Making Sense of Method of Characteristics Solution Geometry

Question

How can I make sense of the following surface as insight into understanding the method of characteristics?

The partial differential equation this initially came from was $2xu_x + u_y = 0$, which has solutions $u = f(xe^{-2y})$ for arbitrary functions $f$. I don't know what boundary conditions produce this particular choice of $f$; I just whimsically picked some functions to graph.

I can almost believe that $\begin{bmatrix}2x \\ 1\end{bmatrix}$ is tangential to the "main" lines on this graph, the lines which hug the planes $x = 0$ (for negative $y$) and $y = 0$, although the main lines do have a slight oscillating pattern which seems out of place. However the lines on the stalagmites and stalactites certainly do not obey this tangency condition; they wrap the entire way around, while $x$ and $y$ vary only slightly in these regions.

Answer 1

That function is oscillating like crazy for $y<0$, and your picture doesn't bear any resemblance to the actual graph, since the resolution is way too low. Try something like Plot3D[Cos[Sinh[x Exp[-2 y]]], {x, -1, 1}, {y, -1, 1}, PlotPoints -> 400] and you might begin to get some feeling for what it really looks like:

Answer 2

In addition to Hans's plot, it might be easier to understand a 2D plot. I've superimposed the characteristics (blue) $xe^{-2y}=k$, $k\in\{-100,-99,\dots,100\}$ with the graph (green) of your chosen function $\cos(\sinh(x))$: Note that $\cos(\sinh(x))=\cos(\sinh(x e^{-2(0)}))$ is the $y=0$ cross-section of $u(x,y) = \cos(\sinh(xe^{-2y}))$. By following the characteristic lines down, you are squeezing this graph into smaller and smaller regions. At $y=-1/2$, the portion of $\cos(\sinh(x))$ that I've plotted ($x\in[-10,10]$) is shrinked into $[-4,4]$, and then it keeps getting smaller exponentially: For example, everything that is above the lowest characteristic curve $ xe^{-2y}=\pm 100$ is squeezed into a tiny interval at $y=-4$. More precisely, the graph, over the region $x\in[-100,100]$, which is already incredibly squiggly, is squeezed into $$ |x|e^8\le 100\quad \text{i.e.}\quad x\in[-0.0335,0.0335].$$ Hopefully, this gives you a feel for why your plot wasn't accurate.