Exact solutions of the modified Gross-Pitaevskii equation in `smart' periodic potentials in the presence of external source
aa r X i v : . [ n li n . PS ] D ec Exact solutions of the modified Gross-Pitaevskii equation in‘smart’ periodic potentials in the presence of external source
Thokala Soloman Raju and Prasanta K Panigrahi Indian Institute of Science Education and Research (IISER), Salt Lake, Kolkata 700106,IndiaE-mail: solomonr [email protected], [email protected]
Abstract.
We report wide class of exact solutions of the modified Gross-Pitaevskii equation(GPE) in ‘smart’ Jacobi elliptic potentials: V ( ξ ) = − V sn( ξ, m), V ( ξ ) = − V cn( ξ, m), and V ( ξ ) = − V dn( ξ, m) in the presence of external source. Solitonlike solutions, singularsolutions, and periodic solutions are found using a recently developed fractional transform: ρ ( ξ ) = A + Bf + Df , where f is the respective Jacobi elliptic function and the amplitude parameters A , B , and D nonzero . These results generalize those contained in (Paul T, Richter K andSchlagheck P 2005
Phys. Rev. Lett. , 020404) for nonzero trapping potential. xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source
1. Introduction
In a mean-field approximation, the dynamics of a dilute-gas Bose-Einstein condensate (BEC)can be captured by the cubic nonlinear Schr¨odinger equation (NLSE) with a trapping potential[1, 2, 3]. The various traps which are used to contain the BEC have spurred the solutions ofthe NLSE with new potentials [4, 5]. We consider the mean-field model of a quasi-one-dimensional BEC trapped in a ‘smart’ potential in the presence of an external source[6] i ∂ψ∂ t = " − ∂ ∂ x + V ( ξ ) + g | ψ | ψ + K exp( i χ ( ξ ) − i ø t ) , (1)where ψ ( x , t ) represents the macroscopic wave function of the condensate and V ( ξ ) is anexperimentally generated macroscopic potential. The parameter g indicates the strength ofatom-atom interactions and it alone decides whether Eq. (1) is attractive ( g = −
1, focussingnonlinearity) or repulsive ( g =
1, defocussing nonlinearity). Here, K and ø are real constantsrelated to the source amplitude and the chemical potential, respectively, and χ ( ξ ) is a realfunction of ξ = α ( x − v t ), α and v being two real parameters. In the field of nonlinear optics,Eq. (1) may describe the evolution of the local amplitude of an electromagnetic wave in thespatial domain, in a two-dimensional waveguide where t becomes the propagation distanceand x becomes the retarded time, and the system is driven by an external plane pump wave.The Jacobi elliptic potential may describe a transverse modulation of the refractive index inthe waveguide.As is well-known, Eq. (1) is not integrable if KV ,
0, and only small classes of explicitsolutions can most likely exist. For V ( ξ ) =
0, Eq. (1) is a cubic NLSE with a source, andexact rational solutions using a fractional transform are found in Ref. [7]. And in Ref. [11]periodic solutions without source have been reported. More recently, in Ref.[8], a class ofexact solutions of Eq. (1) for V ( ξ ) = − V sn ( ξ, m ) have been reported. In particular, therational solutions of the fractional transform: ρ ( ξ ) = A + B f + D f , where f = sn have been reportedfor B =
0. This is due to the form of the trapping potential. Nonetheless, in the present paperwe find rational solutions of the type ρ ( ξ ) = A + B f + D f , where f being the respective Jacobi ellipticfunction with all the amplitude parameters A , B , and D nonzero . These results generalizethose contained in Ref [6] for nonzero trapping potential. The choice of a smart potential V ( ξ ) allows one to construct a large class of exact solutions, as done in a number of works forthe cubic GP equations [9, 10, 11]. In the present work, we consider three di ff erent potentialsin the GP equation: V ( ξ ) = − V sn( ξ, m), V ( ξ ) = − V cn( ξ, m), and V ( ξ ) = − V dn( ξ, m)in the presence of external source, and find exact travelling wave solutions of Eq. (1) with K ,
0. The choice of these three di ff erent ‘smart’ potentials is motivated by the followingfacts. Firstly, the potential V ( ξ ) = − V sn( ξ, m) in the limit m → V ( ξ ) = − V sin( ξ )which is similar to the standard optical lattice potential [12, 13]. Secondly, the choice ofthe potential V ( ξ ) = − V dn( ξ, m) in GP equation mimics[14] the harmonic potential that wasused to achieve BEC experimentally. The third potential, we hope it is relevant to the availableexperimental conditions to achieve BEC.
2. EXACT SOLUTIONS OF THE GPE IN ‘SMART’ PERIODIC POTENTIAL WITHSOURCE
The travelling wave solutions of Eq. (1) with potential V ( ξ ) are taken to be of the form ψ ( x , t ) = ρ ( ξ ) e i χ [ α ( x − v t )] − i ø t . Inserting this expression for ψ ( x , t ) in Eq. (1) and separating the xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source χ ′ = vα + C αρ , (2)where C is a constant of integration. In order that the external phase be independent of ψ , weconsider only solutions with C =
0, to obtain ρ ′′ + ( v + α ) ρ − gα ρ − V ( ξ ) α ρ − K α = . (3)Below we consider three di ff erent ‘smart’ Jacobi elliptic potentials [15] and find exactsolutions.Case(I):- V ( ξ ) = − V sn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α sn ρ − K α = . (4)Substituting ρ ( ξ ) = A + B sn( ξ, m ) (5)in Eq.(4) and equating the coe ffi cients of equal powers of sn( ξ, m ) result in relations amongthe solution parameters A , and B , and the equation parameters V , g , K , α , and ø. We findthat α = + v + m , (6) A = V α √ g m , B = s m α g . (7)From Eq. (7) it follows that V > g > V ( ξ ) = − V cn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α cn ρ − K α = . (8)Substituting ρ ( ξ ) = A + B cn( ξ, m ) (9)in Eq.(8) and equating the coe ffi cients of equal powers of scn( ξ, m ) result in relations amongthe solution parameters A , and B , and the equation parameters V , g , K , α , and ø. We findthat α = + v − m , (10) A = V α √− g m , B = s − m α g . (11)From the positivity of α we conclude that cn solutions exist for V > g <
0. Thecondition g < V ( ξ ) = − V dn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α dn ρ − K α = . (12)Substituting ρ ( ξ ) = A + B dn( ξ, m ) (13) xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source ffi cients of equal powers of dn( ξ, m ) result in relations amongthe solution parameters A , and B , and the equation parameters V , g , K , α , and ø. We findthat α = + v m − , (14) A = V α √− g , B = s − α g . (15)Here we conclude that dn solutions exist only for V > g <
3. Rational solutions
In order to obtain Lorentzian-type of solutions of Eq. (3) we use a fractional transform ρ ( ξ ) = A + B f + D f (16)where f is the respective Jacobi elliptic functions. Again we obtain the Lorentzian-type ofsolutions of Eq. (3) for three di ff erent ‘smart’ potentials.Case(I):- V ( ξ ) = − V sn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α sn ρ − K α = . (17)Substituting ρ ( ξ ) = A + B sn ( ξ, m )1 + D sn( ξ, m ) (18)in Eq.(17) and equating the coe ffi cients of equal powers of sn( ξ, m ) will yield the followingconsistency conditions.2 B + A D + Γ A − gα A − K α = , (19)2 mB D − gα B = , (20)6 mB D + V α B D = , (21)6 mB + B D ( Γ − m − − gα A B + V α B D = , (22) − mA D + B D (2 Γ − m − + V α A D + V α B − K α D = , (23) − B (1 + m ) + A D ( Γ − m − + Γ B − gα A B + V α A D − K α D = , (24) A D (2 Γ + m + + V α A − K α D = . (25)From the above consistency conditions we obtain the following relations. A = mK m ( m + α + m Γ α − V α , (26) B = − α m / V g / , D = − m α V , (27) xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source Γ = v + α . Here, we would like to emphasize that these results generalize thosecontained in Ref [6], for nonzero trapping potential. This stems from the fact that the constant B in expression (18) is nonzero, which follows from the choice of our ‘smart’ potential in Eq.(1).Case(II):- V ( ξ ) = − V cn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α cn ρ − K α = . (28)Substituting ρ ( ξ ) = A + B cn ( ξ, m )1 + D cn( ξ, m ) (29)in Eq.(28) and equating the coe ffi cients of equal powers of cn( ξ, m ) will yield the followingconsistency conditions.2 B (1 − m ) + A D (1 − m ) + Γ A − gα A − K α = , (30) − mB D − gα B = , (31) − mB D + V α B D = , (32) − mB + B D ( Γ + m − − gα A B + V α B D = , (33)2 mA D + B D (2 Γ + m − + V α A D + V α B − K α D = , (34) − B (1 − m ) + A D ( Γ + m − + Γ B − gα A B + V α A D − K α D = , (35) A D (2 Γ − m + + V α A − K α D = . (36)From the above consistency conditions we obtain the following relations A = mK m (1 − m ) α + m Γ α + V α , (37) B = α mV r − m g , D = m α V . (38)Case(III):- V ( ξ ) = − V dn( ξ, m ). Eq. (3) reads as ρ ′′ + ( v + α ) ρ − gα ρ + V α dn ρ − K α = . (39)Substituting ρ ( ξ ) = A + B dn ( ξ, m )1 + D dn( ξ, m ) (40)in Eq.(39) and equating the coe ffi cients of equal powers of dn( ξ, m ) will yield the followingconsistency conditions.2 A D ( m − + Γ A − gα A − K α = , (41) − B D − gα B = , (42) − B D + V α B D = , (43) xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source È Ψ È Figure 1.
Singular solitary wave solution for α = V =
1, and g = − B − B D + B D ( Γ − m + − gα A B + V α B D = , (44)2 A D + B D (2 Γ − m + + V α A D + V α B − K α D = , (45) − B ( m − + B D + A D ( Γ − m + + Γ B − gα A B + V α A D − K α D = , (46) A D (2 Γ + m − + V α A − K α D = . (47)From the above consistency conditions we obtain the following relations A = K α (2 Γ + m − + V α , (48) B = α V s − α g , D = α V . (49) From the consistency conditions that arise from the first two ‘smart’ periodic potentials, weconclude that the limit m = A , B , and D will bezero. On the other hand, for V ( ξ ) = − V dn( ξ, m ) case, only flat background solutions will bepossible for m = Here, in this subsection, we describe the solitonlike solutions that are obtained from thesolutions in sn( ξ, m ) and cn( ξ, m ) in the limit m =
1, in detail. In the limit m = V ( ξ )becomes an array of well separated kink-type of potential barriers: V ( ξ ) = − V tanh( ξ ). Thenwe have the following relations A = K Γ + α − V α , (50) B = α V g / , D = − α V . (51) xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source È Ψ È Figure 2.
Non-singular solitary wave solution for α = V = − g = − K = / And the strength of the source is K = V [6( Γ + α − V ]108 α g / Γ − α − V V . As a special case, if we set α = V =
1, then we get A = K Γ+ , B = / √ g , and D = −
3. This results in a solitonlike solution ρ ( ξ ) = A + B tanh ( ξ )1 − ξ ) . (52)This set corresponds to the singular solution for repulsive case i.e., g >
0. The singularityof the pulse profile may correspond to the beam power exceeding material breakdown dueto self-focussing [16, 17, 18, 19]. Figure (1) depicts a surface plot of this solution for theparameter values given in the figure caption.Another interesting solitonlike solution is obtained from the solution in cn( ξ, m ) for m =
1. In the limit m = V ( ξ ) becomes an array of well separated secant hyperbolicpotential barriers: V ( ξ ) = − V sech( ξ ).Then we have the following relations A = K Γ − α + V α , (53) B = α V p − /g, D = α V . (54)As a special case, if we set α = V = − g = − K = / A = Γ+ , B = − (1 / D = − (1 / ρ ( ξ ) = A + B sech ( ξ )1 − . ξ ) . (55)This set corresponds to the non-singular solution for attractive case i.e., g <
0. The same hasbeen depicted in Fig. 2 for the parameter values given in in the figure caption.
4. Conclusions
In conclusion, we have shown the existence of wide class of exact solutions for the modifiedGP equation in ‘smart’ periodic potentials with an external source. The Lorentzian-type of xact solutions of the modified Gross-Pitaevskii equation in ‘smart’ periodic potentials in the presence of external source ff erence method [7], as the much used numericaltechniques based on fast Fourier transform (FFT) requires the FFT of the source, which iscostly. Acknowledgements
TSR would like to dedicate this paper to the fond memory of his father Shri. Thokala RatnaRaju, for his love and encouragement.
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