In Chapter 2, we introduced the Lagrangian and the Hamiltonian equations of motion. The variational formulation of Chapter 7 describes the Lagrangian as an energy density functional from which it is possible to derive the equations of motion. In the case of wave propagation, the physics of nonlinear wave interactions becomes mathematically tractable when we use the Hamiltonian formalism with the understanding that the classical spin waves can be represented by their complex amplitudes instead of Bose operators that would represent magnons. The Hamiltonian method is specifically suitable for the analysis of weakly interacting and weakly dissipative wave systems, where nonlinear interactions can be treated as higher order corrections to the lowest order wave solutions. The Hamiltonian yields first-order differential equations which are easier to solve than Lagrange’s equations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Cash and D. D. Stancil, ‘Measurement of magnetostatic wave profiles using the interaction with transverse optical guided modes,’ IEEE Trans. Mag., vol. 32, p. 5188, 1996.
D. J. Seagle, S. H. Charap, and J. O. Artman, ‘Foldover in YIG,’ J. Appl. Phys., vol. 57, p. 3706, 1985.
Y. K. Fetisov, C. E. Patton, and V. T. Synogach, ‘Nonlinear ferromagnetic resonance and foldover in yttrium iron garnet thin films – inadequacy of the classical model,’ IEEE Trans. Mag., vol. 35, p. 4511, 1999.
Y. T. Zhang, C. E. Patton, and M. V. Kogekar, ‘Ferromagnetic resonance foldover in single crystal YIG films – sample heating or Suhl instability,’ IEEE Trans. Mag., vol. 22, p. 993, 1986.
A. Prabhakar and D. D. Stancil, ‘Auto-oscillation thresholds at the main resonance in ferrimagnetic films,’ Phys. Rev. B, vol. 57, p. 11483, 1998.
M. Weiss, ‘Microwave and low-frequency oscillation due to resonance instabilities in ferrites,’ Phys. Rev. Lett., vol. 1, p. 239, 1958.
H. Suhl, ‘The theory of ferromagnetic resonance at high signal powers,’ J. Phys. Chem. Solids, vol. 1, p. 209, 1957.
X. Y. Zhang and H. Suhl, ‘Theory of auto-oscillations in high power ferromagnetic resonance,’ Phys. Rev. B, vol. 38, p. 4893, 1988.
A. Prabhakar and D. D. Stancil, ‘Nonlinear microwave-magnetic resonator operated as a bistable device,’ J. Appl. Phys., vol. 85, p. 4859, 1999.
Y. K. Fetisov and C. E. Patton, ‘Microwave bistability in a magnetostatic wave interferometer with external feedback,’ IEEE Trans. Mag., vol. 35, no. 2, pp. 1024–1036, Mar 1999.
H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. Cambridge, MA: Addison-Wesley, 2001.
C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics. New York, NY: Wiley & Sons, 1997.
E. Schlömann, ‘Ferromagnetic resonance at high power levels,’ Raytheon Corporation, Tech. Rep., 1959.
V. S. L’vov, Turbulence Under Parametric Excitation, Applications to Magnets. Berlin: Springer-Verlag, 1994.
T. Holstein and H. Primakoff, ‘Field dependence of the intrinsic domain magnetization of a ferromagnet,’ Phys. Rev., vol. 58, no. 12, pp. 1098–1113, Dec 1940.
H. Benson and D. L. Mills, ‘Spin waves in thin films; dipolar effects,’ Phys. Rev., vol. 178, no. 2, pp. 839–847, Feb 1969.
S. M. Rezende and F. M. Aguiar, ‘Spin-wave instabilities, auto-oscillations, and chaos in yttrium-iron-garnet,’ Proc. IEEE, vol. 78, p. 893, 1990.
P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, ‘Hamiltonian formalism for two magnon scattering microwave relaxation: Theory and applications,’ J. Appl. Phys., vol. 101, p. 083901, 2007.
H. Suhl, ‘Subsidiary absorption peaks in ferromagnetic resonance at high signal levels,’ Phys. Rev., vol. 101, pp. 1437–1438, 1956.
P. H. Bryant, C. D. Jeffries, and K. Nakamura, ‘Spin-wave dynamics in a ferrimagnetic sphere,’ Phys. Rev. A, vol. 38, p. 4223, 1988.
V. E. Zakharov, V. S. L’vov, and S. S. Starobinets, ‘Instability of monochromatic spin waves,’ Sov. Phys. Solid State, vol. 11, p. 2368, 1970.
P. Wigen, Ed., Nonlinear Phenomena and Chaos in Magnetic Materials. Singapore: World Scientific, 1994.
R. Marcelli and S. A. Nikitov, Eds., Nonlinear Microwave Signal Processing: Towards a New Range of Devices. Dordrecht: Kluwer Academic Publishers, 1996.
M. A. Tsankov, M. Chen, and C. E. Patton, ‘Forward volume wave microwave envelope solitons in yttrium iron garnet films: Propagation, decay and collision,’ J. Appl. Phys., vol. 76, p. 4274, 1994.
G. P. Agrawal, Nonlinear Fiber Optics. New York, NY: Academic Press, 2006.
B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, ‘Spin-wave solitons in ferromagnetic films: observation of a modulational instability of spin-waves during continuous excitation,’ JETP Lett, vol. 10, p. 392, 1984.
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. New York, NY: Cambridge University Press, 1991.
S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 5th ed. New York, NY: McGraw-Hill, 2005.
M. J. Lighthill, ‘Contributions to the theory of waves in nonlinear dispersive systems,’ J. Inst. Maths Applics., vol. 1, p. 269, 1965.
B. Kalinikos and A. N. Slavin, ‘Theory of dipole exchange spin-wave spectrum for ferromagnetic films with mixed exchange boundary conditions,’ J. Phys. C: Solid State Phys., vol. 19, p. 7013, 1986.
A. N. Slavin, ‘Thresholds of envelope soliton formation in a weakly dissipative medium,’ Phys. Rev. Lett., vol. 77, p. 4644, 1996.
R. A. Stuadinger, P. Kabos, H. Xia, B. T. Faber, and C. E. Patton, ‘Calculation of the formation time for microwave magnetic envelope solitons,’ IEEE Trans. Mag., vol. 34, p. 2334, 1998.
B. A. Kalinikos, N. G. Kovshikov, and C. E. Patton, ‘Decay free microwave magnetic envelope soliton pulse trains in yttrium iron garnet thin films,’ Phys. Rev. Lett., vol. 78, no. 14, pp. 2827–2830, 1997.
B. Kalinikos, N. V. Kovshikov, and C. E. Patton, ‘Self-generation of microwave magnetic envelope soliton trains in yttrium iron garnet thin films,’ Phys. Rev. Lett., vol. 80, p. 4301, 1998.
B. A. Kalinikos, N. G. Kovshikov, and C. E. Patton, ‘Excitation of bright and dark microwave magnetic envelope solitons in a resonant ring,’ Appl. Phys. Lett., vol. 75, p. 265, 1999.
M. Wu, B. A. Kalinikos, L. D. Carr, and C. E. Patton, ‘Observation of spin wave soliton fractals in magnetic film active feedback rings,’ Phys. Rev. Lett., 2006.
Y. Xu, G. Su, D. Xue, H. Xing, and F. shen Li, ‘Nonlinear surface spin waves on ferromagnetic media with inhomogeneous exchange anisotropies: solitonsolutions,’ Phys. Lett. A, vol. 279, pp. 385–390, 2001.
O. Büttner, M. Bauer, S. O. Demokritov, B. Hillebrands, Y. S. Kivshar, V. Grimalsky, Y. Rapoport, M. P. Kostylev, B. A. Kalinikos, and A. N. Slavin, ‘Spatial and spatiotemporal self-focusing of spin waves in garnet films observed by space- and time-resolved brillouin light scattering,’ J. Appl. Phys., vol. 87, pp. 5088–5090, 2000.
R. Gong, Y. Cheng, and H. Li, ‘Variational analysis of evolution for magnetostatic envelope bright soliton with higher-order dispersion,’ J. Magn. Magn. Matl., vol. 313, pp. 122–126, 2007.
B. Kalinikos and M. P. Kostylev, ‘Parametric amplification of spin wave envelope solitons in ferromagnetic films by parallel pumping,’ IEEE Trans. Mag., vol. 33, no. 5, p. 3445, 1997.
G. Gibson and C. Jeffries, ‘Observation of period doubling and chaos in spin-wave instabilities in yttrium iron garnet,’ Phys. Rev. A, vol. 29, p. 811, 1984.
R. D. McMichael and P. E. Wigen, ‘High power ferromagnetic resonance without a degenerate spin-wave manifold,’ Phys. Rev. Lett., vol. 64, p. 64, 1990.
A. Borovik-Romanov and S. Sinha, Eds., Spin Waves and Magnetic Excitations. Amsterdam: North Holland Physics, 1988.
M. Cottam, Ed., Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices. Singapore: World Scientific, 1994.
G. Srinivasan and A. N. Slavin, Eds., High Frequency Processes in Magnetic Materials. Singapore: World Scientific, 1995.
C. Robinson, Dynamical Systems. Boca Raton, FL CRC Press Inc., 1995.
P. Glendinning, Stability, Instability and Chaos. Cambridge CambridgeUniversity Press., 1994.
J. I. Neimark, Mathematical Models in Natural Science and Engineering. Berlin: Springer, 2003, ch. 4.
T. Y. Li and J. A. Yorke, ‘Period three implies chaos,’ Amer. Math. Monthly, vol. 82, pp. 985–992, 1975.
A. Prabhakar and D. D. Stancil, ‘Variations in auto-oscillation frequency at the main resonance in rectangular YIG films,’ J. Appl. Phys., vol. 79, p. 5374, 1996.
A. Prabhakar and D. D. Stancil, ‘Information dimension analysis of chaotic forward volume spin waves in a yttrium-iron-garnet thin film,’ J. Appl. Phys., vol. 87, p. 5091, 2000.
R. Hegger and H. Kantz, ‘Practical implementation of nonlinear time series methods: The TISEAN package,’ Chaos, vol. 9, no. 2, pp. 413–435, 1999.
H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, ‘The analysis of observed chaotic data in physical systems,’ Rev. Mod. Phys., vol. 65, no. 4, p. 1331, 1993.
M. Kennel, R. Brown, and H. D. I. Abarbanel, ‘Determining embedding dimension for phase-space reconstruction using a geometrical construction,’ Phys. Rev. A, vol. 45, p. 3403, 1992.
I. Procaccia and M. Shapiro, Eds., Chaos and Related Nonlinear Phenomena. New York: Plenum, 1987, ch. Practical Considerations in estimating dimension from time series data.
J. P. Eckmann and D. Ruelle, ‘Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems,’ Physica D, vol. 56, p. 185, 1992.
R. Badii and A. Politi, ‘Statistical description of chaotic attractors: the dimension function,’ J. Stat. Phys., vol. 40, p. 725, 1985.
E. J. Kostelich, ‘Nearest neighbour algorithm,’ personal communication.
H. N. Bertram, V. L. Safonov, and Z. Jin, ‘Thermal magnetization noise, damping fundamentals, and mode analysis: Application to a thin film GMR sensor,’ IEEE Trans. Mag., vol. 38, pp. 2514–2519, 2002.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag US
About this chapter
Cite this chapter
Stancil, D.D., Prabhakar, A. (2009). Nonlinear Interactions. In: Spin Waves. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-77865-5_9
Download citation
DOI: https://doi.org/10.1007/978-0-387-77865-5_9
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-77864-8
Online ISBN: 978-0-387-77865-5
eBook Packages: EngineeringEngineering (R0)