During my physics studies, I ran into the following partial differential equation of the form: $$\frac{\partial^2 \eta}{\partial t^2} - a^2 \frac{\partial^2 \eta}{\partial z^2} - 2 a b\frac{\partial^4 \eta}{\partial z^4} - b^2 \frac{\partial^6 \eta}{\partial z^6} - 2 a c \eta \frac{\partial^2 \eta}{\partial z^2} - 2 b c \frac{\partial^4 \eta}{\partial z^4} = 0$$
I'd like to benefit from the knowledge of all the mathematicians gathered here about it: could you please tell me what you think of or know about this equation, and/or give me references to look at to learn more about it ? Currently, the only thing I know is that it is supposed to generalize in two directions of propagations the KdV equation: $$\frac{\partial \eta}{\partial t} + a \frac{\partial \eta}{\partial z} + b \frac{\partial^3 \eta}{\partial z^3} + c \eta \frac{\partial \eta}{\partial z} = 0$$ and I thus expect it to have soliton solutions, with two possible directions of propagation indeed. But I have no idea of how to prove that actually.
Thanks in advance for the insight you'll provide me on this equation,
Thomas
This looks like a special case of the Boussinesq equation. If the nonlinear equation can be rewritten in terms of a conservation law, then in the case of integrable systems, it can be written in terms of a Hamiltonian system. If there is a Hamiltonian, you need to check that it is bounded and if there are multiple extrema, there is potential for soliton solutions. The level sets of constant energy of the Hamiltonian correspond to the varying types of solitons that exist. Relationship between KDV and Boussinesq is also known. Richard Palais gives a great description in Symmetries of Solitons