Laser Physics Letters       LETTER Elastic collision and breather formation of spatiotemporal vortex light bullets in a cubic- quintic nonlinear medium To cite this article: S K Adhikari 2017 Laser Phys. Lett. 14 065402   View the article online for updates and enhancements. 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Introduction A one-dimensional (1D) bright soliton with cubic nonlinear- ity, capable of moving at a constant velocity [1, 2], has been observed in nonlinear optics [1, 2] in both temporal [3] and spatial [4] varieties. Although a three-dimensional (3D) spa- tiotemporal soliton cannot be formed with a cubic nonlinear- ity due to collapse [1, 5], the soliton can be stabilized in higher dimensions with a saturable [5, 6] or a modified nonlinearity [7], with a cubic-quintic nonlinearity [8], or with a modified dispersion. A two-dimensional (2D) spatiotemporal optical soliton has been observed [9] in a saturable nonlinearity gen- erated by the cascading of quadratic nonlinear processes. A 2D spatial soliton in a cubic-quintic medium has been suggested [10] and realized experimentally [11]. The generation of a stable 2D vortex soliton in a cubic-quintic medium has been suggested [12]. There has also been a study of the dynamics of the vortex pulsed beam in a medium with nonlinearities of opposite sign [13] and of interacting vortices in Bose–Einstein condensates (BEC) [14]. A 3D spatiotemporal optical soliton, commonly known as a light bullet, was realized experimentally in arrays of wave guides [15]. There are many theoretical—numerical and analytical—studies on light bullets using the 3D nonlinear Schrödinger (NLS) equation  [1] with a modified nonlinear- ity [6, 7], nonlinear dissipation [16], and/or dispersion [17]. Dispersion and nonlinearity management can stabilize light bullets in a medium with cubic nonlinearity [18]. Solitons have also been studied in the coupled NLS equation [19]. Recently, we studied [8] the formation of a 3D spatiotemporal light bul- let [5, 6] in a cubic-quintic medium for a defocusing quintic nonlinearity and a focusing cubic nonlinearity. A cubic-quintic medium is also of experimental interest. A study featuring a polydiacetylene paratoluene sulfonate crystal in the wave- length region near 1600 nm shows that the refractive index versus input intensity correlation leads to a cubic-quintic form Laser Physics Letters S K Adhikari Elastic collision and breather formation of spatiotemporal vortex light bullets in a cubic-quintic nonlinear medium Printed in the UK 065402 LPLABC © 2017 Astro Ltd 14 Laser Phys. Lett. LPL 10.1088/1612-202X/aa6c1c 6 Laser Physics Letters Elastic collision and breather formation of spatiotemporal vortex light bullets in a cubic-quintic nonlinear medium S K Adhikari Instituto de Física Teórica, UNESP—Universidade Estadual Paulista, 01.140-070 São Paulo, São Paulo, Brazil E-mail: adhikari44@yahoo.com Received 6 February 2017, revised 31 March 2017 Accepted for publication 5 April 2017 Published 4 May 2017 Abstract The statics and dynamics of a stable, mobile three-dimensional (3D) spatiotemporal vortex light bullet in a cubic-quintic nonlinear medium with a focusing cubic nonlinearity above a critical value and any defocusing quintic nonlinearity are considered. The present study is based on an analytic variational approximation and a full numerical solution of the 3D nonlinear Schrödinger equation. The 3D vortex bullet can propagate with constant velocity. Stability of the vortex bullet is established numerically and variationally. Collision between two vortex bullets moving along the angular momentum axis is considered. At large velocities the collision is quasi-elastic, with the bullets emerging after collision with practically no distortion. At small velocities two bullets coalesce to form a single entity called a breather. Keywords: NLS equation, soliton, vortex bullet (Some figures may appear in colour only in the online journal) Astro Ltd IOP Letter 2017 1612-202X 1612-202X/17/065402+8$33.00 https://doi.org/10.1088/1612-202X/aa6c1cLaser Phys. Lett. 14 (2017) 065402 (8pp) publisher-id doi mailto:adhikari44@yahoo.com http://crossmark.crossref.org/dialog/?doi=10.1088/1612-202X/aa6c1c&domain=pdf&date_stamp=2017-05-04 https://doi.org/10.1088/1612-202X/aa6c1c 2 S K Adhikari of nonlinearity in the NLS equation  [1, 20]. Such a cubic- quintic nonlinearity also arises in a low intensity expansion of the saturable nonlinearity used in the pioneering study of light bullets [6]. In this Letter we demonstrate the stabilization of a 3D spa- tiotemporal vortex (rotating) light bullet in a cubic-quintic medium and study its statics and dynamics, employing varia- tional and numerical solutions of the 3D nonlinear Schrödinger equation. The vortex light bullet is capable of moving with- out deformation with constant velocity. We study the collision between two vortex light bullets moving along the spinning axis. Such a collision in 3D is expected to be inelastic, with loss of energy. In the present numerical simulation of col lision between two vortex light bullets in different parameter domains of nonlinearities and velocities three distinct scenarios are found to take place. At sufficiently large velocities, the col- lision is found to be quasi-elastic: the two bullets emerge after collision with practically no deformation. At small velocities, the collision is inelastic: the bullets form a single bound entity in an excited state, last for ever and execute oscillation. We call this a breather. In a small domain of intermediate velocities, the bullets coalesce to form a single entity that expands indefi- nitely, leading to the destruction of the bullets. We present the 3D NLS equation used in this study in sec- tion 2. In section 3 we present the numerical results for sta- tionary profiles of 3D spatiotemporal vortex light bullets, and numerical tests of stability of the vortex light bullet under a small perturbation. The quasi-elastic nature of collision of two vortex bullets at large velocities and formation of a breather at low velocities are demonstrated by realistic simulation. We end with a summary of our findings in section 4. 2. Nonlinear Schrödinger equation: variational formulation The 3D NLS equation we describe below to study vortex soli- tons has application in two areas: in nonlinear optics, where the soliton is known as a spatiotemporal optical vortex bullet, and in BEC. In nonlinear fiber optics the 3D NLS equation is [1, 21] β β γ κ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + | | − | | = ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦⎥ z x y t A A A x y t i 1 2 2 , , 0, 0 2 2 2 2 2 2 2 2 4 ( ) (1) where the unit of the parameter γ is W −1m, that of κ is W −2m3, that of the intensity | |A 2 is W m−2, that of the disper- sion parameter β2 is ps2 m−1, and that of the propagation con- stant β0 is m−1. We define the diffraction length β ω≡LDF 0 2 and dispersion length τ β≡ | |LDS 2 2/ , where ω is the width of the pulse, and τ is the time scale of the soliton [22]. Now one defines the following dimensionless variables [21] ω ω ω β β γ κ γ = = = = Φ = = = x x y y t t z z L A L P p P q P L , , , , , , . 0 2 DF DF 0 0 0 2 2 DF (2) The scale P0 is chosen to yield unit norm: ∫ |Φ| =x y td d d 1.2 Using dimensionless variables, one obtains the following NLS equation with self-focusing cubic and self-defocusing quintic nonlinearity [1] ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + |Φ| − |Φ| Φ = ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦⎥z x y t p q zri 1 2 , 0, 2 2 2 2 2 2 2 4 ( ) (3) where ≡ x y tr , ,{ }, p and q are the coefficients of cubic and quintic nonlinearities respectively. In (3) x, y denote trans- verse extensions, z the propagation distance, and t the time. The quintic nonlinearity of strength q with a negative sign denotes self-defocusing. The plus sign before |Φ|2 denotes a self-focusing cubic nonlinearity. For a vortex of charge L with circular symmetry in the x − y plane, we can write ( ) ( ) ( )φ ρ θΦ =z t z Lr, , , exp i ,L ρ ρ θ ρ θ= + = =x y x y, sin , cos2 2 , where the function φ ρ t z, ,L( ) is real with the property ( )→ φ ρρ t zlim , ,L0 → ρ .L This generates an optical pulse with a dark spot at the center (ρ = 0) and is called an optical vortex [23]. The wave function Φ zr,( ) is periodic in θ with period π2 (rotational symmetry). Consequently, recalling ρ ρ ρ ρ θ ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂x y 1 1 , 2 2 2 2 2 2 2 2 2 (4) for unit charge L = 1, (3) becomes [23] ( ) ρ ρ ρ ρ φ φ φ ρ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − + | | − | | = ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦⎥ z t p q t z i 1 2 1 1 2 , , 0, 2 2 2 2 2 2 4 (5) where we have dropped the L = 1 index from the wave function. To estimate the order of magnitude of different variables, we consider an infrared beam of wavelength λ = 1 μm in a nonlinear medium of β = −102 2 ps2 m−1, with the time scale τ = 60 fs. Then the beam width ω≈ 239 μm and the disper- sion length =L 36DS cm. These numbers are quite similar to those in an experiment on spatiotemporal optical bullets in a planar glass wave-guide [24]. Here we present the results in dimensionless units, which can be converted to actual exper- imental units using the transformations (2). The analytic model (5) is also applicable to the case of a vortex soliton in BEC [25]. In that case the mean-field Gross– Pitaevskii equation  describing the BEC in the presence of attractive two-body and repulsive three-body interactions is given by [26] ħ ħ ħ ħ ( ) ( ) ( ) ( ) ψ π ψ ψ ψ ∂ ∂ = − ∇ − | | | | + | | ⎡ ⎣⎢ ⎤ ⎦⎥ t t m a N m t N K t t r r r r i , 2 4 , 2 , , , r 2 2 2 2 2 3 4 (6) where m is the mass of each atom of the BEC, ψ tr,( ) is the condensate wave function at space point = x y zr , ,{ } and time t, ρ = +x y2 2 , a is the s-wave scattering length of atoms, K3 is the three-body interaction term, and N is the number of Laser Phys. Lett. 14 (2017) 065402 3 S K Adhikari atoms. Equation  (6) can be written in the following dimen- sionless form after a redefinition of the variables ψ ψ ψ ψ ∂ ∂ = − ∇ − | | + | | ⎡ ⎣ ⎢ ⎤ ⎦ ⎥t t p t q t t r r r ri , 2 , , , ,r 2 2 4( ) ( ) ( ) ( ) (7) where π=p N4 , =q mN K a22 3 4ħ/( ), length is scaled in units of | |a , time in ma2 ħ/ , ψ| |2 in units of | |−a 3. Equations (3) and (7) are mathematically the same, but the interpretation of the various terms in them is distinct. A BEC vortex soliton can be introduced in (7) in a similar fashion as in the case of an opti- cal pulse, viz. (5). In the following we will discuss mostly the case of a spatiotemporal vortex light bullet in cubic-quintic medium. Nevertheless, the similarity of the mathematical models (3) and (7) ensures the the possibility of generating a 3D vortex soliton in a BEC with repulsive three-body and attractive two-body interactions. For an analytic understanding of the formation of a spin- ning light bullet (a vortex soliton of unit charge), we con- sider the Lagrange variational formulation of an optical pulse [27]. In this axially symmetric problem, a convenient analytic variational approximation of the vortex bullet is [27, 28] φ ρ π ρ σ σ ρ σ σ α ρ β = − − + + − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦⎥ t z z z z t z z z t , , exp 2 2 i i , 3 4 1 2 2 2 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) / (8) where ρ= +r t2 2 2, σ z1( ) and σ z2( ) are radial and axial widths, respectively, and α βz z,( ) ( ) are corresponding chirps. The (generalized) Lagrangian density corresponding to (5) is given by ( ) ( ) ( ) ( ) ( ) ( ) L ρ φφ φ φ φ ρ φ ρ ρ φ ρ φ ρ = − + |∇ | + | | − | | + | | ∗ ∗t z t z t z p t z q t z , , i 2 ˙ ˙ , , 2 , , 2 2 , , 3 , , , 2 2 2 4 6 (9) where the overhead dot denotes z-derivative. Equation  (5) can be obtained by extremizing the functional (9) [27]. Consequently, the effective Lagrangian function ∫σ σ α β π ρ ρ ρ≡L t z t, , , 2 , , d d1 2( ) ( )L becomes ( ) ( ) / σ σ α β σ β β σ α α σ σ π σ σ π σ σ = + + + + + − + − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟L p q , , , ˙ 2 2 ˙ 2 1 1 4 8 2 2 81 3 . 1 2 2 2 2 1 2 2 1 2 2 2 3 2 1 2 2 3 1 4 2 2 (10) The variational parameters ν σ σ α β≡ , , ,1 2 are obtained from the Euler–Lagrangian equations ν ν ∂ ∂ = ∂ ∂z L Ld d ˙ . (11) After some straightforward algebra the four Euler–Lagrangian equations  lead to the following dynamical equations  for the widths: σ π σ σ π σ σ σ− + = − −p q1 8 2 4 81 3 ¨ , 1 3 3 2 1 3 2 3 1 5 2 2 1 / (12) σ π σ σ π σ σ σ− + = − −p q1 4 2 8 81 3 ¨ . 2 3 3 2 1 2 2 2 3 1 4 2 3 2 / (13) The stationary profile of the vortex bullet is obtained by setting the z-derivatives on the right-hand sides of (12) and (13) [27]: σ π σ σ π σ σ − + = − −p q1 8 2 4 81 3 0, 1 3 3 2 1 3 2 3 1 5 2 2 / (14) σ π σ σ π σ σ − + = − −p q1 4 2 8 81 3 0. 2 3 3 2 1 2 2 2 3 1 4 2 3 / (15) Equations (14) and (15) correspond to the global minimum of a conserved α- and β-independent effective Lagrangian σ σ σ σ α β≡ = =L L, , , 0, 01 2 1 2( ) ( ): σ σ∂ ∂ = ∂ ∂ =L L 01 2/ / . The function σ σL ,1 2( ) describes the Lagrangian dynamics of the widths σ σ,1 2 and is independent of the generalized veloci- ties σ σ˙ , ˙1 2. 3. Numerical results The 3D NLS equation  (5) is generally solved by the split- step Crank–Nicolson [29] and Fourier spectral [30] methods. The split-step Crank–Nicolson method in Cartesian coordi- nates is employed in the present study. We use r ={x, y, t} step of 0.05–0.016, a z step of 0.0005–0.000 0025 [29] and the number of r discretization points 128–320. There are dif- ferent C and FORTRAN programs for solving the NLS-type equations [29, 31] and one should use the appropriate one. We use both imaginary- and real-z propagation [29] for numerical solution of the 3D NLS equation. The imaginary-z propaga- tion is appropriate to find the stationary state and the real-z propagation for the dynamics. In the imaginary-z propagation the initial state was taken as in (8). A stable bullet corresponds to a global minimum of the conserved effective Lagrangian σ σL ,1 2( ) at a negative value. To demonstrate the appearance of a global minimum, we show in figure 1 the two-dimensional contour plot of the Lagrangian σ σL ,1 2( ) in the σ σ−1 2 plane for (a) p = q = 200, (b) p = 200, q = 400, (c) p = 100, q = 200, and (d) p = q = 100, where we illustrate the region with negative Lagrangian; the Lagrangian is positive outside this region. The Lagrangian σ σL ,1 2( ) goes to zero as σ σ ∞,1 2 → . At the origin, σ σ, 01 2 → , and the Lagrangian σ σ ∞L ,1 2( ) → , which guarantees the absence of a collapsed state at the origin. The repulsive quintic nonlinear- ity contributes positively to the Lagrangian; so do the first two terms on the right-hand side of (10). To make the Lagrangian (10) negative, the cubic nonlinearity coefficient p has to be larger than a critical value, e.g. >p pcrit, when the minimum of the Lagrangian could be negative, correspond ing to a stable vortex bullet. For

p pcrit. (b) Variational (line) and numerical (points) results for rms sizes xrms and trms versus cubic nonlinearity coefficient p for quintic nonlinearity coefficient q = 100 and 200. (c) Variational (line) and numerical (points) results for Lagrangian L versus cubic nonlinearity coefficient p for quintic nonlinearity coefficient q = 100 and 200. Figure 3. Numerical (line) and variational (points) reduced densities δ x1D( ) and δ =x y, 02D( ) for different cubic nonlinearity coefficient p and quintic nonlinearity coefficient q: (a) p = q = 200, (b) p = 200, q = 400, (c) p = 100, q = 200 and (d) p = q = 100. Laser Phys. Lett. 14 (2017) 065402 6 S K Adhikari formation as in figure  6), a distortion of the vortex bullets takes place after collision, with eventual destruction of the vortex bullets. This is illustrated in figure  7, where apart from the two trajectories of the vortex bullets after collision a central peak can be seen at t = 0. On further reduction of the initial velocity, the central peak at t = 0 becomes more pronounced, and the outer tracks less prominent. Eventually, at very small velocities only the central peak corresponding to the formation of a breather after collision prevails, viz. figure 6. Figure 4. Real-z evolution of the vortex bullet with p = 100 and q = 200 by isodensity contour plot of φ| |zr, 2( ) at (a) z = 0, (b) 6, (c) 12. The dimensionless density on the contour is 0.02. (d) The rms sizes xrms and trms versus z during real-z evolution. Figure 5. Collision dynamics of two vortex bullets, each with p = 100, q = 200, placed at =±t 1.45 at z = 0 and set into motion in opposite directions along the t axis with the velocity of v = 43, illustrated by isodensity contours at (a) z = 0, (b) =0.0135, (c) =0.027, (d) =0.0405, (e) =0.054, (f) =0.0675. The density on the contour is 0.02. Laser Phys. Lett. 14 (2017) 065402 7 S K Adhikari 4. Summary To summarize, we demonstrate the formation of a stable 3D spatiotemporal vortex bullet with cubic-quintic nonlinearity, employing a variational approximation and full 3D numer- ical solution of the NLS equation. The statical properties of the bullet are studied by a variational approx imation and a numerical imaginary-z solution of the 3D NLS equation. The cubic nonlinearity is taken as focusing Kerr type above a critical value, whereas the quintic nonlinearity is defocus- ing. The dynamical properties are studied by a real-z solu- tion of the NLS equation. In the 3D spatiotemporal case, the vortex light bullet can move with a constant velocity. At large velocities, the collision between the two spatio- temporal vortex light bullets is quasi-elastic, with no vis- ible deformation of the final bullets. At small velocities, the collision is inelastic with the formation of a breather after collision. At medium velocities the bullets can be destroyed after collision. Acknowledgments We thank the Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil) (Project: 2012/00451-0) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (Brazil) (Project: 303280/2014-0) for support. References [1] Kivshar Y S and Agrawal G 2003 Optical Solitons: From Fibers to Photonic Crystals (San Diego: Academic) [2] Kivshar Y S and Malomed B A 1989 Rev. Mod. Phys. 61 763 Mihalache D 2014 Rom. J. Phys. 59 295 [3] Di Trapani P, Caironi D, Valiulis G, Dubietis A, Danielius R and Piskarskas A 1998 Phys. Rev. Lett. 81 570 [4] Kang J U, Stegeman G I and Aitchison J S 1996 Opt. Lett. 21 189 Kang J U, Stegeman G I, Aitchison J S and Akhmediev N 1996 Phys. Rev. Lett. 76 3699 [5] Silberberg Y 1990 Opt. Lett. 15 1282 [6] Akhmediev N and Soto-Crespo J M 1993 Phys. Rev. Lett. 47 1358 Skryabin D V and Firth W J 1998 Opt. Commun. 148 79 Edmundson D E and Enns R H 1992 Opt. Lett. 17 586 Fibich G and Ilan B 2004 Opt. 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