 Next-nearest-neighbour ferromagnetic magnon dispersion

Positive nearest- and next-nearest-neighbour exchanges give one-magnon energy $2S[J_1(1-\cos k)+J_2(1-\cos 2k)]$. At small momentum this is $S(J_1+4J_2)k^2$. The quadratic-boson result describes the harmonic many-magnon spectrum; higher-order spin-wave interactions remain in the full Hamiltonian.