Modeling Interactions of Soliton Trains. Effects of External Potentials. Part II

Modeling Interactions of Soliton Trains. Effects of External Potentials. Part II Michail Todorov Department of Applied Mathematics and Computer Science Technical University of Sofia, Bulgaria Work done in collaboration with V. Gerdjikov and A. Kyuldjiev from INRNE-BAS, Sofia 6th International Conference AMiTaNS, Albena, Bulgaria, June 26-July 1, 2014

Outline NLS Adiabatic Approximation Vector NLS/Manakov System Variational Approach Perturbed CTC Effects of Polarization Vectors Lax Representation GCTC RTC CTC Asymptotic Regimes Conservative Numerical Method vs Variational Approach Results and Discussion References

NLS Soliton Trains The idea of the adiabatic approximation to the soliton interactions (Karpman&Solov ev (1981)) led to effective modeling of the N-soliton trains of the perturbed scalar NLS eq.: iu t + 1 2 u xx + u 2 u(x, t) = ir[u]. (1) By N-soliton train we mean a solution of the NLSE (1) with initial condition N u(x, t = 0) = u k (x, t = 0), (2) k=1 u k (x, t)=2ν k e iφ k sechz k, z k =2ν k (x ξ k (t)), ξ k (t)=2µ k t+ξ k,0, φ k = µ k ν k z k + δ k (t), δ k (t)=2(µ 2 k + ν2 k )t + δ k,0. Here µ k are the amplitudes, ν k the velocities, δ k the phase shifts, ξ k - the centers of solitons.

NLS Adiabatic Approximation The adiabatic approximation holds if the soliton parameters satisfy the restrictions ν k ν 0 ν 0, µ k µ 0 µ 0, ν k ν 0 ξ k+1,0 ξ k,0 1, (3) where ν 0 and µ 0 are the average amplitude and velocity respectively. In fact we have two different scales: ν k ν 0 ε 1/2 0, µ k µ 0 ε 1/2 0, ξ k+1,0 ξ k,0 ε 1 0. In this approximation the dynamics of the N-soliton train is described by a dynamical system for the 4N soliton parameters. In our previous works we investigate it in presence of periodic and polynomial potentials. Now, we are interested in what follow perturbation(s) by external sech-potentials: ir[u] V (x)u(x, t), V (x) = s c s sech 2 (2ν 0 x y s ). (4) The latter allows us to realize the idea about localized potential wells(depressions) and humps.

Potentials Figure: 1. Single sech-potential vs. composite potential well V (x) = 32 s=0 c ssech 2 (x x s ), c s = 10 1, x s = 16 + sh, h = 1, s = 0,..., 32.

Perturbed Vector NLS Here we generalize to the perturbed vector NLS i u t + 1 2 u xx + ( u, u) u(x, t) = ir[ u]. (5) The corresponding vector N-soliton train is determined by the initial condition u(x, t = 0) = N u k (x, t = 0), u k (x, t) = 2ν k e iφ k sechz k n k, (6) k=1 and the amplitudes, the velocities, the phase shifts, and the centers of solitons are as in Eq.(2). The phenomenology, however, is enriched by introducing a constant polarization vectors n k that are normalized by the conditions ( n k, n k) = 1, n arg n k;s = 0. s=1

Variational approach and evolution We use the variational approach (Anderson and Lisak (1986)) and derive the GCTC model. Like the (unperturbed) CTC, GCTC is a finite dimensional completely integrable model allowing Lax representation. The Lagrangian of the vector NLS perturbed by external potential is: L[ u] = dt i [ ] ( u, u t ) ( u t, u) H, 2 [ H[ u] = dx 1 2 ( u x, u x ) + 1 ] (7) 2 ( u, u) 2 ( u, u)v (x). Then the Lagrange equations of motion: d dt δl δ u t δl = 0, (8) δ u coincide with the vector NLS with external potential V (x).

Variational approach and evolution Next we insert u(x, t) = N k=1 u k(x, t) (see eq. (6)) and integrate over x neglecting all terms of order ɛ and higher. Thus after long calculations we obtain: L = N L k + k=1 N k=1 n=k±1 L k,n, L k,n = 16ν 3 0e k,n (R k,n + R k,n ), R k,n = e i( δ n δ k ) ( n k n n), δk = δ k 2µ 0 ξ k, k,n = 2s k,n ν 0 (ξ k ξ n ) 1, s k,k+1 = 1, s k,k 1 = 1. (9) where ) L k = 2iν k (( n k,t, n k) ( n k, n dξ k k,t) + 8µ k ν k dt dδ k 4ν k dt +... (10)

Variational approach and evolution The equations of motion are given by: d δl δl = 0, (11) dt δp k,t δp k where p k = {δ k, ξ k, µ k, ν k, n k }. Let λ k = µ k + iν k, X k = 2µ k Ξ k + D k and Q k = 2ν 0 ξ k + k ln 4ν 2 0 i(δ k + δ 0 + kπ 2µ 0 ξ k ). (12)

Then: dλ ) k = 4ν 0 (e Q k+1 Q k ( n dt k+1, n k) e Q k Q k 1 ( n k, n k 1) + M k + in k, dq k = 4ν 0 λ k + 2i(µ 0 + iν 0 )Ξ k ix k, dt (13)

Potential s impact where N k [u]= 1 2 Re R[u k ] sech z k e iφ k dz k, M k [u]= 1 2 Im R[u k ] tanh z k sech z k e iφ k dz k, Ξ k [u]= 1 4 Re R[u k ]z k sech z k e iφ k dz k, D k [u]= 1 2ν k Im R[u k ](1 z k tanh z k ) sech z k e iφ k dz k,

Connection with PCTC So, we have a generalization of CTC: dλ ( ) k dt = 4ν 0 e Q k+1 Q k ( n k+1, n k) e Q k Q k 1 ( n k, n k 1) +M k +in k, dq k dt = 4ν d n k 0λ k + 2i(µ 0 + iν 0 )Ξ k ix k, = O(ɛ), dt (14) The explicit form of M k, N k, Ξ k and D k for the potential chosen is given by M k = s 2c s ν k P( k,s ), N k = 0, Ξ k = 0, D k = s c s R( k,s ). (15) where k,s = 2ν 0 ξ k y s and the functions P( ) and R( ) are known explicitly.

Integrals Figure: 2. P and R functions: for a single sech-potential and for the composite potential.

Integrals For the potential well V (x, y s ) = c s [tanh(2ν 0 x y i ) tanh(2ν 0 x y f )] functions and P( ) = sinh cosh sinh 3 R( ) = exp ( 3 ) + (4 2 1) cosh + (3 8 ) sinh 8 sinh 3.

Polarization Vectors Now we have additional equations describing the evolution of the polarization vectors. But note, that their evolution is slow, and in addition the products ( n k+1, n k) multiply the exponents e Q k+1 Q k which are also of the order of ɛ. Since we are keeping only terms of the order of ɛ we can replace ( n k+1, n k) by their initial values ( n k+1, n k) = m0k 2 e2iφ 0k, k = 1,..., N 1 (16) t=0

Lax representation of CTC The CTC is completely integrable model; it allows Lax representation L t = [A.L], where: L = A = N (b s E ss + a s (E s,s+1 + E s+1,s )), s=1 N (a s (E s,s+1 E s+1,s )), s=1 (17) where a s = exp((q s+1 Q s )/2), b s = 1 2 (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.

Lax representation of GCTC The GCTC is also a completely integrable model because it allows Lax representation L t = [Ã. L], where: L = Ã = N (b s E ss + ã s (E s,s+1 + E s+1,s )), s=1 N (ã s (E s,s+1 E s+1,s )), s=1 (18) where ã s = m 0k e iφ 0k a s, b s = µ s,t + iν s,t. Like for the scalar case, the eigenvalues of L are integrals of motion. 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) then their real parts κ s (resp. κ s ) determine the asymptotic velocities for the soliton train described by CTC (resp. GCTC).

RTC and CTC. Asymptotic regimes While for the RTC the set of eigenvalues ζ s of the Lax matrix are all real, for the CTC they generically take complex values, e.g., ζ s = κ s + iη s. Hence, the only possible asymptotic behavior in the RTC is an asymptotically separating, free motion of the solitons. In opposite, for the CTC the real parts κ s Reζ s of eigenvalues of the Lax matrix ζ s determines the asymptotic velocity of the sth soliton.

Effects of the polarization vectors on the soliton interaction 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 regime of the soliton train. Regime (i) κ k κ j ( κ k κ j ) for k j asymptotically separating, free solitons; Regime (ii) κ 1 = κ 2 = = κ N = 0 ( κ 1 = κ 2 = = κ N = 0) a bound state; Regime (iii) group of particles move with the same mean asymptotic velocity and the rest of the particles will have free asymptotic motion. Varying only the polarization vectors one can change the asymptotic regime of the soliton train.

Problem Formulation: Conservation Laws Define mass, M, (pseudo)momentum, P, and energy, E: M def = 1 L2 ( ψ 2 + φ 2) dx, 2β L 1 E def = L2 P def L2 = I(ψ ψ x + φ φ x )dx, L 1 L 1 Hdx, where (19) H def = β ( ψ x 2 + φ x 2) 1 2 α 1( ψ 4 + φ 4 ) (α 1 + 2α 2 ) ( φ 2 ψ 2) 2Γ[R( ψ φ)] is the Hamiltonian density of the system. Here L 1 and L 2 are the left end and the right end of the interval under consideration. The following conservation/balance laws hold, namely dm dt = 0, dp dt = H x=l2 H x= L1, de dt = 0, (20)

Numerical Method To solve the main problem numerically, we use an implicit conservative scheme in complex arithmetic. i ψn+1 i + ψn+1 i i 1 2ψn+1 i + ψ n+1 i+1 + ψn i 1 2ψi n + ψi+1 n ) [ ( α 1 ψ n+1 i 2 + ψi n 2) + (α 1 + 2α 2 ) ( φ n+1 i 2 + φ n i 2)] ψi n = β ( ψ n+1 τ 2h 2 + ψi n 4 1 2 Γ ( φ n+1 i + φ n ) i, i φn+1 i + φn+1 i i 1 2φn+1 i + φ n+1 i+1 + φn i 1 2φ n i + φ n ) i+1 [ ( α 1 φ n+1 i 2 + φ n i 2) + (α 1 + 2α 2 ) ( ψ n+1 i 2 + ψi n 2)] φ n i = β ( φ n+1 τ 2h 2 + φ n i 4 1 2 Γ ( ψ n+1 i + ψi n ).

Numerical Method: Internal Iterations i ψn+1,k+1 i τ + ψn+1,k i i φn+1,k+1 i + φn+1,k i ψ n i + ψi n [ 4 = β ( 2h 2 ψ n+1,k+1 i 1 2ψ n+1,k+1 i ) + ψ n i 1 2ψ n i + ψ n i+1 α 1 ( ψ n+1,k+1 i ψ n+1,k i + ψ n i 2) + ψ n+1,k+1 i+1 +(α 1 + 2α 2 ) ( φ n+1,k+1 i φ n+1,k i + φ n i 2)] φ n i = β ( φ n+1,k+1 τ 2h 2 i 1 2φ n+1,k+1 i + φ n i 1 2φ n i + φ n ) i+1 + φ n [ ( i α 1 φ n+1,k+1 i φ n+1,k i + φ n i 2) 4 +(α 1 + 2α 2 ) ( ψ n+1,k+1 i ψ n+1,k i + ψi n 2)]. + φ n+1,k+1 i+1

Numerical Method: Conservation Properties It is not only convergent (consistent and stable), but also conserves mass and energy, i.e., there exist discrete analogs for (20), which arise from the scheme. M n = E n = N 1 i=2 N 1 i=2 ( ψ n i 2 + φ n i 2) = const, β ( ψ n 2h 2 i+1 ψi n 2 + φi+1 n φi n 2) + α 1 ( ψ n 4 i 4 + φ n i 4) + 1 2 (α 1 + 2α 2 ) ( ψ n i 2 φ n i 2) ΓR[ φ n i ψ n i ] = const, for all n 0. These values are kept constant during the time stepping. The above scheme is of Crank-Nicolson type for the linear terms and we employ internal iterations to achieve implicit approximation of the nonlinear terms, i.e., we use its linearized implementation.

Effects of the external potentials. Numeric checks vs Variational approach The predictions and validity of the CTC and GCTC are compared and verified with the numerical solutions of the corresponding CNSE using fully implicit difference scheme of Crank-Nicolson type, which conserves the energy, the mass, and the pseudomomentum. The scheme is implemented in a complex arithmetics. Such comparison is conducted for all dynamical regimes considered. First we study the soliton interaction of the pure Manakov model (without perturbations, V (x) 0) and with vanishing cross-modulation α 2 = 0; 2- and 3-soliton configurations and transitions between different asymptotic regimes under the effect of well- and hump-like external potential.

Three-soliton configuration in free asymptotic regime corresponding to real parts of eigenvalues of the Lax pair Reζ 1 = 0.0116, Reζ 2 = 0, Reζ 3 = 0.0116 Figure: 3. Free potential behavior (left); External potential well V (x) = 32 s=0 c ssech 2 (x x s ), c s = 10 1, x s = 16 + sh, h = 1, s = 0,..., 32(right).

Three-soliton configuration in bound state regime corresponding to real parts of eigenvalues of the Lax pair Reζ 1 = Reζ 2 = Reζ 3 = 0 Figure: 4. Free potential behavior (left); External potential hump V (x) = 12 s=0 c ssech 2 (x x s ), c s = 10 2, x s = 10 + sh, h = 5/3, s = 0,..., 12 (right).

Three-soliton configuration in mixed asymptotic regime corresponding to real parts of eigenvalues of the Lax pair Reζ 1 = Reζ 2 = 0.00321, Reζ 3 = 0.00642 Figure: 5. Free potential behavior (left); External potential well V (x) = 32 s=0 c ssech 2 (x x s ), c s = 10 2, x s = 16 + sh, h = 1, s = 0,..., 32.

Nine-soliton asymptotic regime: ξ k = 45 + 9k, ν k = 0.4625 + 0.0075(k 1), µ k = 0, δ k = kπ, k = 1,..., 9, θ k+1 = θ k π 10, k = 1,..., 8, θ 1 = 9π 10 Calculated eigenvalues of the potentialfree Lax pair are ζ 1 = 0.005720 + 0.239562i ζ 2 = 0.005720 + 0.239562i ζ 3 = 0.001564 + 0.245551i ζ 4 = 0.001564 + 0.245551i ζ 5 = 0.005720 + 0.260438i ζ 6 = 2.939394 10 19 + 0.250000i ζ 7 = 0.001564 + 0.254449i ζ 8 = 0.001564 + 0.254449i ζ 9 = 0.005720 + 0.260438i

Nine-soliton configuration Figure: 6. Free potential behavior (left); External potential well V (x) = cs [tanh(2ν0 x yi ) tanh(2ν0 x yf )], cs = 0.004, yi = 69.5, yf = 218.5 (right). M. Todorov Multisoliton Interactions for the Manakov System

Nine-soliton asymptotic regime: ξ k = 40 + 8k, ν k = 0.4625 + 0.0075(k 1), µ k = 0, δ k = kπ, k = 1,..., 9, θ k+1 = θ k π 10, k = 1,..., 8, θ 1 = 9π 10 Calculated eigenvalues of the potentialfree Lax pair are ζ 1 = 0.011877 + 0.241165i ζ 2 = 0.011877 + 0.241165i ζ 3 = 0.011877 + 0.258835i ζ 4 = 0.006926 + 0.248312i ζ 5 = 0.006926 + 0.251688i ζ 6 = 8.679330 10 20 + 0.250000i ζ 7 = 0.006926 + 0.248312i ζ 8 = 0.006926 + 0.251688i ζ 9 = 0.011877 + 0.258835i

Nine-soliton configuration Figure: 7. Free potential behavior (left); External potential well V (x) = cs [tanh(2ν0 x yi ) tanh(2ν0 x yf )], cs = 0.004, yi = 40.5, yf = 215.5 (right). M. Todorov Multisoliton Interactions for the Manakov System

Nine-soliton configuration Figure: 8. External potential well V (x) = cs [tanh(2ν0 x yi ) tanh(2ν0 x yf )], cs = 0.004: yi = 40.5, yf = 151.5 (left); yi = 40.5, yf = 115.5 (right). M. Todorov Multisoliton Interactions for the Manakov System

Nine-soliton configuration Figure: 9. External potential well V (x) = c s [tanh(2ν 0 x y i ) tanh(2ν 0 x y f )], c s = 0.004: y i = 40.5, y f = 151.5.

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