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A systematic step-by-step development of nonlinear spin-wave theory within the framework of a Hamiltonian formalism is given. The expansion coefficients in the spin-wave Hamiltonian up to the fourth order in the canonical spin-wave amplitudes for a uniformly magnetized ellipsoidal sample are given in general form. This is done for a general orientation of the external field and for a general wave vector. Applications are given for (A) the second-order Suhl spin-wave instability coupling coefficients, (B) the spin-wave wave-vector-dependent nonlinear frequency shift for an in-plane magnetized thin film, and (C) the uniform magnetization mode (ferromagnetic resonance) nonlinear frequency shift for an obliquely magnetized thin film. The analytical results for (C) are compared with results obtained from a time forward numerical analysis of the torque equation of motion and show a near-perfect match.
Krivošı́k et al. (Tue,) studied this question.