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

problem number 124

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'));
 

\[u \left (x , t\right ) = \frac {2 \operatorname {RootOf}\left (-x \textit {\_c}_{1}^{3}-c_{2} \textit {\_c}_{1}^{3}-2 \sqrt {\textit {\_Z}}\, c_{1} \textit {\_c}_{1}+2 c_{1}^{2} \ln \left (\sqrt {\textit {\_Z}}\, \textit {\_c}_{1}+c_{1} \right )+\textit {\_Z} \textit {\_c}_{1}^{2}\right )}{\textit {\_c}_{1} t +2 c_{3}}\]

with RootOf

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