Dynamical Structure Factors of Quasi-One-Dimensional Magnets of the Type of $CsNiF_3$

By a direct derivation of the equations of motion for the spins in $CsNiF_3$ we show that the extensively used sine-Gordon equation fails to describe the dynamics of this 1D magnet. Instead of this soliton-bearing model we use the spin-wave theory and, without going to the continuum approximation, calculate the dynamic structure factors of the scattering of neutrons on $CsNiF_3$. Complete analytical solutions for the dynamic structure factors in the frequency domain are obtained both within the classical consideration and with quantum corrections.