Any numerical methods for solving a hyperbolic nonlinear PDE?

The solution of any differential equation involves specifying boundary and initial conditions, too-else the problem isn't well defined at all. The most efficient methods for solving such problems these days are pseudo-spectral methods, where the functions are expanded in a basis that satisfies the boundary conditions and fast transforms are used, when evaluating the terms of the system of ordinary differential equations thus obtained. The qualification pseudo- means that the basis need not be the eigenfunctions of the differential operator that contains the spatial derivatives.

I completely agree with Professeur Nicolis: The most efficient methods for solving such problems these days are pseudo-spectral methods. Please see this very good reference from SIAM J. Numer. Analysis:

I found it very informative. Thanks for a very interesting question for the community!

The above answer are correct. However, I would like to emphasize  with intuitive heuristics why the posed problem is particularly difficult and may in fact not be well specified in all cases. A core issue is that this is not in general a purely hyperbolic PDE. Rather, the character changes from hyperbolic to elliptical depending on the sign of the product of f and g. If they are positive it is hyperbolic, but if negative it is elliptical (and if zero - perhaps parabolic). Specifically the PDE you had in mind suffers this complication. I refer you to discussions of the Tricomi equation to start pulling the thread into a deeper investigation. Why does this make it so difficult? Well the condition for the well-posedness of the B.C.'s changes depending on the character of the PDE. You will have trouble solving a wave equation with boundary values and similarly you will have trouble solving an elliptical equation as an initial value problem.  Even worse, for the general non-linear problems (i.e. when f and g depend on u) you will have to use your solution or approximate solution to determine the the boundary between the elliptical and hyberbolic regions - a difficult self-referential chestnut to crack indeed. That said, these sorts of problems have real applications such as mixed sub-sonic and supersonic flows and finding solutions to the solar wind problem.

You have to have boundary conditions; otherwise the problem is ill-posed. Keep in mind your specific PDE u_{xx}u^3-\sin(xt)u_{tt} = u is not always hyperbolic! Nevertheless, if your PDE is hyperbolic then I'd suggest using the finite volume method or the discontinuous Galerkin method. Both methods are suitable for hyperbolic PDEs.

you can use the simple explicit finite difference method and this can be easily applied and it is of second order accuracy in both directions (space and time), the major defect of this method the stability issue (conditionally stable).

To overcome this difficulty implicit schemes  can be used , but in this case we end with a nonlinear tridiagonal system which can be solve by Newton's or fixed point method. 

thanks