Question: Numerical solutions of PDE

Dear experts

I am interested in solving the following differential equation numerically

diff(eta(k,t),t,t) + gamma*diff(eta(k,t),t) + omega^2*eta(k,t) + (3/4)*k*omega*eta(k,t)*eta(k,t) + (3/2)*k*eta(k,t)*(omega*eta(k,t) + (3/4)*k*eta(k,t)*eta(k,t)) + (omega*gamma*eta(k,t) + (3/2)*k*eta(k,t)*(gamma*eta(k,t)))*(diff(eta(k,t),t) + gamma*eta(k,t))/(omega*eta(k,t) + (3/4)*k*eta(k,t)*eta(k,t)) = 0;

where eta(x,t) is the Fourier transform of zeta(x,t) and plot of zeta(x,t) is the goal. I think FFT method can help but I failed in running and applying this method on the equation above.
Initial conditions are

f= 0.28
C = 1/L*0.2/f*tanh(L/f)
L=20
amp = 0.5e-2
zeta0 = amp/cosh(f*x)/cosh(f*x)-C

and u0 as initial velocity can be 0 or an arbitrary function such as 
u0 =2*amp/cosh(f*x)/cosh(f*x)*f*tanh(f*x)

I would appreciate it if one could help me.