First Order PDE Solution Method Issues

Question

Greetings, I have a rather pedantic problem that's honestly been holding me back for over a month & is something neither stackoverflow nor some of my professors nor some graduate students have been able to help me with so I apologize if this question is too elementary but I'm really stuck & this is more a question of understanding man-made formalisms relating to first order partial differential equations than the basic logic of the subject. Links to all references in [ ] brackets below are on my stackexchange page here as I've been restricted to only 5 links on this page.

Just to quickly let you know what I'm asking, the first is about solution methods to first order PDE's & pretty much requires you to have familiarity, by name, with Lagrange's method, Charpit's Method, Jacobi's Method & Cauchy's Method of characteristics & understand the distinctions between them (discussed in [A], [B] & [C]). Although many books unfortunately just take only one of these approaches, such as [1], [2], [3] & [4], other books, such as [5], [6], [7] & [8], at least let you know there are these different methods though I'm a little confused as to the logical differences between them. My second question is basically about the "characteristic equations" $\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)} = \ ... \ $, more specifically about the distinction between $\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$ & $\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \ $, something obviously motivated by, related to, and may even answer, the first question though it seems to have taken on a life of it's own.

The first question is about solution methods for first order PDE's, as I understand it there are basically two methods: Lagrange's method, with Charpit's extension to the nonlinear case, & Cauchy's method of characteristics which is supposed to hold in both the quasilinear & fully nonlinear cases. I've only superficially studied both of these but can't really advance any further because both methods look practically the same & I'd like to know the fundamental difference between them. In [8] they develop Cauchy's Method of characteristics for a nonlinear equation by, at one stage, borrowing from results they'd established using Lagrange's method (i.e. developing something in terms of parameters then implicitly assuming parameter-independent theory), while in [5] they discuss Lagrange's method after developing Cauchy's method & do it entirely in terms of parametrizations (something you wouldn't even know was possible had you only studied from [2] which does everything without mentioning parametrizations). As far as I can tell the only difference is that Cauchy's method applies to both quasilinear & fully nonlinear cases while Lagrange's method applies only to quasilinear equations, though Lagrange & Charpit's method applies to the nonlinear case when you have a function of two independent variables (though [6] & [7] say that Jacobi's method is just a further extension of Charpit's method to functions of n variables though confusingly [8] says Jacobi's method is an extension of Cauchy's method, hence you see why I'm confused). I originally thought the main distinction between these methods was that Cauchy's method required parametrizations whereas the other methods ignore the parameters (i.e. whether you end up solving $\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$ or $\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \ $), but if Cauchy's method is established using non-parameter results at key steps as I've alluded to above, & that the whole method can be established in the Jacobi case with or without parameters, then obviously this can't be the fundamental distinction. I know that Cauchy's method requires the specification of initial conditions, so maybe this is the only distinction? I don't know how to make sense of the initial condition aspect of Cauchy's method if Jacobi's method can be established both with & without initial conditions as my links would have you believe. Basically I'm hoping someone could make sense of this for me.

The second question is about the distinction between $\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$ & $\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \ $.

This book (Page 63) refers to solving $\frac{dx}{P(x,y,z)} = \frac{dy}{Q(x,y,z)}$ as a shorthand for solving $\frac{\frac{dx}{dt}}{P(x,y,z)} = \frac{\frac{dy}{dt}}{Q(x,y,z)} \ $ since they are both meant to represent the definition of a curve as the intersection of two surfaces, however there's also the idea of the first notation as being completely invalid & requiring justification through the formalism of differential forms. The essay Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations talks about the "dishonesty involved" in the first notation & how "one should bear in mind that this misleading notation is just another way of writing an autonomous system of differential equations", which leads me to wonder whether Lagrange's method, or Lagrange & Charpit's method, is not just a 'dishonest' exposition of Cauchy's method of characteristics? If Lagrange's method is just a dishonest exposition of Cauchy's method, then how does it give a general solution not requiring in initial conditions while Cauchy's method only gives a complete integral? If it isn't then what is the distinction between these methods?

Thanks for reading, if you can help I'd really appreciate you taking your time to do so.

Answer 1

Let me work with $n$ dimensions: you want to study the vector field $$ X=\sum_{1\le j\le n} a_j(x)\frac{\partial}{\partial x_j}, \tag {1}$$ and in particular find the so-called first integrals of $X$ i.e. the functions $f$ such that $Xf=0$. You introduce the system of ODE: $$ \dot x(t,y)=a(x(t,y)),\quad x(0,y)=y. \tag {2}$$ The solutions $t\mapsto x(t,y)$ are the integral curves of $X$. You realize easily that a function is a first integral iff it is constant along the integral curves of $X$: just compute $$ \frac{d}{dt}\bigl(f(x(t,y))\bigr)=\sum_{1\le j\le n} \frac{\partial f}{\partial x_j}(x(t,y))a_j(x(t,y))=(Xf)(x(t,y)) $$ It means that solving the PDE (1) is somehow equivalent to solving (2).

Now the notational business. It is tempting to write (2), which is $ \frac{dx_j}{dt}=a_j(x), 1\le j\le n, $ symbolically as $$ \frac{dx_1} {a_1(x)}=\dots=\frac{dx_n} {a_n(x)} $$ since they are all equal to $dt$ ! Well just take this as a symbolic notation which eliminates the presence of the parameter $t$.

Now the Cauchy problem for this autonomous vector field $X$: find an hypersurface $\Sigma$ to which $X$ is transverse, i.e. $X$ is not tangent to $\Sigma$. Then the Cauchy problem $$ \begin{cases} Xu=f,\quad \\ u_{\vert \Sigma}=g \end{cases} $$ has locally a unique solution: this problem is equivalent to the scalar ODE $$ \frac{d}{dt}\bigl( u(x(t,y))\bigr)=f(x(t,y)),\quad u(x(0,y))=u(y)=g(y) \text{ for $y\in \Sigma$}, $$ so that $$ u(x(t,y))= u(y)+\int_0^tf(x(s,y)) ds\quad \text{ for $y\in \Sigma$}. \tag{3}$$ Note that $y$ moves on $\Sigma$ ($(n-1)$ degree of freedom) and $t$ in $\mathbb R$ so that it is a nice choice of coordinates to pick $y\in \Sigma$ and $t\in \mathbb R$.

There are variants of this when the vector field is not autonomous, i.e. is of type $$\frac{\partial}{\partial t}+ \sum_{1\le j\le n} a_j(t,x)\frac{\partial}{\partial x_j}. $$

More comments on the quasi-linear case and the general method of characteristics: the quasi-linear Cauchy problem $$ \frac{\partial u}{\partial t}+\sum_{1\le j\le n} a_j(t,x, u)\frac{\partial u}{\partial x_j}=b(t,x,u),\quad u(0,x)=u_0(x). \tag{4}$$ has a linear companion $$ \frac{\partial F}{\partial t}+\sum_{1\le j\le n} a_j(t,x, v)\frac{\partial F}{\partial x_j}+b(t,x,v)\frac{\partial F}{\partial v}=0,\quad F(0,x,v)=v-u_0(x) \tag{5}$$ where $t,x,v$ are independent variables. It is not difficult to solve using the linear method of characteristics outlined above. Then since $\partial F/\partial v=1$ at $t=0$, the equation $ F(t,x,v)=0 $ determines implicitely $v=u(t,x)$ and the expression of derivatives of $u$ in terms of derivatives of $F$, e.g. $ \partial u/\partial x=-\frac{\partial F/\partial x}{\partial F/\partial v} $ imply that $u$ solves the Cauchy problem (4). Here also the notational industry is working full throttle. People would write $$ \dot x=a(t,x,u)\quad \dot u=b(t,x,u)\quad \text{which is } \frac{dx_j}{a_j}=\frac{du}{b},\quad 1\le j\le n. $$