how to solve this nonlinear partial differential equation?

Question

How to solve this nonlinear partial differential equation? $$\displaystyle\frac{\partial^2}{\partial x^2} f(x,t) +b \frac{\partial^2}{\partial x^2} f(x,t) \cdot \frac{\partial^2}{\partial t^2} f(x,t) + a = 0,$$ where $a,b$ are constant.

Answer 1

You might find some solutions by making the ansatz $f(x,t) = g(x) + h(t)$, i.e., that the solution separates. The equation then reads $$g''(x) + b \, g''(x) h''(t) +a =0,$$ $$g''(x) [1+ b \,h''(t)] = -a.$$ This equation can be solved by setting $g''(x)=c$ and $1+ b \,h''(t) = -a/c$. The solutions read $$g(x) = \frac{c}{2} x^2 + c_1 x + c_2$$ and $$h(t) = -\frac{a+c}{2 b c} t^2 +C_1 t + C_2$$ with $c\neq0$, $c_1$, $c_2$, $C_1$, $C_2$ arbitrary constants.

Answer 2

Also it is possible to look for traveling waves solutions $f(x,t)=g(x-сt)\,$. It leads to ODE $$ g''(x)+bc^2(g''(x))^2+a=0 $$ solutions of which can be written down explicitly: $$ g(x)=C_1+C_2 x+\frac{1\pm\sqrt{1-4 a b c^2}}{4 b c^2}x^2. $$