Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Boussinesq type PDE. Solve for \(u(x,t)\) \[ u_{tt}-u_{xx}-2 \alpha (u u_x)_x - \beta u_{xxtt} = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \frac {1}{6} \left (-12 c_1{}^2 \tanh ^2(c_2 t+c_1 x+c_3)-1+8 c_1{}^2+\frac {c_2{}^2}{c_1{}^2}\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,t),t$2)-diff(u(x,t),x$2)-2*alpha*diff( (u(x,t)*diff(u(x,t),x)) ,x) - beta*diff(u(x,t),x,x,t,t)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[u \left (x , t\right ) = \frac {-12 c_{2}^{2} c_{3}^{2} \beta \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )+\left (8 c_{3}^{2} \beta -1\right ) c_{2}^{2}+c_{3}^{2}}{2 c_{2}^{2} \alpha }\]
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