Solitons in quasi-one-dimensional Bose-Einstein condensates with competing dipolar and local interactions

Solitons in quasi-one-dimensional Bose-Einstein condensates with competing dipolar and local interactions J. Cuevas,1Boris A. Malomed,2P. G. Kevrekidis,3and D. J. Frantzeskakis4 1Departamento de Física Aplicada I, Grupo de Física No Lineal, Escuela Universitaria Politécnica, C/Virgen de África, 7, 41011 Sevilla, Spain 2Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel 3Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA 4Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece 共Received 20 January 2009; published 6 May 2009兲 We study families of one-dimensional matter-wave bright solitons supported by the competition of contact and dipole-dipole 共DD兲interactions of opposite signs. Soliton families are found, and their stability is investigated in the free space and in the presence of an optical lattice 共OL兲. Free-space solitons may exist with an arbitrarily weak local attraction if the strength of the DD repulsion is fixed. In the case of the DD attraction, solitons do not exist beyond a maximum value of the local-repulsion strength. In the system which includes the OL, a stability region for subfundamental solitons is found in the second finite band gap. For the existence of gap solitons 共GSs兲under the attractive DD interaction, the contact repulsion must be strong enough. In the opposite case of the DD repulsion, GSs exist if the contact attraction is not too strong. Collisions between solitons in the free space are studied too. In the case of the local attraction, they merge or pass through each other at small and large velocities, respectively. In the presence of the local repulsion, slowly moving solitons bounce from each other. DOI: 10.1103/PhysRevA.79.053608 PACS number共s兲: 03.75.Lm, 32.10.Dk, 05.45.Yv I. INTRODUCTION Stable localized matter-wave structures in Bose-Einstein condensates 共BECs兲are supported by the interplay between the intrinsic nonlinearity, which is induced by collisions between atoms, quantum pressure, which originates from the kinetic energy of atoms, and external potentials 关1兴. This mechanism has made it possible to create bright solitons in condensates of 7Li and 85Rb atoms confined in cigar-shaped traps 关2兴, where the interatomic interactions are made attractive by means of the Feshbach resonance 共FR兲关3兴.Inthe condensate of 87Rb atoms with repulsive interactions, the introduction of an optical-lattice 共OL兲potential gives rise to gap solitons 共GSs兲, as demonstrated experimentally in Ref. 关4兴; see also review 关5兴. Dark solitons have been created too, by means of various techniques, in the self-repulsive 87Rb condensate 关6兴. In terms of the theoretical description, the limit case of a very deep OL may be mapped, by means of the tightly binding approximation, into a discrete nonlinear Schrödinger equation and, accordingly, GSs are mapped into staggered discrete solitons 关7兴. New possibilities for the formation of matter-wave solitons are suggested by the presence of long-range interactions in dipolar condensates, which may be composed of magnetically polarized 52Cr atoms 关8兴, dipolar molecules 关9兴,oratoms in which electric moments are induced by a strong external field 关10兴. Solitons supported by the dipole-dipole 共DD兲interactions were predicted in two-dimensional 共2D兲 settings. In the isotropic configuration, with moments fixed perpendicular to the plane, the natural DD interaction gives rise to repulsion, which can support delocalized states in the form of vortex lattices 关11,12兴. In principle, the sign of the DD interaction in this configuration may be reversed by means of rapid rotation of the dipoles 关13兴, suggesting a possibility to create isotropic solitons 关14兴, as well as solitary vortices 关12,15兴. On the other hand, stable anisotropic solitons have been predicted assuming the natural DD interaction between dipoles with a fixed in-plane polarization 关16兴. Peculiarities of the collapse in the medium with the DD interactions were investigated too 关17兴. In addition to these results pertaining to the BEC context, it is relevant to mention that stable vortex rings were predicted in an optical model with the nonlocal thermal nonlinearity 关18兴, and elliptically shaped spatial solitons were created in such media experimentally 关19兴. Although one-dimensional 共1D兲configurations may be simpler than their 2D counterparts, 1D matter-wave bright solitons were not yet studied in detail in models of dipolar condensates, except for discrete solitons of the unstaggered type, which were recently predicted in the condensate trapped in a deep OL 关20兴. In those works, both attractive and repulsive signs of the on-site 共contact兲nonlinearity and long-range DD interactions between sites of the respective lattice were considered. The objective of the present work is to study theoretically the various types of bright solitons possible in the continuum BEC model featuring the competition between the contact and DD interactions. The effective strength of the DD interactions in the 1D geometry can be controlled by adjusting the orientation of the dipoles with respect to the axis of the linear trap, while the strength of the contact interactions may be effectively tuned by means of the FR technique, as shown in the condensate of 52Cr atoms 关21兴. The model is formulated both in the free space 共i.e., in the absence of external potentials兲and in the presence of the OL potential, which opens additional possibilities for the creation of stable localized states, including GSs. In the frame- PHYSICAL REVIEW A 79, 053608 共2009兲 1050-2947/2009/79共5兲/053608共11兲©2009 The American Physical Society053608-1 work of the Gross-Pitaevskii equation 共GPE兲with the ordinary local nonlinear term, the concept of GSs was elaborated in detail; see Refs. 关22–24兴and review 关5兴. However, to the best of our knowledge, it was not yet extended to the new physically relevant case when the OL potential acts together with the long-range DD interaction—a situation that we address below. The paper is organized as follows. The model is formulated in Sec. II, which, in Sec. III, is followed by the consideration of solitons supported by competing nonlinearities— attractive local/repulsive DD or vice versa—in the free space. The model which combines the competing nonlinearities of both types and the OL is considered in Sec. IV. In that case, we report results for regular solitons in the semi-infinite gap and for GSs in the two lowest finite band gaps. In Sec. V, we deal with collisions between moving solitons in the absence of the OL. The paper is concluded by Sec. VI. II. MODEL Our aim is to construct soliton states within the framework of the 1D GPE for the mean-field wave function, ␺ 共x,t兲. The scaled equation includes the OL potential, V共x兲 = ⑀ sin2x共where ⑀ is the strength of the OL, while its period is normalized to be ␲ 兲, the local nonlinear term with the respective coefficient, gc, and its nonlocal counterpart, with coefficient gd, which accounts for the DD interactions, i ␺ t=−1 2 ␺ xx + 冋 ⑀ sin2x+gc兩 ␺ 兩2 +gd 冕 −⬁ +⬁ K共x−x⬘兲兩 ␺ 共x⬘兲兩2dx⬘ 册 ␺ ,共1兲 where subscripts denote partial derivatives. The kernel Kof the DD term was considered in two different forms, so as to avoid the divergence at x=x⬘. The first version relies on an explicit cutoff 共CO kernel兲, K共y兲=10 ␲ yc 3共y2+yc 2兲−3/2,共2兲 where ycis a constant. The other choice makes use of the regularized expression deduced in Ref. 关25兴by means of the single-mode approximation 共SMA兲, i.e., the SMA kernel, K共y兲=10 ␲ 冋 共1+2y2兲exp共y2兲erfc共兩y兩兲 −2兩y兩 冑 ␲ 册 ,共3兲 with erfc共y兲being the standard complementary error function. The two kernels are compared in Fig. 1. The most significant difference between them is that the SMA version features a cusp at y=0, whereas the CO kernel has a smooth maximum. We have concluded that the choice of yc= ␲ −1/2in CO expression 共2兲, which makes areas beneath both curves equal, provides for the best proximity of the corresponding results to what has been found using the SMA approximation. Below, we report results obtained with the SMA kernel, as its CO counterpart yields virtually identical findings. We fix the normalizations in Eq. 共1兲by setting gd=⫾1 and then vary gc. In the case of the 52Cr condensate, a characteristic value of the relative strength of the DD and contact interactions, which can be estimated as 兩gd/gc兩,is⯝0.15 关21兴; this value may be altered in broad limits by means of the FR technique. The interactions are repulsive or attractive for gd,gc⬎0 and gd,gc⬍0, respectively. We will focus on the case of competing interactions, with gcgd⬍0, which is the most interesting one; in the case when both nonlinearities have the same sign, results turn out to be very similar to those reported previously in the local model. It is relevant to mention that, in terms of discrete systems, the competition of on-site 共local兲and intersite 共short-range nonlocal兲interactions in 1D and 2D Salerno lattices were considered in Refs. 关26,27兴, respectively. A number of stable discrete-soliton states which are impossible in the absence of the competition were reported in those works, including cuspons and peakons in one dimension and vortex breathers in two dimensions. Stationary solutions to Eq. 共1兲with chemical potential ␮ are sought as ␺ 共x,t兲=⌿共x兲exp共−i ␮ t兲. To construct such solutions, we discretize the resulting equation for ⌿共x兲by means of a finite-difference scheme, which leads to a set of coupled algebraic equations, ␮ ⌿n=− 1 2共⌬x兲2共⌿n+1 +⌿n−1 −2⌿n兲 + 冉 ⑀ sin2xn+gc⌿n 2+gd⌬x兺 m Kn−m⌿m 2 冊 ⌿n,共4兲 with xn=n⌬xand Kn−m=K共兩n−m兩⌬x兲. Solutions to Eq. 共4兲 are sought by dint of the Newton-Raphson scheme. Results were obtained with a reasonable accuracy by choosing ⌬x = ␲ /40. To present soliton families, we will use the norm N and width Wof the soliton, N= 冕 −⬁ ⬁ 兩⌿兩2dx,W=冑N−1 冕 −⬁ ⬁ x2兩⌿兩2dx.共5兲 These definitions were adapted to the finite-difference form of the model as well. −10 −5 0 5 1 0 0 0.5 1 1.5 2 2.5 3 3.5 K ( y ) y FIG. 1. 共Color online兲The solid and dashed lines show the CO kernel, with yc= ␲ −1/2and its SMA counterpart, which are based on Eqs. 共2兲and 共3兲, respectively. CUEVAS et al. PHYSICAL REVIEW A 79, 053608 共2009兲 053608-2 Stability of the solutions was analyzed in the standard way 共see, e.g., Ref. 关28兴兲 by considering a perturbation in the form of ␦ ␺ 共x,t兲=exp共−i ␮ t兲关P共x兲exp共−i␭t兲+Qⴱ共x兲exp共i␭ⴱt兲兴 共where ⴱstands for complex conjugate兲. The linearized equations for the perturbation eigenmodes 共i.e., the Bogoliubov–de Gennes equations兲are written as ␭ 冉 P共x兲 Qⴱ共x兲 冊 = 冉 L ˆ1L ˆ2 −L ˆ2 ⴱ−L ˆ1 ⴱ 冊冉 P共x兲 Qⴱ共x兲 冊 ,共6兲 where we define L ˆ1⬅− ␮ −1 2 ⳵ x 2+V共x兲+2gc兩 ␺ 共x兲兩2 +gd 冕 −⬁ +⬁ dx⬘K共x−x⬘兲关 ␺ ⴱ共x⬘兲 ␺ 共x兲+兩 ␺ 共x⬘兲兩2兴,共7兲 L ˆ2⬅gc ␺ 2共x兲+gd 冕 −⬁ +⬁ dx⬘K共x−x⬘兲 ␺ 共x⬘兲 ␺ 共x兲.共8兲 The instability sets in when there emerges an eigenmode with Im共␭兲⫽0. Stability eigenvalues were obtained from a numerical solution of Eq. 共6兲by means of a standard eigenvalue solver from Fortran-based software package LAPACK 关29兴. −3 −2 −1 0 1 2 3 0 2 4 6 8 10 12 Ψ(x) x gc=−1 gc=−2 gc=−3 −40 −20 0 20 4 0 0 1 2 3 4 5 Ψ ( x ) x gc=3.6 gc=3 gc=2 (b)(a) FIG. 2. 共Color online兲共a兲Profiles of stable solitons in the model with attractive local and repulsive nonlocal interactions. The solitons were found in the numerical form with resolution ⌬x= ␲ /40, which yields accurate results for 兩gc兩⬎0.9. The peak gets sharper as 兩gc兩 decreases. 共b兲Profiles of solitons in the model with repulsive local and attractive nonlocal interactions, obtained with ⌬x= ␲ /10. The compactonlike solution corresponding to 兩gc兩=3.6 is close to the critical point, beyond which 共at larger 兩gc兩兲 solitons do not exist, irrespective of the value of ⌬x. In both panels, all solitons pertain to ␮ =−1. −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 4 6 8 10 12 14 16 18 20 22 gc=−1 gc=−1.5 gc=−2 gc=−2.5 N µ −3 −2.5 −2 −1.5 − 1 0.1 0.15 0.2 0.25 0.3 0.35 0 . 4 µ=−2 µ=−1.5 µ=−1 W g c (b)(a) FIG. 3. 共Color online兲共a兲The norm of solitons in the free space, in the case of attractive local and repulsive nonlocal interactions, versus the chemical potential. 共b兲The width of the solitons, defined as per Eq. 共5兲, versus the strength of the local attraction. In panel 共b兲, the plots are shown for W⬎⌬x= ␲ /40: recall that obtaining numerical results for smaller Wrequires using smaller ⌬x. SOLITONS IN QUASI-ONE-DIMENSIONAL BOSE-…PHYSICAL REVIEW A 79, 053608 共2009兲 053608-3 III. SOLITONS IN THE ABSENCE OF THE OPTICAL LATTICE We start the presentation of results by considering the model with competing nonlinearities in the free space, i.e., with ⑀ =0 in Eq. 共1兲. This implies that solitons may exist with ␮ ⬍0. A. Attractive local and repulsive nonlocal interactions In accordance with what was said above, we first fix gd =1 共the repulsive DD interaction兲and vary negative gc共local attraction兲and negative ␮ . In this case, solitons exist for every gc⬍0; however, due to discretization, the numerical solution of Eq. 共4兲, with the above-mentioned choice of ⌬x = ␲ /40, yields solitons in the region of 兩gc兩⬎0.25. In fact, although very narrow solitons exist at arbitrarily small values of −gc, the soliton’s width becomes comparable to or smaller than this value of ⌬xat 兩gc兩ⱗ0.9, i.e., a better numerical accuracy is required to produce accurate soliton solutions in this range. Note that, for very narrow solitons with amplitude ⌿0, the nonlinear part of Eq. 共4兲, with gd⬅1, takes the form of 共gc+K0⌬x兲⌿0 3. The existence of the soliton demands a negative coefficient in this expression, i.e., 兩gc兩⬎共10/ ␲ 兲⌬x,共9兲 where it was taken into regard that K0=10/ ␲ , as per Eq. 共3兲. In particular, for ⌬x= ␲ /40, condition 共9兲amounts to 兩gc兩 ⬎0.25, as said above. Figure 2共a兲shows typical examples of the solitons in the present case, while Fig. 3represents soliton families in terms of dependences N共 ␮ 兲and W共gc兲for fixed values of gcand ␮ , respectively; note that plots in panel 共b兲of the latter figure 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 35 40 45 50 55 µ=−1 µ=−2 µ=−3 N g c 0 0.5 1 1.5 2 2.5 3 3. 5 0 0.5 1 1.5 2 2.5 3 3.5 4 µ=−1 µ=−2 µ=−3 W g c (b)(a) FIG. 4. 共Color online兲The norm and width of fundamental solitons in the free space, in the case of attractive nonlocal and repulsive local interactions, versus the strength of the local repulsion. −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 5 10 15 20 25 30 35 g c =0 g c =1 g c =2 g c =3 N µ −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 1 2 3 4 5 6 7 gc=0 gc=1 gc=2 gc=3 W µ (b)(a) FIG. 5. 共Color online兲The norm and width of fundamental solitons in the free space, in the case of attractive nonlocal and repulsive local interactions, versus the chemical potential. CUEVAS et al. PHYSICAL REVIEW A 79, 053608 共2009兲 053608-4 are cut at gc=−1, as the numerical accuracy is insufficient to extend them to smaller values of 兩gc兩, as explained above. The stability of the solitons was verified both through the computation of the eigenvalues, using Eq. 共6兲, and by means of direct simulations of the evolution of perturbed solitons. The well-known Vakhitov-Kolokolov criterion, dN/d ␮ ⬍0 关30兴, also suggests the stability of solitons in this case, although the negative slope of the N共 ␮ 兲curve in Fig. 3共a兲is very small. B. Repulsive local and attractive nonlocal interactions Now,wefixgd=−1, varying gc⬎0 and ␮ ⬍0. Contrary to the previous case, solitons 共which are stable兲can be readily found for all values of gcup to gc⬇3.6, the difference between gc=0 共zero local interaction, while the DD attraction is present兲and gc⬎0 amounting to a gradual increase in the soliton’s amplitude and width with gc. As seen in Fig. 2共b兲, close to gc=3.6 the soliton develops a compactonlike shape, and solitons cannot be found at gc⬎共gc兲max⬇3.7. The existence of 共gc兲max can be easily explained. Indeed, in the limit of ⌬x→0 and for a very broad soliton, the nonlinear part of Eq. 共4兲, with gd⬅−1, takes the approximate form of 冋 gc− 冕 −⬁ +⬁ K共y兲dy 册 兩 ␺ 共x兲兩2 ␺ 共x兲.共10兲 The necessary condition for the existence of solitons is that the coefficient in front of 兩 ␺ 共x兲兩2 ␺ 共x兲in this expression must be negative 关cf. the derivation of Eq. 共9兲兴, i.e., gc⬍共gc兲max ⬅ 冕 −⬁ +⬁ K共y兲dy =20 ␲ −3/2⬇3.59, 共11兲 where expression 共3兲was used to perform the integration. For finite ⌬x, the integral in expression 共11兲is replaced by 兺mKm⌬x. In particular, for ⌬x= ␲ /10, this yields 共gc兲max ⬇3.708, which is consistent with the above-mentioned numerical finding. Note also that the vanishing of the coefficient in front of 兩 ␺ 共x兲兩2 ␺ 共x兲in expression 共10兲at 共gc兲max −gc→0 implies that the soliton’s amplitude and norm diverge in this limit, which is corroborated by the numerical results shown in Fig. 4. The dependence of the soliton’s norm and width on the chemical potential is displayed in Fig. 5. Unlike the nearly flat N共 ␮ 兲dependences in Fig. 3, the present ones clearly satisfy the VK stability criterion, dN/d ␮ ⬍0. The full stability of the solitons was confirmed by the computation of eigenvalues using Eq. 共6兲. At this point, it is also worth briefly considering dark solitons, which are known to be stable in BEC with contact repulsive interactions, such as 87Rb condensates 关1,6兴. Dark solitons in three-dimensional dipolar BECs were recently considered in Ref. 关31兴, where it was shown that, for sufficiently strong repulsive DD interactions and a sufficiently deep OL in the soliton’s nodal plane, dark solitons exist and are stable. In the present 1D setup, to investigate the existence and stability of dark solitons, it is first necessary to ensure that the respective background, namely, the constantamplitude state ␺ =冑 ␮ exp共−i ␮ t兲, is modulationally stable. A comprehensive analysis of the modulational instability 共MI兲 of the background in the context of Eq. 共1兲can be performed following the lines in Ref. 关32兴. Here we will briefly consider this issue and provide an example of a stable dark soliton, assuming gc+共20/ ␲ 3/2兲gd⬎0. In this case, the effective nonlinearity for long-wavelength perturbations 共i.e., those with wave numbers k→0兲is self-defocusing, i.e., the MI band cannot start at k=0, as it does in the case of the standard nonlinear Schrödinger equation with the self-focusing non- −100 −50 0 50 10 0 −1 −0.5 0 0.5 1 Ψ ( x ) x FIG. 6. 共Color online兲The profile of a stable dark soliton in the model with attractive local and repulsive nonlocal interactions for gc=5, gd=−1, and ␮ =1. The soliton was found in the numerical form with resolution ⌬x= ␲ /10. −2.5 −2 −1.5 −1 −0.5 00 1 2 3 4 5 6 7 gc µ FIG. 7. 共Color online兲Unshaded and light-gray-shaded areas represent existence regions for solitons in the semi-infinite and two lowest finite gaps in the model combining the optical lattice with competing attractive local and repulsive nonlocal interactions. Dark-gray shading covers Bloch bands, where solitons do not exist. In the gray-shaded parts of the finite band gaps 共intermediate between light and dark gray兲, the solitons do not exist either. In the white part of the second band gap on the left side of the vertical dashed line, only SFSs exist. They are stable between the red 共solid兲 line and the gray area. On the right side of the dashed line, stable fundamental solitons exist in the white region, while unstable SFSs exist both in the white and light-gray areas. SOLITONS IN QUASI-ONE-DIMENSIONAL BOSE-…PHYSICAL REVIEW A 79, 053608 共2009兲 053608-5 linearity. Although the MI band may appear at finite k, this is not expected to happen as long as the background density is small enough because the maximum MI gain is proportional to that density. Thus, in this case, a modulationally stable background may exist, and dark-soliton solutions can be found. As an example, in Fig. 6we show a stable dark soliton 共its stability was verified through the computation of the full spectrum of eigenvalues for small perturbations兲, which was found for ␮ =1 and gc=5, gd=−1. A systematic analysis of the MI of the background and dark-soliton families in the framework of Eq. 共1兲is beyond the scope of this work and is deferred to a separate publication. IV. SOLITONS IN THE OPTICAL LATTICE In the presence of the OL potential, generic results for regular solitons and GSs in the model with the competing interactions can be adequately represented by fixing the OL strength to ⑀ =6, which is adopted below. GSs have been found in the first and second finite band gaps of the OL- induced linear spectrum. For ⑀ =6, the two numerically computed 共with ⌬x= ␲ /40兲band gaps cover, respectively, the following intervals of the chemical potential: 1.61 ⬍ ␮ ⬍4.26, 4.63 ⬍ ␮ ⬍6.02. 共12兲 A. Attractive local and repulsive nonlocal interactions Here, we consider solitons in the case of the competition between local attraction 共gc⬍0兲and repulsive DD interactions 共gd=1兲, varying ␮ and gc. The self-focusing character of the local interaction allows the existence of regular solitons in the semi-infinite gap, which is −⬁⬍ ␮ ⬍1.5810 for ⑀ =6. The numerical solution of Eq. 共4兲, with ⌬x= ␲ /40, yields regular solitons for gc⬍−0.25, similar to the case of ⑀ =0 共see above兲. Apart from the solitons in the semi-infinite gap, GSs have been found in parts of the first and second finite band gaps, 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6. 5 0 1 2 3 4 5 6 7 8 9 A B C D E F N µ FIG. 8. 共Color online兲N共 ␮ 兲curves for gap solitons in the model including the optical lattice, local attraction, and nonlocal repulsion. The strength of the local attraction is fixed to gc=−1. −20 −10 0 10 20 0 0.5 1 Ψ ( x ) x−20 −10 0 10 20 0 0.5 1 Ψ(x) x−20 −10 0 10 20 0 0.5 1 Ψ(x) x−20 −10 0 10 2 0 −1 −0.5 0 0.5 1 Ψ(x) x −20 −10 0 10 20 −1 −0.5 0 0.5 1 Ψ ( x ) x −20 −10 0 10 20 0 0.5 1 1.5 2 Ψ(x) x −20 −10 0 10 20 0 1 2 3 Ψ(x) x −20 −10 0 10 2 0 −2 0 2 Ψ(x) x −20 −10 0 10 20 −2 0 2 4 Ψ(x) x −20 −10 0 10 20 −2 0 2 Ψ(x) x (b) ( a) (c) (d) (f) ( e) (g) (h) ( i ) ( j ) FIG. 9. 共Color online兲Profiles of stable solitons labeled A–F in Fig. 8. Additionally, panels G–J display examples of stable regular solitons found in the semi-infinite gap. The parameters are 共gc=−1.5, ␮ =1兲and 共gc=−1, ␮ =3 and 5.5兲for the solitons in the semi-infinite gap and first and second finite band gaps, respectively. CUEVAS et al. PHYSICAL REVIEW A 79, 053608 共2009兲 053608-6 for gcexceeding a certain critical value, which depends on ␮ , as shown in Fig. 7. Multihumped solitons, which are bound states of fundamental single-humped solitons, can be found too. Numerical results demonstrate that the existence range for the multihumped solitons is slightly broader than for the fundamental ones. In the second band gap, single-humped solitons exist 共and are stable兲at ␮ ⬎ ␮ cr⬇5.1. This boundary value, which corresponds to the vertical dashed line in Fig. 7, is located above the lower edge of the second band gap, ␮ ⬇4.63 关see Eq. 共12兲兴. Another species of stable GSs was found in a part of the second finite band gap in the form of subfundamental solitons 共SFSs兲. These are antisymmetric modes which are squeezed, essentially, into a single cell of the OL. The norm of the SFS is smaller than that of the fundamental GS if the latter one can be found at the same value of ␮ 共hence the name of “subfundamental” 关33兴兲. In a narrow interval adjacent to the lower edge of the second band gap, 4.63ⱕ ␮ ⬍4.71, the SFSs are stable 共see Fig. 7兲, whereas above this interval, they undergo a Hamiltonian-Hopf bifurcation, being unstable for 兩gc兩smaller than a certain critical value. In the region of ␮ ⬎ ␮ cr⬇5.1, SFSs are unstable in their entire existence region. In direct simulations, the destabilized SFSs spontaneously transform themselves into stable fundamental solitons belonging to the first 共rather than second兲band gap; a similar scenario of the instability development of SFSs was found in the local model 关33兴. The existence regions for fundamental solitons 共both regular ones in the semi-infinite gap and GSs in the first two finite band gaps兲and SFSs in the 共 ␮ ,gc兲plane are depicted together in Fig. 7; the existence region for multihumped bound states is not shown separately, as it almost coincides with that of the fundamental solitons. In addition, Fig. 8 shows the norm as a function of ␮ at gc=−1 for families of subfundamental and fundamental GSs and for stable bound states of fundamental GSs 关in the semi-infinite gap, the norm of all types of regular solitons very weakly depends on ␮ ; cf. Fig. 3共a兲兴. Typical examples of solitons of all these types are displayed in Fig. 9. Bound states in the second finite band gap are not shown, as they are completely unstable against oscillatory instabilities 共while they are stable in the semiinfinite and first finite gaps兲. In fact, the same instability of bound states of GSs in the second finite band gap occurs in the local model. B. Repulsive local and attractive nonlocal interactions To adequately represent results in the model featuring the competition between the local repulsion 共gc⬎0兲and DD attraction 共gd=−1兲in the model with the OL, it was sufficient to use a coarser numerical mesh, with ⌬x= ␲ /10 共recall that ⌬x= ␲ /40 was used above兲. The nonlocal self-attraction allows the existence of solitons in the semi-infinite gap. 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 g c µ FIG. 10. 共Color online兲The existence range for solitons in the semi-infinite and two lowest finite gaps in the model with the optical lattice and competing local repulsive and nonlocal attractive interactions. The notation is the same as in Fig. 7. 0 0.5 1 1.5 0 2 4 6 8 10 12 14 16 18 20 G I J H N µ 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 1 2 3 4 5 6 7 8 9 10 A B C D E F N µ (b)(a) FIG. 11. 共Color online兲N共 ␮ 兲curves for regular 共left panel兲and gap 共right panel兲solitons in the model with the optical lattice, local repulsion, and nonlocal attraction. The strength of the local repulsion is fixed to 共a兲gc=1 and 共b兲gc=5. SOLITONS IN QUASI-ONE-DIMENSIONAL BOSE-…PHYSICAL REVIEW A 79, 053608 共2009兲 053608-7 Similar to what was reported above for the same case in the free-space model, the solitons become very broad at gcapproaching the critical value, gc⬇3.6. Other features of the regular solitons found in the semi-infinite gap are also similar to those of their free-space counterparts. The similarity holds also in the case of gc=0, i.e., in the model with the pure nonlocal attractive interactions. The existence range for stable fundamental solitons 共both the regular ones and GSs兲and SFSs in the present version of the model is depicted in Fig. 10. Further, Fig. 11 shows the respective N共 ␮ 兲dependences, including those for boundstate solutions. This figure shows N共 ␮ 兲lines in the semiinfinite gap too, as, on the contrary to the situation for the model with gc⬍0 and gd=1, these lines do not degenerate into N=const. In fact, the opposite signs of dN/d ␮ for regular solitons and GSs are a typical feature, observed in local models as well. Typical profiles of various soliton species belonging to the semi-infinite and finite gaps are displayed in Fig. 12. −20 −10 0 10 20 0 0.2 0.4 0.6 0.8 Ψ ( x ) x−20 −10 0 10 20 −0.5 0 0.5 Ψ(x) x−20 −10 0 10 20 0 0.2 0.4 0.6 0.8 Ψ(x) x−20 −10 0 10 2 0 0 0.2 0.4 0.6 0.8 Ψ(x) x −20 −10 0 10 20 −0.5 0 0.5 Ψ ( x ) x −20 −10 0 10 20 0 0.5 1 Ψ(x) x −20 −10 0 10 20 0 1 2 3 Ψ(x) x −20 −10 0 10 2 0 −2 0 2 Ψ(x) x −20 −10 0 10 20 −2 0 2 Ψ(x) x −20 −10 0 10 20 −2 0 2 4 Ψ(x) x (b) ( a) (c) (d) (f) ( e) (g) (h) (i) ( j ) FIG. 12. 共Color online兲Soliton profiles at points A–J in Fig. 11. The parameters are 共gc=1, ␮ =1兲and 共gc=5, ␮ =3 and 5.5兲for the solitons in the semi-infinite gap and first and second finite band gaps, respectively. x t −100 −50 0 50 100 0 20 40 60 80 100 120 140 160 180 200 0. 5 1 1. 5 2 2. 5 x t −100 −50 0 50 100 0 5 10 15 20 25 30 35 40 0 0.5 1 1.5 2 2.5 (b)(a) FIG. 13. 共Color online兲Soliton-soliton collisions in the free space 共 ⑀ =0兲for gc=1, gd=−1, and ␮ =−1. The initial velocities are 共a兲c =⫾0.2942 and 共b兲c=⫾0.9651. CUEVAS et al. PHYSICAL REVIEW A 79, 053608 共2009兲 053608-8 V. COLLISIONS BETWEEN MOVING SOLITONS We have studied collisions between solitons moving in the free space. In the model with the repulsive local and attractive DD interactions, a usual collision scenario is observed: at small velocities, solitons merge into a bound state, while, at high velocities, they pass through each other, as shown in Fig. 13. The collision scenario is different in the opposite case, with the local attraction and nonlocal repulsion. As shown in Fig. 14, at small velocities the solitons bounce from each other. This feature is easily explained by the fact that the long-range interaction between the solitons is repulsive. The rebound is changed by the merger at intermediate values of the velocities. Finally, fast solitons pass through each other. Dependences of critical values of collision velocities, which separate different outcomes of the collision, on the strength of the local interaction are displayed in Fig. 15. In the presence of the OL, the solitons can be made mobile 共by application of a kick to them兲if the nonlinearity in the model is weak enough; otherwise, the respective Peierls- Nabarro barrier is very high. In the weak-nonlinearity regime, the mobility of GSs in the present model is quite similar to that reported in the local model with the self-repulsive interactions 关34兴, as well as in the discrete model including the DD interactions 关19兴. VI. CONCLUSION This work presents results of the systematic analysis of one-dimensional bright solitons supported by contact and DD interactions of opposite signs in BEC. In the absence and in the presence of the OL, stable soliton families have been found for the cases of local attraction and DD repulsion or vice versa. In particular, free-space solitons can be supported by arbitrarily weak local attraction if the DD repulsion is fixed; in the opposite case, there is a maximum value of the strength of the local repulsion beyond which solitons do not exist 共which was explained in an analytical form兲.Inthe model including the OL, a notable finding is a region of stability of SFSs in the second finite band gap. It is noteworthy too that, as seen in Figs. 7and 10, the GSs exist, in the case of the attractive DD interaction, if the contact repulsion is strong enough, and, in the opposite case of the repulsive DD interaction, GSs exist if the contact attraction is not too strong. Collisions between bright solitons in the free space were considered too. The collision scenario is the usual one in the x t −50 0 50 0 20 40 60 80 100 0.1 0.2 0. 3 0. 4 0.5 0. 6 0. 7 0. 8 x t −50 0 50 0 20 40 60 80 100 0 0. 5 1 1. 5 2 2. 5 3 3. 5 x t −50 0 50 0 5 10 15 20 2 5 0 0. 5 1 1. 5 2 2. 5 3 3. 5 ( b )( a ) ( c ) FIG. 14. 共Color online兲Soliton collisions at gc=−5, gd=1, ␮ =−1, and ⑀ =0. Initial velocities are 共a兲c=⫾0.0990, 共b兲⫾0.1984, and 共c兲 ⫾0.4969. These plots were generated using the mesh with ⌬x=0.1. −5 −4.5 −4 −3.5 −3 −2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ccrit g c Rebound Merger Passage 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.35 0.4 0.45 0.5 0.55 0.6 0 . 6 5 c crit g c Merger Passage (b)(a) FIG. 15. 共Color online兲Critical values of the collision velocity versus the strength of the contact attraction or repulsion for 共a兲gd=1 and 共b兲gd=−1. In both cases, ␮ =−1 and ⑀ =0. SOLITONS IN QUASI-ONE-DIMENSIONAL BOSE-…PHYSICAL REVIEW A 79, 053608 共2009兲 053608-9