Exact Bright and Dark Spatial Soliton Solutions in Saturable Nonlinear Media

Exact Brigh t a nd Dark Spatial Soliton Solutio ns in Saturable No nlinear Media Gabriel F. Calvo, Juan Belmonte-Beitia, and V ´ ıctor M. P ´ erez-Garc ´ ıa Dep artamento de Matem´ atic a s, E. T. S. de Ingenier os Industriales and Instituto de Matem´ atic a Aplic ada a la Ciencia y la Ingenier ´ ıa (IMACI), E. T. S. I. Industr iales, Avda. Camilo Jos ´ e Cela, 3 Universidad de Castil la-L a M ancha 13071 Ciudad R e al, Sp ain. Abstract W e presen t exac t analytica l brigh t an d dark (blac k a nd grey) solitary wa v e solu- tions of a nonlinear Sc hr¨ odinger-t yp e equation describin g the propagation of spatial b eams in media exhibiting a saturable n onlinearit y (su c h as cen tr osym m etric p ho- torefractiv e material s). A qualita tiv e study of th e stationary equation is carried out together with a discussion of the stabilit y of the solutions. Key wor ds: Nonlinear S c hr ¨ odinger equation, bright and dark solitons, saturable nonlinearit y . 1 In t ro duction The Nonlinear Sc hr¨ odinger Equation (NLSE) in its man y v ersions is one of the most imp ortant mo dels in mathematical ph ysics, with applications to n umer- ous fields [1,2] suc h a s, for example, in semiconductor electronics [3,4], pho- tonics [5 ,6 ], plasma ph ysics [7], fundamentals of quan tum mec hanics [8], dy- namics of accelerators [9], mean-field theory of Bose-Einstein condensates [10] or biomolecule dynamics [11], to name o nly a few of them. In some of thes e fields a nd many others, the NLSE a pp ears as an asymptotic limit for a slo wly v arying disp ersiv e w av e en v elop e ev olving in a nonlinear medium [12]. The study of t hese equations has serv ed as a cat a lyzer for the dev elopmen t of new ideas or ev en mathematical concepts s uc h as solitons [13] or singularities in partial differen tial equations [14,15]. Among the many differen t v arian ts of the nonlinear Sc hr¨ odinger equation, one whic h has attra cted a lot of interes t is the so-called lo cal saturable nonlinear Preprint submitted to Elsevier 17 Nov em b er 2018 Sc hr¨ odinger equation, which w e write in the form i ∂ A ∂ z + 1 2 ∂ 2 A ∂ x 2 + σ V ( | A | 2 ) A = 0 , (1) where A is a complex env elop e f unction, and σ V ( | A | 2 ) represen ts the nonlin- earit y . Equation (1 ) describ es the propag a tion of a la ser b eam in a satura ble nonlinear medium in (1+1) dimensions. A ( x, z ) measures the (complex) b eam amplitude with x and z b eing the tr a nsv erse and longitudinal co ordinates, resp ectiv ely . Here, V ( | A | 2 ) includes the light-induced refractiv e index c hange and σ ∈ R is prop ortional to the strength of the nonline arity . F or lo cal sat- urable nonlinearities, V ( | A | 2 ) ∝ (1 + | A | 2 ) − α , with α > 0 b eing the s atur ation index . Equation (1) has sho wn a n umber of applications in plasma phy sics [1 6] and nonlinear optics [17 ,18,19], sp ecifically in photorefractive media. Photore- fractiv e materia ls manifest a w ealth of nonlinear phenomena whic h include the propagation of solitons [20,21,22,23,24], surface w av es [26] and slo w ligh t [2 5], pattern formation [27], ch arge singularit ies [28], and critical enhancemen t [29]. The case with α = 1 (which corresp onds to no n- cen trosymmetric photorefrac- tiv e media) has been throughly studied, and solitary w a v e solutions in the form of bright, blac k and grey solitons ha v e b een found n umerically [20,21,22,30]. It was also shown theoretically (by means of nu merical calculations) [31], and exp erimentally [32,33], t ha t the nonlinearit y with α = 2 also app ears in the, so- called, cen tr osymmetric phot o refractiv e media a nd e xhibits brig ht and blac k spatial solitons. Motiv ated by these studies, here we go b ey ond the previous n umerical studies and obt a in three classes of exact solutions to Eq. (1) in closed form: bright, blac k and gr ey solitons (the latter class b eing addressed here fo r the first time, to the b est of our knowle dge). Although many methods hav e b een used to construct exact solutions to v arious t yp es of nonlinear Sc hr¨ odinger equations suc h as the cubic one in differen t sce- narios [34,35,36,37], the cubic-quin tic [38,39] and others [40], it is remark able that no ex act solutions are av ailable for the saturable nonlinear Schr¨ odinger equation (1) for α = 2 despite its practical a pplicatio ns. The pap er is organized as follo ws. In section 2, w e first imp ose the necess ary conditions to deriv e solitary w av es of a general v ersion of the nonlinear paraxial w av e equation in whic h the nonlinearit y dep ends lo cally on the ligh t in tensity (the latter b eing prop ortional to | A | 2 ). Section 3 presen ts a qualitativ e study of the stationary saturable nonlinear Sc hr¨ odinger equation. Sections 4, and 5, con ta in the main results ab out the study of analytical solutions to Eq. (1): w e obtain exact brig ht, a nd dark solitary w a v es, r esp ective ly , together with their existence curv es, that constrain the amplitude and the width of the wa v es. Stabilit y of brigh t and da r k solutio ns is also discussed. 2 2 Solitary W a v es of the Nonlinear P araxial W ave Equation F or completeness, our first purp ose is to deal with a g eneral lo cal nonlinearit y σ V ( | A | 2 ) to deriv e the c haracteristic w av e equation for solitary w a v es from Eq. (1). It is more con v enien t to express the solutions of Eq. (1) in terms of the real amplitude u ( x, z ) and phase φ ( x, z ) as A ( x, z ) = u ( x, z ) e iφ ( x,z ) . (2) Substitution of Eq. (2) in to (1) yields the system of equations ∂ 2 φ ∂ x 2 + 2 ∂ ln u ∂ x ∂ φ ∂ x + 2 ∂ ln u ∂ z = 0 , (3a) ∂ φ ∂ z + 1 2   ∂ φ ∂ x ! 2 − 1 u ∂ 2 u ∂ x 2   − σ V ( u 2 ) = 0 . (3b) It is clear that solving the coupled system of Eqs. (3a) and (3b) is equiv alen t to solving Eq. (1) . W e now imp ose the fundamental prop ert y o f solitary w a v es: their shap e in- v ariance with resp ect to Ga lilean b o osts u ( x, z ) = u ( x − v z ) , (4) where v can b e regarded as the dimensionles s tr ansve rse sp atial velo city (or the steering angle b etw een the propagation direction and the z − axis). In particu- lar, when v = 0, the profile of solitary w av es remains unc hanged with resp ect to translations alo ng the z − axis. Let ξ = x − v z a nd ζ = z . Then, the system of Eqs. (3 a) a nd (3b) r eads a s ∂ 2 φ ∂ ξ 2 + 2 ∂ ln u ∂ ξ ∂ φ ∂ ξ − 2 v ∂ ln u ∂ ξ = 0 , (5a) ∂ φ ∂ ζ − v ∂ φ ∂ ξ + 1 2   ∂ φ ∂ ξ ! 2 − 1 u ∂ 2 u ∂ ξ 2   − σ V ( u 2 ) = 0 . (5b) Equation (5a) can b e readily in tegrated with resp ect to ξ , and yields φ ( ξ , ζ ) = C ( ζ ) + v ξ + D ( ζ ) Z ξ dτ u 2 ( τ ) , (6) where C ( ζ ) and D ( ζ ) are tw o integration functions that de p end solely on ζ . Also, since φ ( ξ , ζ ) satisfies Eq. (5b), inserting expression (6) into Eq. (5b) o ne 3 finds 2 k ∂ ∂ ζ " C ( ζ ) + D ( ζ ) Z ξ dτ u 2 ( τ ) # − v 2 + D 2 ( ζ ) u 4 = 1 u ∂ 2 u ∂ ξ 2 + 2 σ V ( u 2 ) , (7) that must b e fulfilled f or a ll ξ and ζ . Now , if u ( ξ ) is a non-constan t function, this implies that the only a dmissible solutio ns for C ( ζ ) a nd D ( ζ ) a re C ( ζ ) = Γ 0 ζ + φ 0 , (8a) D ( ζ ) = l 0 , (8b) where Γ 0 is a constan t (propagation constan t), φ 0 is a fixed reference phase, and l 0 is a pa r a meter. Under these conditions, Eq. (7) reduces to the follow ing char acteristic e q uation f or solitary waves d 2 u dξ 2 − 2Γ u − l 2 0 u 3 + 2 σ V ( u 2 ) u = 0 , (9) with Γ ≡ Γ 0 − v 2 2 . If l 0 6 = 0, in order to obtain nonsingular solutions to Eq. (9), it is nec essary that u 6 = 0 for −∞ < ξ < ∞ . Multiplying Eq. (9) by du/dξ and up on in t egration, it follows that 1 2 du dξ ! 2 − Γ u 2 + l 2 0 2 u 2 + σ U ( u ) = E 0 , (10) where U ( u ) ≡ 2 R u V ( τ 2 ) τ dτ and E 0 is a constant (it pla ys the ro le o f an effec- tiv e total mec hanical energy). It is expected that if the nonlinear con tributio n is a bsen t, i.e. if σ U ( u ) = 0, then, the solutio n t o Eq. (10), give n by u 2 ( ξ ) = 1 √ 2Γ      1 4   e ± √ 2Γ( ξ − ξ 0 ) − s 2 Γ E 0 e ∓ √ 2Γ( ξ − ξ 0 )   2 + l 2 0 e ∓ 2 √ 2Γ( ξ − ξ 0 )      , cannot represen t a lo calized w av e for an y v alue o f Γ, l 0 , and E 0 . The presence of σ U ( u ) is th us ess en tial for the existence of solitary w a v es. Henceforth, w e assume σ 6 = 0. Moreov er, for solitary w a v es to exist it is clearly required that E 0 − σ U ( u ) + Γ u 2 − l 2 0 / (2 u 2 ) ≥ 0 . Since spatial solitary w av es are w ell-lo calized excitations in space, w e lo ok for solutio ns suc h that their deriv ativ es of all order v anish at infinit y . Three families of solutions are distinguished according to additional b oundary con- ditions at infinit y and at the orig in ξ = ξ 0 . Let u = u ∞ for | ξ | → ∞ , and u = u 0 , du dξ = u ′ 0 b oth at ξ = ξ 0 . Solitary w av es whose asymptotics tend to zero ( u ∞ = 0), together with u ′ 0 = 0, are the well-kno wn bright solitary w a v es. Those that fulfill u 0 = 0 (with u ∞ 6 = 0 and u ′ 0 6 = 0) are the black solitar y w av es, whereas those that do not v anish in −∞ < ξ < ∞ and satisfy u ′ 0 = 0 4 are gr ey solitary wa v es ( b oth blac k and grey are denoted as dark solitons). In con tra st with brigh t and black solutions, grey solitary w a v es p o ssess nonzero v alues o f l 0 . Using the ab ov e prescrib ed b oundary conditions, form ulae for the constan ts Γ, l 0 , and E 0 can b e fo und for eac h family o f solutions. Brigh t solitary w av es corr esp o nd to 0 ≤ u 2 ≤ u 2 0 , Γ = σ [ U ( u 0 ) − U (0)] u 2 0 , l 0 = 0 , E 0 = σ U (0) , (11) and are the implicit solutions obtained after in t egr a tion of 1 2 σ du dξ ! 2 = U (0) − U ( u ) + [ U ( u 0 ) − U (0)] u 2 u 2 0 . (12) Blac k solitary w av es corresp ond to 0 ≤ u 2 ≤ u 2 ∞ , Γ = σ V ( u 2 ∞ ) , l 0 = 0 , E 0 = σ h U ( u ∞ ) − U ( u ) − u 2 ∞ V ( u 2 ∞ ) i , (13) and ob ey 1 2 σ du dξ ! 2 = U ( u ∞ ) − U ( u ) −  u 2 ∞ − u 2  V ( u 2 ∞ ) . (14) Finally , grey solitary wa v es correspond to 0 < min { u 2 0 , u 2 ∞ } ≤ u 2 ≤ max { u 2 0 , u 2 ∞ } , Γ = σ { [ U ( u 0 ) − U ( u ∞ )] u 2 0 + ( u 2 ∞ − u 2 0 ) V ( u 2 ∞ ) u 2 ∞ } ( u 2 ∞ − u 2 0 ) 2 , (15a) l 2 0 = 2 σ [ U ( u ∞ ) − U ( u 0 ) − ( u 2 ∞ − u 2 0 ) V ( u 2 ∞ )] u 2 0 u 4 ∞ ( u 2 ∞ − u 2 0 ) 2 , (15b) E 0 = σ [( u 4 ∞ + u 4 0 ) U ( u ∞ ) − ( u 4 ∞ − u 4 0 ) V ( u 2 ∞ ) u 2 ∞ − 2 u 2 0 u 2 ∞ U ( u 0 )] ( u 2 ∞ − u 2 0 ) 2 , (15c) and are g ov erned by 1 2 σ du dξ ! 2 = ( u 4 ∞ + u 4 0 ) U ( u ∞ ) − ( u 4 ∞ − u 4 0 ) V ( u 2 ∞ ) u 2 ∞ − 2 u 2 0 u 2 ∞ U ( u 0 ) ( u 2 ∞ − u 2 0 ) 2 − U ( u ) − [ U ( u ∞ ) − U ( u 0 ) − ( u 2 ∞ − u 2 0 ) V ( u 2 ∞ )] u 2 0 u 4 ∞ ( u 2 ∞ − u 2 0 ) 2 u 2 + { [ U ( u 0 ) − U ( u ∞ )] u 2 0 + ( u 2 ∞ − u 2 0 ) V ( u 2 ∞ ) u 2 ∞ } u 2 ( u 2 ∞ − u 2 0 ) 2 . (16) When u ∞ = 0 or u 0 = 0, one reco v ers fro m Eq. (1 6) the bright and dark Eqs. ( 1 2) and (14), resp ectiv ely . Moreo v er, dep ending on the given forms for V ( u 2 ) and U ( u ), t o gether with the particular v alues fo r u ∞ and u 0 , the non- linearit y co efficien t σ can b e p ositiv e or negativ e. 5 In the next sections we carry o ut the analysis for brigh t, dark and grey solitary w av e families when t he nonlinearity is o f the saturable form V ( u 2 ) = 1 + u 2 ∞ 1 + u 2 ! 2 , ( 1 7) and therefore U ( u ) = − (1 + u 2 ∞ ) 2 / (1 + u 2 ). Notice that the cubic-lik e no nlin- earit y (e.g. in Kerr media) corresp onds for Eq. (17) to the limit when u 2 ≪ 1; that is, for V ( u 2 ) = (1 + u 2 ∞ ) 2 (1 − 2 u 2 ) and U ( u ) = (1 + u 2 ∞ ) 2 ( u 2 − u 4 ). 3 Qualitativ e Analysis Before deriving explicit localized solutions t o Eqs . (12 ) , (14), and (1 6) when V ( u 2 ) is g iv en by (17), we will carry out a qualita t ive analysis o f Eq. (9) with the latter form for V ( u 2 ): d 2 u dξ 2 − 2Γ u − l 2 0 u 3 + 2 σ 1 + u 2 ∞ 1 + u 2 ! 2 u = 0 . (18) Let us first examine the case l 0 = 0. One easily v erifies that Eq. (18) has three p ossible equilibrium p oints u = 0 , u ± = ± s (1 + u 2 ∞ ) r σ Γ − 1 , (19) if (1 + u 2 ∞ ) > q Γ /σ . When σ < 0 and Γ < 0, u = 0 is a saddle p o in t and u ± are cen ters [see Fig. 1 (a)], whereas fo r σ > 0 and Γ > 0, u = 0 is a center and u ± are saddle p oin ts [see Fig. 1(b)]. In Fig. 1 (a), the closed o r bits of the phase-plane po r trait ( u, du/dξ ) correspond to p erio dic solutions of Eq. (18). It is appar ent that b etw een the external and in ternal closed o rbits t here exists a homo clinic orbit (bla c k curv e), whic h, will b e iden tified as a bright soliton solution in the next section. With resp ect to Fig. 1(b), w e hav e a hetero clinic orbit (black curv e) in the phase-plane p ortrait, and, as w e shall see in section 5, it will corr esp o nd to a blac k soliton solution. Inside this orbit, there exist closed orbits which, again, represen t p erio dic solutions of Eq. (18). Outside the orbit, the solutions are no longer p erio dic. When l 0 6 = 0, Eq. ( 1 8) has a repulsiv e singularit y and p ossess four nonzero equilibrium p o in ts u j , j = 1 , 2 , 3 , 4 ; they corresp o nd to tw o cente rs u 1 = − u 2 (w e omit their length y expres sions), and t w o saddle p oints u 3 = − u 4 = u ∞ . Tw o homo clinic o r bits a pp ear [see blac k curv es in Fig. 1(c)], whic h will b e in terpreted in section 5 as gr ey solitary w a v es. Similarly with the cases in whic h l 0 = 0, the inner closed orbits Fig. 1(c) are p erio dic solutio ns. 6 - 1 - 0.5 0 0.5 1 u - 1 0 1 du d Ξ H 2 È Σ È L - 1  2 H a L - 1 - 0.5 0 0.5 1 u - 1 0 1 H b L - 1 - 0.5 0 0.5 1 u - 1 0 1 H c L Fig. 1. Ph ase-plane p ortrait of the real solutions to Eq. (18) for (a) σ = 2Γ < 0 and u ∞ = 0; (b) σ = Γ > 0 and u ∞ = 0 . 9; (a) and (b) both with l 0 = 0. (c) l 0 6 = 0, σ = 1 . 2Γ > 0 and u ∞ = 1 . 2. 4 Brigh t Solitary W a ves 4.1 Explicit expr essions for bright solitons F undamen tal brigh t solitary wa v es fulfill u ∞ = u ′ 0 = 0, and, without lo ss of generalit y , w e may assume tha t u 0 > 0. Using Eqs. (11), (12), and (17), the constan t Γ = σ / (1 + u 2 0 ), and the resulting energy equation is du dξ ! 2 = − 2 σ ( u 2 0 − u 2 ) u 2 (1 + u 2 0 )(1 + u 2 ) . (20) No w, since u ( ξ ) is a real function, it is thus necessary that σ < 0. Eq uation (20 ) can b e inte grated exactly and it yields an implicit relation for the en v elop e distribution u ( ξ ) of brigh t solitary w av es v u u t 1 + u 2 0 2 | σ |    arctan   s u 2 0 − u 2 1 + u 2   + 1 u 0 arctanh   v u u t u 2 0 − u 2 (1 + u 2 ) u 2 0      = ± ( ξ − ξ 0 ) . (21) Notice that, in this case, the obta ined solution corresp onds to the homo clinic orbit o r lo op of Fig. 1(a). Bright solitary profiles giv en b y Eq. (21) a r e sho wn in F ig . 2 (a). In the limit u ≤ u 0 ≪ 1 we reco v er from Eq. (21) the well-kno wn profile u ( ξ ) = ± u 0 sec h[ u 0 q 2 | σ | ( ξ − ξ 0 )] o f the one-dimensional brigh t solito n in a nonlinear cubic medium. Moreo v er, the dependence of the full width at half maxim um (FWHM), ∆ ξ , as a function of the p eak amplitude u 0 can b e easily obtained from Eq. (21) b y setting u 2 (∆ ξ / 2) = u 2 0 / 2, a nd reads ∆ ξ ( u 0 ) = v u u t 2(1 + u 2 0 ) | σ |    arctan   v u u t u 2 0 2 + u 2 0   + 1 u 0 arctanh   1 q 2 + u 2 0      . (22) Figure 2(b) displays the fo r m of Eq. (2 2). It is also kno wn as the exi s- tenc e curve for brigh t solitary wa v es b ecause it provides a relation b et we en the FWHM, the p eak amplitude o f the b eam and the nonlinearity strength (through t he parameter σ ) for whic h brigh t solitar y w a v es can exist. Suc h a 7 - 10 - 5 0 5 10 x è !!!!!!!!!! 2 » s » 0 1 2 3 » A H x L » 2 H a L 0 2 4 6 8 10 u 0 0 5 10 15 20 Dx H u 0 L è !!!!!!!!! 2 » s » H b L Fig. 2. (a) T ransverse profiles of brigh t solita ry w av es for u 2 0 = 3 and u 2 0 = 0 . 5, as obtained from E q. (21). (b ) Existence curve of b righ t solitary w a ves, w here the FWHM ∆ ξ is plotted ve rsus the p eak amp litud e u 0 . dep endence is o f ten measured in experimen t s [3 2]. In our presen t case there are sev eral in teresting features. A prop erty of ∆ ξ ( u 0 ) is the so-called ampli tude bistability . This means tha t for a g iv en FWHM there are tw o differen t v alues of u 0 for whic h brig h t solitary wa v es are to b e found. This is depicted in Fig. 2(a), where the tw o exhibited brigh t solito ns are c har a cterized by the same FWHM. It is eviden t that with increasing p eak in tensit y , the brig ht solitary w av es hav e a wid er width. This is b ecause the nonlinearit y b ecomes w eaker for large u 0 , and, therefore, in order t o arrest the broa dening (due to diffrac- tion), ∆ ξ ( u 0 ) has to b e larger. All these feat ures are inheren t of brigh t solitons in many satura ble nonlinearities and constitute an ingredien t for stability o f the solutio ns [22]. The minim um of the existence curv e can b e ev aluated b y setting ∂ ∆ ξ /∂ u 0 = 0 in Eq. (22), a nd is obtained f o r u 0 ≃ 1 . 0 59 (whic h giv es q 2 | σ | ∆ ξ ≃ 3 . 33 8 ). These v alues are in exact agreemen t with those calculated n umerically in Ref. [3 1 ]. 4.2 Stability So fa r , w e hav e b een able to dev elop a f ull analytical approach a llo wing us t o determine t he exact profiles and the existence curv e of brigh t solitary w av es. Ho wev er, an imp ortant asp ect still remains op en: Ar e the found bright solitary waves line arly stabl e with r esp e ct to s mal l p erturb ations? Using a w ell-kno wn stabilit y criterion for f undamen tal brig h t solitary wa v es [41,42], w e a nsw er t o this question. All we need to do is to calculate the dep endence of the p ow er P s 8 - 1 - 0.8 - 0.6 - 0.4 - 0.2 0 g 10 −3 10 −1 10 1 10 3 10 5 è !!!!!!!!! 2 » s » P s H g L Fig. 3. Dep endence of the p ow er P s on th e p r opagation constan t, w here γ = Γ / | σ | . of t he brig ht solitary wa v es on the propagatio n constant Γ 0 . More explicitly , P s = Z ∞ −∞ u 2 ( x ) dx = 2 Z ∞ 0 u 2 ( x ) dx = − 2 Z 0 u 0 u 2 du  du dx  = v u u t 2(1 + u 2 0 ) | σ | Z u 0 0 s 1 + u 2 u 2 0 − u 2 u 2 du = v u u t 1 + u 2 0 2 | σ | Z u 2 0 0 s 1 + η u 2 0 − η dη , (23) where w e ha v e emplo y ed the symmetry prop ert y o f the squared amplitude u 2 ( − ξ ) = u 2 ( ξ ) and Eq. (20). After integration, the r esult is P s ( u 0 ) = v u u t 1 + u 2 0 2 | σ | h u 0 + (1 + u 2 0 ) arctan u 0 i . (24) Using Γ = σ / (1 + u 2 0 ), the p ow er P s can b e cast in the form P s ( γ ) = 1 | γ | q 2 | σ |    q 1 − | γ | + 1 q | γ | arctan " s 1 | γ | − 1 #    , (25) where γ = Γ / | σ | . F undamen tal brigh t solitary w a v es a re stable if ∂ P s /∂ Γ 0 > 0, and unstable ot herwise. Figure 3 sho ws that P s is an increasing function of the propagation constant Γ 0 for v 2 / 2 − | σ | < Γ 0 < v 2 / 2. Accordingly , the found brigh t solitary w av es a r e all of them stable. 9 5 Dark Solitary W a v es 5.1 Black Solitary Waves F undamen tal blac k solita r y w a v es fulfill u 0 = 0 ( u ∞ 6 = 0 a nd u ′ 0 6 = 0). Us ing Eqs. (13) , (14), and (17) w e find that Γ = σ and the energy equation is given b y du dξ ! 2 = 2 σ ( u 2 ∞ − u 2 ) 2 1 + u 2 . (26) Again, since u ( ξ ) is a real f unction, it is necessary no w t ha t σ > 0. Equation (2 6) can b e in tegrated exactly and it yields an implicit relation for the en ve lop e distribution of blac k solitary wa v es q 1 + u 2 ∞ u ∞ arctanh   u q 1 + u 2 ∞ u ∞ √ 1 + u 2   − arcsinh( u ) = ± q 2 | σ | ( ξ − ξ 0 ) . (27) The obtained solution corresp onds to the hetero clinic orbit of Fig. 1(b). Bla ck solitary profiles give n b y Eq. (27) are sho wn in Fig. 4(a ) . In the limit | u | ≤ u ∞ ≪ 1 we r eco ve r from Eq. (27) the expres sion u ( ξ ) = ± u ∞ tanh[ u ∞ √ 2 σ ( ξ − ξ 0 )] corresponding t o the one-dimensional blac k soliton o f the cubic NSE. Moreo ver, the dependence of the F WHM, ∆ ξ , as a function of the amplitude a t infinit y u ∞ can b e easily o bta ined from Eq. (27) by setting u 2 (∆ ξ / 2) = u 2 ∞ / 2, and reads as ∆ ξ ( u ∞ ) = s 2 | σ |   q 1 + u 2 ∞ u ∞ arctanh   v u u t 1 + u 2 ∞ 2 + u 2 ∞   − arcsinh u ∞ √ 2 !   . (28) As depicted in Fig. 4(b), the amplitude bistability prop ert y of bright solitary w av es is absen t here. F or increasing v alues of u ∞ , the FWHM tends to the nonzero constan t ln(2) / q 2 | σ | . Regarding the stability of the fo und blac k solitary w av es, they are all stable since the satura ble nonlinearit y ( 1 7) b elongs to the class of lo cal nonlinearities for whic h t he resulting solutions ar e stable for any ve lo city v [43 ]. 5.2 Gr ey Solitary Wave s F undamen tal grey solitary wa v es fulfill u 0 6 = 0, u ∞ 6 = 0, and u ′ 0 = 0. F rom Eqs. (15a), (15 b), (16), and (17 ) w e find that Γ = σ / (1 + u 2 0 ) and l 2 0 = (2 σ u 2 0 u 4 ∞ ) / (1 + u 2 0 ). This last expression impo ses that σ > 0. The energy 10 - 10 - 5 0 5 10 x è !!!!!!!!!! 2 » s » 0 1 2 » A H x L » 2 H a L 0 2 4 6 8 10 u • 0 5 10 15 20 Dx H u • L è !!!!!!!!! 2 » s » H b L Fig. 4 . (a) T ransv erse profiles of blac k solitary w av es for u 2 ∞ = 2 and u 2 ∞ = 1 / 2 as obtained from Eq. (27). (b) Existence cu r v e of b lac k solitary wa v es, wh ere the FWHM ∆ ξ is plotted ve rsus the amplitud e at infi nit y u ∞ . equation is g iv en no w by du dξ ! 2 = 2 σ ( u 2 − u 2 0 )( u 2 ∞ − u 2 ) 2 (1 + u 2 )(1 + u 2 0 ) u 2 . (29) Notice that the structure of Eq. (29) precludes the existence of solutions sat- isfying u 2 ∞ ≤ u 2 ≤ u 2 0 . This means that t here cannot b e h ump-like wa v es on a constan t no nzero bac kground; and so w e lo ok for dip-lik e w av es (i.e. u 2 0 ≤ u 2 ≤ u 2 ∞ ) in a constant nonzero bac kground. Up on inte gration of Eq. (29) w e deriv e the following implicit relation fo r the en v elop e distribution u ( ξ ) v u u t 1 + u 2 ∞ u 2 ∞ − u 2 0 arctanh   v u u t (1 + u 2 ∞ )( u 2 − u 2 0 ) (1 + u 2 )( u 2 ∞ − u 2 0 )   − arcsinh   v u u t u 2 − u 2 0 1 + u 2 0   = ± v u u t 2 | σ | 1 + u 2 0 ( ξ − ξ 0 ) . (30) The obtained solution corr esp o nds to the homo clinic or bit s of Fig. 1(c). Fig- ure 5(a) illustrates the profiles of tw o grey solitary w a v es ha ving the same FWHM. The existence prop erties of g r ey-type solutions are describ ed by an existenc e surfac e , whic h can b e easily found from Eq. (30) b y setting u 2 (∆ ξ ) = ( u 2 0 + u 2 ∞ ) / 2. Figure 5(b) represen ts f our slices of this surface for v arious r a tios of u 0 /u ∞ when v aryin g u ∞ . Th e amplitude bistabilit y feature dep ends no w on this ratio and disapp ears as u 0 /u ∞ → 0. Moreo ver, in con t r a st with brig ht and blac k solutions, the pha se structure of gr ey solitary w av es is non t r ivial. T he term R ξ dτ /u 2 ( τ ) in Eq. (6 ) can b e calculated exactly b y re- sorting to Eq. (29) together with a metho d similar to the one employ ed to ev aluate the p o w er P s of brig h t solitary w av es [see Eq. (23)]. The resulting analytical form of the phase is 11 - 10 - 5 0 5 10 Ξ  2 È Σ È 0 1 2 3 4 È A H Ξ L È 2 H a L 0 2 4 6 8 10 u ¥ 0 5 10 15 20 DΞ H u ¥ L  2 È Σ È H b L Fig. 5. (a) T ransv erse pr ofi les of grey solitary w a ves for u 2 ∞ = 4 and u 2 0 = 1 (u pp er curv e), and u 2 ∞ = 2 . 6 and u 2 0 = 0 . 64 (low er curv e), as obtained from Eq. (30). (b) Existence curves f or grey solitary w a ves, where the FWHM ∆ ξ is p lotted v ersus the amplitude at infin it y u ∞ for u 0 /u ∞ = 0 . 1 , 0 . 4 , 0 . 7 , and 0 . 9 (lo we r to upp er curv es). φ ( ξ , ζ ) = φ 0 + σ 1 + u 2 0 + v 2 2 ! ζ + v ξ − arcsin   u 0 u ( ξ ) v u u t 1 + u 2 ( ξ ) 1 + u 2 0   + u 0 v u u t 1 + u 2 ∞ u 2 ∞ − u 2 0 arctanh   v u u t (1 + u 2 ∞ )[ u 2 ( ξ ) − u 2 0 ] ( u 2 ∞ − u 2 0 )[1 + u 2 ( ξ )]   , (31) where u = u ( ξ ) is giv en implicitly b y Eq. (30). In the limit u 2 0 ≤ u 2 ≤ u 2 ∞ ≪ 1, one retriev es fro m Eq. (30) t he pro file for the one-dimensional grey soliton in a nonlinear cubic medium, which reads as u ( ξ ) = ± s u 2 ∞ − ( u 2 ∞ − u 2 0 ) sec h 2  q 2 | σ | ( u 2 ∞ − u 2 0 ) ( ξ − ξ 0 )  , (32) and t he cor r esp o nding phase dep endence follows straigh t aw a y fro m Eq. (31 ). With resp ect to the the stabilit y of the f o und grey solitary w av es, it is an op en problem to determine the regions defined by u 0 and u ∞ for whic h t hese solutions are stable. This issue will b e addressed in a subsequen t w or k. 6 Conclusions In this pa p er we ha ve constructed explicit analytical expressions for solitary w av e solutions of the nonlinear Schr¨ odinger equation with a satura ble non- linearit y of the form ∝ (1 + | A | 2 ) − 2 arising, for instance, in the propagation of nonlinear optical b eams in cen trosymmetric photorefractiv e materials. The brigh t and black soliton solutions had only b een found nu merically previously in the contex t of optical applications of the mo del equation. In the case of brigh t solito ns solutions w e ha v e also p erfor med a study of their stability . The 12 analysis of t he stabilit y o f gr ey solitons is not so simple and could b e the goal of f uture researc h in this area. Our w ork can b e extended in sev eral directions. The first one, by consider- ing more complicated saturable nonlinearities suc h as those a rising in media with elec tromagnetically induced transparency [44]. Secondly , by studying sat- urable v ector media with s ev eral fields inv olv ed. Finally , another in teresting extension could b e the analysis of the mo del with space-dependen t parameters, resorting to the t ec hnique of Lie symmetries [36 ,3 7]. Ac kno wledgemen ts W e w ould like to thank I. A. Molo tk ov for discussions . 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