Change of coordinates in PDEs

Motivation for change of coordinates

Consider the simple example of a linear PDE:

$$a{u}_x+b{u}_y=0.$$

This PDE says that the directional derivative of $u$ in the direction $(a,b)$ is 0. This means that $u$ is constant along lines in these directions. The lines in the direction $(a,b)$ are the characteristic curves of this PDE. Notice that these lines partition the plane.

Now we would like coordinates say $x^{\prime },y^{\prime }$ that has the property where of these coordinates has derivative 0. The intuitive idea is to let one of these somehow represent moving along characteristic curves, and to capture the idea that moving between characteristic curves does not change $U$. What we can do is let $y^{\prime }=bx-ay$, and $x^{\prime }=ax+by$. It turns out that $x^{\prime }$ picks out position along some characteristic curve, and $U_{x^{\prime }}$ is 0 since we are moving along the curve and the value shouldn’t change along the curve, since $u$ doesn’t. Now $y^{\prime }$ picks out which curve we are on. This indeed does pick out which curve we are on: If we move from a point $(x_0,y_0)$ on a curve to $(x_0,y_0)+t(a,b)$, $y^{\prime }$ does not change.

We check that these are in fact true: working out the chain rule gives us

$$u_x=a{U}_{x^{\prime }}+b{U}_{y^{\prime }},u_y=b{U}_{x^{\prime }}-a{U}_{y^{\prime }}$$ $$a{u}_x+b{u}_y=(a^2+b^2)U_{x^{\prime }}+(ab-ab)U_{y^{\prime }}=(a^2+b^2)U_{x^{\prime }}.$$

Hence the PDE s converted to $(a^2+b^2)U_{x^{\prime }}=0$, so $U=f(y^{\prime })$ for some arbitrary function. Then just substitute in $y^{\prime }=bx-ay$.

In general, you would just want to pick one coordinate such that the derivative of the transformed function with respect to that coordinate is 0.

”Proof”-ish

To see why change of coordinates works, consider the general form of a PDE in 2 variables (this should generalize to as many variables as we want in a similar way but it gets annoying to write out):

$$F(x,t,u,u_x,u_t,u_{xx},u_{tt},u_{xt})=0$$

Let $\Phi$ be an invertible $C^2$ function such that $\Phi (x,t)=(\xi ,\eta )$. If we define $U=u\circ {\Phi }^{-1}$, we see that the various derivatives of $u$ are just combination of derivatives of $U$ (see Superposition for inhomogeneous wave equations). Then we can form a new PDE:

$$G(\xi ,\eta ,U,U_{\xi },U_{\eta },U_{\xi \xi },U_{\eta \eta },U_{\xi \eta })=0.$$

For each curve $\Gamma$ that the initial conditions are defined on, $\Phi [\Gamma ]$ is a new curve which has the initial conditions, i.e. say $u=g$ on $\Gamma$, then $U=u\circ {\Phi }^{-1}=g\circ {\Phi }^{-1}$ on $\Phi [\Gamma ]$. Example: $u(x,0)=g$ has curve $\Gamma =\{(x,0):x\}$.