Pattern dynamics of parametrically excited spin waves near the instability threshold

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Abstract
The dynamics of dissipative patterns built up by parametrically excited spin waves in an insulating ferromagnetic film driven by out-of-plane parallel pumping is studied theoretically. Crystal-field anisotropies and surface pinning of spins are neglected whereas the dipolar field is fully included. Near the instability threshold the dynamics are governed by amplitude equations for the slowly time-varying amplitudes of pairs of unstable spin waves. Because of rotational symmetry an infinite number of spin waves having the same wavelength but propagating in different directions become unstable simultaneously. The dynamics of the amplitude equations is mainly determined by a nonlinear coupling coefficient which is not an even function of the difference between the angles of propagation. A detailed description of the numerical method of the calculation of the coefficients of the amplitude equations is given. From the amplitude equations stationary solutions and their stability are calculated. The only stable patterns are squares and three-wave patterns; hexagons are unstable. For frequencies of the pump field above some threshold all stationary patterns are unstable. In this regime the dynamical behavior is dominated by switching between squares and/or three-wave patterns. The pattern switching is extremely sensitive to noise which leads to a weak noise-induced turbulence.