When we study 1D model for 3D Euler equations, we encounter an interesting 1+1D nonlinear PDE for dependent variable $\gamma(\alpha,t)$: $$\gamma_{\alpha}\gamma_{tt}=b(\alpha),\quad \alpha\in[0,1]\tag{1}$$ $$\gamma(\alpha,0)=\alpha,\quad \gamma_t(\alpha,0)=c(\alpha)\tag{2}$$.
***Question 1: *** What is the name of this nonlinear PDE?
***Question 2: *** Is the method of separation variable the only method to solve this kind of PDE?
Just noticed that in addition to the method of separation variables with $\gamma(\alpha,t)=A(\alpha)T(t)$, we can also try self similar solutions like $\gamma(\alpha,t)=a_1 B(x(t_1-t)^\lambda)$.