Continuations of NLS solutions beyond the singularity Gadi Fibich Tel Aviv University ff
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Transcript of Continuations of NLS solutions beyond the singularity Gadi Fibich Tel Aviv University ff
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Continuations of NLS solutionsbeyond the singularity
Gadi Fibich Tel Aviv University
ff
• Moran Klein - Tel Aviv University• B. Shim, S.E. Schrauth, A.L. Gaeta - Cornell
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NLS in nonlinear optics
Models the propagation of intense laser beams in Kerr medium (air, glass, water..)
Competition between focusing Kerr nonlinearity and diffraction
z“=”t (evolution variable)
focusing nonlinearitydiffraction
2 0, ,z xx yyi z x y
r=(x,y)
z
z=0Kerr Medium
Input Beam
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Self Focusing
Experiments in the 1960’s showed that intense laser beams undergo catastrophic self-focusing
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Kelley (1965) : Solutions of 2D cubic NLS can become singular in finite time (distance) Tc
2 2
0 4 x ye
Finite-time singularity
20, , 0, 0, , ,t xx yyi t x y x y x y
Beyond the singularity
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?• No singularities in nature
• Laser beam propagates past Tc
• NLS is only an approximate model• Common approach: Retain effects that were neglected in NLS
model: Plasma, nonparaxiality, dispersion, Raman, …• Many studies• …
Compare with hyperbolic conservation laws Solutions can become singular (shock waves) Singularity arrested in the presence of viscosity
Huge literature on continuation of the singular inviscid solutions: Riemann problem Vanishing-viscosity solutions Entropy conditions Rankine-Hugoniot jump conditions Specialized numerical methods …
Goal – develop a similar theory for the NLS
Continuation of singular NLS solutions
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Tc
?NLS
Continuation of singular NLS solutions
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Tc
NLSNLSno ``viscous’’ terms
Continuation of singular NLS solutions
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Tc
NLSNLS
``jump’’ condition
no ``viscous’’ terms
Continuation of singular NLS solutions
2 key papers by Merle (1992) Less than 10 papers
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Tc
NLSNLS
``jump’’ condition
no ``viscous’’ terms
Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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General NLS
d – dimension, σ – nonlinearity
2, 0ti t x
( )1 1
1,
...d d
d
x x x x
x xx ==¶ + +¶D
Definition of singularity:
Tc - singularity point
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0 , limc
Ht THy y
®Î =¥
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Classification of NLS
global existence
(no blowup)
Subcritical σd<2
blowup Critical σd=2
blowup Supercritical
σd>2
( )1, dx xx =
2, 0ti t x
σd= 2 Physical case considered earlier (σ=1,d=2) Since 2σ= 4/d, critical NLS can be rewritten as
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4/, 0dti t x
Critical NLS (focus of this talk)
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Solutions of the form
The profile R is the solution of
Enumerable number of solutions Of most interest is the ground state:
Solution with minimal power (L2 norm)
4/ 11 0, 0, 00ddR R R R R Rrr
Solitary waves
,ite R r r x
d=2Townes profile
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Critical power for collapse
Thm (Weinstein, 1983): A necessary condition for collapse in the critical NLS is
Pcr - critical power/mass/L2-norm for collapse
2 20 2 2
,cr crP P R
Explicit blowup solutions
Solution width L(t)0 as tTc
ψR,αexplicit becomes singular at Tc
Blowup rate of L(t) is linear in t
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t
2( )explicit 4
R, /2
2
0
1( , ) ,( ) ( )
( ) ), ( )(
tL ri t iL
d
t
c
rt r RL t L t
L t t L s dsT
e
Minimal-power blowup solutions
ψR,αexplicit has exactly the critical power
Minimal-power blowup solution
ψR,αexplicit is unstable, since any perturbation that
reduces its power will lead to global existence 18
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R,
1( ) ( )d
rR
L t L t
2
2crP R
Thm (Weinstein, 86; Merle, 92) The explicit blowup solutions ψR,α
explicit are the only minimal-power solutions of the critical NLS.
Stable blowup solutions of critical NLSFraiman (85), Papanicolaou and coworkers (87/8) Solution splits into a singular core and a regular tail Singular core collapses with a self-similar ψR profile
Blowup rate is given by
Tail contains the rest of the power ( ) Rigorous proof: Perelman (01), Merle and Raphael (03)
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2( ) ( )
log logc
c
tTL ttT
loglog law
R / 2
1( , ) ~ ( , )( ) ( )d
rt r t r Rt L tL
20 2 crP
Bourgain-Wang solutions (1997)
Another type of singular solutions of the critical NLS Solution splits into a singular core and a regular tail Singular core collapses with ψR,α
explicit profile Blowup rate is linear
ψB-W are unstable, since they are based on ψR,αexplicit
(Merle, Raphael, Szeftel; 2011) Non-generic solutions
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Continuation of NLS solutions beyond the
singularity
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Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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Explicit continuation of ψR,αexplicit (Merle, 92)
Let ψε be the solution of the critical NLS with the ic
Ψε exists globally Merle computed rigorously the limit
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explicit0 R,( ) (1 ) ( 0, ), 0 1r t r
0lim ( , ), 0t r t
2 2 2
220 (1 ) crPR
• Before singularity, since
• After singularity
Thm (Merle 92)
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explicit
0lim ( , ) ( , ) , 0c ct r t r tT T
explicitR,( 0) (1 ) ( 0),t t
explicitR,0
lim ( , ) ( , ), 0 ct r t r t T
• Before singularity
• After singularity
Thm (Merle 92)
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explicitR,0
lim ( , ) ( , ) , 0c ct r t r tT T
explicitR,0
lim ( , ) ( , ), 0 ct r t r t T
• Before singularity
• After singularity
Thm (Merle 92)
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explicitR,0
lim ( , ) ( , ) , 0c ct r t r tT T
0lim ( )
0
cL t tT
t
explicitR,0
lim ( , ) ( , ), 0 ct r t r t T
NLS is invariant under time reversibility
Hence, solution is symmetric w.r.t. to collapse-arrest time Tε
arrest
As ε 0, Tεarrest Tc
Therefore, continuation is symmetric w.r.t. Tc
Jump condition
Symmetry Property - motivation
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*andt t
Thm (Merle 92)
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• Symmetry property: Continuation is symmetric w.r.t. Tc
• Phase-loss Property: Phase information is lost at/after the singularity
• After singularity
explicit *R,0
For any , there exists a sequence 0 such that
lim ( , ) ( , ) , 0n
n
n
ic ct r t r tT Te
Phase-loss Property - motivation
Initial phase information is lost at/after the singularity
Why?
For t>Tc, on-axis phase is ``beyond infinity’’
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explicit 2R,
0
arg ( ,0) ( ) ( ) , ( ) )(
lim ( )
t
c
t Tc
t t L s ds L t tT
t
Merle’s continuation is only valid for Critical NLS Explicit solutions ψR,α
explicit
Unstable Non-generic
Can this result be generalized?
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Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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Sub threshold-power continuation (Fibich and Klein, 2011)
Let f(x) ∊H1
Consider the NLS with the i.c. ψ0 = K f(x) Let Kth be the minimal value of K for which the NLS solution
becomes singular at some 0<Tc<∞ Let ψε be the NLS solution with the i.c. ψ0
ε= (1-ε)Kth f(x) By construction,
0<ε≪1, no collapse -1≪ε<0, collapse
Compute the limit of ψε as ε0+ Continuation of the singular solution ψ(t, x; Kth) Asymptotic calculation (non-rigorous)
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• Before singularity
• Core collapses with ψR,αexplicit profile
• Blowup rate is linear• Solution also has a nontrivial tail
Conclusion: Bourgain-Wang solutions are ``generic’’, since they are the ``minimal-power’’ blowup solutions of ψ0 = K f(x)
Proposition (Fibich and Klein, 2011)
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0( , ) ( , ), 0lim cB Wt r t r t T
• Before singularity
• After singularity
• Symmetry w.r.t. Tc (near the singularity)
• Hence,
Proposition (Fibich and Klein, 2011)
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0( , ) ( , ), 0lim cB Wt r t r t T
0lim ( , ) ( , ) , 0 1c cB Wt r t r tT T
0lim ( ) , 1c cL t t t TT
Proposition (Fibich and Klein, 2011)
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*B-W0
For any , there exists a sequence 0 such that
lim ( , ) ( , ) , 0 1n
n
n
c cit r t r tT Te
• Phase information is lost at the singularity
• Why?
• After singularity
explicitR,lim arg lim argB Wt Tc t Tc
Simulations - convergence to ψB-W
Plot solution width L(t; ε)
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0lim ( ) cL t tT
2
40, 0, (1 )
1.481395tht xx
th
xi t x KK
e
Simulations – loss of phase
How to observe numerically? If 0<ε≪1, post-collapse phase is ``almost lost’’ Small changes in ε lead to O(1) changes in the phase
which is accumulated during the collapse Initial phase information is blurred
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Simulations - loss of phase
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O(10-5) change in ic lead to O(1) post-collapse phase changes
Simulations - loss of phase
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0lim arg( ( ) )ct T
O(10-5) change in ic lead to O(1) post-collapse phase changes
Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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NLS continuations So far, only within the NLS model:
Lower the power below Pth , and let PPth- Different approach: Add an infinitesimal perturbation to
the NLS Let ψε be the solution of
If ψε exists globally for any 0<ε≪1, can define the ``vanishing –viscosity continuation’’
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2
``viscosity''
, 0[ ]ti t F x
0
, : lim , , 0continuation t t t
x x
NLS continuations via vanishing -``viscosity’’ solutions
What is the `viscosity’? Should arrest collapse even when it is infinitesimally small Plenty of candidates:
Nonlinear saturation (Merle 92) Non-paraxiality Dispersion …
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Nonlinear damping
``Viscosity’’ = nonlinear damping Physical – multi-photon absorption Destroys Hamiltonian structure
Good!
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Critical NLS with nonlinear damping
Vanishing nl damping continuation : Take the limit δ0+
Consider ψ0 is such that ψ becomes singular when δ=0 if q≥ 4/d, collapse arrested for any δ>0 If q< 4/d, collapse arrested only for δ> δc(ψ0)>0
Can define the continuation for q≥ 4/d
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4/
10
, 0, 0 1
0,
d qti t i
H
x
x x
0
, : lim , , 0continuation t t t
x x
Explicit continuation
Critical NLS with critical nonlinear damping (q=4/d) Compute the continuation of ψR,α
explicit as δ0+
Use modulation theory (Fibich and Papanicolaou, 99) Systematic derivation of reduced ODEs for L(t) Not rigorous
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4/
explicit0 R,
, (1 ) 0
0,
dti t i
t r
x
Asymptotic analysis Near the singularity
Reduced equations given by
Solve explicitly in the limit as δ0+
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~
3 2( ) , ( )tt tt tL
L L
R / 2
1( , ) ~ ( , )( ) ( )d
rt r t r Rt L tL
Asymptotic analysis Near the singularity
Reduced equations given by
Solve explicitly in the limit as δ0+
Asymmetric with respect to Tc Damping breaks reversibility in time
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~
3 2( ) , ( )tt tt tL
L L
R / 2
1( , ) ~ ( , )( ) ( )d
rt r t r Rt L tL
0
( ) 0lim ( ; ) 1.614
( ),c c
c c
t tT TL tt tT T
• Before singularity
• After singularity
• Phase information is lost at the singularity• Why?
Proposition (Fibich, Klein, 2011)
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explicitR,0
lim
* explicitR,0
For any , there exists a sequence 0 such that
lim 1.61( , ) , ), 4(n
n
ic ct r t rT Te
explicitR,lim arg
t Tc
Simulations – asymmetric continuation
L=α)Tc-t( L=κα(t-Tc)
Simulations – loss of phase
Nonlinear-damping continuation of loglog solutions
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240
6
Example:
, 0, 1.6 xt xxi t x i e
5 4......... 2.5 4____ 1 5
ee
e
• Highly asymmetric• Slope ±∞
0lim ( ; ) c
tc
t TtL t T
Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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Collapse in linear optics
Can solve explicitly in the geometricaloptics limit (k0∞)
Linear collapse at z=F
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2
0
0
2
2
2 , 0,
( 0, )
z
ikF
i zk
z e e
x
x
xx
2
2( )/ 2
1 , ( ) 1( )
zLd
zL zz FL e
x
GO
• z=t• k0 is wave #
Continuation of singular GO solution consider the linear Schrödinger with k0<∞ Global existence Can solve explicitly (without GO approx) Compute the limit as k0∞
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lim ( ) 1 , 0k
zL z zF
GO
*/2
GO
( , ),lim ( , )
(2 , ),idk
z z Fz
F z F ze
xx
x
No phase loss after singularity Why?
Post-collapse phase loss is a nonlinear phenomena
Continuation of singular GO solution
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GOlim arg 0 ( )z F
GO
*/2
GO
( , ),lim ( , )
(2 , ),idk
z z Fz
F z F ze
xx
x
Talk plan
1. Review of NLS theory2. Merle’s continuation3. Sub-threshold power continuation4. Nonlinear-damping continuation 5. Continuation of linear solutions6. Phase-loss property
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Universality of loss of phase
All NLS continuations have the Phase-loss Property Phase of all known singular NLS solutions becomes
infinite at the singularity Hence, any continuation of singular NLS solutions will
have the Phase-loss Property When collapse-arresting mechanism is small but not
zero, post-collapse phase is unique. But, the initial phase information is blurred by the large sensitivity to small perturbations of the phase accumulated during the collapse. Initial phase is ``almost lost’’
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Simulations – NLS with nonlinear saturation
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42, , 0, 5 4ti t x y e
Simulations – NLS with nonlinear saturation
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2 10, , 0, 0.3 8ti t x y e
Experiments (Shim et al., 2012) Laser beam after propagation of 24cm in water ``Correct’’ physical continuation is not known
Post-collapse loss-of-phase is observed
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Simulations of propagation in water NLS with dispersion, space-time focusing, multiphoton
absorption, plasma … Input-power randomly chosen between 240 -260 MW On-axis phase after propagation of 24cm
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Importance of loss of phase NLS solution is invariant under multiplication by eiθ
Multiplication by eiθ does not affect the dynamics But, relative phase of two beams does affect the dynamics
62collapse no
collapse
Importance of loss of phase NLS solution is invariant under multiplication by eiθ
Multiplication by eiθ does not affect the dynamics But, relative phase of two beams does affect the dynamics
63collapse no
collapse
Importance of loss of phase NLS solution is invariant under multiplication by eiθ
Multiplication by eiθ does not affect the dynamics But, relative phase of two beams does affect the dynamics
64collapse no
collapse
Importance of loss of phase NLS solution is invariant under multiplication by eiθ
Multiplication by eiθ does not affect the dynamics But, relative phase of two beams does affect the dynamics
65collapse no
collapse
Post-collapse chaotic
interactions
Experiments (Shim et al., 2012) Interaction between two ``identical’’ crossing beams after
propagation of 24cm in water - seven consecutive shots
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Experiments (Shim et al., 2012) Interaction between two ``identical’’ crossing beams after
propagation of 24cm in water - seven consecutive shots
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Experiments (Shim et al., 2012) Interaction between two parallel beams with initial
π phase difference, after propagation of 24cm in water - five consecutive shots
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Summary
1. Sub-threshold power continuation ψ0
ε= (1-ε)Kthf(x) Generalization of Merle (92) Limiting solution is a Bourgain-Wang sol., before and after
the singularity
2. Vanishing nonlinear-damping continuation Vanishing-viscosity approach Viscosity = nonlinear damping Explicit continuation of ψR,α
explicit Asymmetric w.r.t. Tc
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Summary: Properties of continuations
Loss of phase at/after singularity Universal feature Leads to post-collapse chaotic interactions Observed numerically and experimentally
Symmetry with respect to Tc
Jump condition Only holds for time-reversible continuations Not a universal feature
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Open problems
What is the `correct’’ continuation? Additional properties of continuations?
``Entropy’’ conditions? ``Riemann Problems’’? …
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References
G. Fibich and M. Klein Continuations of the nonlinear Schrödinger equation beyond the singularity Nonlinearity 24: 2003-2045, 2011
G. Fibich and M. Klein Nonlinear-damping continuation of the nonlinear Schrödinger equation- a numerical study Physica D 241: 519-527, 2012
B. Shim, S.E. Schrauth, , A.L. Gaeta, M. Klein, and G. FibichLoss of phase of collapsing beams Physical Review Letters 108: 043902, 2012
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