial data, datum curve gamma is contained on that surface. Of course we need some assumptions to assert that there is such a function U. We have already seen that this question arose already in the case of linear and semilinear equations when we try to solve Cauchy problems. Some assumptions need to be made and then we will show such a function exists. This will be the final result at the end of implementing step 3 of the method of characteristics. Now few points about importance of mathematical precision. Before that let us state this theorem. This theorem we proved in the last lecture, lecture 2.5. So this said that given a function U defined on a domain D in omega 2 which is a C1 function and S denotes the surface Z equal to UXY that is the graph of this function in R3 then saying that this S is an integral surface is same as saying that S is a union of characteristic curves for Cauchy linear equations. This is what we discussed in lecture 2.5 we proved this theorem. Now a remark about this theorem. Theorem is of great help in the search for a solution to Cauchy linear equations namely the Cauchy problem for Cauchy linear equation Cp for Cauchy linear equations. The assertion 2 implies 1. What is 2? 2 says the surface S is a union of characteristic curves. That implies that surface S is an integral surface. Of course here U is in the background which is fixed in the background. So the assertion 2 implies 1 suggests that an integral surface if exists in other words a surface Z equal to UXY where your solution if exists may be constructed as a union of all characteristic curves. However, theorem does not assert that the geometric object which is formed as a union of all characteristic curves is necessarily a surface. In other words, if you take union of characteristic curves it is not necessary that the third component is expressible as a function of first two coordinates it does not say that. The theorem says you give me a function U and look at the graph of U. Now it is a comment about the graph of U. The graph of U is a surface it S denoted by S, S is an integral surface if and only if it is union of characteristic curves that U has to be brought beginning. So therefore 2 implies 1 suggests that an integral surface may be constructed. It does not assert that geometric object which is formed as a union of characteristic curves is necessarily surface that means there is a U hidden behind that such that the third component is a U of first two components XY and moreover even if it is a surface it is not clear whether it is going to be an integral surface. So here surface means Z equal to UXY for some function U that is what we mean in both of this. Now mathematical analysis takes over in deciding whether the geometrically constructed surface is an integral surface or not. So we will come across this analysis in a future lectures maybe in the next one. Important thing is this precision mathematical precision is very important. So I call this mind the language. So mathematical statements when put in plain English sentences or any other language sentences could lead to imprecise expressions which could be misunderstood. The statement if two integral surfaces intersect at a point if two integral surfaces intersect at a point then they intersect along the entire characteristic curve through P. This statement is a classic case of such a sentence. What does this sentence say? Suppose you have two integral surfaces they intersect at a point P then they intersect along the entire characteristic curve through P. What do you mean by entire characteristic curve through P? The longest possible characteristic passing through the point P what is it? So a corrected formulation of the sentence is stated in the following corollary and it follows immediately from theorem. What is the corollary? Let S1 and S2 be two integral surfaces for quasi-linear equation QL such that their intersection is non-empty. Of course that is not a big deal because here we are saying it intersects at a point P therefore we have not made much difference. The sentence is as it is non-empty. Let P be a point in the intersection so that takes care of the first condition the conditional statement if two integral surfaces intersect at a point P that is captured in 1 and 2. Now we have to see how we are capturing the conclusion they intersect along the entire characteristic curve. Then some part of the characteristic curve passing through P lies on both S1 and S2. So we can only say some part near P alright there is a big characteristic curve passing through P but entire curve why will it be there and both of them. Some part is there that is reasonable to believe and that is what is true. Now second point another example of a misleading statement that is widely in use is intersection of two integral surfaces is a characteristic curve because we believe that if two surfaces intersect imagine two planes intersect it is a line is that true always. If two intersection of two planes is not a line always it could be a plane it could be the same plane right it need not be a line all the time. So that is in fact happens which is a counter example to this statement so we have to be careful. The next example illustrates that intersection of two distinct integral surfaces is not necessarily a curve forget about characteristic curve. Any curve on the intersection need not be a characteristic curve even that is true that simply because the example we are going to see the two integral surfaces intersect and give a surface. Now however that through every point of intersection there passes a characteristic curve which lies on both integral surfaces. So a corrected formulation of the original sentence is stated as next corollary. So intersection of two integral surfaces is a characteristic curve this is not precise it is not correct as we usually understand this. The standard meaning of this turns out to be that the statement is incorrect. So corrected formulation we are going to give before that let us do this example. Suppose u is a solution to ql defined on d then so is another function v defined on d1. Now how do I define this v it is going to be using u therefore d1 I will take it to be a subset of d it is a proper subset of d ok. Of course we remain solution to the ql and look at the integral surfaces SU are right to denote that this surface is defined using the function u this is used this sv this surface z equal to vxy is defined using the function v. They are different because domains of the functions u and v are different ok it is true that sv is a subset of SU but they are different. Two functions are different the moment their domains are different. So but intersection of SU and sv is sv which is an integral surface so it is not a curve it is an integral surface through every point of sv we can find a curve which is not a characteristic curve and another that is a characteristic curve. This will not be the case if the two integral surfaces intersect without touching and that is what is the content of the next curve. So when is the intersection of two planes is a straight line when they do not match right some the two planes it is not a line if and only if they are the same planes for a surface the planar approximation will be the tangent plane. So if we say that tangent planes are not same then it will be a curve that is what is the next corollary. So two surfaces in R3 are said to touch each other if at each of the points which are common that means wherever intersection whichever point is in intersection are the two surfaces at those points the tangent planes are the same. So our theorem is going to be for surfaces which do not touch each other. If two integral surfaces intersect without touching each other and the intersection is a curve then it is a characteristic curve this is the correct corollary. If the touch each other it is like tangent planes are one and the same whenever planes are same we got the intersection to be plane right for planes. So here we do not expect straight line there for planes therefore same thing here two surfaces if they intersect by touching each other then we are not making any statement but if they do not touch each other and intersect that means points are in common then the intersection if it is a curve so which means it may not be curve as well it can be a point is a curve then gamma is a characteristic curve. So to prove that gamma is a characteristic curve what we need to do we have to prove that the tangential direction at any point on the curve is the characteristic direction let us prove that. So the tangential direction to gamma at p belongs to the tangent plane to both the surfaces S1 and S2 at that point p as gamma is lying on both of them if the direction of the tangent is not along the characteristic direction then it follows that the direction of the tangent to gamma at p and the characteristic direction apbpcp they form a linearly independent set in a two dimensional tangent space this implies that both the tangent spaces are the same all the directions in the tangent planes for both S1 and S2 at the point p are the same and hence tangent planes coincide which means they touch each other therefore we have assumed they do not touch each other therefore this contradiction proves that the tangent to gamma at p is not independent of characteristic direction it is proportional to the characteristic direction at p which means it has a character the curve gamma has characteristic direction at p and p is arbitrary point that means the curve is a characteristic curve. Now in the proof we have not used the theorem therefore why do we use a word corollary therefore this question arises terming it as a corollary is acceptable as one can prove corollary using theorem you can use the theorem and prove the corollary that is left as an exercise to you we have seen one proof other proof uses theorem directly the statement of theorem and that is left as an exercise. Now the question is do integral surfaces as in the corollary exists that is two integral surfaces which intersect but do not touch each other whether such surfaces exist corollary is concerned with the two integral surfaces which intersect without touching having a curve in common and corollary asserts that such a curve is necessarily a characteristic curve for quazilinear equation. We came across such integral surfaces when a question problem has more than one solution look at this example ux equal to u ok this equation we are always considering partial differential equations in two independent variables unless otherwise stated so this is a function of two variables x and y and the equation is ux equal to u so this is like a ODE in the x variable and we are given initial condition ux0 equal to e power x if you want Cauchy data what is it x equal to s y equal to 0 z equal to e power s s belongs to r this problem has infinitely many solutions we saw this already there are the form u of x y equal to e power x into ty where t is a c 1 function t of 0 should be 1 therefore as many c 1 functions as you have with the property t0 equal to 1 you have so many solutions clearly infinitely many. Now consider two integral surfaces z equal to u x y and z equal to u tilde of x y two integral surfaces defined by this formulae one is e power x plus y other one is e power x minus y both are solutions to this Cauchy problem the two integral surfaces intersect all along the datum curve that is s0 e power s that is intersection in figure on the next slide the surface s corresponding to u e power x plus y is depicted in black the one for s tilde is depicted in blue and datum curve is in red color here so 3D picture ok so intersection is shown here that is the datum curve blue is one integral surface the other one is another integral surface. Yeah I am getting these integral surfaces how did I get this example we use the idea that if two integral surfaces touch each other what happens the tangent planes are same the directions in the tangent planes are same and u x u y minus 1 for both integral surfaces would be the same because tangent plane is same normal has to be same plus or minus ok we are not insisting that normal has unit length etc. So therefore direction is direction any other normal will be proportional to this they will be same so in these examples we made sure that normal are not the same therefore tangent planes will not be the same that was what was done and please do a few more examples of pairs of integral surfaces as above ok now let us discuss method of characteristics for quasi linear equations first we start with an inspiration for this method what inspired this method of course it is no secret because we just saw one theorem at the beginning of today's lecture and even in lecture 2.5 that is the inspiration integral surface as a union of characteristic curves that is the idea so this is the theorem ok this is inspiration it is not saying that you construct a union of characteristic curves that is automatically an integral surface it does not say that it still requires you that the surface is given by z equal to u x y we still have to do some work but inspiration it works. So in this example u x equal to 0 u x y equal to sin y and integral surface is blue and that can be obtained as union of this black lines and this magenta is the datum curve that is 0 y sin y 0 s sin s ok this is that one. So the method relies on using characteristic curves associated to quasi linear equations to find a solution to Cauchy problem it believes that this implication 2 implies 1 yields a solution from characteristic curves it believes that ok let us take the union of characteristic curves somehow we can get that function u and it will be alright that is what the method believes. In the figure on the last slide the curve in magenta is a datum curve characteristic curves passing through points of gamma are in black and we saw that the blue thing is a union of black lines the integral surface may be obtained as a union of all characteristic curves passing through points of gamma. So it is working in that example of course another point that we knew the answer beforehand the function but inspiration is fine. So what are the main steps first step passing characteristic curves through points of gamma these are strategy we tried for linear and semi linear equations in the last lecture. Defining a candidate solution u using inverse function theorem this is a key step step 3 is establishing that the function u defined in step 2 is actually a solution to the Cauchy problem question yeah you have given 3 steps can we always implement those 3 steps successfully that is a question step 1 is ok what is the step 1 it is to pass characteristic curves through points of gamma that is a step 1 passing characteristic curves through points of gamma that is ok because A, B, C are 7 functions in omega 3 Picard's theorem will give you that characteristics pass through every point of gamma. Second step this is where we encounter difficulty this is where the problem lies because we need to apply inverse function theorem inverse function theorem requires some conditions to be met. So we will be forced to impose compatibility conditions between the PDE and gamma even then the integral surface may not contain entire gamma or a solution may not be defined on whole of omega 2 these are the 2 things we somehow want I want Cauchy problem so that the integral surface contains the entire datum curve which is given to me and it is defined on whole of omega 2 omega 2 being the projection of omega 3. So this is desirable but neither of these 2 may happen we will see using examples step 3 is a cakewalk one step 2 is carried out let us discuss step 1 passing characteristic curves through points of gamma finding an integral surface containing the datum curve now we have decided a piece of datum curve means what does that mean we need to view a surface around gamma gamma is given to us and we need to find a surface something like that a surface yes such that the resulting surface is an integral surface for the equation a surface could be woven around gamma by actually take a point of gamma take a characteristic through that through that through that okay like that repeat this at every point of gamma by passing curves through each point of gamma further if these curves are characteristic curves for QL then the surface is expected to turn out to be an integral surface you have to be very careful I use the word expected usually people mean expected means it will happen no this may not happen hoped maybe that is a correct word then the surface is hoped to be turned out to be an integral surface by the theorem so weaving an integral surface yeah this is a computer generated picture of what I have just written so this is the datum curve and you pass characteristic curves through the each point and then hopefully you will get a surface and that surface is expressed like z equal to x y and u is a solution to the Cauchy to the PDE it will be a solution to the Cauchy problem the sense Cauchy data will be satisfied that is how you are getting the solution okay how do we implement the step one take a point P on the datum curve it looks like FS GSHS for some SNI the characteristic curve through P is the image or trace of solutions to characteristic ODEs this is the system of characteristic ODEs dx by dt equal to a d by by dt equal to b dz by dt equal to Ca what are ABC ABC are in QL what is QL AUX plus BUY equal to C satisfying the initial conditions I need the solutions of this ODEs to pass through this point so at t equal to 0 X of 0 is FS Y of 0 is GS Z of 0 equal to HS by Cauchy Lipschitz Picard theorem we are assuming ABC are C1 function they are locally Lipschitz IVP will have a unique solution let the solution be represented by this notation X equal to capital X of TS Y equal to capital Y of TS Z equal to capital Z of TS of course ODEs were in the variable t then why are we writing S here it is because the characteristic ODE we have solved using initial conditions which depend on S the initial conditions depend on S to remember that we write X of TS Y of TS Z of TS so it is defined for t belonging to some interval now you see JS the interval may change from S to S for some years it may be one interval for another S it may be a different interval only thing that we can assure is that 0 belongs to JS now recall a lemma that we did on reparameterization of characteristic curves we may take JS equal to R according to that lemma the reason being let me recall the reason the system of characteristic ODEs a system of autonomous equations and for that if you are looking only at the trajectories you can always change change the independent variable that is namely t there to make that the interval JS is actually equal to R trace will be the same now remaining steps in the method of characteristics will be carried out in lecture 2.7 the next lecture let us summarize what we did we understood that mathematical statement should be written with as much precision as possible full precision okay in mathematical statement it has to be fully precise and everyone should understand the same meaning of the sentence we should clearly write or understand the mathematical meaning of statements made in non-mathematical languages using the connection between integral surface and characteristic curves we hope to solve Cauchy problems for Cauchy linear equations to achieve this goal we proposed three steps out of which the first step was carried out successfully thank you