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Lazar Friedland Hebrew University of Jerusalem

12. EMERGENCE AND CONTROL OF MULTI-PHASE WAVES BY RESONANT PERTURBATIONS. Lazar Friedland Hebrew University of Jerusalem. OUTLINE. I. Autoresonance phenomenon: past and present research II. Autoresonance of traveling nonlinear waves

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Lazar Friedland Hebrew University of Jerusalem

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  1. 12 EMERGENCE AND CONTROL OF MULTI-PHASE WAVES BY RESONANT PERTURBATIONS Lazar FriedlandHebrew University of Jerusalem OUTLINE I. Autoresonance phenomenon: past and present research II. Autoresonance of traveling nonlinear waves III. Excitation and control of multiphase nonlinear waves (KdV, DNLS)

  2. 11 I. AUTORESONANCE PHENOMENON Autoresonance: a property of driven nonlinear systems to stay in resonance when parameters vary in time and/or space Example: Cyclotron resonance: B Cyclotron autoresonance: oscillating E-field of frequency (acceleration)

  3. 10 PAST AND PRESENT RESEARCH Present Phase Stability Principle McMillan,Veksler Various acceleration schemes 1946 45 years Dynamic autoresonance Atomic, molecular, astrophysical 1990 Simplest traveling waves KdV, SG, NLS 1991-92 Shagalov Fajans Meerson Friedland & students 2D Vortex states 1999 Multiphase KdV, Toda, NLS, SG 2003 Vlasov-Poisson, BGK modes 2005

  4. 9 II. WAVE AUTORESONANCE Suppose the medium supports a slowly varying TRAVELING wave Fast phase Slow amplitude Slow Lowest order dispersion relation: Drive by a small amplitude eikonal wave Fast phase Slow amplitude Slow

  5. WHY PHASE-LOCKING MEANS CONTROL? 8 Assume continuing phase-locking (autoresonance) yields Then nonlinear dispersion The driven wave is fully controlled by the driving wave QUESTIONS:1. How to phase-lock? 2. Is the phase-locking stable?

  6. 7 THEORY OF AUTORESONANT WAVES 1. PHASE LOCKING by passage through a LINEAR resonance 2. STABILITY via Whitham’s averaged variational principle Typical set of slow equations (1D, spatially periodic) Phase mismatch Nonlinear wave frequency Slow driving frequency Stability is guaranteed for small near or unless one hits another resonance

  7. 6 F. MULTIPHASE NONLINEAR WAVES Q: How to excite multiphase waves of, say, the KdV equation? A: Periodic KdV case Friedland, Shagalov (2004) 3-phase wave

  8. 5 IST Diagnostics Associated linear eigenvalue problem - Main IST spectrum EXAMPLE 1 EXAMPLE 2 A two-phase solution Two open gaps degenerate pairs

  9. 4 Synchronization is seen via spectral IST analysis 3-phase KdV wave Evolution of main IST spectrum Frequency locking

  10. 4 Synchronization is seen via spectral IST analysis 3-phase KdV wave Evolution of main IST spectrum Frequency locking

  11. Integrable DNLS: • Diagonal DNLS: Multiphase solutions of the periodic IDNLS: 3 Discrete nonlinear Schroedinger systems

  12. 2 Autoresonant multiphase DNLS waves Gofer, Friedland (2005) Successively apply driving waves passing through different system’s resonances. Example: a 4-phase solution

  13. 1 IST diagnostics • Main spectrum - N pairs of complex conjugate values. • A degenerate pair – dormant phase. • Open pair - excited phase. • 3-phase IDNLS (N=5) solution

  14. 1 SIMULATION – 4-phase solution

  15. SUMMARY 0 (1) The paradigm: How to excite and control a nontrivial wave by starting from zero and using weak forcing? A general solution:Sinchronization by passage through resonances.

  16. SUMMARY 0 (1) The paradigm: How to excite and control a nontrivial wave by starting from zero and using weak forcing? A general solution:Sinchronization by passage through resonances. • (2) We have discussed emergence and control of traveling and multiphase nonlinear waves (KDV, DNLS) by passage through resonances.

  17. SUMMARY 0 (1) The paradigm: How to excite and control a nontrivial wave by starting from zero and using weak forcing? A general solution:Sinchronization by passage through resonances. • (2) We have discussed emergence and control of traveling and multiphase nonlinear waves (KDV, DNLS) by passage through resonances. • (3)Mathematical methods for autoresonant waves: • (1) Whitham’s averaged variational principle for traveling waves • (work exists for a general case with application to SG, KdV, and NLS). • (2) Spectral IST approach for diagnostics of autoresonant multiphase waves (work exists for KdV, Toda, NLS, DNLS).

  18. SUMMARY 0 (1) The paradigm: How to excite and control a nontrivial wave by starting from zero and using weak forcing? A general solution:Sinchronization by passage through resonances. • (2) We have discussed emergence and control of traveling and multiphase nonlinear waves (KDV, DNLS) by passage through resonances. • (3)Mathematical methods for autoresonant waves: • (1) Whitham’s averaged variational principle for traveling waves • (work exists for a general case with application to SG, KdV, and NLS). • (2) Spectral IST approach for diagnostics of autoresonant multiphase waves (work exists for KdV, Toda, NLS, DNLS). • (4) Main open questions: • (1) General IST theory of multiphase autoresonant waves? • (2) Interaction of resonances and stability of driven waves? • (3) Autoresonance in nonintegrable systems. Controlled transition to chaos. • (4) Higher dimensionality.

  19. SUMMARY 0 (1) The paradigm: How to excite and control a nontrivial wave by starting from zero and using weak forcing? A general solution:Sinchronization by passage through resonances. • (2) We have discussed emergence and control of traveling and multiphase nonlinear waves (KDV, DNLS) by passage through resonances. • (3)Mathematical methods for autoresonant waves: • (1) Whitham’s averaged variational principle for traveling waves • (work exists for a general case with application to SG, KdV, and NLS). • (2) Spectral IST approach for diagnostics of autoresonant multiphase waves (work exists for KdV, Toda, NLS, DNLS). • (4) Main open questions: • (1) General IST theory of multiphase autoresonant waves? • (2) Interaction of resonances and stability of driven waves? • (3) Autoresonance in nonintegrable systems. Controlled transition to chaos. • (4) Higher dimensionality. • (5)Applications in astrophysics, atomic/molecular physics, plasmas, fluids and nonlinear waves: www.phys.huji.ac.il/~lazar

  20. Main spectrum: Auxiliary spectrum:

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