Bright solitons from defocusing nonlinearities Olga V. Borovkova, - - PowerPoint PPT Presentation

Olga V. Borovkova, Yaroslav V. Kartashov, Lluis Torner

ICFO-Institut de Ciencies Fotoniques, Castelldefels (Barcelona), Spain

Boris A. Malomed

Department of Physical Electronics, School of Electrical Engineering Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Bright solitons from defocusing nonlinearities

Part 1: Introduction

A commonly known principle of the soliton theory is that self-focusing nonlinearity is necessary to create bright solitons in uniform media. Alternatively, stable gap solitons of the bright type can be supported by self-defocusing nonlinearity in the combination with a periodic potential (an optical lattice in BEC,

• r a Bragg grating/photonic crystal in
• ptical media):
• V. A. Brazhnyi and V. V. Konotop, Mod. Phys.
• Lett. B 18, 627 (2004); O. Morsch and M.

Oberthaler, Rev. Mod. Phys. 78, 179 (2006).

Recently, attention was drawn to possibilities of supporting bright solitons by means of nonlinear lattices, i.e., spatially periodic modulations of the local strength of the nonlinearity (or, more generally, by non-periodic modulations of the nonlinearity). It is easy to support stable solitons by nonlinear lattices in 1D. Under special conditions (a sharp modulation profile, rather than a smooth

• ne), this is possible too (although not easy) in 2D:
• Y. V. Kartashov, B. A. Malomed, and L. Torner,

Solitons in nonlinear lattices, Rev. Mod.

• Phys. 83, 247 (2011).

More recently, solitons supported by nonlinear lattices were also studied in the framework of the Salasnich’s nonpolynomial 1D equation:

• L. Salasnich and B. A. Malomed, Quasi-one-

dimensional Bose-Einstein condensates in nonlinear lattices, J. Phys. B: At. Mol. Opt.

• Phys. 45, 055302 (2012).

Nevertheless, it is commonly believed that the self-defocusing nonlinearity alone, unless it is combined with a linear potential (such the one created by an optical lattice), cannot support bright solitons in principle. The objective of the talk is to demonstrate that this seemingly obvious “prohibition” can be circumvented. Namely, it is possible to support stable bright solitons by the self-defocusing spatially modulated nonlinearity if its local strength grows fast enough at |r | → ∞.

Publications on the topic:

• O. V. Borovkova, Y. V. Kartashov, L. Torner, and
• B. A. Malomed, Bright solitons from

defocusing nonlinearities, Phys. Rev. E 84, 035602(R) (2011).

• O. V. Borovkova, Y. V. Kartashov, B. A. Malomed,

and L. Torner, Algebraic bright and vortex solitons in defocusing media, Opt. Lett. 36, 3088 (2011).

• Y. V. Kartashov, V. A. Vysloukh, L. Torner, and
• B. A. Malomed, Self-trapping and splitting of

bright vector solitons under inhomogeneous defocusing nonlinearities, Opt. Lett. 36, 4587 (2011).

Related publications:

• O. V. Borovkova, Y. V. Kartashov, V. A. Vysloukh,
• V. E. Lobanov, B. A. Malomed, and L. Torner,

Solitons supported by spatially inhomogeneous nonlinear losses, Opt. Exp. 20, 2657 (2012);

• V. E. Lobanov, O. V. Borovkova, Y. V. Kartashov,
• B. A. Malomed, and L. Torner, Stable bright

and vortex solitons in photonic crystal fibers with inhomogeneous defocusing nonlinearity, Opt. Lett. 37, 1799 (2012).

The structure of the talk:

Part 2: The model and analytical results for the steeply modulated local nonlinearity Part 3: Numerical results for the same model Part 4: A model with a mild modulation of the local nonlinearity Part 5: A two-component model Part 6: Conclusions

Part 2:

The model with the steeply modulated local nonlinearity: analytical results

To introduce the concept, we start with the following 3D NLS equation for wave amplitude q:

2 2 2 2 2 2 2 2 2

1 ( ) . 2 Here is the propagation distance (in optics) or time (in ), ( , , ) is the set of transverse coordinates, / / / is the transverse diffraction/dispersion ope q i q q q σ ξ ξ η ζ τ η ζ τ ∂ =− ∇ + ∂ = ∇ =∂ ∂ +∂ ∂ +∂ ∂ r BEC r

2 2 2 2

rator, and the local strength

• f the defocusing nonlinearity,

, is taken, , in a very steep "anti-Gaussian" form, , w ith . 1 ( ) exp , 2 2 for the time be r i r r ng r σ σ σ σ σ →∞ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ = + ≥ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ ⎝ ⎠ growing at

Such a steep growth of the local nonlinearity coefficient at a necessary condition for the existence

• f stable bright solitons in models of this type. In fact,

it is to adopt the r →∞ suffi is not cient following

• f the growth of the local strength of the nonlinearity

at : ( ) const , with , where is t he . any positive r r r

ε

σ ε

+

→∞ = ⋅

D

asymptotic form spatial dimension D

In optics, inhomogeneous nonlinearity landscapes can be created by means of a nonuniform density distribution of resonant dopants enhancing the local nonlinearity:

• J. Hukriede, D. Runde, and D. Kip, J. Phys. D:
• Appl. Phys. 36, R1 (2003).

In BEC, spatially modulated profiles of the nonlinearity can be induced, via the Feshbach resonance, by nonuniform external magnetic or laser fields:

• I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod.
• Phys. 80, 885 (2008); C. Chin, R. Grimm, P.

Julienne, and E. Tiesinga., ibid. 82, 1225 (2010).

In the case when the resonant dopants are used, the trend to the divergence of the local nonlinearity coefficient at r →∞ may be achieved not necessarily through the steep increase of the density of the dopants, but rather by gradually tuning the dopant to the exact resonance at r →∞. In a similar way, the divergence may be achieved by means

• f the Feshbach resonance in BEC.

The model with the steep anti-Gaussian spatial modulation is chosen here for the presentation because it admits a particular exact solution for fundamental solitons in the form of a Gaussian, at a single value of propagation constant b, in any dimension D:

( )

( )

( )

2 2 2

( , ) 1/2 exp /4 , (1/4) / . q r ib r b D ξ σ ξ σ σ = − =− +

This solution does not exist at σ2 = 0, i.e., for the modulation profile but in that case it is possible to find exact solutions for a 1D dipole (twisted) soliton, and for a 2D vortex soliton with topological charge m = 1:

2

1 ( , ) exp 4 2 2 (the , , appears only in ,and is replaced by in azimuthal coordinate ), with propagation constant (1/2)(1 /2). r q r r ib i r x b D ξ ξ φ σ φ ⎛ ⎞ ⎟ ⎜ = + − ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ =− + 2D 1D

( )

2 2

( ) exp /2 , r r r σ σ =

In the general case (for arbitrary values of the propagation constant, b < 0), a family of approximate analytical solutions can be constructed by means of the variational method. ( ) ( )

2 /2 2 2

The corresponding is taken in the simplest form, with amplitude , , exp . 4 The

• f the ansatz is

( ) 2 . The norm is tre n at

• rm

variational paramete ed as r a

D

A r q r A ib ansatz Gaussi U an q A ξ ξ π ⎛ ⎞ ⎟ ⎜ = − ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ≡ =

∫

r dr

2 2 2 2

. Here we again take the simplest version of the model, with =0, i.e., 1 exp . 2 2 q r i q q q σ σ ξ ⎛ ⎞ ∂ ⎟ ⎜ =− ∇ + ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ∂

( ) ( ) The variational approximation yields the following relation between the and

• f the soliton:

. Comparison of these results with numerical findings will be presente π

D/2

U=- 2 b+ norm propagation const D ant /4 d below. In the , it is also possible find an for the soliton's at , which is , analytical asymptotic ex as it

• n the dimension ( ) and

propagati p

• n

ression r →∞ general case universal does not depen D d tail ( )(

) ( )

constant , neither on the type of the soliton ( , , etc.): Note that the asymptotic expression is obtained in the equation, i.e., t fundamental vortica b l ξ ≈

3/2 2

q exp ib r/2 exp -r /4 . keeping the nonlinear term he equation of this type is at , for the . r →∞ nonlineari soliton ta ble ils za

Particular exact solutions in 1D can be also found for other forms of the nonlinearity modulation, with the local strength growing exponentially (rather than as the anti-Gaussian) at r →∞. First, an exact fundamental soliton can be found for the following 1D equation:

( )

( )

2 2 2 2

1 sinh ( ) . 2 A particular is available f 1

• r

: 1 sech( ),with . 2 1 1

ib

ex q q act sol i a q q e a q b a ution a a

ξ

η ξ η η ∂ ∂ =− + + ∂ ∂ + = =− − − <

2 2 2 2

fundament At 1, when t al soliton he equation takes the form of 1 cosh ( ) , 2 the above solution for the does not exist. However, in this exact solutio case an can be found for n a a q q i q q η ξ η = ∂ ∂ =− + ∂ ∂ 1D dipo

2

: 3 sinh( ) 5 ,with . 2 cosh ( ) ( )

ib

twist q ed e b

ξ

η η = =− le soliton

Moreover, it is possible to formulate a model with the local strength of the nonlinearity diverging at finite values of the coordinate (in 1D), η = ±1, (instead of the divergence at | η |→∞), hence the solutions exist in finite domain, η2 < 1 (a nonlinear counterpart of a quantum dot):

( ) ( ) ( )

2 2 2 2 2 3 2 2 2

1 3 1 1 . 2 4 1 A particular for the in this model can be found for propagation constant

• 9/4:

3 exact solutio 1 . n

ib

q q i q q b q e

ξ

η ξ η η η + ∂ ∂ =− + ∂ ∂ − = = − ground state

Part 3:

Numerical results for the model with the anti- Gaussian modulation of the local nonlinearity

Recall the form of the model: Results: In the 1D case, all the fundamental solitons are stable. The higher-order 1D solitons with k nodes (zeros) are also completely stable for k = 1,2. Instability regions appear only for k ≥ 3. Stable higher-order 1D solitons were found even for k = 10. 2 2 2

1 exp . 2 2 q r i q q q σ ξ ⎛ ⎞ ∂ ⎟ ⎜ =− ∇ + ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ ∂

Stationary profiles (w = |q|) and stability results for the fundamental and higher-order 1D solitons with k nodes. Red curves: the nonlinearity-modulation profile, σ(η). In (c), variational and numerical U(b) curves completely overlap for k = 0 (the green segments for k = 4 are unstable). In (d), shown for k = 5, the gray regions are unstable, alternating with white stability regions.

Stable solitons, if kicked by q(r) → q(r) exp(iθη), perform regular oscillations around η = 0. An effective equation of motion can be derived for the

• scillations of the coordinate of the soliton’s center

(R) around the origin, for D = 1 and 2:

( )

( )

( ) ( )

2 /2 2 2 2 /2 2 2

2 exp (recall is the norm of the

• dimensional soliton).

It follows from here that the , including a , is = 2 3/4 .

D D D D D

nonlinear correction d U R d U D U

• scillation freque

y R R nc π ξ θ ω π θ

− −

=− + ∼

Unstable solitons spontaneously carry over into breathers, which remain tightly localized. Examples of the evolution of stable and unstable 1D solitons are displayed here (top row), along with examples of oscillations of 1D kicked solitons (bottom row; the kick is θ = 1.5; recall k is the soliton’s order).

Profiles and stability of 2D solitons with vorticity m. In (b), m = 2. In (d), gray regions are unstable for m = 2. All vortices with m = 1 are stable.

Examples of the stable evolution of a perturbed 2D vortex with m = 2 (a), and spitting of unstable vortices with m = 2 (b) and m = 3 (c) into rotating sets of unitary vortices:

In the 3D model (which cannot be applied to optical media, but is relevant to BEC), the family

• f fundamental solitons is completely stable.

The variational U(b) dependence for them almost exactly coincides with the numerical one (not shown here). Radial profiles of the stable fundamental 3D solitons:

An example of the relaxation of a perturbed stable fundamental 3D soliton with b = -10. Snapshots are displayed at ξ = 0, 300, 600.

Part 4:

Analytical and numerical results for 1D and 2D solitons in the model with the mild modulation of the local nonlinearity

The same model, but with a slow growth of the local nonlinearity at r →∞: ( )

2 2 2

1 1 , 2 with 0.In the case, is replaced by | |. can be found, for any , by means

• f the

( ) approximation, dropping the diffraction term: | | / Fundamental q i solito q r q q r D q b ns

α

ξ α η ∂ =− ∇ + + ∂ > ≈− Thomas-Ferm F i 1D T

( )

( ) 1 . The corresponding norm of the soliton is 2 | | . sin / This formula shows that the fundamental solitons exist for ,i.e., if the local strength of the self-defocusing grows at

D

r b r D U D

α

π α π α α + = ∞ > →

TF

at any ra .

D

te faster than r

Examples of 1D and 2D fundamental and vortical solitons. (a) 2D, α = 5, b = -10, for vorticities m = 0,1,2; (b) 2D, m = 1, b = -5,-10,-20 (curves 1,2,3); (c) 1D, b = 10, α = 5; k = 0 – fundamental, k = 2 – second-order solitons. TF profiles for the 2D and 1D fundamental solitons are indistinguishable from their numerical counterparts.

Numerical and analytical results for families of solitons. (a): 2D, α = 5; (b): 2D, b = -10; (c): 1D, b = -10 [red curves in (b) and (c) are the TF predictions]. In the exact agreement with TF, αcr = D.

All the fundamental 1D and 2D solitons are stable. Examples of the evolution: unstable vortices with m = 1, α = 3.5, b = -3 (a) and m = 2, α = 4, b = -20 (b) – transformation into fundamental solitons; (c) self-healing

• f two stable vortices with m = 2, α = 5.

Part 5:

A two-component 1D model

The system of two 1D equations with the steep anti-Gaussian modulation, coupled by the repulsive XPM terms with coefficient C:

2 1,2 1,2 2 2 2 1,2 2,1 1,2 2

1 exp( )( ) . 2 q q i q C q q η η ξ ∂ ∂ =− + + ∂ ∂

Examples of solitons with overlapping (a,b) and separated (c,d) centers (w1,2 = |q1,2|). C = 2, b1 = -5 in all panels, and b2 = -8 in (a), -3.1 in (b), -5 in (c), -7.2 in (d). The soliton is unstable in (a), and stable in the other cases.

The most essential property of the two-component solitons is the transition from the overlapping to separated components in symmetric solitons, with b1 = b2 ≡ b. The transition leads to destabilization of the overlapping solitons. It can be predicted analytically by means of a variational approximation, based on ansatz

2 2 1,2 thr

1 1 1 exp , 2 2 where 2 is the separation between the centers of the two components. The is that solitons with the separated components exist at 1/4 = 3/ q A ib b C C b ζη η ξ η ζ ⎛ ⎞ ⎛ ⎞ ⎟ ⎟ ⎜ ⎜ = ± − − ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ ⎠ − > + result

2 thr thr thr

1/4 ,

• .

4 1 For instance, at 5 the formula yields ,while the numerical result is .A "naive" result for unifo the rm spa is 1 ce . b A C b C C C + = + =− = = ≡ 1.23 1.19

Part 6: Conclusion

The local modulation of the purely self- defocusing nonlinearity, with the local strength diverging at r →∞, can easily support stable fundamental solitons in the space of any dimension, as well as multipoles and vortices in 1D and 2D. These conclusions pertain to single- and two-component solitons alike. The necessary growth rate of the local nonlinearity strength may be anything faster than r D.

The setting can be implemented in optics, using resonant dopants with a nonuniform density, and in BEC by means

• f the Feshbach resonance controlled by

nonuniform external fields. Many results have been obtained in the analytical form – sometimes, as exact

• nes, and, in the general case, by means
• f the variational and Thomas-Fermi

approximation. A remaining challenging problem is to construct stable 3D solitons with embedded vorticity.