Generalized Separation of Variables for Nonlinear Equation

We consider the equation u t t = F ( u ) u x x + a F ′ ( u ) u x 2 , where F(u) is an arbitrary function and a ≠ 0 is a constant. The problem in question is for which functions F(u) this equation admits the ansatz t   =   w 1 ( x ) d ( u )   +   w 2 ( x ) reducing it to a system of two ordinary differential equations with unknown functions w1(x) and w2(x). New classes of exact solutions with generalized separation of variables were constructed for these equations, which cannot be obtained by the method of classical group analysis.