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On Modeling Adiabatic N-Soliton Interactions ... - TCPA Foundation

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Gravity, Astrophysics, and Strings’04eds. P.P.Fiziev and M.D.TodorovSt. Kliment Ohridski University Press, Sofia, 2005On modelling adiabatic N-soliton interactions andperturbations. Effects of external potentialsV. S. Gerdjikov 1 , B. B. Baizakov 21 Institute of Nuclear Research and Nuclear Energy, Bulgarian Academyof Sciences, Sofia 1784, Bulgaria2 Department of Physics “E. R. Caianiello”, Salerno University, I-84081Baronissi, ItalyAbstract. We analyze several perturbed versions of the complex Toda chain(CTC) in an attempt to describe the adiabatic N-soliton train interactions of theperturbed nonlinear Schrödinger equation (NLS). Particular types of perturbations,including quadratic and periodic external potentials are treated by bothanalytical and numerical means. We show that the perturbed CTC model providesa good description for the N-soliton interactions in the presence of a weakexternal potential.1. IntroductionThe N-soliton train interactions for the nonlinear Schrödinger equation (NLS) and itsperturbed versionsiu t + 1 2 u xx + |u| 2 u(x, t) = iɛR[u], (1)started with the pioneering paper [1] by now has been extensively studied (see [2]–[6]and references therein). Several other nonlinear evolution equations (NLEE) were alsostudied, among them the modified NLS equation [7]–[11], some higher NLS equations[6] and others.Below we concentrate on the perturbed NLS Eq. (1). By N-soliton train we mean asolution of the (perturbed) NLS fixed up by the initial condition:u(x, t = 0) =N∑k=1u 1sk (x, t = 0),u 1sk (x, t) = 2ν ke iφ kcosh z k, (2)z k (x, t) = 2ν k (x − ξ k (t)), ξ k (t) = 2µ k t + ξ k,0 , (3)138


On modelling adiabatic N-soliton interactions and perturbations. Effects of ... 139φ k (x, t) = µ kν kz k + δ k (t), δ k (t) = W k t + δ k,0 . (4)Each soliton has four parameters: amplitude ν k , velocity µ k , center of mass position ξ kand phase δ k . The adiabatic approximation uses as a small parameter ε 0 ≪ 1 the solitonoverlap which falls off exponentially with the distance between the solitons. Then thesoliton parameters must satisfy [1]:|ν k − ν 0 | ≪ ν 0 , |µ k − µ 0 | ≪ µ 0 , |ν k − ν 0 ||ξ k+1,0 − ξ k,0 | ≫ 1, (5)where ν 0 = 1 N∑ Nk=1 ν k, and µ 0 = 1 N∑ Nk=1 µ k are the average amplitude and velocityrespectively. In fact we have two different scales:|ν k − ν 0 | ≃ ε 1/20 , |µ k − µ 0 | ≃ ε 1/20 , |ξ k+1,0 − ξ k,0 | ≃ ε −10 .One can expect that the approximation holds only for such times t, for which the set of4N parameters of the soliton train satisfy (5).Eq.(1) finds a number of applications to nonlinear optics and for R[u] ≡ 0 is integrablevia the inverse scattering transform method [12, 13]. The N-soliton train dynamics in theadiabatic approximation is modelled by a complex generalization of the Toda chain [14]:d 2 Q jdt 2 = 16ν02 (eQ j+1−Q j− e Qj−Qj−1) , j = 1, . . . , N. (6)The complex-valued Q k are expressed through the soliton parameters by:Q k (t) = 2iλ 0 ξ k (t) + 2k ln(2ν 0 ) + i(kπ − δ k (t) − δ 0 ), (7)where δ 0 = 1/N ∑ Nk=1 δ k and λ 0 = µ 0 + iν 0 . Besides, we assume free-ends conditions,i.e., e −Q0 ≡ e Q N+1≡ 0.Note that the N-soliton train is not an N-soliton solution. The spectral data of thecorresponding Lax operator L is nontrivial also on the continuous spectrum of L. Thereforethe analytical results from the soliton theory can not be applied. Besides, we want totreat solitons moving with equal velocities and also to account for the effects of possiblenonintegrable perturbations R[u].The present paper extends the results of several previous ones: see Refs. [2]–[5],[11], [15]. Recently with the studies of Bose-Einstein condensates it became importantto explore the NLS equation with additional potential term iR[u] = V (x)u(x, t), see[16, 17]. We continue the analysis in [6] of the corresponding perturbed CTC (PCTC)model for quadratic and periodic potentials V (x). Our results give additional confirmationof the stabilization properties of the periodic potentials observed in [18, 19] in a differentphysical setup.2. The CTC Model and Its ImportanceThe fact [14] that the CTC, like the (real) Toda chain (RTC), is a completely integrableHamiltonian system allows one to analyze analytically the asymptotic behavior of the N-soliton train. However unlike the RTC, the CTC has richer variety of dynamical regimes[20] such as:


140 V.S. Gerdjikov and B.B. Baizakov• asymptotically free motion if v j ≠ v k for j ≠ k; this is the only dynamical regimepossible for RTC;• N-s bound state if v 1 = · · · = v N but ζ k ≠ ζ j for k ≠ j;• various intermediate (mixed) regimes; e.g., if v 1 = v 2 > · · · > v N but ζ k ≠ ζ j fork ≠ j then we will have a bound state of the first two solitons while all the otherswill be asymptotically free;• singular and degenerate regimes if two or more of the eigenvalues of L becomeequal, e.g., ζ 1 = ζ 2 . . . and ζ j ≠ ζ k for 2 < j ≠ k.By ζ k = v k + iw k above we have denoted the eigenvalues of the Lax matrix L in theLax representation L τ = [M, L] of the CTC where:L =N∑k=1b k ≡ − 1 dQ k2 dτN−1∑b k E kk + a k (E k,k+1 + E k+1,k ), (8)k=1= 1 2 (µ k + iν k ), a k = 1 2 exp((Q k+1 − Q k )/2),and the matrices E jk are defined by (E jk ) sp = δ js δ kp . The eigenvalues ζ k of L aretime independent and complex-valued along with the first components η k = ⃗z (k)1 of thenormalized eigenvectors of L:L⃗z (k) = ζ k ⃗z (k) , (⃗z (k) , ⃗z (m) ) = δ km . (9)The set {ζ k = v k + iw k , η k = σ k + iθ k } provides the action-angle variables of the CTC.Using the CTC model one can determine the asymptotic regime of the N-soliton train.Given the initial parameters µ k (0), ν k (0), ξ k (0), δ k (0) of the N-soliton train one can calculatethe matrix elements b k and a k of L at t = 0. Then solving the characteristic equationfor L| t=0 one can calculate the eigenvalues ζ k and to determine the asymptotic regimeof the N-soliton train [2, 5]. Another option is to impose on ζ k a specific constraint, e.g.,that all ζ k be purely imaginary, i.e., all v k = 0. This will provide a set of algebraic conditionson L| t=0 , and on the initial soliton parameters µ k (0), ν k (0), ξ k (0), δ k (0), whichcharacterize the region in the soliton parameter space responsible for the N-soliton boundstates.3. The Perturbed CTC ModelBelow we consider several specific choices R (p) [u] of perturbations, p = 1, 2, . . . inEq.(1). In the adiabatic approximation the dynamics of the soliton parameters can bedetermined by the system (see [1] for N = 2 and [2, 5] for N > 2):dλ k(= − 4ν 0 eQ k+1 −Q k− e Q )k−Q k−1 (p) + Mk+ iN (p)kdt(10)dξ kdt =2µ k + Ξ (p)k , dδ kdt = 2(µ2 k + νk) 2 + X (p)k , (11)


On modelling adiabatic N-soliton interactions and perturbations. Effects of ... 141where λ k = µ k +iν k and X (p)kare determined by R (p)kN (p)k= 1 2M (p)k= 1 2Ξ (p)k= 1[u] through:∫ ∞−∞∫ ∞−∞∫ ∞4νk2 −∞∞D (p)k= 12ν k∫−∞= 2µ k Ξ (p)kdz(kRecosh z kdz k sinh z(kcosh 2 Imz kdz k z(kRecosh z kInserting (10), (11) into (7) we derive:+D(p) k. The right hand sides of Eqs. (10)–(11)R (p)k[u]e−iφ kR (p)k[u]e−iφ kR (p)k[u]e−iφ kdz k (1 − z k tanh z k )(Imcosh z k), (12)), (13)), (14)R (p)k[u]e−iφ k). (15)dQ kdt= −4ν 0 λ k + 2k N (p)0ν 0() (+ 2iξ k M (p)0 + iN (p)0 + i 2λ 0 Ξ (p)k− X (p)k)− X (p)0 , (16)N (p)0 = 1 NN∑j=1N (p)j , M (p)0 = 1 NN∑j=1M (p)j , X (p)0 = 1 NN∑j=1X (p)j .In deriving Eq.(16) we have kept terms of the order ∆ν k ≃ O( √ ɛ 0 ) and neglectedterms of the order O(ɛ 0 ). The perturbations result in that ν 0 and µ 0 may become timedependent.Indeed, from (10) we get:dµ 0dt= M (p)0 , dν 0dt = N (p)0 . (17)The small parameter ɛ 0 can be related to the initial distance r 0 = |ξ 2 −ξ 1 | t=0 betweenthe two solitons. Assuming ν 1,2 ≃ ν 0 we find:ɛ 0 =∫ ∞−∞dx ∣ ∣u 1s1 (x, 0)u 1s2 (x, 0) ∣ ∣ ≃ 8ν 0 r 0 e −2ν 0r 0. (18)In particular, Eq.(18) means that ɛ 0 ≃ 0.01 for r 0 ≃ 8 and ν 0 = 1/2.We assume that initially the solitons are ordered in such a way that ξ k+1 − ξ k ≃ r 0 .One can check [3, 11] that N (p)k≃ M (p)k≃ exp(−2ν 0 |k−p|r 0 ). Therefore the interactionterms between the k-th and k±1-st solitons will be of the order of e −2ν 0r 0; the interactionsbetween k-th and k ± 2-nd soliton will of the order of e −4ν 0r 0≪ e −2ν 0r 0.The terms Ξ (0)k, X(0) kare of the order of r0 a exp(−2ν 0 r 0 ), where a = 0 or 1. Howeverthey can be neglected as compared to ˜µ k and ˜ν k , where˜µ k = µ k − µ 0 ≃ √ ɛ 0 , ˜ν k = ν k − ν 0 ≃ √ ɛ 0 , (19)


142 V.S. Gerdjikov and B.B. BaizakovThe corrections to N (p)k, . . . , coming from the terms linear in u depend only on theparameters of the k-th soliton; i.e., they are ‘local’ in k. The nonlinear in u terms presentin iR (p) [u] produce also ‘non-local’ in k terms in N (p)k , . . . .3.1. Nonlinear gain and second order dispersionConsider the NLS Eq.(1) withR[u] = c 0 u + c 2 u xx + d 0 |u| 2 u, (20)where c 0 , c 2 and d 0 are real constants, see [11]. Another important factor is the order ofmagnitude of the perturbation coefficients c 0 , c 2 and d 0 in (20). If we take them to be ofthe order of ɛ 0 we find that the N-soliton train evolves according to:where for µ 0 = 0 we get U 00 = − 8iν2 03CTC (21) can be solved exactly:d 2 Q kdt 2 = U 00 + 16ν02 (eQ k+1 −Q k− e Q )k−Q k−1 , (21)( )3c0 + 8ν0d 2 0 − 4c 2 ν02 . This form of perturbedQ k (t) = 1 2 U 00t 2 + V 00 t + Q (0)k(t),where Q (0)k(t) is a solution of the unperturbed CTC and V 00 is an arbitrary constant. Inthis case the effect of the perturbation will be an overall motion of the center of massof the N-soliton train. The relative motion of the solitons will remain the same. Forlarger values of the coefficients c 0 , c 2 and d 0 , e.g., of the order of √ ɛ 0 the correspondingdynamical system is more complicated and has to be treated separately.3.2. Quadratic and periodic potentialsLet iR[u] = V (x)u(x, t). Our first choice for V (x) is a quadratic one:Skipping the details we get the results:V (1) (x) = V 2 x 2 + V 1 x + V 0 . (22)N (1)k= 0, M (1)k= −V 2 ξ k − V 1(Ξ (1)k= 0, D (1) π2k= V 248νk2 − ξk22 , (23a))− V 1 ξ k − V 0 , (23b)and X (1)k= D (1)k. As a result the corresponding PCTC takes the form [6]:d(µ k + iν k )dtdQ kdt(= −4ν 0 eQ k+1 −Q k− e Q )k−Q k−1 − V2 ξ k − V 12 , (24)= −4ν 0 (µ k + iν k ) − iD (1)k− i N∑D (1)j .N(25)j=1


On modelling adiabatic N-soliton interactions and perturbations. Effects of ... 143If we now differentiate (25) and make use of (24) we get:d 2 Q kdt 2 = 16ν02 (eQ k+1 −Q k− e Q )k−Q k−1(+ 4ν 0 V 2 ξ k + V )1− i dD(1) k2 dt− i NN∑j=1dD (1)j. (26)dtIt is reasonable to assume that V 2 ≃ O(ɛ 0 /N); this ensures the possibility to have theN-soliton train ‘inside’ the potential. It also means that both the exponential terms andthe correction terms M (1)kare of the same order of magnitude. From Eqs.(24) and (25)there follows that dν 0 /dt = 0 and:dµ 0dt= −V 2 ξ 0 − V 12 , dξ 0dt = 2µ 0, (27)where µ 0 is the average velocity and ξ 0 = 1 ∑ NN j=1 ξ j, is the center of mass of the N-soliton train. The system of equations (27) for V 2 > 0 has a simple solution√2µ 0 (t) = µ 00 cos(Φ(t)), ξ 0 (t) = µ 00 sin(Φ(t)) − V 1, (28)V 2 2V 2where Φ(t) = √ 2V 2 t + Φ 0 , and µ 00 and Φ 0 are constants of integration. Therefore theoverall effect of such quadratic potential will be to induce a slow periodic motion of thetrain as a whole.Another important choice is the periodic potentialV (2) (x) = A cos(Ωx + Ω 0 ), (29)where A, Ω and Ω 0 are appropriately chosen constants. NLS equation with similar potentialsappear in a natural way in the study of Bose-Einstein condensates, see [16].For two interacting solitons the corresponding Karpman-Solov’ev system was derivedin [19]. For N > 2 we obtain the PCTC where the integrals for N k , M k , Ξ k and D k areequal to [6]:N (2)k= 0, M (2)k= πAΩ2 1sin(Ωξ k + Ω 0 ),8ν k sinh Z k(30)Ξ (2)k= 0, D (2)k= − π2 AΩ 2 cosh Z k16νk2 sinh 2 cos(Ωξ k + Ω 0 ),Z k(31)where Z k = πΩ/(4ν k ). These results allow one to derive the corresponding perturbedCTC models. Again we find that dν 0 /dt = 0.4. Numerical Verification of the PCTCHere we present the numerical verification of the PCTC model. The perturbed NLS Eq.(1)is solved by the operator splitting procedure using the fast Fourier transform [21]. In thecourse of time evolution we monitor the conservation of the norm and energy of the N-soliton train. The corresponding PCTC equations are solved by the Runge-Kutta schemewith the adaptive step-size control [22].


146 V.S. Gerdjikov and B.B. Baizakov300800250600t400200150t1000-80 -60 -40 -20 0 20 40 60 80x200-30 -20 -10 0 10 20 30 0 50xFigure 2: Harmonic oscillations of a N-soliton train initially shifted relative to the minimum ofthe quadratic potential V (x) = V 2 x 2 . Left panel: 9-soliton train, V 2 = 0.00005. Right panel:3-soliton train, V 2 = 0.001. Parameters of solitons in the train are the same as in Figure 1. Inboth panels solid lines correspond to direct simulations of the NLS equation (1), and dashed linesto numerical solution of the PCTC equations (33) - (36).wheredδ kdt = 2(µ2 k + ν 2 k) + D k (ν k ) cos(Ωξ k + Ω 0 ), (41)M k (ν k ) =πAΩ2 , D k (ν k ) = − π2 AΩ 2 cosh Z k8ν k sinh Z k 16νk2 sinh 2 ,Z kZ k = πΩ4ν k. (42)Let the initial soliton parameters be as in (32) and let us choose the initial positions ofthe solitons to coincide with the minima of the periodic potential V (x) = A cos(Ωx+Ω 0 );i.e., r 0 = 2π/Ω. Then each soliton of the train experiences confining force of the periodicpotential and repulsive force of neighboring solitons. Solitons placed initially at minimaof the periodic potential (Figure 3, left panel) perform small amplitude oscillations aroundthese minima, provided that the strength of the potential is big enough to keep solitonsconfined (Figure 3, right panel).In contrast to the quadratic potentials, the weak periodic potential is unable to confinesolitons, and repulsive forces between neighboring solitons (at ν k (0) = 1/2, δ k (0) = kπ)induces unbounded expansion of the train similar to what was shown in the left panel ofFigure 1.


On modelling adiabatic N-soliton interactions and perturbations. Effects of ... 147|u(x,0)| 2 , V(x)1.00.80.60.40.20.0-0.2-40 -30 -20 -10 0 10 20 30 40x1000800600400200-40 -20 0 20 40 0xtFigure 3: Left panel: Initial state of the 7-soliton train (solid lines) in a confining periodic potentialV (x) = A cos(Ωx + Ω 0 ) (dashed lines) with A = 0.1, Ω = 1, r 0 = 2π. Right panel: Oscillationsof the 5-soliton train in a moderately weak periodic potential, A = 0.0005, Ω = 2π/9, r 0 = 9.Solid and dashed lines correspond, respectively, to numerical solution of the NLS equation (1) andPCTC system (38) - (41). Solitons have equal amplitudes 2η = 1, the phase difference is π, andΩ 0 = 0 in both panels.5. Discussion and ConclusionsLike any other model, the predictions of the CTC should be compared with the numericalsolutions of the corresponding NLEE. Such comparison between the CTC and the NLShas been done thoroughly in [2, 3, 5] and excellent match has been found for all dynamicalregimes. This means that the CTC may be viewed as an universal model for the adiabaticN-soliton interactions for several types of NLS.For the perturbed CTC equations such comparison has been just started; the goodagreement shown in the figures above supports the hope that the region of applicability ofPCTC can be widened.More detailed investigation of the N-soliton train interactions under different types ofexternal potentials and for different types of initial soliton parameters will be publishedin subsequent papers [23].Acknowledgements. We are grateful to Professor M. Salerno for useful discussions.One of us (VSG) is grateful to Professors M. Boiti, F. Pempinelli and B. Prinari for warmhospitality at the University of Lecce, where part of this work was done. Financial supportin part from Gruppo collegato di INFN at Salerno, Italy is gratefully acknowledged. BBBthanks the Department of Physics at the University of Salerno, Italy, for a research grant.


On modelling adiabatic N-soliton interactions and perturbations. Effects of ... 149[20] Moser, J. (1975) in Dynamical Systems, Theory and Applications, Lecture Notes in Physics,vol.38, Springer Verlag, page 467; Gerdjikov, V. S., Evstatiev, E. G., Ivanov, R. I. (1998) Thecomplex Toda chains and the simple Lie algebras – solutions and large time asymptotics, J.Phys. A: Math & Gen. 31 8221.[21] Taha, T. R. and Ablowitz, M. J. (1984) J. Comp. Phys. 55 203.[22] Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P. (1996) Numerical Recipes.The Art of Scientific Computing, Cambridge University Press.[23] Gerdjikov, V. S., Baizakov, B. B., Salerno, M., Modelling adiabatic N-soliton interactionsand perturbations, in Proc. of the workshop “Nonlinear Physics: Theory and Experiment. III”,Gallipoli, 2004, M. Ablowitz, M. Boiti, F. Pempinelli, B. Prinari (Eds) (in press).

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