2.7.1 \(u_{xxx} + u_t -6 u u_x = 0\)

problem number 98

From Mathematica symbolic PDE document.

Korteweg-deVries (waves on shallow water surfaces) with no initial conditions

Solve for \(u(x,t)\)

\[ u_{xxx} + u_t -6 u u_x = 0 \]

Reference https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], {x, 3}] + D[u[x, t], {t}] - 6*u[x, t]*D[u[x, t], {x}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\[\left \{\left \{u(x,t)\to \frac {12 c_1{}^3 \tanh ^2(c_2 t+c_1 x+c_3)-8 c_1{}^3+c_2}{6 c_1}\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),x$3)+ diff(u(x,t),t)-6*u(x,t)* diff(u(x,t),x)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = \frac {12 c_2^{3} \tanh \left (c_2 x +c_3 t +c_1 \right )^{2}-8 c_2^{3}+c_3}{6 c_2}\]