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.


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


\[u \left (x , t\right ) = \frac {2 H c_{2}^{2} \tanh \left (c_{3} t +c_{2} x +c_{1}\right )-c_{3}}{c_{2}}\]