Wave-Particle Duality in PDE?

I am reading Arnold's Lectures on Partial Differential Equations. It is definitely a good book, yet sometimes I am a little bit confused.

One theme of the first chapter seems to be

From the physical point of view this case is the duality that occurs in describing a phenomenon using waves or particles. The field satisfies a certain first-order partial differential equation, the evolution of the particles is described by ordinary differential equations, and there is a method of reducing the partial differential equation to a system of ordinary differential equations; in that way one can reduce the study of wave propagation to the study of the evolution of particles.

When I first read it, it sort of makes sense. If one understands the evolution of each particle, then globally one should be able to describe the wave. However, when I read on, it starts to confuse me.

For instance, in the $x-y$ plane one has the following equation \begin{equation} (\frac{\partial u}{\partial x})^2+(\frac{\partial u}{\partial y} )^2=1. \end{equation} It can be shown that if one has a convex closed curve, the define a function on the 'outside' region by mapping a point to its distance to the curve, then this function solves that equation. Conversely, any function $u$ solving that equation is the distance to a certain curve.

Then the author asks us to understand the wave-particle duality in this case, which is something quite puzzling to me.

In another case, the author talks about Newton's equation and Euler's equation for a particle moving freely on a line:

\begin{equation} \frac{d^2}{dt^2}x=0, \end{equation} which says the acceleration is $0$; and \begin{equation} u_t+u_x u=0, \end{equation} where $u(t,x)$ is the velocity at location $x$ at time $t$. Then again he asks us to understand the duality in this case.

So could someone give a hint on what the author exactly means by this duality? How one should understand it (in the examples above as well as in other cases)?

Thanks!

This is something I'm still trying to figure out & just wish I knew fully so as to be able to answer this question properly - as far as I understand it at this moment any first order linear or non-linear PDE may be (equivalently) re-expressed in the form of a Hamilton-Jacobi equation by a change of variables, (No. 37), which in turn can be interpreted as providing the (first order partial) differential equation for a family of 'geodesically equidistant' surfaces (wave-fronts!), along each of which the action-functional that generates the Hamiltonian that generated the H-J equation is constant (in physics the action is constant along those physically-realizable wave-fronts, & we're assuming already has the equations of motion for this action inside it), & the characteristics for those surfaces are Hamilton's equations (i.e. the Euler-Lagrange equations in a more symmetric choice of coordinates, hence the notion of geodesic equidistance & amazingly the characteristics for this PDE we derived from the action functional has for it's characteristics the very same equations as those which minimize the action functional directly!).

In other words, from a mechanics point of view the calculus of variations gives us a means to determine the equations of motion directly by extremizing an action functional, resulting in solving ode's for the path of a particle. Hamilton-Jacobi theory instead derives a PDE for the action functional itself (which assumed to already contain the solutions to those ode's), thus you're implicitly associating wave surfaces to the particles inside the action by by turning an ode problem into an equivalent pde problem. This is the origin of the classical wave-particle duality as far as I understand it. I find it really interesting that the Lagrange-Charpit characteristics for this surface turn out to be the same equations of motion for the action functional directly, so it seems like the method offers nothing new mathematically apart from providing the possibility of applying separation of variables.

This is the kind of thinking Schrodinger uses in his first two papers when he extended this to a quantum form of the wave-particle duality. As far as I see it (at the moment) he began from the Hamilton-Jacobi equation (which encodes definite particle paths), then instead of solving it directly he solved it 'on average' over all of space (I guess (?) motivated by the fact that the Hamiltonian is interpreted as energy) & got experimentally-validated solutions, thus giving people the idea that definite particle paths do not exist, they only exist on average...

Some starter references I can give if you're interested are Schrodinger's papers, the Chester book referred to in that link as well as Rund's Hamilton-Jacobi Theory book. If you do get side-tracked into this world as I have I'd love to talk about it all properly & flesh out ideas.

He is using the wave/particle duality as a metaphor for characteristics. The method of characteristics, as he describes later, is to associate a vector field to each partial differential equation, and from that vector field look at motion of particles (I.e. the integral curves).

At the page you are on, he has not yet defined characteristics, so you are supposed to use intuition. He mentions the gradient vector as being associated to the partial equation. Thus, the vector field is the gradient of $u$, and particles move in the direction of steepest descent.

The function $u$ is 0 along some level curve, so if $u$ represents distance from anything, it must be the distance from that curve. To show that $u$ is in fact the dtance from some curve, note that the distance $d$ from the level curve given by $u=0$ satisfies the same boundary equation and differential equation as $u$.