On Asymptotic Dynamical Regimes of Manakov N -soliton Trains in Adiabatic Approximation
aa r X i v : . [ n li n . S I] J u l On Asymptotic Dynamical Regimes of Manakov N -solitonTrains in Adiabatic Approximation Vladimir S. Gerdjikov , Michail D. Todorov National Research Nuclear University MEPHI, 115409 Moscow, Russian FederationInstitute of Mathematics and Informatics Bulgarian Academy of Sciences, 1113 Sofia, BulgariaInstitute for Advanced Physical Studies, New Bulgarian University, 1618 Sofia, Bulgaria,email: [email protected] Dept of Mathematics and Statistics, San Diego State University, 92182-0005 San Diego, CA, USAFaculty of Applied Math. and Informatics, Technical University of Sofia, 1000 Sofia, Bulgaria,e-mail: mtod@tu-sofia.bg
Abstract
We analyze the dynamical behavior of the N -soliton train in the adiabatic approximation ofthe Manakov model. The evolution of Manakov N -soliton trains is described by the complex Todachain (CTC) which is a completely integrable dynamical model. Calculating the eigenvalues ofits Lax matrix allows us to determine the asymptotic velocity of each soliton. So we describesets of soliton parameters that ensure one of the two main types of asymptotic regimes: thebound state regime (BSR) and the free asymptotic regime (FAR). In particular we find explicitdescription of special symmetric configurations of N solitons that ensure BSR and FAR. Wefind excellent matches between the trajectories of the solitons predicted by CTC with the onescalculated numerically from the Manakov system for wide classes of soliton parameters. Thisconfirms the validity of our model. Keywords:
Manakov model, soliton interactions, adiabatic approximations complex Todachain
The solitons and their interactions find numerous applications of in many areas of today nonlinearphysics, such as hydrodynamics, nonlinear optics, Bose-Einstein condensates, etc. [28, 2, 27, 3, 34, 1, 7].This explains why it is important to study their interactions. The first results on soliton interactionswere obtained by Zakharov and Shabat [36, 35]. There they proved that the nonlinear Schr¨odingerequation iu t + 12 u xx + | u | u ( x, t ) = 0 . (1)1an be integrated by the inverse scattering method (ISM). Then they constructed the N -solitonsolution of (1) and calculated their limits for t → ∞ and t → −∞ , assuming that all solitons havedifferent velocities. Comparing the asymptotics they concluded that the soliton interactions are purelyelastic, i.e., no new solitons can be created. In addition the solitons preserve their amplitudes andvelocities, and the only effect of the interactions are relative shifts of the center of masses and phases.Later Karpman and Solov’ev proposed another approach to the soliton interactions based on theadiabatic approximation [26, 25]. They proposed to model the N -soliton trains of the NLS eq. (1).By N -soliton train they meant a solution of the NLS eq. with initial condition: u ( x, t = 0) = N X k =1 ~u k ( x, t = 0) , u k ( x, t ) = 2 ν k e iφ k cosh( z k ) ,z k = 2 ν k ( x − ξ k ( t )) , ξ k ( t ) = 2 µ k t + ξ k, ,φ k = µ k ν k z k + δ k ( t ) , δ k ( t ) = 2( µ k + ν k ) t + δ k, . (2)The adiabatic approximation holds true if the soliton parameters satisfy [26]: | ν k − ν | ≪ ν , | µ k − µ | ≪ µ , | ν k − ν || ξ k +1 , − ξ k, | ≫ , (3)where ν = N P Nk =1 ν k , and µ = N P Nk =1 µ k are the average amplitude and velocity respectively. Infact we have two different scales: | ν k − ν | ≃ ε / , | µ k − µ | ≃ ε / , | ξ k +1 , − ξ k, | ≃ ε − . In this approximation the dynamics of the N -soliton train is described by a dynamical system for the4 N soliton parameters. What Karpman and Solov’ev did was to derive the dynamical system for thetwo soliton interactions: a system of 8 equations for the 8 soliton parameters. They were able also tosolve it analytically.Later their results were generalized to N -soliton trains [17, 24, 16]. The corresponding model canbe written down in the form : dλ k dt = − ν (cid:0) e Q k +1 − Q k − e Q k − Q k − (cid:1) ,dQ k dt = − ν λ k , (4)where λ k = µ k + iν k and Q k = − ν ξ k + k ln 4 ν − i ( δ k + δ + kπ − µ ξ k ) ,ν = 1 N N X s =1 ν s , µ = 1 N N X s =1 µ s , δ = 1 N N X s =1 δ s . (5)Obviously the system (4) becomes the Toda chain with free ends for the complex variables Q k : d Q k dt = − ν dλ k dt = 16 ν (cid:0) e Q k +1 − Q k − e Q k − Q k − (cid:1) k = 2 , . . . , N − ,d Q dt = 16 ν e Q − Q , d Q N dt = − ν e Q N − Q N − . (6)2hich is known as the complex Toda chain (CTC).It is well known that the standard (real) Toda chain is an integrable system [31, 28, 9]. In thecase of (6), which is known as Toda chain with open ends, it was possible to write down its solutionsexplicitly [31]. An important fact is that these solutions depend analytically on their parameters andcan be easily generalized to the CTC.In fact some time ago a special configurations of soliton trains that are modeled by the real Todachain [4, 5]. To this end we must choose solitons with equal amplitudes (i.e., ν k = ν ), vanishing initialvelocities ( µ k = 0, and out-of phase δ k +1 − δ k = π . It is easy to see that under these assumptions Q k become real valued and (6) become the standard Toda chain.The adiabatic approach of Karpman and Solov’ev has a drawback: it is an approximate methodwhose precision is determined by ε . On the other hand it has the advantages: first, it is not limitedonly to solitons with different velocities, and second, it can take into account possible perturbationsof the NLS [17, 24, 16].Another important generalization of the NLS equation is known as the Manakov model [28] (vectorNLS): i~u t + 12 ~u xx + ( ~u † , ~u ) ~u ( x, t ) = 0 . (7)The corresponding vector N -soliton train is determined by the initial condition: ~u ( x, t = 0) = N X k =1 ~u k ( x, t = 0) , ~u k ( x, t ) = 2 ν k e iφ k cosh( z k ) ~n k ,z k = 2 ν k ( x − ξ k ( t )) , ξ k ( t ) = 2 µ k t + ξ k, ,φ k = µ k ν k z k + δ k ( t ) , δ k ( t ) = 2( µ k + ν k ) t + δ k, , (8)where the constant polarization vector ~n k is normalized by ~n k = (cid:18) cos( θ k ) e iγ k sin( θ k ) e − iγ k (cid:19) , ( ~n † k , ~n k ) = 1 , n X s =1 arg ~n k ; s = 0 . Therefore each Manakov soliton is parametrized by 6 parameters.It was natural to extend the Karpman-Solov’ev method to the Manakov model. The result isknown as the generalized CTC (GCTC) [10, 11, 14, 12]. Of course later the GCTC was also adaptedto treat the effects of several types of perturbations on solitons [13, 33, 19, 30, 32, 8].The advantage of the integrability of the CTC and GCTC is in the fact that knowing the initialset of soliton parameters one can predict the asymptotic regime of the soliton train [17, 24, 16]. Onthe other hand it is possible to find the set of constraints on the soliton parameters that would ensuregiven asymptotic regime. These constraints were derived and analyzed for 2 and 3-soliton trains;for larger number of solitons only fragmentary results such as the quasi-equidistant propagation ofsolitons [16].The aim of the present paper is to reinvestigate these results and to demonstrate several configura-tion of multisoliton trains for which one can predict that they will go into bound state regime (BSR)or into free asymptotic regime (FAR). In Section 2 we outline the derivation of the GCTC model, see3q. (16) below which now depends also on the polarization vectors ~n k and models the behavior ofthe N -soliton train of the vector NLS. We also formulate the Lax representation for the GCTC andexplain how it can be used to determine the asymptotic regime of the soliton train. In Section 3 weformulate two classes of explicit constraints on the soliton parameters that are responsible for BSRand FAR. The first class are generic conditions that ensure that the Lax matrix becomes either real orpurely imaginary. The second class are based on special explicit constraints on the soliton parametersthat make the eigenvalues of the Lax matrix proportional to each and easier to establish if they arereal ir purely imaginary. The Lagrangian of the vector NLS perturbed by external potential is: L [ ~u ] = Z ∞−∞ dt i h ( ~u † , ~u t ) − ( ~u † t , ~u ) i − H, H [ ~u ] = Z ∞−∞ dx (cid:20) −
12 ( ~u † x , ~u x ) + 12 ( ~u † , ~u ) (cid:21) . (9)Then the Lagrange equations of motion: ddt δ L δ~u † t − δ L δ~u † = 0 , (10)coincide with the vector NLS with external potential V ( x ).Next we insert ~u ( x, t ) = P Nk =1 ~u k ( x, t ) (see eq. (8)) and integrate over x neglecting all terms oforder ǫ and higher. In doing this we assume that ξ < ξ < · · · < ξ N at t = 0 and use the fact, thatonly the nearest neighbor solitons will contribute. All integrals of the form: Z ∞−∞ dx ( ~u † k,x , ~u p,x ) , Z ∞−∞ dx ( ~u † k , ~u p ) , (11)with | p − k | ≥ Z ∞−∞ dx ( ~u † k , ~u p )( ~u † s , ~u l ) , where at least three of the indices k, p, s, l have different values. In doing this key role play thefollowing integrals: J ( a ) = Z ∞−∞ dz e iaz z = πa aπ ,K ( a, ∆) ≡ Z ∞−∞ dz e iaz z cosh( z + ∆) = π (1 − e − ia ∆ )2 i sinh(∆) sinh( πa/ , (12)4hus after long calculations we obtain: L = N X k =1 L k + N X k =1 X n = k ± e L k,n , L k,n = 16 ν e − ∆ k,n ( R k,n + R ∗ k,n ) ,R k,n = e i ( e δ n − e δ k ) ( ~n † k ~n n ) , e δ k = δ k − µ ξ k , ∆ k,n = 2 s k,n ν ( ξ k − ξ n ) ≫ , s k,k +1 = − , s k,k − = 1 , (13)where L k = − iν k (cid:16) ( ~n † k,t , ~n k ) − ( ~n † k , ~n k,t ) (cid:17) + 8 µ k ν k dξ k dt − ν k dδ k dt − µ k ν k + 8 ν k ddt δ L δp k,t − δ L δp k = 0 , (15)where p k stands for one of the soliton parameters: δ k , ξ k , µ k , ν k and ~n † k . The corresponding systemis a generalization of CTC: dλ k dt = − ν (cid:16) e Q k +1 − Q k ( ~n † k +1 , ~n k ) − e Q k − Q k − ( ~n † k , ~n k − ) (cid:17) ,dQ k dt = − ν λ k , d~n k dt = O ( ǫ ) , (16)where again λ k = µ k + iν k and the other variables are given by (5). Now we have additional equationsdescribing the evolution of the polarization vectors. But note, that their evolution is slow, and inaddition the products ( ~n † k +1 , ~n k ) multiply the exponents e Q k +1 − Q k which are also of the order of ǫ .Since we are keeping only terms of the order of ǫ we can replace ( ~n † k +1 , ~n k ) by their initial values( ~n † k +1 , ~n k ) (cid:12)(cid:12)(cid:12) t =0 = m k e iσ k , k = 1 , . . . , N − . (17)We will consider most general form of the polarization vectors: | ~n k i = (cid:18) cos( θ k ) e iγ k sin( θ k ) e − iγ k (cid:19) , h ~n † k +1 | ~n k i = cos( θ k +1 cos( θ k ) e − i ( γ k +1 − γ k ) + sin( θ k +1 sin( θ k ) e i ( γ k +1 − γ k ) = ρ k e iσ k ,ρ k = cos ( γ k +1 − γ k ) cos ( θ k +1 − θ k ) + sin ( γ k +1 − γ k ) cos ( θ k +1 + θ k ) .σ k = − arctan (cid:18) tan( γ k +1 − γ k ) cos( θ k +1 + θ k )cos( θ k +1 − θ k ) (cid:19) ,a k = ν ρ k exp( − ν ( ξ k +1 − ξ k )) exp( − i ( δ k +1 − δ k − σ k + π ) / . (18)In our previous papers we considered configurations for which | ~n k i are real, i.e., γ k = 0. Note thatthe effect of the polarization vectors could be viewed as change of the distance between the solitonsand between the phases. 5he system (16) was derived for the Manakov system n = 2 by other methods in [18]. There theGCTC model it was tested numerically and found to give very good agreement with the numericalsolution of the Manakov model. However the tests were done only for real values of the polarizationvectors, i.e., all γ k = 0, k = 1 , . . . , N . Below we will take into account the effect of γ k onto thedynamical regimes of the solitons. We first briefly remind the main results concerning the CTC model [17, 24, 16, 14, 18]. The CTC iscompletely integrable model; it allows Lax representation L t = [ A.L ], where: L = N X s =1 ( b s E ss + a s ( E s,s +1 + E s +1 ,s )) , A = N X s =1 ( a s ( E s,s +1 − E s +1 ,s )) , (19)where a s = exp(( Q s +1 − Q s ) / b s = µ s,t + iν s,t and the matrices E ks are determined by ( E ks ) pj = δ kp δ sj . The eigenvalues of L are integrals of motion and determine the asymptotic velocities.The GCTC derived in [10, 11, 14, 18, 12] is also a completely integrable model. It allows Laxrepresentation just like the standard real Toda chain [9, 31, 29] ˜ L t = [ ˜ A. ˜ L ], where:˜ L = N X s =1 (cid:16) ˜ b s E ss + ˜ a s ( E s,s +1 + E s +1 ,s ) (cid:17) , A = N X s =1 (˜ a s ( E s,s +1 − E s +1 ,s )) , (20)where ˜ a s = m s e iσ s a s , b s = µ s + iν s . Like for the scalar case, the eigenvalues of ˜ L are integrals ofmotion. If we denote by ζ s = κ s + iη s (resp. ˜ ζ s = ˜ κ + i ˜ η s ) the set of eigenvalues of L (resp. ˜ L ) thentheir real parts κ s (resp. ˜ κ s ) determine the asymptotic velocities for the soliton train described byCTC (resp. GCTC). Thus, starting from the set of initial soliton parameters we can calculate L | t =0 (resp. ˜ L | t =0 ), evaluate the real parts of their eigenvalues and thus determine the asymptotic regimeof the soliton train. Regime (i) κ k = κ j (resp. ˜ κ k = ˜ κ j ) for k = j , i.e., the asymptotic velocities are all different. Thenwe have asymptotically separating, free solitons, see also [4, 17, 24, 16] Regime (ii) κ = κ = · · · = κ N = 0 (resp. ˜ κ = ˜ κ = · · · = ˜ κ N = 0), i.e., all N solitons move withthe same mean asymptotic velocity, and form a “bound state.” Regime (iii) a variety of intermediate situations when one group (or several groups) of particlesmove with the same mean asymptotic velocity; then they would form one (or several) boundstate(s) and the rest of the particles will have free asymptotic motion.
Remark 1
The sets of eigenvalues of L and ˜ L are generically different. Thus varying only thepolarization vectors one can change the asymptotic regime of the soliton train. Let us consider several particular cases.
Case 1 ~n = · · · = ~n N . Since the vector ~n is normalized, then all coefficients m ok = 1 and σ k = 0.Then the interactions of the vector and scalar solitons are identical.6 ase 2 ( ~n † s +1 , ~n s ) = 0. Then the GCTC splits into two unrelated GCTC: one for the solitons { , , . . . , s } and another for { s + 1 , s + 2 , . . . .N } . If the two sets of soliton parameters are suchthat both groups of solitons are in bound state regimes, then these two bound states. Case 3 h n † k +1 | ~n k i = m e iϕ – effective change of distance and phases of solitons. In this case we canrewrite ˜ a s = exp(( ˜ Q s +1 − ˜ Q s ) / Q s +1 − ˜ Q s = Q s +1 − Q s + ln m + iϕ , (21)i.e., the distance between any two neighboring vector solitons has changed by ln( m / ν ); sim-ilarly have the phases. N -Soliton Trains with N ≥ The asymptotic regimes for scalar solitons and for small values of N are known for long time now, see[17, 24, 16]. Obviously for N = 2 we have only two possibilities: BSR and FAR. For N = 3 for thefirst time there appears MAR when two of the solitons form a bound state while the third one goesaway off them. For N > N -soliton trains wasdeveloped in [10, 14, 18, 12]. Roughly speaking we have to use the characteristic polynomial of L N whose generic form is: P ( z ) = det( L N − z ) = N X k =0 p k ( ~a,~b ) z k = N Y k =1 ( z − z k ) . (22)Next we have to analyze the roots z k and formulate the conditions on the soliton parameters for whichi) Re z k = 0; ii) Im z k = 0; (23)Formally condition i) in (23) ensures the BSR, while condition ii) in (23) is responsible for the FAR.However each soliton now has 6 parameters, so 3, 4 and 5 solitons will be parametrized by 18, 24and 30 parameters respectively. The large number of parameters makes it difficult to derive explicitanalytical results, or to do an exhaustive numerical studies. Of course some configurations of Manakovsolitons behave just like the scalar ones. This happens if all ~n k are equal. Naturally our aim is considermore interesting cases and demonstrate the important role that the polarization vectors play for thesoliton interactions. Indeed m k in (17) take any value from 0 to 1, i.e., they ‘regulate’ the strength ofthe interaction. In particular, if the polarization vectors of two neighboring solitons are orthogonal,then they do not interact. In addition the phases σ k modify the phase difference of the solitons whichis a substantial factor in their interaction.Situations when we have 2, 3 and 4 solitons are easier because we can write down explicit formulaefor z k in terms of the soliton parameters in the generic case. For two and three solitons most of thisanalysis for scalar solitons were done [17, 24, 16]. For bigger values of N such formulas are not done7ven for the scalar case, in which the number of the soliton parameters are 4 N . For N = 4 alreadythe formulae for z k are involved; in addition the number of the parameters is 4 N = 16. Thereforefor N ≥ N .Our aim here will be: first to revisit the particular cases considered before and, second, to proposespecial soliton configurations responsible for the BSR and FAR for any number of solitons. We willillustrate our results by several figures. Let us now outline some effective ways of choosing soliton parameters that would ensure given asymp-totic behavior of the solitons. The soliton parameters of the Manakov N -soliton train are 6 N anddetailed study of the regions in which the solitons will develop given asymptotic regime does not seempossible. However we will outline several ways to effectively pick up configurations ensuring BSR orFAR asymptotic regimes.Let us also remind several important issues that one needs to consider. First we need to specifywhat we will consider as asymptotic state. Obviously we need a criterium that would ensure us thatwe are in the asymptotic region. In our case we have two scales: ǫ / and ǫ that are fundamental forthe adiabatic approximation. It is reasonable to assume that the asymptotic times must be of theorder of 1 /ǫ . Our choices of soliton parameters are such that ǫ ≃ − . So one could expect that theasymptotic times would be of the order of ǫ − ≃ t as ≃ t >
300 we see thatthe bound state of 5 solitons in fact transforms into a MAR. The first and the fifth solitons ‘peel off’and go freely away, and the other three still stay in a BSR. It seems that increasing the differencesbetween the amplitudes stabilizes the BSR.The general criterium that ensures FAR or BSR is based on the following well known propositioncoming from linear algebra.
Proposition 1
Let L be symmetric L = L T matrix with real-valued matrix elements. Then itseigenvalues z j will be real and different, i.e., z j = z k for k = j . Corollary 1
Let L be symmetric (not hermitian) L = L T matrix with purely imaginary matrixelements. Then its eigenvalues z j will be purely imaginary and different, i.e., z j = z k for k = j . Proof 1
Follows directly from the Proposition if we consider L = iL .
8n addition below we will assume that ν = 0 . µ = 0. T i m e Position 9609008007006005004003002001000 3020100-10-20-30 T i m e Position
Figure 1: Left panel: FAR with initial conditions r = 7 . µ = 0 . ν = 0 . g = 9; Rightpanel: BSR with initial conditions r = 8 . µ = 0 . ν = 0 . g = 9. The rest of the parametersare defined by eqs. (26) and (28) respectively. These configurations are characteristic for the real Toda chain solved by Moser [31, 29, 9].In what follows we choose the polarization vectors ~n k by setting: θ k = kπ , γ k = kπg . (24)where g = 8, or g = 9.For the CTC using the Proposition we obtain:Im b k | t =0 = 0 , Im a k | t =0 = 0 , (25)which means that ν k | t =0 = 0 . , b k | t =0 = µ k | t =0 = µ k , θ k = kπ , γ k = kπg ,ξ k = ( k − r , µ k = ( k − µ , ν k = 0 . k − ν ,δ , = 0 , δ ,k +1 − δ ,k = σ k , (26)9ndeed, from the Proposition the eigenvalues of L will be real and different, which is FAR. Aparticular case of (26) as configuration ensuring FAR for scalar solitons was noticed long ago, namelychoosing solitons with equal amplitudes (i.e., ∆ ν k = 0) and and out-of phase δ k +1 − δ k = π [4].However, eq. (26) provides more general configurations, in which the solitons may have non-vanishinginitial velocities, see Figure 1. T i m e Position 9609008007006005004003002001000 3020100-10-20-30 T i m e Position
Figure 2: Left panel: FAR with initial conditions r = 8 . µ = 0 . ν = 0 . g = 4; Rightpanel: BSR with initial conditions r = 8 . µ = 0 . ν = 0 . g = 4. The rest of the parametersare defined by eqs. (26) and (28) respectively. Here we use the Corollary and impose on L the conditions:Re b k | t =0 = 0 , Re a k | t =0 = 0 , (27)which means that b k | t =0 = iν k | t =0 = iν k , θ k = kπ , γ k = kπg ,ξ k = ( k − r , µ k = 0 . , ν k = 0 . k − ν ,δ , = 0 , δ ,k +1 − δ ,k = σ k + π, (28)This is also rather general and simple condition on the soliton parameters that fixes the initial velocitiesto be 0, but does not put restrictions (except the adiabatic ones) on the amplitudes and on the initialpositions of the solitons. 10 .4 Symmetric Configurations of Soliton Parameters In addition to these we find other configurations of soliton parameters that provide FAR or BSR. Tothis end we use special symmetric constraints on L described below. These constraints will leave onlyone of ν k and a k independent. As a result the characteristic polynomial of L will factorize and wewill find that all roots are proportional to each other.Let us give few examples of them. We will provide the corresponding Lax matrix, its characteristicpolynomial and eigenvalues. • N = 3, P = z ( z − a + b )): L = b √ a √ a √ a √ a − b ,z , = ± p a + b , z = 0; (29) • N = 4, P = ( z − a − b )( z − a + b )) L = b √ a √ a b a
00 2 a − b √ a √ a − b ,z , = ± p a + b , z , = ± p a + b ; (30) • N = 5, P = z ( z − a − b )( z − a + b )) L = b √ a √ a b a a √ a
00 0 √ a − b √ a √ a − b z , = ± p a + b , z , = ± p a + b , z = 0; (31) • N = 6, P = ( z − a − b )( z − a + b ))( z − a + b )): L = b √ a √ a b √ a √ a b a a − b √ a
00 0 0 √ a − b √ a √ a − b ,z , = ± p a + b , z , = ± p a + b , z , = ± p a + b . (32)11 T i m e Position 9609008007006005004003002001000 3020100-10-20-30 T i m e Position
Figure 3: Left panel: FAR with initial conditions µ = 0 . ν = 0 . g = 4; Right panel: BSRwith initial conditions µ = 0 . ν = 0 . g = 4. The rest of the parameters are defined by eqs.(38) and (39) respectively.Such examples can be found for any value of N ; from algebraic point of view they are related to thethe maximal embedding of sl (2) as a subalgebra of sl ( N ).In order to ensure FAR or BSR we need to impose on a and b the condition thatFAR a + b > , BSR a + b < . (33)Initial conditions for BSR of 5 scalar solitons: ξ = − r + ln 62 ν , ξ = − r + ln 32 ν , ξ = 0 , ξ = r − ln 32 ν , ξ = 2 r − ln 62 ν ,ν k = 0 . − k ) ν , µ k = 0 , δ k = kπ, k = 1 , . . . , . (34)Initial conditions for FAR of 5 scalar solitons: ξ = − r + ln 62 ν , ξ = − r + ln 32 ν , ξ = 0 , ξ = r − ln 32 ν , ξ = 2 r − ln 62 ν ,ν k = 0 . , µ k = (3 − k ) µ , δ k = kπ , k = 1 , . . . , . (35)12 k left panel right panel k = 1 0.0 0.0 k = 2 2.868037 -0.273554 k = 3 -0.405708 -0.405708 k = 4 2.781038 -0.360554 k = 5 -0.150741 -0.150741 δ k left panel right panel k = 1 0.0 0.0 k = 2 2.484841 -0.656751 k = 3 -1.006917 -1.006917 k = 4 2.258187 -0.883405 k = 5 -0.354039 -0.354039Table 1: Initial phases for Fig. 1 and Fig. 2For Manakov solitons the initial positions are determined by: ξ = − r − ν ln m m m m , ξ = − r − ν ln m m m m ,ξ = − ν ln m m m m ,ξ = r + 12 ν ln m m m m , ξ = 2 r + 12 ν ln m m m m , (36)For the numerics we again fix the polarization vectors as in (24) and evaluate ξ k by the formula(36). The result is: ξ = − ..., ξ = − ... (37)In order to have FAR we choose the amplitudes, velocities and the phases of the solitons by: ν k = 0 . , µ k = ( k − µ , k = 1 , , . . . , δ = 0 , δ = δ + σ + π, δ = δ + σ + σ + π,δ = δ + σ + σ + σ + π, δ = δ + σ + σ + σ + σ + π, (38)For the BSR we choose the amplitudes, velocities and the phases of the solitons by: ν k = 0 . k − ν , µ k = 0 , k = 1 , , . . . , δ = 0 , δ = δ + σ , δ = δ + σ + σ ,δ = δ + σ + σ + σ , δ = δ + σ + σ + σ + σ , (39) In the Tables we list the numeric values for m k and σ k for the two typical choices of θ k and γ k usedabove. Conclusions and Discussion
The above analysis can be extended to any number of solitons. As we mentioned above, the symmetricLax matrices are realizations of the maximal embedding of the sl (2) algebra as a subalgebra of sl ( N ).13eft panel right panel δ k ξ k δ k ξ k k = 1 0.0 -15.154654 0.0 -15.154654 k = 2 2.484841 -7.133487 -0.656751 -7.133487 k = 3 -1.006917 .140982 -1.006917 0.140982 k = 4 2.258187 7.305540 -0.883405 7.305540 k = 5 -.354039 15.154654 -0.354039 15.154654Table 2: Initial phases and positions for Fig. 3In this case we effectively reduce the N -soliton interactions to an effective 2-soliton interactions.Therefore the symmetric configurations studied above allow only two asymptotic regimes: BSR andFAR. We make the hypothesis that it would be possible to construct more general symmetric Laxmatrices that would be responsible for effective 3-soliton interactions. In this paper we includednumerical tests only for 3 soliton interactions. However previously we have run test starting with 2-solitons and ending with 9-soliton configurations. Our results are that the CTC models adequately notonly the purely solitonic interactions, but also the effects of external potentials and other perturbationson them.An interesting question is how long should we wait for the asymptotic regime. This question isdirectly related to the other one: What are the limits of applicability of CTC? In our simulations wehave chosen ε ≃ .
01 which means that the asymptotic time must be of the order of 1 /ε ≃ Acknowledgements
MDT was supported by Fulbright – Bulgarian-American Commission for Educational Exchange underGrant No 19-21-07.
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