2.15.20 Korteweg de Vries (KdV) \(u_t + (u_x)^3+ 6 u u_x = 0\)

problem number 129

Added December 27, 2018.

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

Korteweg de Vries (KdV). Solve for \(u(x,t)\)

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] + D[u[x, t], {x,3}] + 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),t)+ diff( u(x,t),x$3 ) + 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'));
 

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

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