I can transform it to:

R(t)*(U_t)^2=1+(U_x)^2  where R(t)=[Q(t)]^(-1)

Transformations: dT=dt*Q(t), U_x=u ->u_T=f''(u)u_x,f(u)=1/(1+u^2)^{1/2}, v_T=v v_x - Hopf equation, where v=f'(u). The well-known solution of Hopf equation is v=g(x+vT) where g - any function.

If it is for something explicite, and if you want do something easy, just use the dichotomie at every steps.

Hello Meguenni Bouhadjar, Why don't you solve the PDE with an explicit finite difference scheme such as

u(xi,tj+dt) = u(xi, tj) + Q(tj)/sqrt[1 + k2] 

where k = (u(xi+dx, tj)- u(xi-dx, tj))/(2dx) .  The time increment should be kept very small to ensure stability. Also you need to specify the initial condition (values of u) at time t = 0. The time increment dt is related to the spatial increment dx. However, the relationship between dt and dx can only be determined from a stability analysis which can be a difficult task for this nonlinear PDE.