2.15.15 Hunter Saxton \(\left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2\)

problem number 125

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Hunter Saxton. Solve for \(u(x,t)\) \[ \left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[D[u[x, t], t] + u[x, t]*D[u[x, t], x], x] == (1*D[u[x, t], x]^2)/2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

Failed

Maple

restart; 
pde := diff( (diff(u(x,t),t)+ u(x,t)* diff(u(x,t),x)) , x) = 1/2* (diff(u(x,t),x))^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
 

\[u \left ( x,t \right ) =2\,{\frac {\RootOf \left ( -{\it \_C2}\,{{\it \_c}_{{1}}}^{3}-x{{\it \_c}_{{1}}}^{3}-2\,{\it \_C1}\,\sqrt {{\it \_Z}}{\it \_c}_{{1}}+2\,{{\it \_C1}}^{2}\ln \left ( \sqrt {{\it \_Z}}{\it \_c}_{{1}}+{\it \_C1} \right ) +{\it \_Z}\,{{\it \_c}_{{1}}}^{2} \right ) }{{\it \_c}_{{1}}t+2\,{\it \_C3}}}\] with RootOf

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