Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Sawada Kotera. Solve for \(u(x,t)\) \[ u_t + 45 u^2 u_x + 15 u_x u_{xx} + 15 u u_{xxx} + u_{xxxxx} = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], t] + 45*u[x, t]^2*D[u[x, t], x] + 15*D[u[x, t], x]*D[u[x, t], {x, 2}] + 15*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\begin {align*} & \left \{u(x,t)\to -\frac {4}{3} c_1{}^2 \left (-2+3 \tanh ^2\left (-16 c_1{}^5 t+c_1 x+c_3\right )\right )\right \}\\& \left \{u(x,t)\to \frac {-30 c_1{}^{5/2} \tanh ^2(c_2 t+c_1 x+c_3)+20 c_1{}^{5/2}+\sqrt {20 c_1{}^5-5 c_2}}{15 \sqrt {c_1}}\right \}\\& \left \{u(x,t)\to \frac {20 c_1{}^{5/2}-\sqrt {20 c_1{}^5-5 c_2}}{15 \sqrt {c_1}}-2 c_1{}^2 \tanh ^2(c_2 t+c_1 x+c_3)\right \}\\ \end {align*}
Maple ✓
restart; pde := diff(u(x,t),t)+45* u(x,t)^2* diff(u(x,t),x)+ 15* diff(u(x,t),x)*diff(u(x,t),x$2)+15*u(x,t)*diff(u(x,t),x$3)+diff(u(x,t),x$5); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',PDEtools:-TWSolutions(pde,u(x,t))),output='realtime'));
\begin {align*} & \left \{ u \left ( x,t \right ) ={\it \_C4} \right \} \\& \left \{ u \left ( x,t \right ) =-4\, \left ( \tanh \left ( -16\,{{\it \_C2}}^{5}t+{\it \_C2}\,x+{\it \_C1} \right ) \right ) ^{2}{{\it \_C2}}^{2}+{\frac {8\,{{\it \_C2}}^{2}}{3}} \right \} \\& \left \{ u \left ( x,t \right ) =-2\, \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}{{\it \_C2}}^{2}-{\frac {1}{15\,{\it \_C2}} \left ( -20\,{{\it \_C2}}^{3}+\sqrt {20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}} \right ) } \right \} \\& \left \{ u \left ( x,t \right ) =-2\, \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}{{\it \_C2}}^{2}+{\frac {1}{15\,{\it \_C2}} \left ( 20\,{{\it \_C2}}^{3}+\sqrt {20\,{{\it \_C2}}^{6}-5\,{\it \_C3}\,{\it \_C2}} \right ) } \right \} \\ \end {align*}
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