Checking nonlinear hyperbolic PDE

Question

We know the inviscid burger equation $$ u_t+u u_x=0 $$ is a nonlinear example of hyperbolic PDE. But I cannot verify the $B^2-4AC>0$ test for the above.

Answer 1

There is a difference between classification linear and nonlinear PDEs. The above inviscid Burgers' eqaution is a nonlinear PDE, so you can't prove it as hyperbolic by using the formula ,$B^2 - 4AC > 0$, which is meant for linear PDEs.

To define the type of nonlinear PDEs for hyperbolic type, you need to find eigenvalues and corresponding eigenvectors of this equation and prove that eigenvalues are real. I suggest you to have a look at the book "Time Dependent Problems and Difference Methods" by Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger. Especially for Definition 4.3.1.

Consider the initial value problem $$u_{t} = Au_{x}$$ $$u(x, 0) = f_{x}$$

Then Definition 4.3.1 is stated as

"The system in the above equation is called symmetric hyperbolic if A is a Hermittian matrix. It is called strictly hyperbolic if the eigenvalues are real and distinct, it is called strongly hyperbolic if the eigenvalues are real and there exists a complete system of eigenvectors, and finally it is called weakly hyperbolic if eigenvalues are real."