Stability and instability of solitons in inhomogeneous media

Stability and instability of solitons in inhomogeneous media Yonatan Sivan, Tel Aviv University, Israel now at Purdue University, USA G. Fibich, Tel Aviv University, Israel M. Weinstein, Columbia University, USA B. Ilan, UC Merced, USA

Motivation laser x z Nonlinear medium Light propagation in a nonlinear medium z is the direction of propagation transverse coordinate x = ( x,, x ), d = 1,2,3 Look for solitons waves that maintain their shape along the propagation in z 1 d

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 x laser Nonlinear medium z E field In practice, soliton must be stable (robust) under perturbations E field

Stability analysis KEY question is the soliton stable? hundreds of papers Typical answer yes (stable)/no (unstable) What is the instability dynamics? In this talk, a different approach Qualitative approach: characterize instability dynamics Quantitative approach: quantify strength of stability/instability

Outline of the talk Solitons in Nonlinear Schrödinger (NLS) Eq. Stability theory Qualitative approach Quantitative approach

Paraxial propagation NLS model laser Nonlinear medium ia (, z x) + A + F( A ) A = 0 z A amplitude of electric field Initial condition: z is a time -like coordinate x 2 2 diffraction focusing nonlinearity A( z = 0, x) = A ( x) 2 2 2 Competition of diffraction ( = x + + ) and 1 x d focusing nonlinearity (F) z 0

Typical (self-) focusing nonlinearities ia (, z x) + 2 A + F( A ) 2 A = 0 z n= n + F A 2 0 ( ) Cubic (Kerr) nonlinearity Cubic-quintic nonlinearity F( A ) = 2 2 A F( A ) = A γ A 2 2 4 Saturable nonlinearity F A 2 ( ) = A 1 + γ 2 A 2

Physical configurations slab/planar waveguide 2 iaz(, z x) + A ( ) 0 xx + F A A = diffraction x = x, d = 1 no dynamics in y direction 1+1 dimensions (x,z) y nonlinearity

Physical configurations bulk medium iaz (, z x, y) + A + F( A ) A = 0 x = ( x, y), d = 2 2+1 dimensions (x+y,z) 2 2 diffraction nonlinearity x laser z Nonlinear medium z=0

Physical configurations pulses in bulk medium 2 2 iaz(, z x, y,) t + A β2 Att + F( A ) A = 0 diffraction dispersion nonlinearity x = ( x, y, t), d = 3 β 2 < 0, anomalous group velocity dispersion (GVD) 3+1 dimensions (x+y+t,z) Spatio-temporal soliton = Light bullets

Eq. for solitons ia (, z x) + 2 A + F( A ) 2 A = 0 z Solitons are of the form (, ) i z A z x = e ν u x;, 2 u x; + F( u 2 ) u νu= 0 ( ν) ( ν ) Do not change their shape during propagation (in z) Exhibit perfect balance between diffraction and nonlinearity

Example 1D solitons in a Kerr medium z ia ( z, x) + A + A A = 0 Explicit solution ( ) ν (propagation const.) proportional to amplitude inverse width z i ν z xx 2 Kerr ( cubic) nonlinearity ( ν) ( ν ) 2 3 A(, x) = e u x;, u x; + u νu= 0 u x; ν = 2ν sech( νx) ~ ν ~ x ~1/ ν

Outline of the talk Solitons in Nonlinear Schrödinger Eq. Stability theory Qualitative approach Quantitative approach

Stability theory Vakhitov-Kolokolov (1973): necessary condition for stability of is the slope condition iν z A = e u( x, ν ) P( ν ) ν 2 > 0, P( ν) = u ( x; ν) dx soliton power

Example: homogeneous Kerr medium ν = ν = ν 2 d 2 2 P( ) u ( x; ) dx C( d) d = 1, P > 0 ν d = 2, P = 0 ν d = 3, P < 0 ν stability instability

Stability in a d=1 Kerr medium Az ( = 0, x) = (1 + 0.05) 2ν sech( ν x) u( x) Incident beam is a perturbed d=1 soliton A Azx= (, 0) 2 0 0 20 z Solution stays close to the soliton 1D Solitons are stable!

Instability in a d=2 Kerr medium Incident beam is a perturbed d=2 soliton A z A( z) 50 ( = 0) = (1+ 0.02) u( x, y) amplitude A( z) 1 0.2 width 0 0 1.2 z collapse at a finite distance! 2D Solitons are unstable 0 0 1.2 z

Vakhitov-Kolokolov (1973): Slope condition is necessary for stability Is it also sufficient?

Rigorous stability theory (u>0) Weinstein (1985-6), Grillakis, Shatah, Strauss (1987-9): 1. Slope (VK) condition 2. Spectral condition: the operator 2 2 L ν + = + G( u ) must have only one negative eigenvalue Two conditions are necessary and sufficient for stability

Spectrum of L + 2 2 L ν + = + G( u ) λ min < 0 λ0 ν Only one negative eigenvalue λ min Spectral condition is satisfied

Summary - stability in homogeneous medium Spectral condition is always satisfied Stability determined by slope (VK) condition P( ν ) ν 2 > 0, P( ν) = u ( x; ν) dx soliton power

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, )

Inhomogeneous media Varying linear refractive index Waveguide arrays / photonic lattices n= n ( z) + n A 0 2 2 n= n ( x) + n A 0 2 2

Inhomogeneous media Varying nonlinear refractive index Novel materials n= n + n ( z) A 0 2 2 n= n + n ( x) A 0 2 2

NLS in inhomogeneous media This study modulation in only x Refractive index = Potential

This study modulation in only Refractive index = Potential ( ) NLS in inhomogeneous media ia z, x A 1 F( A ) A V() x A 0 z + 2 + ( V () x ) 2 = nl l arbitrary potentials (V nl, V l ) periodic/disordered potentials, periodic potentials with defects, single/multi-waveguide potentials etc. any nonlinearity F any dimension d x

Inhomogeneities in BEC Same equation (Gross-Pitaevskii) in BEC ia t, x A 1 F( A ) A V() x A 0 t + 2 + V 2 = nl l ( ) ( () x ) Dynamics in time (not z) Inhomogeneities created by x = ( xyz,, ), x= ( xy, ), x= x Magnetic traps Feshbach resonance Optical lattices

How do inhomogeneities affect stability?

Applied approach Rigorous approach Slope (VK) condition Numerics Ignore spectral condition Slope (VK) condition Spectral condition L L V ( x) G( A ) V( x) ( V ) 2 + = + + nl + l 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. Stability theory Qualitative approach Slope condition Spectral condition Quantitative approach

Instability due to violation of slope condition

Violation of slope condition d=2 homogeneous Kerr medium P( ν ) = Const Slope = 0, i.e., instability A( z = 0) = (1 + 0.02) u( x, y) Collapse A power 5 100 ν 50 0 0 1.2 z A( z = 0) = (1 0.02) u( x, y) Total diffraction A 1 0 0 10 z

Violation of slope condition cont. d=2 Kerr medium with potential Stable and unstable branches unstable power homogeneous medium stable A( z = 0) = (1+ 0.01) u( x, y) 5 100 ν 1% perturbation width decrease by factor of 15 30 amplitude 1 width A( z ) A( z ) 1 2 0 5 z 0 0 5 z

Violation of slope condition cont. Conclusion: violation of slope condition focusing instability 30 2 0 5 z

Instability due to violation of spectral condition

The operator L Spectral condition iν z Azx (, ) = e u( x), u> 0 V x) u ) V ( x) ( V ) 2 ν 2 + = + (1 nl ( ) G( u+ must have only one negative eigenvalue l ( ) + + ( ) No potential L V = L : spectral condition satisfied ν λ min < 0 λ0

Spectral condition cont. λ < ( V ) min 0 With potential: V λ ( ) λ ( V ) 0 remains negative min continuous spectrum remains positive ( V ) only can become negative λ0 ( V ) Spectral condition determined by λ0 Spectral condition not automatically satisfied

Generic families of solitons Soliton at lattice min. Soliton at lattice max.

Sign of λ 0 (V) Numerical/asymptotic/analytic observation λ 0 (V) > 0 for solitons at a lattice min. (potential well) spectral condition satisfied λ ( V ) 0 0 ν

Sign of λ 0 (V) Numerical/asymptotic/analytic observation λ 0 (V) > 0 for solitons at a lattice min. (potential well) spectral condition satisfied λ 0 (V) < 0 for solitons at a lattice max. (potential barrier) spectral condition violated λ ( V ) 0 λ ( V ) 0 0 0 ν ν

Physical intuition 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

Numerical demonstration Input beam: Center of mass : Soliton centered at a lattice min. stays at lattice min. lateral stability <x> A( z = 0, x) = u( x δ ) x = xa Soliton centered at a lattice max. moves from lattice max. to lattice min. drift instability 2 <x> shift Center of x mass x max. min. 0 25 z min. max. 0 25 z

Conclusion: violation of spectral condition drift instability <x> min. max. 0 25 z

Mathematical intuition Eigenfunction associated with λ 0 (V) is odd eigenfunction + = x

Mathematical intuition Eigenfunction associated with λ 0 (V) is odd soliton eigenfunction + = x x x

Mathematical intuition Eigenfunction associated with λ 0 (V) is odd Its growth causes lateral shift of beam center soliton eigenfunction + = x x 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

Example V Created experimentally by optical induction

Example V Created experimentally by optical induction Consider solitons centered at a shallow local maximum

Example y x

Example Narrow soliton centered at a lattice max. y x

Example Narrow soliton centered at a lattice max. y x Wide solitons: soliton centered at a lattice max. y x

Example Narrow soliton centered at a lattice max. y x Wide solitons: soliton centered at a lattice max. y x Drift instability <x> <y> global max. global max. shallow max. 0 1.3 z

Example Narrow soliton centered at a lattice max. y x Wide solitons: soliton centered at a lattice max. y x Drift instability Stability <x> <y> global max. global max. shallow max. <x> <y> global max. global max. shallow max. 0 1.3 z 0 20 z

Example Narrow solitons: centered at a lattice max. y Drift instability x Wide solitons: effectively centered at a lattice max. min. Stability y x <x> <y> global max. global max. shallow max. <x> <y> global max. global max. shallow max. 0 1.3 z 0 20 z

Example Narrow solitons: centered at a lattice max. y x Wide solitons: effectively centered at a lattice max. min. y x At which width is the transition between drift instability and stability?

Example Narrow solitons: centered at a lattice max. y x Wide solitons: effectively centered at a lattice min. λ 0 (V) y x λ ( V ) 0 0 0.06 1.5 Width

Example Narrow solitons: centered at a lattice max. y x Wide solitons: effectively centered at a lattice min. λ 0 (V) y x λ ( V ) 0 0 transition 0.06 1.5 Width

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 Stability theory Qualitative approach Slope condition width instability Spectral condition drift instability Quantitative approach

Motivation: 2d nonlinear lattices ( ) ( V y ) 2 2 nl ia z, x, y + A+ 1 (, x ) A A= 0 z Can the nonlinear potential stabilize the solitons? spectral condition satisfied only at potential min. slope condition satisfied only for narrow solitons specially designed potential

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 A( z = 0, x, y) = 1.0001 uxy (, ) collapse instability! amplitude 0 20 z

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 Fixed input beam A( z = 0) = 1.0001 u( x, y) Change lattice slope = 0.01 (stable) amplitude 0 80 z

Strength of stability: nonlinear lattices Fixed input beam A( z = 0) = 1.0001 u( x, y) Change lattice slope = 0.01 (stable) slope = 0.006 (less stable) amplitude 0 80 z

Strength of stability: nonlinear lattices Fixed input beam A( z = 0) = 1.0001 u( x, y) Change lattice slope = 0.01 (stable) slope = 0.006 (less stable) slope = 0.002 (unstable) amplitude 0 80 z

Strength of stability: nonlinear lattices Fixed input beam A( z = 0) = 1.0001 u( x, y) Change lattice slope = 0.01 (stable) slope = 0.006 (less stable) slope = 0.002 (unstable) amplitude slope = 0.0002 (even more unstable) 0 80 z

Narrow solitons in 2d nonlinear lattices Original soliton was weakly stable Slope = 0.01 A( z = 0) = 1.0001 u( x, y) amplitude 0 80 z

Narrow solitons in 2d nonlinear lattices Original soliton was weakly stable Slope = 0.01 A( z = 0) = 1.0001 u( x, y) A( z = 0) = 1.00004 u( x, y) Stability for a smaller perturbation 0 80 z Soliton is theoretically stable, but practically unstable amplitude

Old approach check only sign of slope New quantitative approach check also the magnitude of the slope.

Slope condition: old approach Discontinuous transition between stability and instability width instability stability 0 Slope

Slope condition: quantitative approach Continuous transition between stability and instability width instability stability 0 Slope

Spectral condition: old approach Discontinuous transition between stability and instability drift instability stability 0 λ 0 (V)

Spectral condition: quantitative approach Continuous transition between stability and instability drift instability stability 0 λ 0 (V)

Spectral condition: quantitative approach Continuous transition between stability and instability drift instability stability 0 λ 0 (V) Can we get an analytical prediction for the drift λ 0 (V) rate as a function of?

Narrow solitons ε = soliton width/potential period <<1 Can use perturbation analysis! Expand potential as Solve for x = xa 1 V( x) = V(0) + V ''(0) ( ε x) 2 + 2 2

Narrow solitons cont. 1 V( x) V(0) V ''(0) x 2 2 2 = + ε + Compute effective force d dz 2 2 2 2 x = 2d A V 2 dv ''(0) ε A x, 2 2 Ehrenfest law Ω x 2 d 2 2 2 x =Ω x, Ω = 2 dv ''(0) ε 2 dz Center of mass obeys an oscillator equation!

Narrow solitons - cont. Use perturbation analysis to compute eigenvalue λ ( V ) 0 2 V ''(0) ε ν + V(0) V(0) ( ) Combine with Ω 2 = - 2dV''(0)ε 2 and get: 2 C 2 λ ( V ), C 2 0 Ω = = 4d ( ν + V ) (0) V(0)

Conclusion: 2 d dz Narrow solitons cont. x =Ω 2 x, Ω 2 = C 2 λ ( V ) 2 0 ( V) 2 λ0 > 0 Ω < 0 oscillatory solutions stability (Lattice min.) λ < 0 Ω > 0 exponential solutions instability ( V) 2 0 (Lattice max.) Analytical quantitative relation between spectral condition ( λ (V) 0 ) and dynamics (Ω)

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 d dz Valid for 2 x =Ω 2 x, Ω 2 = C 2 λ ( V ) 2 0 ( V) ( V) f, f C = > 0, L f = λ f f, L f 2 ( V) ( V) ( V) ( V) 1 0 ( V) ( V) + ( V) ( ) any soliton width any potential (periodic/non-periodic, single/multi waveguide, ) any nonlinearity (Kerr, cubic-quintic, saturable, ) any dimension

Examples of lateral dynamics (1) Soliton slightly shifted from lattice max. A(0, x) = u( x δ ) shift <x> 1 max min numerics max 0 0 40 20 z/lz diff

Examples of lateral dynamics (1) Soliton slightly shifted from lattice max. Solution of oscillator equation Ω is the drift (=instability) rate x = δ cosh( Ωz) A(0, x) = u( x δ ) shift <x> 1 max min theory numerics max 0 0 20 0 40 z/lz diff agreement up to the lattice min. up to many diffraction lengths

Drift rates Excellent agreement between analysis and numerics d=1, weak lattice Ω 3 theory numerics 0 17 10 100 300 ν d=2, strong lattice Ω 0 0 50 100 ν

Examples of lateral dynamics (2) So far, studied instability dynamics Oscillator equation good also for stability dynamics soliton moving at an angle from a lattice min. max min θ max Solution of oscillator equation is x z tgθ = sin( Ωz) Ω Ω determines the maximal deviation (=strength of stability)

Examples of lateral dynamics (2) d=2, periodic lattice Kerr (cubic) medium Cubic-Quintic medium <x> max min max min theory x tgθ = sin( Ωz ) Ω numerics max 0 10 z / L diff max 0 200 z / L diff

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

Qualitative approach Summary 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