How to solve this coupled nonlinear equation?

[blenx]

Here is the equation I don't know how to solve:
$$\begin{aligned} \left( {\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{t^2}}} + \beta _1^2} \right){u_1} = {g_1}u_2^{}{u_3} \\ \left( {\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{t^2}}} + \beta _2^2} \right){u_2} = {g_2}u_1^{}{u_3} \\ \left( {\frac{{{{\rm{d}}^2}}}{{{\rm{d}}{t^2}}} + \beta _3^2} \right){u_3} = {g_3}u_2^{}{u_1} \\ \end{aligned}$$
where $${\beta _i},{g_i}$$ are constants.
Is there an exact solution to this problem? If not, how to solve it approximately or numerically?

 

[swraman]

I don't know whether or not there is a exact solution to this, but MATLAB can solve it numerically.