Matrix analogues of elliptic functions (Q752922)

The second order matrix equation with cubic nonlinearity \[ (1)\quad \ddot Q+a_ 0Q^ 3+a_ 1Q^ 2+a_ 2Q+a_ 3=0,\quad a_ j\in {\mathbb{C}},\quad Q\in gl(n,{\mathbb{C}}), \] appears in connection with the investigation of the solutions of the equations of motion of one- dimensional systems of interacting particles in an external field with a Hamiltonian of the form \[ H=\frac{1}{2}\sum^{n}_{j=1}p_ j^ 2+\sum^{n}_{j>k}(x_ j-x_ k)^{-2}+\sum^{n}_{j=1}(\frac{a_ 0}{4}x_ j^ 4+\frac{a_ 1}{3}x_ j^ 3+\frac{a_ 2}{2}x_ j^ 2+a_ 3x_ j). \] The object of this paper is the construction of the solutions of (1) for matrices of arbitrary dimension and general initial positions; they can be understood as the matrix analogues of the elliptic functions which satisfy (1) for \(n=1\), ie. \(Q\in {\mathbb{C}}\).