Abstract
Large amplitude spin waves exhibit a variety of nonlinear phenomena. One approach to general nonlinear problems in spin wave dynamics is to transform the Landau–Lifshitz torque equation of motion into a scalar Hamiltonian in terms of canonical variables. We begin with the Hamiltonian formalism for spin wave interactions in Problems 9.1–9.3. We then expand the Hamiltonian to include higher order nonlinear terms, and diagonalize the Hamiltonian using the Bogoliubov transformation in Problem 9.4. A semi classical expansion in Problem 9.5 allows us to derive the various coefficients in the nonlinear Hamiltonian. Problems 9.6–9.8 explore the conditions necessary for soliton formation, and show that they are valid solutions to the nonlinear Schrödinger equation.
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Notes
- 1.
Here \(a_{\mathbf {k}}^*\equiv (a_{\mathbf {k}})^*= (a^*)_{-\mathbf {k}}\).
- 2.
We will see in Problem 9.4 that \(u_{\mathbf {k}}\) can be chosen to be real without loss of generality.
- 3.
This condition ensures that the transformation is symplectic.
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Stancil, D.D., Prabhakar, A. (2021). Nonlinear Interactions. In: Spin Waves. Springer, Cham. https://doi.org/10.1007/978-3-030-68582-9_9
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DOI: https://doi.org/10.1007/978-3-030-68582-9_9
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