Boris A. MalomedDepartment of Physical Electronics,

School of Electrical EngineeringFaculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Bright solitons from defocusing nonlinearities

In collaboration withOlga V. Borovkova, Yaroslav V. Kartashov,

Valery E. Lobanov, and Lluis TornerICFO-Institut de Ciencies Fotoniques, Castelldefels

(Barcelona), Spain

Jianhua ZengDepartment of Physics, Tsinghua University, Beijing

100084, China

Yingji HeSchool of Electronics and Information, Guangdong

Polytechnic Normal University, Guangzhou, China

Part 1: IntroductionA 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 brighttype can be supported by self-defocusing nonlinearity in the combination with a periodic potential (it may represent an optical lattice in BEC, or a grating or photonic crystal in optical 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).

Nevertheless, it was considered obvious that the self-defocusing nonlinearity alone, unless combined with a linear potential,could not support bright solitons.

The main point of this talk is to demonstrate that this seemingly obvious “prohibition” can be circumvented. It is possible to support stablebright solitons in the D-dimensional space by a purely self-defocusing spatially modulatednonlinearity if its local strength grows fast enough at |r | → ∞, namely, at any rate faster than |r|D.

Publications on the topic of this talk: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);

J. Zeng and B. A. Malomed, Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity, Phys. Rev. E 86, 036607 (2012).

Y. V. Kartashov, V. E. Lobanov, B. A. Malomed, and L. Torner, Asymmetric solitons and domain walls supported by inhomogeneous defocusing nonlinearity, Opt. Lett. 37, 5000 (2012).

Y. He and B. A. Malomed, Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity, Phys. Rev. A 87, 053812 (2013).

The structure of the talk:Part 2: The model and analytical results for the steeply modulated local nonlinearityPart 3: Numerical results for the same model with the steeply modulated local nonlinearityPart 4: Analytical and numerical results for the model with mild modulation of the local nonlinearityPart 5: A two-component modelPart 6: A generalization for a model with nonlocal repulsive interactionsPart 7: Conclusions

Part 2: The model and analytical results for the model with steeply modulated local nonlinearity

To introduce the concept, we start with the following 3D NLS equation:

22

2 2 2 2 2 2 2

Here is the propagation distance (in optics) or time

(in ), ( , , ) is the set of transverse coordinates,

/ / / is the transverse

diffraction/dispersion

1( )

op

.2

e

qi q q q

ξη ζ τ

η

ξ

ζ τ

σ

=

∇ =∂ ∂ +∂ ∂ +

∂=− ∇ +

∂

∂ ∂

B

r

EC r

2 2 2 2

20 2

rator, = , and the

local strength of the defocusing nonlinearity,

, is taken, , in the for

gro

m of the

- ,with the pre-exponential factor

wing at

1( )

:

2

r

anti G

fo

aussian

r the time ber i g

r r

n

σ σ σ

η ζ τ+ +

→∞

⎛⎜= +⎝

2

0 2exp , 02

, with . r

σ σ⎛ ⎞⎞ ⎟⎟ ⎜ ≥⎟⎟ ⎜⎜ ⎟ ⎟⎜⎠ ⎝ ⎠

Such a ( ) growth at

a necessary condition for the existence of

with a (convergent)

norm in the model. It is to adopt the

follo

-

wing

anti Gaussian r→∞is notstable bright soliton

steep

suffs fini

c ntte

i ie of the growth of the

at :

( ) const ,

with , where is the spatial

dimensio

local strength of the nonlineari

ty

n.

D

any

r

r r

Dpositive

εσε

+

→∞

= ⋅

asymptotic form

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 a nonuniform external magnetic or laser field: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 dopant is used (in optics), the trend to the divergence of the local nonlinearity coefficient at r →∞ may be achieved not necessarily through the increase of the density of the dopant, but rather by gradually tuning the dopant to the exact resonance at r →∞. In a similar way, the divergence may be achieved by means of tuning the Feshbach resonance in BEC.

Stationary solutions with a real propagation constant, b, are looked for in the usual form:

( )

2 2

( , ) exp ( ),

with real propagation constant , and function

satisfying the stationary equation:

1( )| | .

2

q ib w

b w

bw w w w

ξ ξ

σ

=

− =− ∇ +

r r

r

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

( ) ( )( )

22

0 2

( , ) 1/2 exp /4 ,

(1/4) / .

q r ib r

b D

ξ σ ξσ σ

= −=− +

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

2

0

1( , ) exp

42 2(the , , appears only

at 2), with propagation constant

(1/2)(1 /2).

Recall that the profile of the modulation

of the

azimuth

self-defocusing nonl

al coordinate

rq r r ib i

D

b D

ξ ξ φσ

φ

⎛ ⎞⎟⎜= + − ⎟⎜ ⎟⎜⎝ ⎠

==− +

( )22

0

inearity in this

case ( =0) is .( ) exp /2r rσ σσ =

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 approximation.

( )

( )

2

/22 2

The corresponding is taken in the simplest

form, with amplitude ,

, exp .4

The of the ansatz is ( ) 2 .

Here we again t

n

ake the simplest version of th

m

e

orD

A

rq r A ib

ansatz Gaussi

U

an

q A

ξ ξ

π

⎛ ⎞⎟⎜= − ⎟⎜ ⎟⎜⎝ ⎠

≡ =∫ r dr

222

0

model, i.e.,

1exp .

2 2q r

i q q qσξ

⎛ ⎞∂ ⎟⎜=− ∇ + ⎟⎜ ⎟⎜⎝ ⎠∂

( ) ( )

The variational approximation yields the following relation

between the and of the soliton:

. Comparison of this prediction with

numerical findings will be presen

πD/2

U=- 2 b+

norm propagation const

D

ant

/4ted below.

In the , it is possible find an

for the soliton's at , which

is , as it on the dimension ( ) and

propagation

analytical

asymptotic expressi

o

on

c n

r

D

→∞general case

universal does not dependtail

( )( ) ( )

stant , nor on the type of the soliton

( , , etc.):

Note that this asymptotic expression is obtained

in the equation, i.e., the equa

fundamental v

b

ortical

ξ≈ 3/2 2q exp ib r/2 exp -r /4 .keeping the

nonlinear term tion of this type is

at , for the . In fact,

it is the of the tail which makes the existence

of the solitons possible in this type of the non

r→∞nonlinearizablenonlin

decaying soliton tearizab

ailsility

linear media.

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 →∞. The fundamental soliton can be found for the

following 1D equation:

( )

( )

222

2

1sinh ( ) ,

2for :

1sech( ),with .

2 11

1ib

q qi a q q

e aq b

aa

aξ

ηξ η

η

∂ ∂=− + +

∂ ∂

+= =−

−−

<

222

2

At 1, when the equation takes the form of

1 cosh ( ) ,

2the above exact solution for the

does not exist. However, in this case an

fundamental solit

ex

can be fou

act solutio

nd for

n

on

a

a

q qi q qη

ξ η

=

∂ ∂=− +

∂ ∂

1

2

:

3 sinh( ) 5,with .

2cosh

)

( )

(

ib

twist d

q b

e

e ξη

η= =−

dipole solitonD

Part 3:Numerical results for the model with the anti-Gaussian (steep) modulation of the local nonlinearityRecall the form of the model:

In the 1D case, all the fundamental solitons are stable. The higher-order 1D solitons with k nodes are also completely stable for k = 1,2. Instability regionsappear for k ≥ 3. Stable higher-order 1D solitonswere found even for k = 10.

222

01

exp .2 2

q ri q q qσ

ξ⎛ ⎞∂ ⎟⎜=− ∇ + ⎟⎜ ⎟⎜⎝ ⎠∂

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

Unstable solitons spontaneously transform into breathers, which remain tightly localized. Typical examples of the evolution of stable and unstable 1D solitons are displayed here (top row), along with examples of oscillations of kicked1D stable solitons (bottom row; the kick is θ = 1.5).

Profiles and stability of 2D solitons with vorticity m. All vortices with m = 1 are stable. Unstable vortices appearstarting from m = 2. In (b), m = 2. In (c), variational and numerical U(b) curves completely overlap for fundamental solitons, k = 0. In (d), gray regions are unstable for m = 2.

Examples of the stable evolution of a 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 may be relevant to BEC), the family of fundamental solitons is completely stable. An example of the relaxation of a perturbed stablefundamental 3D soliton with b = -10 is presented here. Snapshots are displayed at ξ = 0, 300, 600.

Part 4:Analytical and numerical results for 1D and 2D solitons in the model with the mildmodulation of the local nonlinearity

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

( ) 2211 ,

2with 0. In the case, is replaced by | |.

In thismodel, can be found, for any ,

by mea Thomas-Fns of the ( ) approximation,

dropping the Laplacian:

| |

ermi

fundamental solito

qi q r q q

r

Dns

q

α

ξα η

∂=− ∇ + +

∂> 1D

TF

( )

( )

2

TF

/ 1 .

The corresponding norm of the soliton is

2 | |.

sin /This formula shows that the fundamental solitons exist for

, i.e. (as mentioned above), if the local strength of the

self-defocus

D

b r

bU

D

D

α

πα π α

α>

≈− +

=

ing grows at .r→∞ Dat any rate 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. The TF profiles for the 2D and 1D fundamentalsolitons 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) display 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 a fundamental soliton in (a), and into a stable vortex with m = 1 in (b); (c) self-healing of two stable vortices with m = 1, α = 5, or m = 2, α = 10.

Part 5: A two-component 1D model

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

21,2 1,2 2 22

1,2 2,1 1,22

1exp( )( ) .

2

q qi q C q qη

ηξ

∂ ∂=− + +

∂∂

Examples of solitons with overlapping (a,b) and separated(c,d) centers (w1,2 = |q1,2|) of the components. C = 2, b1 = -5in 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 separatedcomponents in symmetric solitons, with b1 = b2 ≡ b. The transition leads to destabilization of the overlapping solitons.The transition can be predicted analytically by means of a variational approximation, based on ansatz

2 21,2

2thr

1 11 exp ,

2 2where 2 is the separation between centers of the two components.

The is that solitons with the separated components exist at

1/4= ,

3/4

q A ib

bC C A

b

ζη η ξ η

ζ

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= ± − −⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

−>

+

result

thr

thr thr

1/4- .

1For instance, at 5 the formula yields 1.23,while the numerical

result is 1.19.A "naive" re unifosult for the rm is 1.

space

b

Cb C

C C

+=

+=− =

= ≡

Part 6: 1D solitons in the model with the spatially modulated nonlocal self-defocusing nonlinearity

2

:

10,

2( )| | .

Here is the squared correlation length of the nonlocal

response, - is a local perturbation of the refractive index,

due to heating by the light wave, assuming

z xx

xx

iu u mu

m dm x u

d

m

σ

+ − =

− =

The model

that the

the local value of refractive index.

( ) is the local density of the absorptive dopant.

xσlocal heating decreases

Stationary solutions with propagation constant band integral power (norm) P are looked for as

2

2

( , ) ( ), ( ),

with ( ) and ( ) obeying equations

10,

2( ) ,

and ( ) .

ibxU x z e U x m M x

U x M x

bU U MU

M dM x U

P U x dx

σ+∞

−∞

= =

′′− + − =

′′− =

≡∫

2 22 0 2

0

20

22

2

An example of an can

be obtained for ( ) cosh ( ) sech ( ),

2 1with

( )

1 , namely3 4

.

The co

sech( ), ( ) sech ( )

efficients of the exact sol

x ax

U x A ax M x M M x

ax

daa

σ σ σ σ

σ σ

−

=

= + −

⎛ ⎞⎟⎜=

=

− ⎟⎜ ⎟⎜⎝−

⎠

exact analytical solution

ution are

22 0

2

02

20

2

2

Another example, of an

solution, is available

( )

for ( ) cosh ( ) :

,

1 1with = 1 , and coefficie

sinh( )sech ( ), ( ) s

nts3 4

ech ( )U x B a

x a

x ax M x M M a

x

da

x

σ σ σ

σ σ

−

−

= +

⎛ ⎞⎟⎜ + ⎟⎜ ⎟⎜⎝ ⎠

= = −

exact twisted (dipole -shaped)

The power-vs.-propagation-constant curve for the family of fundamental solitons at d = a = σ2 = 1. The red dot indicates the particular exact solution:

The shape of the exact soliton solution and its stability in direct simulations:

The power-vs.-propagation-constant curve for the family of twisted solitons at d = a = σ2 = 1. The red dot indicates the particular exact solution:

The shape of the exactly found soliton and its stability in direct simulations:

Part 6: Conclusion

The local modulation of the purely self-defocusing nonlinearity, with the local strength diverging at r →∞, can support stable fundamental solitons in the space of any dimension, as well as multipolesand vortices in 1D and 2D. The necessary growth rate of the localnonlinearity strength is anything fasterthan r D.

The setting can be implemented in optics, using resonant dopants with a nonuniform density, and in BEC by means of the Feshbach resonance controlled by nonuniform external fields.

Many results have been obtained in the analytical form – exact, in some particular cases, and, in the general case, by means of the variational and Thomas-Fermiapproximations.

Stable 1D solitons, of the fundamental and twisted types, were also found in the model with the nonlocal self-defocusing nonlinearity, whose strength grows fast enough towards |x| → ∞.