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It is well-known that the dynamics of biaxial ferromagnets with a strongeasy-plane anisotropy is essentially governed by the Sine-Gordon equation. Inthis paper, we provide a rigorous justification to this observation. Moreprecisely, we show the convergence of the solutions to the Landau-Lifshitzequation for biaxial ferromagnets towards the solutions to the Sine-Gordonequation in the regime of a strong easy-plane anisotropy. Moreover, weestablish the sharpness of our convergence result.  This result holds for solutions to the Landau-Lifshitz equation in high orderSobolev spaces. We first provide an alternative proof for local well-posednessin this setting by introducing high order energy quantities with bettersymmetrization properties. We then derive the convergence from the consistencyof the Landau-Lifshitz equation with the Sine-Gordon equation by usingwell-tailored energy estimates. As a by-product, we also obtain a furtherderivation of the free wave regime of the Landau-Lifshitz equation.

The Sine-Gordon regime of the Landau-Lifshitz equation with a strong easy-plane anisotropy