clk08-sivan

Yonatan Sivan, Tel Aviv University, Israelnow at Purdue University, USA

• G. Fibich, Tel Aviv University, Israel

• M. Weinstein, Columbia University, USA

• B. Ilan, UC Merced, USA

Stability and instability of solitons in inhomogeneous

media

Motivation

• Light propagation in a nonlinear medium– z is the direction of propagation– transverse coordinate

• Look for solitons – waves that maintain their shape along the propagation in z

z

Nonlinear medium

x

1( , , ), 1,2,3dx x x d= =…

laser

Solitons

• Appear in– Nonlinear optics– Cold atoms – Bose-Einstein Condensation (BEC) – Solid state– Water waves– Plasma physics

• Applications – communications, quantum computing, …

Soliton stability• Ideally – get same soliton at the other end

• In practice, soliton must be stable (robust) under perturbations

z

Nonlinear medium

xlaser

E field

E field

Stability analysis

• In this talk, a different approach– Qualitative approach: characterize instability dynamics– Quantitative approach: quantify strength of

stability/instability

• KEY question – is the soliton stable?– hundreds of papers…

• Typical answer – yes (stable)/no (unstable)– What is the instability dynamics?

Outline of the talk

• Solitons in Nonlinear Schrödinger (NLS) Eq.

• Stability theory

• Qualitative approach

• Quantitative approach

Paraxial propagation – NLS model

• A – amplitude of electric field • Initial condition:

– z is a “time”-like coordinate • Competition of diffraction ( ) and

focusing nonlinearity (F)

2 2( , ) (| | ) 0zdiffraction focusing

nonlinearity

iA z x A F A A+ ∇ + =

0( 0, ) ( )A z x A x= =

zNonlinear medium

xlaser

1

2 2 2dx x∇ = ∂ + + ∂

Typical (self-) focusing nonlinearities

• Cubic (Kerr) nonlinearity

• Cubic-quintic nonlinearity

• Saturable nonlinearity

2 2( )F A A=

2 2 4( )F A A Aγ= −

22

2( )1

AF AAγ

=+

2 2( , ) (| | ) 0ziA z x A F A A+∇ + =2

0 ( )n n F A= +

Physical configurations – slab/planar waveguide

•• no dynamics in y direction• 1+1 dimensions (x,z)

2( , ) (| | ) 0z xx

diffraction nonlinearity

iA z x A F A A+ + =

y

, 1x x d= =

Physical configurations – bulk medium

•• 2+1 dimensions (x+y,z)

z

z=0

Nonlinear medium

2 2( , , ) (| | ) 0zdiffraction nonlinearity

iA z x y A F A A+ ∇ + =

xlaser

( , ), 2x x y d= =

Physical configurations – pulses in bulk medium

•• β2 < 0, anomalous group velocity dispersion (GVD)• 3+1 dimensions (x+y+t,z)• Spatio-temporal soliton = “Light bullets”

2 22( , , , ) (| | ) 0z tt

diffraction dispersion nonlinearity

iA z x y t A A F A Aβ+ ∇ − + =

( , , ), 3tx x y d= =

• Solitons are of the form

• Do not change their shape during propagation (in z)• Exhibit perfect balance between diffraction and

nonlinearity

( ) ( )2 2( , ) ; , ( ) 0;i zA x x xe u u F u u uz ν ν νν= ∇ + − =

Eq. for solitons2 2( , ) (| | ) 0ziA z x A F A A+∇ + =

• Explicit solution• (propagation const.) proportional to

– amplitude– inverse width

Example – 1D solitons in a Kerr medium

( ); 2 sech( )u x xν ν ν=

2

( )

( , ) 0z xxKerr cubicnonlinearity

iA z x A AA+ + =

( ) ( )2 3( , ) ; , 0;i zA e u ux x x u uz ν ν νν= ∇ + − =

ν

~1/√ν~√ ~ν

x

Outline of the talk

• Solitons in Nonlinear Schrödinger Eq.


• Qualitative approach


Stability theory

• Vakhitov-Kolokolov (1973): necessary condition for stability of

is the slope condition

2( ) 0, ( ) ( ; )soliton power

P P u x dxν ν νν

∂> =

∂ ∫

( , )i zA e u xν ν=

Example: homogeneous Kerr medium

•

1, 0

2, 0

3, 0

Pd

Pd

Pd

ν

ν

ν

∂= >

∂∂

= =∂∂

= <∂

⇒

⎫⇒⎬

⎭ins

stabili

tabi

ty

lity

22 2( ) ( ; ) ( )

d

P u x dx C dν ν ν−

= =∫

Stability in a d=1 Kerr medium

• Incident beam is a perturbed d=1 soliton

• Solution stays close to the soliton• 1D Solitons are stable!

( )

( 0, ) (1 0.05) 2 sech( )u x

A z x xν ν= = +

| |A

0 200

2

z

| ( , 0) |A z x =

Instability in a d=2 Kerr medium

•

• collapse at a finite distance!• 2D Solitons are unstable

( 0) (1 0.02) ( , )A z u x y= = +

• Incident beam is a perturbed d=2 soliton

0 1.20

0.2

0 1.20

50

z z

| ( ) |A z 1| ( ) |A z −

amplitude width

• Vakhitov-Kolokolov (1973): Slope condition is necessary for stability

• Is it also sufficient?

• Weinstein (1985-6), Grillakis, Shatah, Strauss (1987-9):

1. Slope (VK) condition

2. Spectral condition: the operator

must have only one negative eigenvalue• Two conditions are necessary and sufficient for

stability

Rigorous stability theory (u>0)

2 2( )L G uν+ = −∇ + −

• Only one negative eigenvalue λmin

• Spectral condition is satisfied

continuous spectrummin 0λ < 0λ

0 ν

Spectrum of L+

2 2( )L G uν+ = −∇ + −

Summary - stability in homogeneous medium

• Spectral condition is always satisfied • Stability determined by slope (VK) condition

2( ) 0, ( ) ( ; )soliton power

P P u x dxν ν νν

∂> =

∂ ∫

Inhomogeneous media

Light propagation in inhomogeneous medium

• Since late 1980’s: interest in inhomogeneous media– E. Yablonovitch, S. John (1986) – photonic crystals– Christodoulides & Joseph (1988) – discrete solitons– ...

• Goals:– Stabilize beams in high dimensions (“Light bullets”)– Applications – communications (switching, routing, …)

• Varying linear refractive index– Waveguide arrays / photonic lattices

Inhomogeneous media

02

2( ) | |n xn n A= +02

2( ) | |n zn n A= +

• Varying nonlinear refractive index– Novel materials

Inhomogeneous media

20 2 ( ) | |n n n x A= +2

0 2 ( ) | |n n n z A= +

• This study – modulation in only• Refractive index = – Potential

NLS in inhomogeneous media

x

• This study – modulation in only• Refractive index = – Potential

– arbitrary potentials (Vnl, Vl)• periodic/disordered potentials, periodic potentials with defects,

single/multi-waveguide potentials etc.

– any nonlinearity F– any dimension d

( )

NLS in inhomogeneous media

( )2 2( ), 1 (| ( )| ) 0lz nlV xiA z x A A V x AF A+∇ + − − =

x

Inhomogeneities in BEC

• Same equation (Gross-Pitaevskii) in BEC

• Dynamics in time (not z)•• Inhomogeneities created by

– Magnetic traps– Feshbach resonance– Optical lattices

( ) ( )2 2( ), 1 (| ( )| ) 0lt nlV xiA t x A A V x AF A+∇ + − − =

( , , ), ( , ),x x y z x x y x x= = =

How do inhomogeneitiesaffect stability?

“Applied” approach

• Slope (VK) condition• Numerics• Ignore spectral condition

Rigorous approach

• Slope (VK) condition• Spectral condition

) 2( ( ) (| | ) ( )nl lVL L V x G A V x+ + += +

Typical result – soliton stable (yes)/unstable (no)

Qualitative approach

• Characterize the instability dynamics• Key observation: instability dynamics depends

on which condition is violated– Look at each condition separately

• Results for ground state solitons only (u>0)

Outline of the talk

• Solitons in Nonlinear Schrödinger Eq.


• Qualitative approach– Slope condition– Spectral condition


Instability due to violation of slope condition

Violation of slope condition

• d=2 homogeneous Kerr medium –– Slope = 0, i.e., instability

–• Collapse

–• Total diffraction

( )P Constν =

0 1.20

50

z

| |A

0 100

1

z

| |A( 0) (1 0.02) ( , )A z u x y= = +

−5 100ν

( 0) (1 0.02) ( , )A z u x y= = −

power

• d=2 Kerr medium with potential– Stable and unstable branches

–– 1% perturbation width decrease by factor of 15

Violation of slope condition – cont.

| ( ) |A z

−5 100ν

unstable stable

homogeneous medium

( 0) (1 0.01) ( , )

power

A z u x y= = +

0 52

30

z0 5

0

1

z

amplitude

1| ( ) |A z −

width

Violation of slope condition – cont.

Conclusion:violation of slope condition focusing instability

0 52

30

z

Instability due to violation of spectral condition

Spectral condition

• The operator

must have only one negative eigenvalue

• No potential : spectral condition satisfied

( ) 2 2((1 ) () () )Vnl lV x u xL VG uν+ = −∇ + − − +

( )( , ) , 0i zA z x e u x uν= >

continuous spectrummin 0λ < 0λ

0 ν( )( )VL L+ +=

• With potential:– remains negative– continuous spectrum remains positive– only can become negative

• Spectral condition determined by • Spectral condition not automatically satisfied

( )0Vλ

0

continuous spectrum

( )m in 0Vλ < ( )

0Vλ

?

( )minVλ

Spectral condition – cont.

( )0Vλ

lattice

Generic families of solitons

soliton

Soliton at lattice min. Soliton at lattice max.

Sign of λ0(V)

( )0Vλ

• Numerical/asymptotic/analytic observation

λ0(V) > 0 for solitons at a lattice

min. (potential well)spectral condition satisfied

ν

0

Sign of λ0(V)

( )0Vλ

• Numerical/asymptotic/analytic observation

λ0(V) < 0 for solitons at a lattice

max. (potential barrier)spectral condition violated

λ0(V) > 0 for solitons at a lattice

min. (potential well)spectral condition satisfied

ν

0 0

( )0Vλ

ν

• Stability only at potential min. – solitons are more “comfortable” at a potential min. (well) than at a potential max. (barrier)– stay near potential min.– tend to move from potential max. to potential min.

• Different type of instability– Lateral location rather width

x

Physical intuition

• Soliton centered at a lattice min. – stays at lattice min.– lateral stability

0 25

<x>

z

max.

min.

0 25

<x>

z

min.

max.

• Soliton centered at a lattice max. – moves from lattice max.

to lattice min.– drift instability

x

Numerical demonstration

Center of

mass

( 0 , ) ( )sh ift

A z x u x δ= = −Input beam:

x

2x x A= ∫“Center of mass”:

Conclusion:violation of spectral condition drift instability

0 25

<x>

z

min.

max.

Mathematical intuition

• Eigenfunction associated with λ0(V) is odd

+ =

eigenfunction

x



+ =

eigenfunctionsoliton

xx x



• Its growth causes lateral shift of beam center

+ =

eigenfunctionsoliton

xx x

Qualitative approach – summary

• Characterization of instabilities– Violation of slope condition focusing instability– Violation of spectral condition drift instability

• Distinction between instabilities is useful for more complex lattices

• Created experimentally by optical induction

V

Example

• Created experimentally by optical induction• Consider solitons centered at a shallow local

maximum

V

Example

Example

x

y

• Narrow solitoncentered at a lattice max.

x

y

Example


x

y

Example

x

y

• Wide solitons:solitoncentered at a lattice max.

x

y

Example

Drift instability

0 1.3z

<x> <y>

global max.

global max.

shallow max.

x

y



x

y

Example

Drift instability Stability

0 1.3z

<x> <y>

global max.

global max.

shallow max.

0 20z

<x> <y>

global max.

global max.

shallow max.

x

y



• Narrow solitons: centered at a lattice max.

• Wide solitons:effectivelycentered at a lattice max. min.x

y

Example

Drift instability Stability

0 1.3z

<x> <y>

global max.

global max.

shallow max.

0 20z

<x> <y>

global max.

global max.

shallow max.

x

y


• Wide solitons:effectivelycentered at a lattice max. min.x

y

Example

x

y

At which width is the transition between drift instability and stability?

x

y

Example

0.06 1.5Width

λ0(V)

0( )0Vλ

• Wide solitons:effectivelycentered at a lattice min.


x

y

x

y

Example

0.06 1.5Width

λ0(V)

0( )0Vλ

• Wide solitons:effectivelycentered at a lattice min.


x

y

transition

Conclusion

• Dynamics “deciphered” using the qualitative approach– “effective centering” determines violation/satisfaction

of spectral condition – In turn, determines dynamics of “center of mass”– Dynamics is determined by slope condition

Outline of the talk

• NLS and solitons - review


• Qualitative approach– Slope condition – width instability– Spectral condition – drift instability


Motivation: 2d nonlinear lattices

• Can the nonlinear potential stabilize the solitons?– spectral condition satisfied only at potential min. – slope condition satisfied only for

• narrow solitons• specially designed potential

( ) ( )2 2, , 1 | |( , ) 0nz lViA z A xx yy A A+∇ + − =

Narrow solitons in 2d nonlinear lattices• u(x,y) is a stable narrow soliton• Test stability of u(x,y) numerically:

– add extremely small perturbation

collapse instability!

( 0, , ) 1.0001 ( , )A z x y u x y= =

0 20

ampl

itude

z

Narrow solitons in 2d nonlinear lattices• u(x,y) is a stable narrow soliton• Test stability of u(x,y) numerically:

– add extremely small perturbation

collapse instability!

( 0, , ) 1.0001 ( , )A z x y u x y= =

0 20

ampl

itude

z

What is going on?

Strength of stability• Slope = 0 instability• Slope > 0 stability

– What happens for a very small positive slope?• Strength of stability determined by magnitude of

slope– small slope leads to weak stabilization– stronger stability for larger slope

Strength of stability: nonlinear lattices

0 80

ampl

itude

z

• Fixed input beam

• Change lattice – slope = 0.01 (stable)

( 0) 1.0001 ( , )A z u x y= =

0 80

ampl

itude

z




– slope = 0.006 (less stable)

( 0) 1.0001 ( , )A z u x y= =

0 80

ampl

itude

z





– slope = 0.002 (unstable)

( 0) 1.0001 ( , )A z u x y= =

0 80

ampl

itude

z





– slope = 0.0002 (even more unstable)

( 0) 1.0001 ( , )


A z u x y= =

0 80

ampl

itude

z






– slope = 0.0002 (even more unstable)

• Strength of stability is determined by magnitude of slope

( 0) 1.0001 ( , )A z u x y= =

Narrow solitons in 2d nonlinear lattices• Original soliton was weakly stable

– Slope = 0.01( 0) 1.0001 ( , )A z u x y= =

0 80

ampl

itude

z

Narrow solitons in 2d nonlinear lattices• Original soliton was weakly stable

– Slope = 0.01

– Stability for a smaller perturbation • Soliton is “theoretically” stable, but “practically”

unstable

( 0) 1.0001 ( , )A z u x y= =0 80

ampl

itude

z( 0) 1.00004 ( , )A z u x y= =

• “Old” approach – check only sign of slope• “New” quantitative approach – check also the

magnitude of the slope.

Slope condition: “old” approach

instability stability

Discontinuous transition between stability and instability

0 Slope

width


Continuous transition between stability and instability

0 Slope

Slope condition: quantitative approach

width

Spectral condition: “old” approach


Discontinuous transition between stability and instability

0 λ0(V)

drift



λ0(V)

0

Spectral condition: quantitative approach

drift



λ0(V)

0Can we get an analytical prediction for the drift rate as a function of ?

Spectral condition: quantitative approach

drift

λ0(V)

• soliton width/potential period <<1

• Can use perturbation analysis!

• Expand potential as

• Solve for

( )21( ) (0) ''(0)2

V x V V xε= + +

Narrow solitons

soliton

ε =

2x x A= ∫

Narrow solitons – cont.

• Center of mass obeys an oscillator equation!

2

22 22

2

Ehrenfest law

2 2 ''(0) ,x

d x d A V dV A xdz

εΩ

= − ∇ −∫ ∫

2 21( ) (0) ''(0)2

V x V V xε= + +

• Compute effective force

22 2 2

2 , 2 ''(0)d x x dVdz

ε= Ω Ω = −

Narrow solitons - cont.

• Use perturbation analysis to compute eigenvalue

• Combine with Ω2 = - 2dV''(0)ε2 and get:

( )2 2 ( ) 2

04,(0) (0)

V dC CV V

λν

Ω = − =+

( )2

( )0

''(0)(0) (0)

V VV V

ελν

≈+

Narrow solitons – cont.• Conclusion:

22 2 2 ( )

02 , Vd x x Cdz

λ= Ω Ω = −

( ) 20

( ) 20

stability0 0 oscil

ins

latory solutions

0 0 expone tabntial solution is il ty

V

V

λ

λ

⎧ > ⇒ Ω < ⇒ ⇒⎪⎨⎪ < ⇒ Ω > ⇒ ⇒⎩

(Lattice min.)

(Lattice max.)

Analytical quantitative relation between spectral condition ( ) and dynamics (Ω)λ0

(V)

• But so far only for narrow beams • Can we compute the drift rate also for wider

solitons?

Calculation of drift rate in a general setting

• Valid for– any soliton width – any potential (periodic/non-periodic, single/multi waveguide, …)– any nonlinearity (Kerr, cubic-quintic, saturable, …)– any dimension

( )

( ) ( )2 ( ) ( ) ( ) ( )

01( ) ( ) ( )

,0,

,

V VV V V V

V V V

f fC L f f

f L fλ+−

−

= > =

22 2 2 ( )

02 , Vd x x Cdz

λ= Ω Ω = −

• Soliton slightly shifted from lattice max.

0 200

1

z

<x>

Examples of lateral dynamics (1)(0, ) ( )

shift

A x u x δ= −

max

min

max numerics

z/Ldiff400

• Soliton slightly shifted from lattice max.• Solution of oscillator equation

– Ω is the drift (=instability) rate

– agreement up to the lattice min.– up to many diffraction lengths

Examples of lateral dynamics (1)

cosh( )x zδ= Ω

0 20

0

1

z

<x>

max

min

maxtheorynumerics

z/Ldiff400

(0, ) ( )shift

A x u x δ= −

Drift rates

• Excellent agreement between analysis and numerics

d=1, weak lattice

d=2, strong lattice

10 100 3000

3

ν

Ω

0 50 1000

17

ν

Ω

theorynumerics

• So far, studied instability dynamics• Oscillator equation good also for stability

dynamics


max

max

min

z

– soliton moving at an angle from a lattice min.

sin( )tgx zθ= Ω

Ω• Solution of oscillator equation is

– Ω determines the maximal deviation (=strength of stability)

θ


• d=2, periodic lattice

Kerr (cubic) medium Cubic-Quintic medium

0 200

max

max

min

z / Ldiff

0 10

<x>

z / Ldiff

max

max

min

theory

numerics

sin( )tgx zθ= ΩΩ

Implications of quantitative study

• Experimental example (Morandotti et al. 2000): – experiment in a slab waveguide array – soliton centered at a lattice max. does not drift to a

lattice min. over 18 diffraction lengths• Explanation: absence of observable drift due to

small drift rate• Theoretical instability but practical stability

Summary• Qualitative approach

– Slope condition focusing instability– Spectral condition drift instability

• Quantitative approach– continuous transition between stability and instability– analytical formula for lateral dynamics– still open: find analytical formula for width dynamics

• “Theoretical” vs. “practical” stability/instability• Results valid for any physical configuration of

lattice, dimension, nonlinearity

clk08-sivan

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