Did you tried Fourier transforming the equation? I guess you might need to invoke separation of variables first.
The Benney-Luke equation is nonlinear Partial differential equation , it is difficult separation of variable.
I just googled it. Many forms showed up! Could you please share a picture or latex code of the equation you are interested in?
I believe for nonlinear PDE's the words dispersion relation refer to the behaviour of the plane wave solutions of the linearised form of the equation. So in that sense i believe Vikash's idea to be helpful.
I have handled nonlinear DEs before. I would suggest as mentioned before, first do a series expansion and take into account the first order terms. And then Fourier transform accordingly. Nonlinear PDEs are still an open problem, most solutions obtained take into account first or second order corrections.
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