Can You Solve This Non-linear PDE with Variable Separation?

[aminfar]

I am new to non-linear PDEs. So I tried to solve it, but I stuck in the beginning.

U^2_xU_t - 1 = 0

U(x, 0) = x

 

[gato_]

The equation admits a separation of variables, and the solution can be written as U(x,t)=f(x)+g(t). Inserting this, you get f'(x)^2=1/g'(t)=constant. The solution can then be checked to be U=x+t