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 TMF, 2014, Volume 178, Number 2, Pages 230–254 (Mi tmf8560)

Vector acoustic solitons from the coupling of long and short waves in a paramagnetic crystal

S. V. Sazonova, N. V. Ustinovb

a National Research Center Kurchatov Institute, Moscow, Russia

Abstract: We investigate the propagation of a longitudinal–transverse elastic pulse in a statically deformed crystal containing paramagnetic impurities and placed in an external magnetic field. We derive a system of three nonlinear wave equations describing the interaction of the pulse with the paramagnetic impurities in the quasiresonance approximation in the Faraday geometry. We assume that the transverse components of the pulse, which cause quantum transitions, have carrier frequencies and are short-wave (acoustic), while the longitudinal component has no carrier frequency and is long-wave. We show that in the case of an equilibrium initial distribution of populations of quantum levels of paramagnetic impurities, the coupling between the longitudinal and transverse components is weak, the pulse is therefore strictly transverse, and its dynamics are described by the Manakov system. With a nonequilibrium initial distribution of populations, conditions of effective interaction between all components of the elastic pulse can be reached, and their nonlinear dynamics are described by a vector generalization of the Zakharov equations. In the case of a unidirectional propagation of the pulse, these equations reduce to the Yajima–Oikawa vector system. We show that the obtained system of equations and its version with an arbitrary number of short-wave components can be integrated using the inverse scattering transform. We construct infinite hierarchies of solutions of the Yajima–Oikawa vector system (including a solution on a nontrivial background. We consider stationary (complex-valued Garnier system) and self-similar reductions of that system, also admitting a representation in the form of compatibility conditions.

Keywords: soliton, nonlinear integrable equations, resonance of long and short waves

DOI: https://doi.org/10.4213/tmf8560

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English version:
Theoretical and Mathematical Physics, 2014, 178:2, 202–222

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Document Type: Article

Citation: S. V. Sazonov, N. V. Ustinov, “Vector acoustic solitons from the coupling of long and short waves in a paramagnetic crystal”, TMF, 178:2 (2014), 230–254; Theoret. and Math. Phys., 178:2 (2014), 202–222

Citation in format AMSBIB
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• https://doi.org/10.4213/tmf8560
• http://mi.mathnet.ru/eng/tmf/v178/i2/p230

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This publication is cited in the following articles:
1. K. Sakkaravarthi, T. Kanna, M. Vijayajayanthi, M. Lakshmanan, “Multicomponent long-wave-short-wave resonance interaction system: bright solitons, energy-sharing collisions, and resonant solitons”, Phys. Rev. E, 90:5 (2014), 052912
2. G. T. Adamashvili, “Vector soliton of self-induced transparency of a generalized love wave”, Acoust. Phys., 63:5 (2017), 517–523
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