# Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions

$$\nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3)$$

Without the cubic term, this gives a linear operator $\mathcal{L} = \nabla^2 + \kappa^2$. In this case I can get a solution via the Green's function $G=\exp{(i\kappa r)}(4\pi r)^{-1}$. In my equation however, the presence of $\phi^3$ does not give me a linear operator. Is anything known about the solution to this equation?

Context: The Poisson-Boltzmann equation can be put into the functional form of $\nabla^2 \phi = -\kappa^2 \sinh \phi$. Expanding sinh to first order gives the Helmholtz equation as mentioned above. The second order term is zero and the third order term gives the equation in question.

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This is a cross-post from MSE. The question has had sufficient time with no comments or answers so I thought I'd try the physics side for some different insight. If this cross-post behavior is not allowed, please let me know and I'll delete. –  Hooked Mar 25 at 21:15