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

Bright solitons from defocusing nonlinearities 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

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 , or a Bragg grating/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).

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 one), 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 : ∂ =− ∇ q 1 2 2 + σ r i q ( ) q q . ∂ ξ 2 ξ Here is the propagation distance (in optics) or time = η ζ τ (in BEC r ), ( , , ) is the set of transverse coordinates, 2 2 2 2 2 2 2 ∇ =∂ ∂ η +∂ ∂ ζ +∂ ∂ τ / / / is the transverse diffraction/dispersion ope rator, and the local strength →∞ of the defocusing nonlinearity, growing at r , is taken, for the time be i ng , in a very steep "anti-Gaussian" form, ⎛ ⎞ 2 ⎛ ⎞ 1 r ⎟ ⎟ ⎜ ⎜ 2 σ = σ + σ σ σ ≥ ( ) r r exp , w ith , 0 . ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ 0 2 0 2 ⎝ ⎠ ⎝ ⎠ 2 2

Such a steep growth of the local nonlinearity coefficient r →∞ is not at a necessary condition for the existence of stable bright solitons in models of this type. In fact, it is suffi cient to adopt the following asymptotic form of the growth of the local strength of the nonlinearity →∞ at r : + ε D σ = ⋅ ( ) r const r , ε D with any positive , where is t he spatial dimension .

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 of 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 ξ = σ ξ − q r ( , ) 1/2 exp ib r /4 , 2 =− ( + σ σ ) b (1/4) D / . 0 2

This solution does not exist at σ 2 = 0 , i.e., for the modulation profile ( ) 2 2 σ = σ ( ) r r exp r /2 , 0 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 r ⎟ ⎜ ξ = ξ + φ − q r ( , ) r exp ib i ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ 4 σ 2 2 0 φ (the azimuthal coordinate , , appears only 2D r x 1D in ,and is replaced by in ), =− + with propagation constant b (1/2)(1 D /2).

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 . The corresponding ansatz is taken in the simplest Gaussi an form, with amplitude , A ⎛ ⎞ 2 r ⎟ ⎜ ( ξ ) = ξ − q , r A exp ib . ⎟ ⎜ ⎟ ⎜ ⎝ ⎠ 4 D /2 ∫ 2 2 ≡ = ( π ) The n orm of the ansatz is U q ( ) r dr 2 A . The norm is tre at ed as a variational paramete r . Here we again take the simplest version of the model, σ with =0, i.e., 2 ⎛ 2 ⎞ ∂ q 1 r ⎟ ⎜ 2 2 =− ∇ + σ i q exp q q . ⎟ ⎜ ⎟ ⎜ 0 ⎝ ⎠ ∂ ξ 2 2

The variational approximation yields the following relation between the norm and propagation const ant of the soliton: D/2 π U=- 2 ( ) ( b+ D /4 ) . Comparison of these results with numerical findings will be presente d below. In the general case , it is also possible find an analytical r →∞ asymptotic ex p ression for the soliton's tail at , which universal as it does not depen d on the dimension ( ) and D is , propagati on constant , neither on the type of the soliton b ( fundamental vortica , l , etc.): ) ( ) ( ) ≈ ξ 3/2 2 q exp ib ( r/2 exp -r /4 . Note that the asymptotic expression is obtained keeping the nonlinear term in the equation, i.e., t he equation of this r →∞ type is nonlineari za ble at , for the soliton ta ils .

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 ∂ ∂ q 1 q ( ) 2 2 =− + + η i a sinh ( ) q q . ∂ ξ 2 2 ∂ η A particular ex act sol ution is available < f or a 1 : ξ ib + e 1 a = η =− q sech( ),with b . − ( ) 2 1 a − 1 a

= At a 1, when t he equation takes the form of 2 ∂ ∂ q 1 q 2 2 =− + η i q q cosh ( ) , ∂ ξ 2 ∂ η 2 the above solution for the fundament al soliton does not exist. However, in this case an exact solutio n can be found for a 1D dipo le ( twist ed ) soliton : η 3 sinh( ) 5 ξ ib = =− q e ,with b . 2 η 2 cosh ( )