On a complex beam–beam interaction model with random forcing

In recent years, many studies have been devoted to complex differential equations (CDE), which appear in very important applications in physics and engineering. This paper aims to investigate one such CDE, containing a random forcing term: (∗)z̈+ωo2z+ε2λg(ż)+ε2f(z,z̄)p(ωot)=αn(t),where z(t)=x(t)+iy(t) a complex function, i=−1, n(t) is a broad-band process with zero mean and ε2 and λ small real parameters. In particular, we use Eq. (∗) to model the interaction between two colliding beams in particle accelerators, setting g(ż)=ż, f(z,z̄)=z|z|2 and p(ωot)=cosωot and extend the work we had started in an earlier publication (Mahmoud, Physica A 216 (1995) 445). We apply the stochastic averaging method to derive a Fokker–Planck–Kolmogorov equation for this equation and obtain analytically the exact stationary probability density function and the first and second moments in the amplitude of the solutions. Numerical simulations are carried out to compare with the theoretical ones and excellent agreement is found.