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Lax Pair and First Integrals for Two of Nonlinear Coupled Oscillators

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Advanced Technologies in Robotics and Intelligent Systems

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 80))

Abstract

The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse scattering transform is one of the most powerful methods for solving the Cauchy problems of partial differential equations. To solve the Cauchy problem for nonlinear differential equations we can use the Lax pair corresponding to this equation. The Lax pair for ordinary differential or systems or for system ordinary differential equations allows us to find the first integrals, which also allow us to solve the question of integrability for differential equations. In this report we present the Lax pair for the system of coupled oscillators. Using the Lax pair we get two first integrals for the system of equations. The considered system of equations can be also reduced to the fourth-order ordinary differential equation and the Lax pair can be used for the ordinary differential equation of fourth order. Some special cases of the system of equations are considered.

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Acknowledgements

The reported study was funded by RFBR according to the research Project No. 18-29-10025.

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Authors and Affiliations

  1. National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 31 Kashirskoe Shosse, 115409, Moscow, Russia

    N. A. Kudryashov

Authors
  1. N. A. Kudryashov
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Correspondence to N. A. Kudryashov .

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Editors and Affiliations

  1. Institute of Cyber Intelligence Systems, National Research Nuclear University MEPhI, Moscow, Russia

    Sergey Yu. Misyurin

  2. LS2N, Institut National des Sciences Appliquées, Rennes, France

    Vigen Arakelian

  3. Institute for System Programming, Russian Academy of Sciences, Moscow, Russia

    Arutyun I. Avetisyan

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Kudryashov, N.A. (2020). Lax Pair and First Integrals for Two of Nonlinear Coupled Oscillators. In: Misyurin, S., Arakelian, V., Avetisyan, A. (eds) Advanced Technologies in Robotics and Intelligent Systems. Mechanisms and Machine Science, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-030-33491-8_3

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