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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
III. Excitation and control of multiphase nonlinear waves (KdV, DNLS)
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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:
c 
q, m
oscillating
E-field
of frequency
B
Cyclotron autoresonance:
c 

qB

mc
 (t )
qB
  (t )
mc
m(t )
(acceleration)
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PAST AND PRESENT RESEARCH
Phase Stability Principle
McMillan,Veksler
Present
Various acceleration schemes
1946
45 years
Dynamic autoresonance
Atomic, molecular, astrophysical
1990
Shagalov
Fajans
Meerson
Friedland
& students
1991-92
Simplest traveling waves
KdV, SG, NLS
2D Vortex states
1999
Multiphase KdV, Toda, NLS, SG
2003
Vlasov-Poisson, BGK modes
2005
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II. WAVE AUTORESONANCE
Suppose the medium supports a slowly varying TRAVELING wave
u ( x, t )  u ( , A)
Nˆ (u )  0
Slow amplitude A( x, t ),
Fast phase
 ( x, t )
Slow k ( x, t )   x ,  ( x, t )   t
Lowest order dispersion relation:
D( k , , A)  0
Drive by a small amplitude eikonal wave
Nˆ (u )   ( x, t ) cos  d
Slow amplitude  ( x, t ),
Slow
k d ( x, t )   / x,
Fast phase  d ( x, t )
d ( x, t )   d / t
WHY PHASE-LOCKING MEANS CONTROL?
8
Assume continuing phase-locking (autoresonance)
 ( x, t )   d ( x, t )
k ( x, t )  k d ( x, t ),  ( x, t )   d ( x, t )
Then nonlinear dispersion D( k , , A)  0
yields
D k d ( x, t ),  d ( x, t ), A   0
A  A( x, t )
u  u ( d , A)
The driven wave is
fully controlled by
the driving wave
QUESTIONS: 1. How to phase-lock?
2. Is the phase-locking stable?
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
    d
At  V ( A, k ) sin 
 t  ( A, k )  d (t )  O( )
Nonlinear wave frequency
Slow driving frequency
 tt  V ( A, k ) A sin   dd / dt
Stability is guaranteed for small d d / dt
unless one hits another resonance
near
  0 or   
F. MULTIPHASE NONLINEAR WAVES
F (k1 x 1t , k2 x  2t , ...)
Q: How to excite multiphase waves of, say, the KdV equation?
A:
ut  uux  uxxx    n cos[kn x    n (t)dt]
Periodic KdV case
u( x, t )  u( x  L, t )
Friedland, Shagalov (2004)
3-phase wave
6
5
IST Diagnostics
Associated linear eigenvalue problem
 xx  [ E  u( x, t )]  0
 (0)  1,  ( x  L)   ( x)
u ( x, t )
E - Main IST spectrum
EXAMPLE 1
u ( x, t )  0
EXAMPLE 2
A two-phase
solution
E
  cos(kx)
Ek
2
k  L n
n  0,1,2,...
E  const (t )
degenerate
pairs
E
Two
open
gaps
4
Synchronization is seen via spectral IST analysis
3-phase KdV wave
Evolution of main IST spectrum
Frequency locking
4
Synchronization is seen via spectral IST analysis
3-phase KdV wave
Evolution of main IST spectrum
Frequency locking
3
Discrete nonlinear Schroedinger systems
• Diagonal DNLS:
i
2
dqn
1
 2 (qn 1  qn 1  2qn )  2 qn qn
dt

• Integrable DNLS:
i
2
dqn 1
 2 (qn 1  qn 1  2qn )   qn (qn 1  qn 1 )
dt 
Multiphase solutions of the periodic IDNLS:
qn (t )  W(1 ,  2  m ) exp{i 0 },
 i   i n  i t , 0  m  N
qn  qn (t ),
n  1 N,
qn  N  qn ,
  1
2
Autoresonant multiphase DNLS waves
Gofer, Friedland (2005)
Successively apply driving waves  i exp[ i ( ki n   i (t ) dt )]
passing through different system’s resonances.
Example:
a 4-phase solution
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
qn  W( ,  ) exp{i 0 }, i   i n  it
1
SIMULATION – 4-phase solution
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.
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.
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).
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.
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
 n   ( xn )
  L / N;
[0, L] {xn }1N
 n1  Fn ( z ) n ,
 z n 
 ,
Fn ( z )   *
 n  1/ z 
M ( z) 
i
ze .
N
 F ( z)
n
n 1
1
( z )  (M11  M 22 ) 2  M 21M12  0
4
Main spectrum:
P
Auxiliary spectrum:
P 2 N 1 ( z )  M12  0
4N
1
  log z
i