= Solution
Repeat the <linear spin-wave approximation> for bonds of lengths one and two. The polarized energy is $E_0=-NS^2(J_1+J_2)$ and each bond length contributes its own lattice Laplacian. Thus
$$
\boxed{\varepsilon_k=2S[J_1(1-\cos k)+J_2(1-\cos2k)]
=4S[J_1\sin^2(k/2)+J_2\sin^2k].}
$$
The <next-nearest-neighbour ferromagnetic magnon dispersion> remains nonnegative for the stated positive exchanges and has its only zero at $k=0$ modulo $2\pi$ when $J_1>0$. At small $k$, $\varepsilon_k\simeq S(J_1+4J_2)k^2$: longer-range ferromagnetic exchange increases the stiffness.
At this harmonic order, a bosonic occupation configuration has energy $E_0+\sum_k\varepsilon_kn_k$. The one-magnon dispersion is also exact. General multi-magnon energies acquire the interactions omitted in the large-$S$ expansion; the formula does not claim an exact diagonalization of the full interacting many-spin Hamiltonian.
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