Lazar Friedland Hebrew University of Jerusalem

12

Lazar FriedlandHebrew University of Jerusalem

I. Autoresonance phenomenon: past and present researchII. Autoresonance of traveling nonlinear wavesIII. Excitation and control of multiphase nonlinear waves (KdV, DNLS)

OUTLINE

EMERGENCE AND CONTROL OF MULTI-PHASE WAVESBY RESONANT PERTURBATIONS

I. AUTORESONANCE PHENOMENON

Example:

B

Cyclotron resonance:

Cyclotron autoresonance:

(acceleration)

oscillatingE-field

of frequency

11

mc

qBc

)(tmc

qBc

mq,

)(t )(tm

Autoresonance: a property of driven nonlinear systems to stay in resonance when parameters vary in time and/or space

PAST AND PRESENT RESEARCH

1946

1990

1991-92

Phase Stability PrincipleMcMillan,Veksler Various acceleration schemes

Dynamic autoresonanceAtomic, molecular, astrophysical

Simplest traveling wavesKdV, SG, NLS

1999

2003Multiphase KdV, Toda, NLS, SG

2D Vortex states

10

Present

ShagalovFajansMeersonFriedland& students

2005

Vlasov-Poisson, BGK modes

45 years

Suppose the medium supports a slowly varying TRAVELING wave

Drive by a small amplitude eikonal wave

0)(ˆ uN

Slow amplitude ),,( txA Fast phase ),( tx

Slow ,),( xtxk ttx ),(

dtxuN cos),()(ˆ

Slow amplitude ),,( tx Fast phase ),( txd

Slow ,/),( xtxkd ttx dd /),(

),(),( Autxu

9

II. WAVE AUTORESONANCE

Lowest order dispersion relation: 0),,( AkD

Assume continuing phase-locking (autoresonance)

),,(),( txktxk d ),(),( txtx d

Then nonlinear dispersion 0),,( AkD yields

0),,(),,( AtxtxkD dd

),( txAA

),(),( txtx d

),( Auu d

The driven wave isfully controlled bythe driving wave

QUESTIONS: 1. How to phase-lock? 2. Is the phase-locking stable?

WHY PHASE-LOCKING MEANS CONTROL? 8

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)

)()(),(

sin),(

OtkA

kAVA

dt

t

d

Phase mismatch

Stability is guaranteed for small near or 0 unless one hits another resonance

THEORY OF AUTORESONANT WAVES 7

dtdkAV dAtt /sin),(

Nonlinear wave frequency Slow driving frequency

dtd d /

F. MULTIPHASE NONLINEAR WAVES

ut uux uxxx n cos[kn x n (t)dt]

Q: How to excite multiphase waves of, say, the KdV equation?

A:

3-phase wave

6

Friedland, Shagalov (2004)

),(),( tLxutxu Periodic KdV case

...),,( 2211 txktxkF

IST Diagnostics

),( txu

Associated linear eigenvalue problem0)],([ txuExx

E - Main IST spectrum )(tconstE

A two-phase solution

Twoopengaps

E E

degeneratepairs

5

EXAMPLE 2

)()(,1)0( xLx

0),( txu

,...2,1,0

2

n

nk

kE

L

EXAMPLE 1

)cos(kx

Synchronization is seen via spectral IST analysis

3-phase KdV wave

4

Evolution of main IST spectrum Frequency locking

Discrete nonlinear Schroedinger systems

n N

( ),

1 ,

,

1

n n

n

q q t

n N

q q

Multiphase solutions of the periodic IDNLS:

Nm0 ,

},exp{ ),W()( 021

tn

itq

iii

mn

2

1 1 1 12

1( 2 ) ( )n

n n n n n n

dqi q q q q q q

dt

• Integrable DNLS:

2

1 12

1( 2 ) 2n

n n n n n

dqi q q q q q

dt

• Diagonal DNLS:

3

Autoresonant multiphase DNLS waves

Successively apply driving waves passing through different system’s resonances.

)])((exp[ dttnki iii

Example:a 4-phase solution

2

Gofer, Friedland (2005)

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

tnqn iii0 },exp{i ),W(

1

SIMULATION – 4-phase solution

1

(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.

SUMMARY 0



(2) We have discussed emergence and control of traveling and multiphase nonlinear waves (KDV, DNLS) by passage through resonances.

SUMMARY 0




(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).

SUMMARY 0







(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.

SUMMARY 0







(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

SUMMARY 0

http://www.phys.huji.ac.il/~lazar

http://www.phys.huji.ac.il/~lazar

)(;/}{],0[ 1 nnN

n xNLxL

.,/1

)(,)( *1

i

n

nnnnn ez

z

zzFzF

N

nn zFzM

1

)()(

Main spectrum:

Auxiliary spectrum:

0)(4

1)( 1221

22211

4 MMMMzP N

0)( 1212 MzP N

zi

log1

Lazar Friedland Hebrew University of Jerusalem

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