Solving phi-fourth theory using Fourier analysis

[spaghetti3451]

The equation of motion of $\phi^4$ theory is $(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}$.

Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?

 

[stevendaryl]

The equation of motion of $\phi^4$ theory is $(\partial^{2}+m^{2})\phi = -\frac{\lambda}{3!}\phi^{3}$.

Why can't this equation be solved using Fourier analysis? Can't we simply write the equation in Fourier space and take it from there?

Well, the right-hand side of the equation is nonlinear, so performing a Fourier transform will produce a mess---a triple integral. You end up replacing a nonlinear differential equation by a nonlinear integral equation. That might be a good first step, but it doesn't solve the equation. You would have to resort to perturbation theory in any case.