2.15.3 Benjamin Ono \(u_t+H u_{xx} +u u_x = 0\)

problem number 112

Added December 27, 2018.

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

Benjamin Ono. Solve for \(u(x,t)\)

\[ u_t+H u_{xx} +u u_x = 0 \]

Important note. \(H\) above is meant to be Hilbert transform. https://en.wikipedia.org/wiki/Benjamin%E2%80%93Ono_equation However, here in the code below it is taken as just a scalar. Need to correct this when I have time.

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] + h*D[u[x, t], {x, 2}] + u[x, t]*D[u[x, t], x] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

\[\left \{\left \{u(x,t)\to 2 c_1 h \tanh (c_2 t+c_1 x+c_3)-\frac {c_2}{c_1}\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),t)+H*diff(u(x,t),x$2)+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'));
 

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

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