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Driven autoresonant three-oscillator interactions

Driven autoresonant three-oscillator interactions. Oded Yaakobi 1,2 Lazar Friedland 2 Zohar Henis 1 1 Soreq Research Center, Yavne, Israel. 2 The Hebrew University of Jerusalem, Jerusalem, Israel. Email: yaakobio@yahoo.com. O. Yaakobi, L. Friedland and Z. Henis, Phys. Rev. E (accepted).

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Driven autoresonant three-oscillator interactions

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  1. Driven autoresonantthree-oscillator interactions Oded Yaakobi1,2 Lazar Friedland2 Zohar Henis1 1Soreq Research Center, Yavne, Israel. 2 The Hebrew University of Jerusalem, Jerusalem, Israel. Email: yaakobio@yahoo.com O. Yaakobi, L. Friedland and Z. Henis, Phys. Rev. E (accepted).

  2. Three waves interactions Frequencies matching (energy): Wave vectors matching (momentum): • Plasma physics • Laser plasma interactions: • Stimulated Brillouin Scattering (SBS) • Stimulated Raman Scattering (SRS) • Nonlinear optics • Optical Parametric Amplifier/Generator (OPA/OPG) • Brillouin scattering, Raman scattering • Hydrodynamics • Acoustic waves Controlling three waves interactions is an important goal of both basic and applied physics research.

  3. Three oscillators interactions

  4. Three oscillators interactions Research goal: Study a control scheme of three oscillators interactions using an external drive. Definitions:

  5. Threshold phenomena

  6. Adiabatic approximation Definitions: Approximated equations neglecting : Nonlinear frequency shift

  7. Small nonlinear frequency shift Small nonlinear frequency shift Assumption: Definitions: Range of validity Approximated equations:

  8. Small nonlinear frequency shift & quasi steady state Autoresonant quasi steady state Assumptions: Quasi steady state:

  9. Constraint: Threshold analysis Quasi-steady-state asymptotic result: Asymptotic phase mismatches:

  10. Threshold analysis Dimensional equations: Necessary condition:

  11. Computed threshold (numerical) Threshold analysis Necessary condition for autoresonant quasi steady state:

  12. Linear frequencies mismatch

  13. Dissipation Small nonlinear frequency shift: Necessary condition: Exponential decay:

  14. Phase mismatches deviations

  15. Linearization Exact equations: Quasi-steady-state: Assumptions: Expansion:

  16. Differentiating with respect to : Linearization Linearized equations:

  17. Numerical results comparison

  18. WKB approximation

  19. WKB approximation First order terms satisfy:

  20. Singular value decomposition

  21. Singular value decomposition First order terms satisfy: Multiplying with :

  22. Asymptotic form of matrices

  23. Quasi steady state stability Small deviations from the quasi-steady-state do not increase with time.

  24. Quasi steady state stability Small deviations from the quasi steady state do not increase with time.

  25. Small nonlinear frequency shift & quasi steady state Large nonlinear frequency shift Assumptions:

  26. Conclusions • Controlling three oscillators interactions using autoresonance is demonstrated. • Analytic expressions for autoresonant time dependent amplitudes are obtained. • Conditions for autoresonant trapping are analyzed in terms of coupling parameter, driving parameter, dissipation and linear frequencies mismatch. • The autoresonant quasi-steady state is linearly stable.

  27. Outlook • Generalization of the theory to driven three-wave interactions is of interest.

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