Lie Symmetries, qualitative analysis and exact solutions of nonlinear Schr"odinger equations with inhomogeneous nonlinearities

Lie Symmetries, qualitativ e analysis and exact solutions of nonlinear Sc hr¨ odinger equations with inhomogeneous nonlinearities Juan Belmonte-Beitia 1 , 3 , ∗ V. M. P´ erez-Gar c ´ ıa 1 , 3 , † V. V eks lerchik 1 , 3 , ‡ and P . J. T o rres 2 § 1 Dep artamento de Matem´ atic as, E. T. S. de Ingenier os Industriales, Universidad de Cast il la-L a Mancha 13071 Ciudad R e al, Sp ain. 2 Dep artamento de Matem´ atic a Aplic ada. F acultad de Ciencias. Universidad de Gr anada Campus de F uentenueva s/n, 18071 Gr anada, Sp ain. 3 Instituto de Matem´ atic a Aplic ada a la Ciencia y l a Ingenier ´ ı a (IMACI), Universidad de Castil la-L a Mancha, 13071 Ciudad R e al, Sp ai n (Dated: Octob er 30, 2018) Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schr¨ odinger equations wi th spatially inhomogeneous nonlinearities. W e presen t the general theory , use it to study d ifferent examples and u se the q ualitativ e theory of dynamical systems to obtain some properties of these sol utions. P ACS n umbers: 35Q51, 35Q55, 34C14. I. INTRO DUCTION The Nonlinear Sc hr¨ odinger Equation (NLSE) in its ma ny versions is o ne of the most impor tant mo dels of mathe- matical ph ysics, with applica tions to differen t fields [33] as fo r example in semico nductor electronics [6, 21], nonlinear optics [1 8], photonics [17], plasma ph ysics [10], fundamen tation of quant um mechanics [28], dynamics of accelerators [13], mean-field theo ry of Bose-E ins tein condensa tes [8, 35] or biomo le c ule dynamics [9] to cite only a few examples. In some of these fields and many others, the NLSE a ppe a rs as an asy mptotic limit for a slowly v a r ying disp ersive wa v e en velope pro pagating in a nonlinear medium [3 0]. Moreov er, the rang e of applica bilit y is la rge be c ause o f the well-kno wn univ ersality of this equation [2]. The study of these equations has served as a catalyz e r of the developmen t of new ideas o r even mathematica l concepts such as solitons [36] or sing ularities in pa rtial differential equations [1 5, 31]. In the last years there has b een an increased interest in a one-dimensional no nlinear Schr¨ odinger equation with inhomogeneous nonlinea rity (INLSE): iψ t = − ψ xx + V ( x ) ψ + g ( x ) | ψ | 2 ψ , (1) with x ∈ R , V ( x ) is a n external p otential and g ( x ) descr ibe s the spatial mo dulation of the nonlinea r ity . This equation arises in different ph ysical co ntexts such as nonlinea r optics and dy namics o f Bose-E instein condensates. But it is in the later field wher e the p ossibility of using the F esc hbac h resonance manage ment tec hniques to mo dify spa tially the collisional in teractions betw een atoms [1, 11, 16, 25, 26, 27, 32, 34] the one whic h has motiv ated a lot of theoretical resear ch in the last few years fo cusing o n questions of direct a pplicability to e xp e riments. Different asp ects o f the dynamics of so litons in these contexts have been studied such as the emission o f solito ns [27, 34] and the propa gation of s olitons when the spac e mo dulation of the nonlinearity is a r andom [1], per io dic [25, 29], linear [32] or lo calized function [26]. Although many ex act solutions of the NLSE with spatially homo geneous nonlinearities and without po tent ials ( V = 0) hav e b een known for a long time, the problem o f finding exact solutions even of the NLS with homogeneous nonlinea rities and g eneral p otentials is a very difficult one. In this pap er, using Lie symmetries w e find gener al classes of po tent ials V ( x ) and nonlinearity functions g ( x ) for which exa ct solutions can be constructed by combining s o lutions of the integrable NLS and solv a ble po tentials V ( x ). The basic idea of the Lie symmetries metho d is to study the inv aria nc e prop erties of given differential equations under contin uous gr oups of transformatio ns. This method has been applied s uccessfully to different equa tions, s uch as, for example, differential equa tions that model anha rmonic oscillators [19, 20] and Madelung fluid equations [3]. In Ref. [4] we have presented so me examples of this metho dolog y o f sp ecific physical int erest. Here we co mplemen t that ∗ Electronic address: juan.b elmonte @uclm.es † Electronic address: victor.p erezgarcia@uclm.es ‡ Electronic address: v adym@v ekslerch ik.uclm.es ; URL: http://mat ematicas.uc lm.es/nlwaves § Electronic address: ptorres@ugr.es 2 analysis by presenting the g eneral theory , provide more examples , study the ca se of a symmetric so lutions and use qualitative theory of dyna mical systems to provide a muc h more c o mplete ana lysis of the metho d and its applications to equations of physical r elev ance. The pap er is o rganized as follows. In Section 2 , we introduce the gener al theory of the Lie symmetry a na lysis for o rdinary differential equations and particulariz e it for our mo del pro blem: the nonlinear Schr¨ odinger eq ua tion with an inhomog eneous nonlinea rity . In Section 3, we study the canonical tr ansformatio ns of the INLSE. In Section 4, we present the connection b etw een the NLSE and the INLSE. In Section 5, we use the metho d to co nstruct explicit so lutions of the s tationary nonlinear Sc hr¨ odinger e quation with an inhomogeneous no nlinea rity and study the qualitative behaviour of the NLSE and its q ua litative c o nnection with the INLSE in differen t exa mples. Finally , in Section 6, we prese n t asymmetr ic solutions of the INLSE. T o our knowledge, this is the first time that such so lutio ns are ca lc ulated. II. GENERAL THEOR Y OF LIE SYMMETRIES In this paper w e will lo o k for lo calized stationary solutio ns of Eq. (1), which ar e of the form ψ ( x, t ) = u ( x ) e − iλt , which satisfy the following nonlinear eig env alue problem − u xx + V ( x ) u + g ( x ) u 3 = λu, (2a) lim | x |→∞ u ( x ) = 0 . (2b) By definition [5, 23], a second-o rder differen tial equation A ( x, u , u ′ , u ′′ ) = 0 p ossess es a Lie gro up of po int transfor - mations or Lie point symmetry of the for m M = ξ ( x, u ) ∂ /∂ x + η ( x, u ) ∂ /∂ u, if the action of the second extension of M , i.e. M (2) on A is equal to zer o, i.e. M (2) A ( x, u, u ′ , u ′′ ) =  ξ ( x, u ) ∂ ∂ x + η ( x, u ) ∂ ∂ u + η (1) ( x, u ) ∂ ∂ u ′ + η (2) ( x, u ) ∂ ∂ u ′′  A ( x, u, u ′ , u ′′ ) = 0 , with η ( k ) given by η ( k ) ( x, u, u ′ , u ′′ , ..., u k ) = D η ( k − 1) D x − u ( k ) D ξ ( x, u ) D x , k = 1 , 2 , ... where η (0) = η ( x, u ) and D /D x is the total de r iv ative, i.e. D D x = ∂ ∂ x + u ′ ∂ ∂ u + u ′′ ∂ ∂ u ′ + .... + u ( n +1) ∂ ∂ u ( n ) + ... F o r example, η (1) is equa l to η (1) = η x + [ η u − ξ x ] u x − ξ u ( u x ) 2 . In our case, A ( x, u, u x , u xx ) is g iven by A ( x, u, u x , u xx ) = − u xx + f ( x, u ) , where f ( x, u ) = V ( x ) u + g ( x ) u 3 − λu , and the action of the opera tor M (2) on A ( x, u, u x , u xx ) leads to a p olynomial equation in u x . By equating co efficients of p ow ers of u x , one obtains ξ uu = 0 , (3a) η uu − 2 ξ ux = 0 , (3b) 2 η xu − ξ xx − 3 f ξ u = 0 , (3c) η xx − ξ f x − η f u + η u f − 2 ξ x f = 0 . (3d) 3 Int egra ting Eqs. (3a ) and (3b), we get ξ ( x, u ) = a ( x ) u + b ( x ) , η ( x, u ) = a ′ ( x ) u 2 ( x ) + c ( x ) u + d ( x ) . (4) Substituting thes e expressio ns in to Eq. (3c) we obta in 2 c ′ ( x ) = b ′′ ( x ) , a ( x ) = 0 . (5) Finally , substituting Eqs. (4) a nd (5) in Eq. (3d), w e get ξ ( x, u ) = b ( x ) , (6a) η ( x, u ) = c ( x ) u, (6b) c ′′ ( x ) − b ( x ) V ′ ( x ) − 2 b ′ ( x ) ( V ( x ) − λ ) = 0 , (6c) 2 c ( x ) g ( x ) + b ( x ) g ′ ( x ) + 2 b ′ ( x ) g ( x ) = 0 . (6d) Also the s ubstitution of Eq. (5) in Eq. (6d) giv es g ( x ) = g 0 b − 3 ( x ) e − 2 C R x 0 1 /b ( s ) ds . where g 0 and C are arbitrary cons tants. Summarizing the prev ious calculations, the Lie p oint symmetry is of the form M = b ( x ) ∂ ∂ x + c ( x ) u ∂ ∂ u , (7) where g ( x ) = g 0 b − 3 ( x ) e − 2 C R x 0 1 /b ( s ) ds , (8a) c ( x ) = 1 2 b ′ ( x ) + C, (8b) c ′′ ( x ) − b ( x ) V ′ ( x ) − 2 b ′ ( x ) ( V ( x ) − λ ) = 0 . (8c) Eqs. (8) allow us to co ns truct pairs { V ( x ) , g ( x ) } for which a Lie po int symmetry exis ts . Thus given either g ( x ) o r V ( x ), in principle we ca n choose the other in or der to sa tisfy Eqs. (8). In what follows, we will study the implica tions of the existence of this Lie symmetry . II I. CANONICAL TRANSFORMA TIONS AND INV ARIANTS. It is known [19], that the in v a riance of the energ y is asso ciated to the tr a nslational inv arianc e . The generato r of such a transformation is of the form M = ∂ /∂ X . T o use this fact, we define the transfor mation from v ar ia bles ( x, u ) to new v ariables ( X , U ) X = h ( x ) , U = n ( x ) u, (9) where h ( x ) and n ( x ) will be deter mined b y requiring that a conserv ation law o f energy t yp e M = ∂ /∂ X exists in the canonical v a riables. In fact, using Eq. (9), w e get ∂ ∂ u = n ( x ) ∂ ∂ U , (10) ∂ ∂ x = n ′ ( x ) u ∂ ∂ U + h ′ ( x ) ∂ ∂ X . (11) Inserting the expressions (10) a nd (11) in E q. (7) and assuming the condition M = ∂ /∂ X , one finds h ′ ( x ) b ( x ) = 1 , b ( x ) n ′ ( x ) + c ( x ) n ( x ) = 0 . (12) By inserting Eq. (8b) into (12), a nd integrating, we o btain h ( x ) = Z x 0 1 b ( s ) ds, n ( x ) = b ( x ) − 1 / 2 e − C R x 0 1 /b ( s ) ds . 4 W e can now write Eq. (2a) in terms of the canonical co ordinates U and X , − d 2 U dX 2 − 2 C dU dX + g 0 U 3 − E U = 0 , (13) with E = ( λ − V ( x )) b ( x ) 2 − 1 4 b ′ ( x ) 2 + 1 2 b ( x ) b ′′ ( x ) + C 2 . (14) Equation (13) is the so -called Duffing equation which arises a s a mo del o f damp ed nonlinear oscillations [14]. It follows from Eqs. (8b)-(8c) that the quantit y E given b y Eq . (14) is a constant of motion. When C = 0 the pr evious tra nsformations pr eserve the Hamiltonian structure, bec ause the ca nonical tra nsformation is symplectic. In that case, Eq. (13) b ecomes − d 2 U dX 2 + g 0 U 3 = E U. (15) As E is constant, this means that in the new vari ables we obtain the nonline ar S chr¨ odinger e quation (N LSE) without external p otent ial and with an homo gene ous nonline arity . Of cour se not all choices of V ( x ) and g ( x ) lead to the existence of a Lie s y mmetry o r an appr opriate canonica l transformatio n, since they a re linked by Eq s . (8). This fact imp oses some obvious restrictions, for ins ta nce b ( x ) must be smo oth and positive. Note that Eq. (15) is a stationa ry homogeneous NLSE without external po tent ial, which can be reduced to quadratures a nd for whic h ma ny so lutions are known. So , we o btain X − X 0 = Z U U 0 dG q 2( N + 1 2 E G 2 + 1 4 g 0 G 4 ) , with N a co nstant o f in tegra tion. Moreover, the e ner gy of the system is given by H = 1 2  dU dX  2 + 1 2 E U 2 − 1 4 g 0 U 4 . (16) Many s olutions of E q. (1 5) are known. In this paper we will use the following ones U 1 ( X ) = η 1 cosh( µX ) ,  E = − µ 2 , g 0 = − 2 µ 2 η 2  , (17a) U 2 ( X ) = η tanh( µX ) ,  E = 2 µ 2 , g 0 = 2 µ 2 η 2  , (17b) U 3 ( X ) = η sn( µX , k ) dn( µX, k ) ,  E = µ 2 (1 − 2 k 2 ) , g 0 = − 2 µ 2 k 2 (1 − k 2 ) η 2  , (17c) U 4 ( X ) = η dn( µX, k ) ,  E = µ 2 ( k 2 − 2) , g 0 = − 2 µ 2 η 2  , (17d) with 0 ≤ k ≤ 1. T able I summar izes the parameter v alues required for the existence o f the solutions listed in Eq s. (17). U E g 0 H U 1 ( X ) n egativ e negative 0 U 2 ( X ) p ositive p ositive positive U 3 ( X ) b oth n egativ e p ositive U 4 ( X ) n egativ e negative negative T ABLE I: Conditions on th e parameters E , g 0 and energy H for th e existence of the solutions U i , i = 1 .. 4 of Eq. (15) listed in Eqs. (17). 5 IV. CONNECTION BETWEEN THE NLSE A ND INLSE V IA THE LSE Setting C = 0 and eliminating c ( x ) in Eqs. (8) we get g ( x ) = g 0 /b ( x ) 3 , (18) plus an equation relating b ( x ) and V ( x ) b ′′′ ( x ) − 2 b ( x ) V ′ ( x ) + 4 b ′ ( x ) λ − 4 b ′ ( x ) V ( x ) = 0 . (19) W e notice that the simplest form to gener ate solutions for our pro blem, which in volv es constructing solutions pa irs ( b ( x ) , V ( x )) of Eq. (19) is to fix b ( x ) a nd then, to calcula te V ( x ), since then we must s o lve a linea r first or de r equation. Although we can eliminate b ( x ) and obtain a nonlinear equation for the pairs g ( x ) and V ( x ) for which there is a Lie symmetry , it is mo re conv enient to work with (19), which is a linear equation. Alternatively , we can define ρ ( x ) = b 1 / 2 ( x ) and get an Ermakov-Pinney equatio n [12, 24] ρ xx + ( λ − V ( x ) ) ρ = E /ρ 3 , whose solutio ns can b e constructed as ρ =  αϕ 2 1 + 2 β ϕ 1 ϕ 2 + γ ϕ 2 2  1 / 2 , with α, β , γ constant and ϕ j ( x ) b eing tw o linea rly indepe ndent s olutions of the Sc hr¨ odinger equation ϕ xx + ( λ − V ( x ) ) ϕ = 0 . This choice leads to E = ∆ W 2 with ∆ = αγ − β 2 and W be ing the (constan t) W ronskia n W = ϕ ′ 1 ϕ 2 − ϕ 1 ϕ ′ 2 . Thu s, given any arbitra r y so lution o f the line ar Schr¨ odinger e qu ation (20) we c an c onstru ct solutions of the n online ar sp atial ly inhomo gene ous pr oblem Eq. ( 2a) fr om t he known solutions of Eq. (15) . Th us, using the h uge amo unt of knowledge on the linea r Schr¨ odinger equation we can g et po tent ials V ( x ) for which ϕ 1 and ϕ 2 are known and construct b ( x ), the c anonical tra nsformations h ( x ) , n ( x ), the no nlinea rity g ( x ) and the explicit solutions u ( x ). V. QUALIT A TIVE ANAL Y SIS AND EXA CT SOLUTIONS In this section, we will calculate exac t solutio ns of Eq. (2a) for different sp ecific choices o f the nonlinear coefficient g ( x ) and the externa l p otential V ( x ), using the metho d desc rib ed in the pr evious sections, for C = 0. Mor eov er, using qualitative ana lysis, we will descr ibe proper ties of the solutions of E q . (2a), on the bas is of the qualitative b ehaviour of Eq. (15). W e b egin b y ca lculating the equilibrium p oints of this equation. One e a sily finds that the Eq. (15) has three p ossible equilibrium p oints, depending of the sig ns of E and g 0 : U ± = ± p E /g 0 , U = 0 . Then, we distinguish four cases: 1. F or E < 0 , g 0 < 0, w e get thr ee equilibrium points. U = 0 is a saddle p oint and U ± are centers, Fig. 1 (a). 2. When E < 0 , g 0 > 0, w e obtain that U = 0 is the only equilibrium p oint, which is a saddle p oint, Fig. 1(b). 3. When E > 0 , g 0 > 0, w e get thr ee equilibr ium p oints. U ± are saddle p oints and U = 0 is a center, Fig. 1(c). 4. The last cas e cor resp onds to E > 0 , g 0 < 0 . F or this case, the only e q uilibrium p oint is the trivial solution U = 0, which is a global center, Fig. 1(d). Using Eq. (16), we can draw the pha s e p ortrait of Eq. (15), as we ca n see in Fig. 1. In what follows we will prese nt three examples as applications o f our theor y: Example 1. Let us take b ( x ) = cosh( x ). By using Eqs. (19) and (14), for C = 0, we obtain V ( x ) = λ + 1 4 +  1 4 − E  1 cosh 2 ( x ) . 6 (a) d U / d X 1 0 -1 -1.5 U 0 1.5 (b) d U / d X 1 0 -1 -1 U 0 1 -1.5 0 1.5 -1.5 0 1.5 U (c) d U / d X -1 0 1 -1 0 1 (d) U d U / d X FIG. 1: [Color online] Phase p ortrait of the real solutions of Eq. ( 15) for (a) E < 0, g 0 < 0, (b ) E < 0, g 0 > 0 (c) E > 0, g 0 > 0 and (d) E > 0, g 0 < 0 0 3.5 0 1.6 U X 0 (a) 0 1.6 -20 20 0 u x (b) FIG. 2: [Colo r online] Solutions of (a) Eq. (15) and (b) Eq. (2a), for E = 0 . 15, g 0 = − 1 (solid blue line) and E = − 0 . 75, g 0 = − 1 (d ashed red line) in b oth cases. W e apply the transformation (21) to the solutions of Eq. (15) shown in Fig. 2(a) to obtain the solutions of Eq. (2a) shown in Fig 2(b ). Moreov er, using Eq. (18), g ( x ) is g iven by g ( x ) = g 0 cosh 3 ( x ) , with X ( x ) being cos X ( x ) = − tanh x, where 0 ≤ X ≤ π , s ub ject to the Dirichlet b oundar y co nditions U (0) = U ( π ) = 0. An y solution U o f Eq. (1 5) g ives a solution u ( x ) = b 1 / 2 ( x ) U ( X ( x )) , 7 -50 0 50 -0.5 0.5 U X (a) -0.6 0.6 -30 0 30 0 x (b) u FIG. 3: [Color online] Example b lac k solitons solutions of (a) Eq. (15) and (b) Eq. (2a) with g 0 = 1, λ = 1 / 4 and (i) α = 0 . 1 (dashed-dot green line), (ii) α = 0 . 4 (soli d blue line) and (iii) α = 0 . 7 (dashed red line).The solutions sho wn in Fig. 3 (b) are obtained from those show n in Fig. 3(a) through the transforma tions (9). of the or iginal equatio n (2a). If E < 0 and g 0 < 0 , w e ar e in the first case , Fig. 1(a). The p er io dic solution (17 c) of Eq. (15 ) is a closed orbit of the phase p or trait ( U, dU / dX ), co rresp onding to one of the external closed or bits to the homo clinic or bits, shown in Fig. 1(a). By an elementary applica tion of L’Hopital rule, it is e asy to verify that the solution u ( x ) = b 1 / 2 ( x ) U 3 ( X ( x )) , (21) is a homoclinic orbit to zer o (bright soliton) of the original equation (2a ). If E > 0 and g 0 < 0, we are in the four case, where U = 0 is a cen ter, Fig. 1(d). Again, u ( x ) = b 1 / 2 ( x ) U 3 ( X ( x )) is a homo clinic orbit (brigh t soliton). The so lutions o f E q. (2a) for E = 0 . 15, g 0 = − 1 a nd E = − 0 . 7 5, g 0 = − 1 a re drawn in Fig. 2 (b). The case E = 1 / 4, where V ( x ) is a co nstant, was studied in [4]. Example 2. Let us take V ( x ) = 0. Then Eq. (1 9) b ecomes b ′′′ ( x ) + 4 λb ′ ( x ) = 0. F o r λ > 0, the solution c a n b e written as b ( x ) = 1 + α cos(2 √ λx ) . Using Eq. (18), we obtain a per io dic nonlinear ity g ( x ) = g 0 (1 + α co s(2 √ λx )) − 3 . F o r small α , this nonlinear it y is a pproximately harmonic g ( x ) ≃ g 0 (1 − 3 α cos(2 √ λx )) , α ≪ 1 . W e can construct o ur canonica l transformation by using E q s. (9) and obtain X ( x ) = 1 p λ (1 − α 2 ) arctan r 1 − α 1 + α tan( √ λx ) ! . (22) Using any solution of Eq. (15) with E = λ  1 − α 2  this tr ansformatio n pr ovides so lutions of E q . (2a) with g ( x ) given by (22). F or example, when g 0 > 0 we can use U 2 as defined by (1 7b), which in the phase po rtrait is the hetero clinic orbit shown in Fig. 1(c). So, the solution of E q. (2a) is of the for m u ( x ) = s λ (1 − α 2 ) g 0  1 + α cos(2 √ λx )  × tanh " r λ (1 − α 2 ) 2 X ( x ) # . (23) As w e can see in Fig. 3(b), u ( x ) is a hetero clinic connection betw een perio dic solutions of Eq. (2 a). It is importa nt to note that in the a symptotic regions the profile of u ( x ) is close to b 1 / 2 ( x ) multiplied by a constant. Therefor e , the canonical tr ansformation (9), in this c a se, tra nsforms a hetero clinic orbit in the phase p or tr ait ( U, dU /dX ) into a 8 -10 x 0 10 (a) 0 V 300 0 1.5 (b) x -10 10 0 u FIG. 4: [Color online] (a) Quasi-harmonic potential for M = 1, λ = 1 and (i) β = 0 . 5 (solid blue line) and (ii) β = 2 . 5 (d ashed red line). (b) Solutions of Eq. (2a) for α = 1, g 0 = − 1, M = 1 and (i) β = 0 . 5 (solid blue line) and (ii) β = 2 . 5 (dashed red line) hetero clinic connection, Eq . (23). On the o ther hand, any closed or bit U inside the hetero clinic lo o p o f the pha se po rtrait ( U, dU /dX ) provides a new heteroclinic connection of the original equation (2 a). Example 3. The last example is the so-calle d quas i-harmonic confinement V ( x ) ∼ x 2 . If we c ho ose b ( x ) = α/ p 1 + β x 2 , with α, β > 0, then, we o btain g ( x ) = g 0 α 3 (1 + β x 2 ) 3 / 2 , and V ( x ) = M (1 + β x 2 ) + 1 4 3 β x 2 − 2 β + 4 λ + 8 λβ x 2 + 4 λβ 2 x 4 (1 + β x 2 ) 2 , with M a p os itive constant. Although the expression of V ( x ) is co mplicated this p o tential V ( x ) is a qua si-harmonic po tent ial and sa tisfies V ( x ) ∼ x 2 for lar ge x . Moreov er V ( x ) is a ha rmonic p otential w ith a b ounded per turbative term (see Fig. 4(a)). As to the nonlinear term, it satisfies, g ( x ) ∼ x 2 for | x | ≪ 1, and g ( x ) ∼ x 3 for | x | ≫ 1. Using Eq. (14) w e get E = − α 2 M . T aking g 0 < 0, w e obtain the nonlinea r Schr¨ odinger equa tion with nonlinear attrac tive term, Eq. (15). As E < 0 and g 0 < 0, all the solutio ns of Eq. (15) are bo unded, (see Fig. 1(a)). If U ( X ) is one of these so lutions, it is clear that u ( x ) = b ( x ) 1 / 2 U ( X ( x )) , is a homoclinic orbit (bright soliton) of the original equation. In particula r, the s olution given by E q. (17a) is U 1 ( X ) = s 2 E g 0 1 cosh( p | E | X ) . As X ( x ) = x p 1 + β x 2 / (2 α ) + sinh − 1 ( √ β x ) / (2 α √ β ), we get u ( x ) = b ( x ) 1 / 2 U 1 ( X ( x )) . In Fig. 4(b), we draw the solutions o f Eq. (2a) for differen t v alues of the parameter β . In this example, we hav e used the solution U 1 ( X ) of Eq. (15). Another p oss ibilit y is to choo s e the clo sed p erio dic orbits inside the homo clinic lo op (see Fig. 1(a)) given b y U 4 ( X ) in Eq. (1 7d). W e note that, for this case, Eq. (16) satisfies H ≤ 0, as one can see in T able I. The ana lytical express ion of the closed per io dic o rbits outside the homo clinic lo op is given by U 3 ( X ), with k > 1 / √ 2. Such orbits satisfy H > 0 (see T able I). As X is a bijective map on the real line, b ( x ) is a p ositive function and U 3 ( X ) is a p er io dic function with infinite no des o n the r eal line , the function u ( x ) = b ( x ) 1 / 2 U 3 ( X ) also has infinite no des on the real line, as it is shown in Fig. 5. It is immediate to c heck that the solution u → 0 when x → ±∞ . Mor eov er, the zero es of φ accumulate for large v alues of x , since the distance b etw een tw o consecutive zero es is given by x n +1 − x n ∼ √ n + 1 − √ n . Other lo calize d solutions with infinite no des have b een studied in a differen t context in Ref. [7]. 9 3 -3 -10 x 0 10 u FIG. 5: Solution of Eq . (2a) with infinite n o des for α = 1, β = 2 . 5, g 0 = − 1 and k = 3 / 4 VI. ASYMMETRIC MODES OF THE INLSE In the previous sectio n, we hav e explored the case C = 0 in the canonica l transformation (9). In this case, the original equatio n (2a) is r educed to a No nlinear Schr¨ odinger E quation. In this section, we will study the case C > 0. If we take g 0 < 0 and E > 0, the resulting equa tion is d 2 U dX 2 + 2 C dU dX + | g 0 | U 3 + E U = 0 . (24) In gener al, this equation is not int egra ble and the energy is not a co nserved quantit y . How ev er, exa ct solutions of E q. (24) can b e constructed a nalytically in pa rticular cases . An exa ct integrabilit y condition w as given in [22] E = 8 9 C 2 . In that case a family o f exac t analytical solutions of Eq. (2 4) is given b y the expression U n ( X ) = µ n p 2 | g 0 | e − B X sn  µ n B (1 − e − B X ) , √ 2 / 2  dn  µ n B (1 − e − B X ) , √ 2 / 2  , n = 1 , 2 , 3 , ... where µ n and B are re la ted to the boundary conditions of the problem . W e are g oing to solve Eq. (2a ), using the s olutions of Eq. (24). Cho o s ing b ( x ) = cosh( x ) and using Eqs . (19) and (14), we can calcula te the p otential V ( x ): V ( x ) = λ + 1 / 4 +  1 4 + C 2 9  1 cosh 2 ( x ) . The nonlinea r term is g ( x ) = g 0 cosh − 3 ( x ) e − 2 C X ( x ) . So, w e can calculate the solutions of E q. (2a) for the case C 6 = 0 and to compare them with the so lution obtained for the cas e C = 0, ex ample 1. In this w ay , we can construct our canonical transforma tio n b y using Eqs. (9) and obtain cos X ( x ) = − tanh x. Then, 0 ≤ X ≤ π and using the bo undary co nditions for u , lim | x |→∞ u ( x ) = 0 , one has to impose U (0) = U ( π ) = 0. Using these b oundar y conditions, we obta in the v alue of the amplitude µ n as a function of the v alue of the integer n µ n = 4 C K ( √ 2 / 2) 3(1 − e − 2 C π/ 3 ) n, and B = 2 C / 3. Thus, the solutions of Eq. (24) are U n ( X ) = µ n p 2 | g 0 | e − 2 C X / 3 sn  2 nK ( √ 2 / 2)(1 − e − 2 C X / 3 ) / (1 − e − 2 C π/ 3 ) , √ 2 / 2  dn  2 nK ( √ 2 / 2)(1 − e − 2 C X / 3 ) / (1 − e − 2 C π/ 3 ) , √ 2 / 2  , n = 1 , 2 , 3 , ... 10 16 0 -20 0 x 20 (a) u (b) 40 -40 -20 0 20 x u 80 (c) -60 -20 0 x 20 u FIG. 6: Asymmetric solutions of Eq. (2a) with C = 4 for (a) n = 1, (b) n = 2 and (c) n = 3. where K ( k ) is the elliptic in tegral of the first kind, K ( k ) = Z π / 2 0 dθ q 1 − k 2 sin 2 ( θ ) . Then, the solutions of Eq. (2a) are u n ( x ) = b 1 / 2 ( x ) e C X ( x ) U n ( X ( x )) , n = 1 , 2 , ... (25) By using L’Hopital’s r ule, u n ( x ) → 0 when | x | → ∞ in (25). So, the solutio ns (25) a r e lo calized solutions o f our problem as it can be seen in Fig. 6. These so lutions are asymmetric solutions of Eq. (2a). Mor eov er, ea ch of those solutions has exac tly n − 1 zer o es. In Fig. 6 , w e plot some o f them corresp onding to n = 1 , 2 , 3 . The pictur e in Fig. 6(a) s hows clearly the difference betw een the p ositive solution given by (25), for C 6 = 0, and the p ositive so lution plotted in Fig. 2(b) and given by Eq. (21 ), for C = 0. VII. CONCLUSIONS In this pap er, we hav e used the metho d of Lie symmetries to find exact s olutions o f the INLSE. W e ha ve int ro duced the general framework of the Lie’s theory a nd pres ented differ ent ex amples as application to the theory . By us ing the qualitative theory o f the dyna mical s y stems, we can show the prop erties o f the solutions of the INLSE and to classify such so lutions. Finally , we hav e calculated a symetric solitons o f the inhomogeneous nonlinear Schr¨ odinger eq ua tion. 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