= Solution
Expand around the $+z$ polarized <ground state>. With $n_j=a_j^\dagger a_j$ fixed as $S\to\infty$, the <Holstein–Primakoff transformation> gives $S_j^+=\sqrt{2S}a_j+O(S^{-1/2})$, $S_j^-=\sqrt{2S}a_j^\dagger+O(S^{-1/2})$ and $S_j^z=S-n_j$. The longitudinal exchange contributes
$$
-J(1+u)\sum_j(S-n_j)(S-n_{j+1})=-LJ(1+u)S^2+2JS(1+u)\sum_jn_j+O(S^0).
$$
Use
$$
\begin{aligned}
S_j^xS_{j+1}^x+S_j^yS_{j+1}^y&=\tfrac12(S_j^+S_{j+1}^-+S_j^-S_{j+1}^+),\\
(\mathbf S_j\times\mathbf S_{j+1})_z&=\tfrac{i}{2}(S_j^+S_{j+1}^--S_j^-S_{j+1}^+).
\end{aligned}
$$
The transverse exchange then gives $-JS(a_{j+1}^\dagger a_j+a_j^\dagger a_{j+1})$, and the <Dzyaloshinskii-Moriya interaction> gives $-iDS(a_{j+1}^\dagger a_j-a_j^\dagger a_{j+1})$. Discarding the constant $-LJ(1+u)S^2$ and all $O(S^0)$ interactions yields
$$
\boxed{H_{\rm LSW}=\sum_j\left[Aa_{j+1}^\dagger a_j+A^*a_j^\dagger a_{j+1}+Bn_j\right],\quad A=-S(J+iD),\quad B=2JS(1+u).}
$$
Here <periodic boundary conditions> make each $n_j$ occur twice in the longitudinal sum. The approximation is the <linear spin-wave approximation>, controlled by small local <magnon> occupation compared with $S$.
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