  We present a comprehensive study of the magnetic properties of the long-range
ordered quasi-one dimensional $J_{1}$-$J_{2}$ systems with a newly developed
torque equilibrium spin-wave expansion approach, which can describe the spin
Casimir and magnon decay effects in a unified framework. While the framework
does not lose the generality, our discussion will be restricted to two
representative types of inter-chain coupling systems: $J_{3}$- or
$J_{4}$-system respectively. In spite of the long-range spiral order, the
dynamical properties of these systems turn out to be highly nontrivial due to
the incommensurate noncollinear spin configuration and the strong quantum
fluctuation effects enhanced by the frustration and low-dimensionality. Both
the systems show prominent spin Casimir effects induced by the vacuum
fluctuation of the spin waves and related modification of the ordering vector,
Lifshitz point's position and sublattice magnetization. Significant and
spontaneous magnon decay effects are manifested in the quantum corrections to
the excitation spectrum. By adjusting the strength of magnetic anisotropy and
varying the approximation scheme, it is revealed that these striking distinct
features are quite robust and have deep connection with both the spin Casimir
and the magnon decay effects. Thus these special consequences of the
inter-chain coupling on the spin wave dynamics may be served as a set of probes
for different types of inter-chain couplings in experiments. At last, to guide
experimental measurements such as inelastic neutron scattering in realistic
materials and complement our theoretical framework, we develop the analytical
theory of the dynamical structure factor within the torque equilibrium
formulism and provide the explicit results of the quasi-one dimensional
$J_{1}$-$J_{2}$ systems.
