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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.

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    I think we should give this problem a better go, or at least explain a bit more about the difficulty we are really facing with this type of problem instead of breaking it down to something solvable but not really representative of $f_{xx}+bf_{xx}f_{tt}+a=0$.2012-05-20

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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.

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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. $