Scaling a system

Question

I can't understand a scaling that has been made: Consider the 1D R-D-system $$\epsilon\frac{\partial u}{\partial t}=\epsilon^2 \frac{\partial^2 u}{\partial x^2}+f(u,v)\\\epsilon\frac{\partial v}{\partial t}=D \frac{\partial^2 v}{\partial x^2}-f(u,v)$$ with $u(x,t), v(x,t)$. Introduce a stretched coordinate $\xi:=\frac{x-\phi(t)}{\epsilon}$ with some function $\phi(t)$. Assume $$u(x,t) = U(\xi, t), v(x,t)=V(\xi,t)$$ By substituting the new scaling into the system above one obtains $$\epsilon\frac{\partial U}{\partial t}-\frac{\partial \phi}{\partial t}\frac{\partial U}{\partial \xi}= \frac{\partial^2 U}{\partial \xi^2}+f(U,V)\\\epsilon\frac{\partial V}{\partial t}-\frac{\partial \phi}{\partial t}\frac{\partial V}{\partial \xi}= D\epsilon^{-2}\frac{\partial^2 V}{\partial \xi^2}-f(U,V).$$ How? I hope someone can help me!

Answer 1

Using the chain rule, we get \begin{align} \frac{\partial u}{\partial x} &= \frac{\partial\xi}{\partial x}\frac{\partial U}{\partial \xi} = \frac{1}{\varepsilon} \frac{\partial U}{\partial\xi}, \\ \frac{\partial^2 u}{\partial x^2} &= \frac{1}{\varepsilon^2} \frac{\partial^2 U}{\partial \xi^2}, \\ \frac{\partial u}{\partial t} &= \frac{\partial \xi}{\partial t}\frac{\partial U}{\partial\xi} + \frac{\partial U}{\partial t} = -\frac{1}{\varepsilon}\frac{\partial\phi}{\partial t}\frac{\partial U}{\partial \xi} + \frac{\partial U}{\partial t}. \end{align} Plugging these formulas in the PDE for $u$ we obtain $$ \varepsilon \frac{\partial U}{\partial t} - \frac{\partial\phi}{\partial t}\frac{\partial U}{\partial\xi} = \frac{\partial^2 U}{\partial\xi^2} + f(U,V).$$ Can you figure out the equation for $V$ by yourself from here?