Reflection and Splitting of Channel-Guided Solitons in Two-dimensional Nonlinear Schrödinger Equation
aa r X i v : . [ n li n . PS ] S e p Reflection and Splitting of Channel-Guided Solitons in Two-dimensional NonlinearSchr¨odinger Equation
Hidetsugu Sakaguchi and Yusuke Kageyama
Department of Applied Science for Electronics and Materials,Interdisciplinary Graduate School of Engineering Sciences,Kyushu University, Kasuga, Fukuoka 816-8580, Japan
Solitons confined in a channel are studied in the two-dimensional nonlinear Schr¨odinger equation.When a channel branches into two channels, a soliton is split into two solitons, if the initial kineticenergy exceeds a critical value. The branching point works as a pulse splitter. If it is below thecritical value, the soliton is reflected. The critical kinetic energy for splitting is evaluated by avariational method. The variational method can be applied in the design of channel systems withreduced reflection.
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The one-dimensional nonlinear Schr¨odinger equation is a typical soliton equation. Optical solitons in optical fibersare described by the nonlinear Schr¨odinger equation.[1] The Bose-Einstein condensates (BECs) can be described bythe Gross-Pitaevskii equation, which is equivalent to the nonlinear Schr¨odinger equation with a potential term.[2]Bright and dark solitons have been observed in several experiments of the Bose-Einstein condensates.[3, 4] The one-dimensional nonlinear Schr¨odinger equation has been sufficiently studied, but solitons in two- or three-dimensionalnonlinear Schr¨odinger equations are not so intensively studied. We studied solitons in guided channels in the two-dimensional nonlinear Schr¨odinger equation.[5] In optical systems, refraction-index-guiding channels are used to con-fine optical pulses. Matter-wave solitons in BECs are created in cigar-shaped traps, which work as guiding channels.Solitons can propagate along the guiding channel with an arbitrary velocity if the guiding channel is uniform and thenorm of the soliton is below a critical value for the collapse.In this paper, we study the motion of solitons in inhomogeneous guiding channels. In particular, we study themotion of a soliton when a channel branches into two channels. Such a branching point is important as a pulsesplitter in optical systems and a beam splitter for atomic waves.[6] There has been no direct numerical simulationof the splitting of two-dimensional solitons in the branching waveguide up to now. We will perform a variationalapproximation using the Lagrangian to understand the numerical results. The approximation is a generalizationapplied to the one-dimensional nonlinear Schr¨odinger equation under an external potential.[7]The model equation is written as i ∂φ∂t = − ∇ φ − | φ | φ + U ( x, y ) φ, (1)where U ( x, y ) denotes the potential, which represents a channel. For a straight channel, U ( x, y ) = − U for − x ≤ x ≤ x and U ( x, y ) = 0 for other regions. The width of the channel is denoted as 2 x and the depth of the potentialis denoted as U . A stationary solution to eq. (1) can be numerically obtained from the time evolution of theGinzburg-Landau-type equation with a feedback term: ∂φ∂t = 12 ∇ φ + | φ | φ − U ( x, y ) φ + µφ,dµdt = α ( N − N ) , (2)where µ is an additional variable corresponding to the chemical potential, N = R R | φ | dxdy is the total norm, and N is the target value of the total norm. By a long-time evolution of eq. (2), a two-dimensional soliton with norm N is obtained as an attractor of the dynamical system. Figure 1(a) displays a 3D plot of φ for N = 5 , U = 5, and x = 3. Figure 1(b) shows the profile | φ ( x p , y ) | in the section of x = x p = 0, and Fig. 1(c) shows | φ ( x, y p ) | in thesection of y = y p , where ( x p , y p ) is the peak position of the soliton solution.The Lagrangian corresponding to the nonlinear Schr¨odinger equation (1) is expressed as L = Z Z [(1 / { i ( ∂φ/∂t ) φ ∗ − i ( ∂φ ∗ /∂t ) φ } − (1 / |∇ φ | + 1 / | φ | − U ( x, y ) | φ | ] dxdy. (3)If φ is approximated as φ = A sech( y/a ) exp {− x / (2 b ) } exp( − iµt ), the Lagrangian is calculated as L e = N {− b − a + N √ πab + U erf( x /b ) } , (4) xy | (cid:131) (cid:211) | | (cid:131) (cid:211) | (a) (b) (c) x FIG. 1: (a) 3D plot of | φ | at x = 3 , U = 5, and N = 5. (b) Profile of | φ | in the section of x = 0. The dashed curve isan approximate solution obtained by the variational method, which well overlaps the numerical result. (c) Profile of | φ | in thesection of y = y p , where (0 , y p ) is the peak position of | φ ( x, y ) | shown in (a). The dashed curve is the approximate solutionobtained by the variational method. where N = R R | φ | dxdy = 2 A √ πab is the total norm and erf( x ) = 2 / √ π R x exp( − z ) dz is the error function. Thevariational principle ∂L e /∂a = ∂L e /∂b = 0 yields coupled equations for a and b as a = 2 √ πb/N, / − N / (24 π ) + 2 U / √ π exp( − x /b )( − x b ) = 0 . (5)The approximate values of a = 1 . , b = 1 . A = 0 .
876 are obtained by the variational method for N = 5 , U = 5, and x = 3. The dashed curve in Fig. 1(b) denotes φ = A sech { ( y − ξ ) /a } where ξ = 20, and thedashed curve in Fig. 1(c) denotes φ = A exp {− x / (2 b ) } . The variational method is a good approximation for thetwo-dimensional soliton in the channel.Next, we study a problem in which the channel width changes in the y -direction as one of the simplest inhomogeneoussystems. We assume that the channel width x ( y ) becomes narrow as x ( y ) = x for y < y , x ( y ) = x + ( x − x ) × ( y − y ) / ( y − y ) for y < y < y , and x ( y ) = x for y > y . The center line of the channel is fixed to x = 0.We have performed direct numerical simulation from the initial condition φ ( x, y ) = φ ( x, y ) exp( iky ), where φ ( x, y )is the stationary solution shown in Fig. 1 for N = 5 , x = 3, and U = 5, and k is the initial wave number in the y direction. If the channel width is constant, the two-dimensional soliton propagates with the velocity v y = k . Thechannel width changes smoothly, since y − y is much larger than x − x . The numerical simulation was performedby the pseudospectral method with 128 × × k = 0 . , x = 3 , x = 2 , y = 40, and y = 100. The initial peak position of the soliton is (0 , k = 0 .
74 for the same channel. The solitoncannot penetrate into the narrow channel and it is reflected. The threshold wave number k c is evaluated as k c = 0 . k c for various values of x for x = 3 , U = 5 , y = 40 and y = 100. The rhombi in Fig. 2(c) show the relation of k c vs x by direct numerical simulation. That is, k c decreaseswith x .To understand this phenomenon, we performed a variational method by assuming φ ( x, y, t ) = A sech { ( y − k c x (c) t(cid:13) y(cid:13) 0(cid:13)50(cid:13)100(cid:13)150(cid:13)200(cid:13)0(cid:13) 40(cid:13) 80(cid:13) 120(cid:13) 160(cid:13) t(cid:13) y(cid:13) (a) (b) FIG. 2: (a) Time evolution of | φ ( x, y, t ) | in the section of x = 0 at k = 0 .
77. (b) Time evolution of | φ ( x, y, t ) | in the section of x = 0 fat k = 0 .
74. (c) Critical value k c for reflection. The dashed curve denotes the critical curve estimated by the variationalmethod. ξ ( t )) /a } exp {− x / (2 b ) } exp { ip ( y − ξ ( t )) − iµt } . For the assumption, the effective Lagrangian is expressed as L eff = N { pξ t ( t ) − p / − b − a + N √ πab + U erf( x /b ) } . (6)The variation principle d/dt ( ∂L eff /∂ξ t ) = ∂L eff /∂ξ and ∂L eff /∂p = 0 yield dξ ( t ) dt = p, dpdt = − ∂U eff ∂ξ , (7)where U eff = 1 / (4 b ) + 1 / (6 a ) − N/ (6 √ πab ) − U erf( x /b ). The parameters a and b are evaluated using thevariational principle at each point y = ξ ( t ). Equation (7) is equivalent to Newton’s equation of motion with theeffective potential U eff ( y ), which increases monotonically between y and y . The threshold value k c of the wavenumber k is evaluated from k c / U eff ( x ) − U eff ( x ). The dashed curve in Fig. 2(c) shows the threshold valueevaluated by the variational method. The agreement with the direct numerical simulation is fairly good, but thetheoretical curve is slightly larger than the numerically obtained values.The main problem in this study is a problem in which one channel branches into two channels. The potential U ( x, y ) is assumed to be U ( x, y ) = − U for − x < x < x when y < y , U ( x, y ) = − U for − x < x < − x and x < x < x , U ( x, y ) = − U for − x < x < x when y < y < y , and U ( x, y ) = − U for − x < x < − x and x < x < x when y > y , where x = x + ( x − x ) × ( y − y ) / ( y − y ), x = x × ( y − y ) / ( y − y ),and U = U × ( y − y ) / ( y − y ). U ( x, y ) = 0 in the other region. We introduced the U component in thepotential U ( x, y ) for the potential to change smoothly in the y -direction. Figure 3(a) shows the channel region with U ( x, y ) = − U for x = 3 , x = 3 , x = 6 , y = 40, and y = 100. The initial condition is the same as before: φ ( x, y, t ) ∼ A sech { ( y − /a } exp {− x / (2 b ) } exp { ik ( y − } . Figure 3(b) shows the time evolution of | φ (0 , y, t ) | in the section of x = 0 at k = 0 .
4. The other parameters are U = 5 and N = 5. The soliton is reflected at thebranching point. Figure 3(c) shows the time evolution of | φ ( x, y p ) | in the y -section including the peak position of themodulus | φ | at k = 0 .
8. The soliton with a single peak is split into two solitons propagating along the two channels.We investigated the threshold value k c for the transition from reflection to splitting, and obtained k c ∼ .
74. Figure3(d) shows the trajectory at the peak position for k = 0 . x, y ) space. The peak position is located at x = 0for y < . y > .
8. The peak positions continuously separate away from x = 0.For the variational method applied to this splitting process, φ is assumed to be φ = ( A/ { ( y − ξ ( t )) /a } [exp {− ( x − η ) / (2 b ) } + exp {− ( x + η ) / (2 b ) } ] exp { ip ( y − ξ ( t )) − iµt } . The variational principle yields,for this ansatz, dξ ( t ) dt = p, dpdt = − ∂U eff ∂ξ , (8)where U eff = 16 a + 14 b − ηe − η /b b (1 + e − η /b ) − N √ πab e − η /b + 4 e − η / (2 b ) (1 + e − η /b ) , − U erf { ( x − η ) /b } + erf { ( x + η ) /b } + erf { ( η − x ) /b } − erf { ( η + x ) /b } e − η /b ) , y (cid:13) x(cid:13) 0(cid:13)50(cid:13)100(cid:13)150(cid:13)0(cid:13) 20(cid:13) 40(cid:13) 60(cid:13) 80(cid:13)y(cid:13) t t(cid:13) x(cid:13) 20(cid:13)40(cid:13)60(cid:13)80(cid:13)100(cid:13)120(cid:13)-6(cid:13) -4(cid:13) -2(cid:13) 0(cid:13) 2(cid:13) 4(cid:13) 6(cid:13) y (cid:13) x(cid:13) (a) (b) (c) (d) FIG. 3: (a) Branching channel with x = 3 , x = 3 , x = 6 , y = 40, and y = 100. (b) Time evolution of | φ ( x, y, t ) | at thesection of x = 0 for k = 0 .
4. (c) Time evolution of | φ ( x, y, t ) | at the x -section passing through the peak position ( x p , y p ) for k = 0 .
8. (d) Trajectory of the peak position ( x p ( t ) , y p ( t )) at k = 0 .
8. The dashed curve is obtained by the variational method. − U e − η /b { erf( x /b ) − erf( x /b ) } e − η /b ) , − U erf { ( x + η ) /b } − erf { ( η − x ) /b } + 2 e − η /b erf( x /b )2(1 + e − η /b ) , (9)where x = 0 , x = x for y < y , x = x , and x = x for y < y < y , and x = x , and x = x for y > y .In Fig. 3(d), the peak positions evaluated by the variational method are drawn with dashed curves. The trajectory iswell approximated by the variational method. The threshold value obtained by the variational method is k c = 0 . k c ∼ .
74, but the theoretical estimate is slightly larger than the numericalone, which is similar to the case of the tapering channel.The reflection of the pulse at the branching point might be undesirable for applications to pulse splitting. We candesign a system including a branching point with reduced reflection using the variational method. The difference U eff ( y ) − U eff ( y ) of the effective potential was the origin of the reflection. The reflection is expected to disappearor be reduced, if the channel system is designed, in which the difference | U eff ( y ) − U eff ( y ) | is minimized. We cancontrol both the channel width x − x and the depth of potential U ( y ). We have designed the value of U ( y ) usingan evolution equation: dU ( y ) /dτ = − β ( U eff ( y ) − U eff ( y )) with β > U eff ( y ) approaches U eff ( y ) at each point y > y . The U component is assumed to be the same as before: U ( y ) = 5 × ( y − y ) / ( y − y ). Figure 4(a) showsthe designed value of U ( y ) for x = 3 , x = 2 , y = 40, and y = 100. Figure 4(b) shows the time evolution of thesoliton with the initial wave number k = 0 .
2. Even for this small wave number, the reflection is suppressed and thesoliton is split into two solitons, that penetrate into the two channels.In summary, we have studied the motion of two-dimensional solitons confined in guided channels. We have foundthat solitons with small wave number are reflected at the branching point of the channel. The variational methodcan predict the reflection and splitting phenomena fairly well. The method can be applied in the design of channelsystems with reduced reflection. [1] L. F. Mollenauer and J. P. Gordon:
Solitons in Optical Fibers (Academic Press, San Diego, 2006).[2] C. J. Pethick and H. Smith:
Bose-Einstein Condensates in Dilute Gases (Cambridge University Press, Cambridge, 2002).[3] J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson,W.P. Reinharts, S. L. Rolston, B. I. Schneider, and W. D. Phillips: Science (2000) 97.[4] L. Kaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon: Science (2002)1290.[5] H. Sakaguchi and B. A. Malomed: Phys. Rev. A (2007) 063825.[6] D. Cassettari, B. Hessmo, R. Folman, T. Maier, and J. Schmiedmayer: Phys. Rev. Lett. (2000) 5483.[7] H. Sakaguchi and M. Tamura: J. Phys. Soc. Jpn. (2004) 503. U y t x (a) (b) FIG. 4: (a) Depth of potential designed for no reflection by the variational method. (b) Time evolution of | φ ( x, y, t ) | forthe designed potential in the x -section passing through the peak position at k = 0 .
2. (c) Time evolution of | φ ( x, y, t ) | for thedesigned potential at k = 0 . xx