interaction of narrow dark solitons with impurities … · 2 interaction of narrow dark solitons...

Romanian Reports in Physics, Vol. 65, No. 2, P. 374–382, 2013

INTERACTION OF NARROW DARK SOLITONS WITH IMPURITIES IN NONLINEAR LATTICES*

M. T. PRIMATAROWA and R. S. KAMBUROVA

Georgi Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria, E-mail: [email protected]

Received September 12, 2012

Abstract. We study the interaction of discrete (narrow) dark solitons with point defects in the integrable Ablowitz-Ladik lattice model. On-site and inter-site (bond) defects are considered. The stability of the analytical solutions for solitons bound to the defect is investigated. A comparison with the standard discrete nonlinear Schrödinger equation is made. Scattering of the dark solitons from point defects of different types is studied numerically. The model plays an important role in numerous physical systems, especially when the corresponding elementary excitations obey Pauli statistics.

Key words: dark solitons, nonlinear lattices, Ablowitz-Ladik equation, soliton-impurity-interaction.

1. INTRODUCTION

The nonlinear dynamics of spatially extended physical systems involve continuous media, so that nonlinear coherent excitations (solitons) are naturally described as solutions of partial differential equations, e.g. the nonlinear Schrödinger (NLS) equation. However, models describing microscopic phenomena in solid-state physics are inherently discrete, with the lattice spacing between the atomic sites being a fundamental physical parameter. For these systems, an accurate microscopic description involves a large set of coupled ordinary differential equations. Defects and discreteness effects may modify drastically the dynamics of the localized nonlinear excitations, even in the framework of the simplest models. Spatially localized modes of the discrete nonlinear lattice equations, called discrete solitons, have appeared in many areas of physics, such as biophysics, nonlinear optics, solid state physics, and more recently in the studies of Bose-Einstein condensates in optical lattices and photonic-crystal waveguides. Widely investigated are the standard discrete nonlinear Schrödinger (DNLS) equation, as well as the completely integrable discrete Ablowitz-Ladik (AL)

* Paper presented at the 8th General Conference of Balkan Physical Union, July 5–7, 2012, Constanta, Romania.

2 Interaction of narrow dark solitons with impurities

375

equation [1–7]. Although the two equations have the same linear properties and yield the same NLS equation in the continuum limit, their nonlinear properties are different. This leads to differences in the dynamics of narrow discrete solitons (bright or dark) for the two models.

An interesting problem due to its theoretical and practical importance is the interaction of solitons with impurities. Extensively studied is the interaction of bright solitons with different point defects (linear, nonlinear and bond) for the DNLS equation [2,8,9] and for the AL model [2,5,10]. The interaction of dark solitons with impurities is also important especially for excitations which obey the Pauli statistics, like spin waves [11–13] or electron excitons [14].

In the present work, we investigate in detail the interaction of dark solitons with impurities in an AL chain. Analytical bound soliton-defect solutions are derived and their stability is analysed. The scattering of solitons from defects is studied numerically for different soliton velocities and impurity parameters.

2. LATTICE NLS EQUATIONS

We start with a generalized set of ordinary differential equations describing a model of intramolecular excitations with the amplitude ( )n tα in a discrete chain with nearest-neighbour interactions:

2 2 2 21 1 1 1i ( )(1 | | ) (| | | | ) | | .

2 2n

n n n n n n n nM J g

t + − + −∂α

= α + α + γ α + α + α α + α α∂

(1)

For methodological reasons, we have included three different nonlinear terms which correspond to different physical systems. The lattice constant equals unity. For molecular crystals, the case g = 0, γ = –1 describes electronic excitations obeying Pauli statistics [12], where M is the intermolecular interaction and J the nonlinear dynamic interaction. There are two competing nonlinear terms – the dynamic (~ J) and the kinematic (~ M). If the dynamic interaction is small and can be neglected, then Eq. (1) turns into the AL equation which is completely integrable and has a bright soliton solution for γ = 1

1 i( )( ) sinh sech e ,

1 1sin sinh , cos cosh ,

nn vt kn tt

L L

v ML k M kL L

− −ωα =

= − ω = (2)

and a dark soliton solution for γ = – 1

2

1 i( )( ) tanh tanh e ,

1 1sin tanh , cos sech .

nn vt kn tt

L L

v ML k M kL L

− − ωα =

= − ω = (3)

M.T. Primatarowa, R.S. Kamburova 3

376

L, v, k, and ω are the width, velocity, wavenumber and frequency of the soliton, respectively.

For J = γ = 0, Eq. (1) describes the dynamics of Bose-type excitations with a nonlinear constant g. In this case, (1) has the form of the standard DNLS equation which is nonintegrable and can have stable soliton solutions only for wide solitons

)1( >>L . Important differences between the solutions of the AL and DNLS equations are that in the AL equation the velocity depends on L and that the type of soliton solution of the AL equation remains the same in the whole Brillouin zone (0 ≤ k ≤ π), while the solution of the DNLS equation changes from a bright to a dark soliton or vice-versa at k = π/2.

In the continuum limit ( ) ( , )n t x tα →α , (1) turns into the nonlinear Schrödinger equation

2 2

i | | ,22

MM Gt x

∂α ∂ α= α + + α α

∂ ∂ (4)

with a composite nonlinear constant ( )G M J g= γ + + , which is completely integrable and, depending on the sign of M/G, has bright or dark soliton solutions.

Equations similar to (1), containing a number of nonlinear terms, have been derived for magnetic chains (see e.g. [11-13]).

In what follows, we shall consider a chain containing a guest molecule with different parameters at site n = 0. This will lead to additional terms in Eq. (1) corresponding to linear, nonlinear and bond defects. Thus, we concentrate on the following equation (J = 0):

,0 ,0 1,0 1

2 2,0 –1,0 –1

1i {[ ( )]2

[ ( )] }(1 | | ) | | ,

nn n n n n

n n n n n n

Mt

M g

+ +∂α

= εδ α + + µ δ + δ α +∂

+ + µ δ + δ α + γ α + α α (5)

where ε and µ characterize the impurity. For 0, 0g = γ ≠ , Eq. (5) corresponds to the perturbed AL equation, and for 0, 0g ≠ γ = to the perturbed DNLS equation. Note that ε corresponds to a linear on-site defect, while µ introduces two neighbouring inter-site linear and nonlinear defects.

We shall investigate the interaction of solitons with localized defects with nonzero ,ε µ . Positive values of the defect parameters correspond to repulsion, while negative values correspond to attraction.

3. BOUND SOLITON-DEFECT SOLUTIONS

Now, we investigate the static case 0k v= = . For the Ablowitz-Ladik model with linear on-site defects ( 0, 0ε ≠ µ = ), Eq. (5) possesses exact bound soliton-defect solutions.


377

For γ = 1, g = 0 the solution is of the bright-soliton type:

–i

.

1 | |( ) sinh sech e ,

1 1cosh , tanh / sinh

tn

ntL L

M ML L

ω α = + ∆

ω = ∆ = ε

(6)

For γ = – 1, g = 0 the solution is of the dark-soliton type:

–i

2

1 | |( ) tanh tanh e

1 2 1sec h , sinh 2 tanh

tn

ntL L

MML L

ω α = + ∆

ω = ∆ = −ε

(7)

If 0>∆ , the function | ( ) |n tα has a single maximum for (6) (minimum for (7)) at 0n = , and if 0∆ < there are two maxima for (6) (two minima for (7)) at n L= ±∆ .

|αn|

2

|αn|2

|αn|2

|αn|2

(a) (b)

(c) (d)

-100 0 100

n

0

3

t

-50 0 50

n

0

3

t

-50 0 50

n

0

3

t

-100 0 100

n

0

3

t

Fig. 1 – Evolution of a dark-soliton-defect solution (9) with L = 5 and M = –2 for: a) ε = 0.05, ∆0 = ∆; b) ε = –0.05, ∆0 = ∆; c) ε = –0.05, ∆0 = 0.8 ∆; d) ε = –0.05, ∆0 = 0.7 ∆. The time is in units of 103/|M|.


378

Similar bound soliton-defect solutions hold for the perturbed DNLS equation with linear on-site defects in the wide soliton limit )1( >>L .

For γ = 0, g = M the solution of the bright-soliton type is:

–iω

2

1 | |( ) sech ,

, tanh / .2

tn

nt eL L

MM L ML

α = + ∆

ω = + ∆ = ε

(8)

For γ = 0, g = –M the solution of the dark-soliton type is:

–iω

2

1 | |( ) tanh ,

2, sinh 2 .

tn

nt eL L

M MML L

α = + ∆

ω = − ∆ = −ε

(9)

The solutions of the bright-soliton type (6) and (8) are extensively studied in [10], while here we concentrate on the investigation of the dark-soliton solutions. In what follows we present results of the numerical solution of Eq. (5) for solitons with the initial form (7) or (9) for M < 0. Figure 1 shows the dark soliton behavior for the two types of discrete equation with a linear impurity. As L = 5 discrete effects do not occur and the results for the two lattices coincide. We analyze the stability of the initial form with a ∆0 which differs from the exact ∆ according to (7) resp. (9). For ∆0 = ∆ the solutions are stable both for ε positive or negative (Fig. 1a, b). If ∆0 ≠ ∆ for ε > 0 the single-peak solution is still stable, while for ε < 0 the double-peak solution is unstable. It begins to oscillate for small perturbations (∆0 = 0.8∆, Fig. 1c) or splits into two solitons which propagate with opposite velocities for larger perturbations (∆0 = 0.7∆, Fig. 1d).

Fig. 2 shows the evolution of a narrow dark soliton for the two types of discrete equation with a linear impurity. As can be expected, the solutions (7) for the AL model are stable (Fig. 2b, b'), while the two-peak solution for the DNLS model (9) exhibits oscillations due to the violation of the wide-soliton limit (Fig. 2a). The single-peak solutions for ε > 0, ∆ > 0 (Fig. 2a', b') correspond to soliton-defect repulsion, while the double-peak solutions with ε < 0, ∆ < 0 (Fig. 2a, b) correspond to attraction. We like to note that for bright solitons (6) the single-peak solution which holds for ε < 0, ∆ > 0 corresponds to soliton-defect attraction, while the double-peak solution with ε > 0, ∆ < 0 corresponds to repulsion [10].


379

|αn|2 |αn|

2

|αn|2 |αn|

2

(a')(a)

(b) (b')

Fig. 2 – Evolution of a narrow dark soliton-defect solution (L = 2.5) of the DNLS equation (9) (a, a') and of the AL equation (7) (b, b'). ε = –0.1 corresponds to attraction (a, b) and ε = 0.1 corresponds to

repulsion (a', b'). All other parameters are the same as in Fig. 1.

|αn|2 |αn|

2

|αn|2 |αn|

2

(a) (a')

(b) (b')

Fig. 3 – Evolution of a narrow dark soliton-defect solution (9) for the DNLS equation (a, a')] and (7) for the AL equation (b, b')] with two neighboring bond defects (ε = 2µ). µ = –0.05 corresponds to attraction (a, b) and µ = 0.05 corresponds to repulsion (a', b'). All other parameters are the same as in Fig. 2.


380

Now, we study the influence of bond defects ( 0, 0ε = µ ≠ ) on the soliton solutions. For bright solitons, the problem was considered in [10, 15] in detail. Here we concentrate on narrow dark solitons (Fig. 3). The input solution is chosen in the form (7) for the AL model (Fig. 3b, b') or (9) for the DNLS model (Fig. 3a, a') with ε = 2µ (the guest molecule alters the interaction energy with the two neighbors). When the bond defect is attractive, the soliton maxima oscillate as the oscillation period and amplitude for the AL model (Fig. 3b) are greater than those for the DNLS model (Fig. 3a). A comparison with the results for the linear defect ε (Fig. 2) shows that the interaction with the inter-site defect µ compensate the discreteness effects and the approximation ε = 2µ is a good one even for narrow solitons. The small oscillations for the AL model (Fig. 3b) are due to the appearance of nonlinear inter-site defects in Eq. (5) with 0µ ≠ . For repulsive defects, the soliton bound state is stable for both models (Fig. 3a', b').

4. SCATTERING OF DARK SOLITONS FROM DEFECTS

Of considerable interest is the scattering of solitons from defects. The scattering of wide dark solitons from point defects was investigated in [13]. In this section we consider the interaction of AL solitons of the form (3) with defects. This allows us to study narrow solitons with an arbitrary wavenumber k in the Brillouin zone (0 ≤ k ≤ π). The evolution depends strongly on the initial soliton velocity which is related to k through (3) and on the sign of the defect. The more interesting case is when the defect is repulsive ( , 0ε µ > ).

First we present the dynamics for the dark solitons of the DNLS equation (Fig. 4). For a given initial velocity (wavenumber) there exists a critical value of the defect εc, µc below which the soliton is transmitted and above which it is reflected. As can be seen from Fig. 4 for slow solitons the two modified intermolecular bonds are equivalent to a linear point defect with strength 2µc and we have εc = 2µc.

On Fig. 5 we present the dynamics for the dark solitons of the AL equation. In this case εc ≠ 2µc. Obviously µc > εc/2. As a whole, linear on-site defects with comparable strength to bond defects induce stronger perturbations. This is owing the different velocity dependence of the corresponding perturbing terms for narrow solitons. With increase of the velocity the effect becomes stronger. The difference in the scattering pattern of narrow solitons from on-site and bond defects vanishes for wide and slow solitons.


381

-400 0 400

n

0

3

6

t

-400 0 400

n

0

3

6

t

|αn|2 |αn|

2

|αn|2 |αn|

2

(a) (a')

(b) (b')

-400 0 400

n

0

3

6

t

-400 0 400

n

0

3

6

t

Fig. 4 – Scattering of a DNLS dark soliton with k = 0.0157 from a repulsive point defect with ε (a, a') and from a comparable bond-defect with µ = ε/2 (b, b'). Transmission for ε = 0.0051 (a, b). Repulsion

for ε = 0.0052 (a', b'). All other parameters are the same as in Fig. 1.

-400 0 400

n

0

3

6

t

-400 0 400

n

0

3

6

t

-400 0 400

n

0

3

6

t

-400 0 400

n

0

3

6

t

|αn|2 |αn|

2

|αn|2 |αn|

2

(a) (a')

(b) (b')

Fig. 5 – Scattering of an AL dark soliton with k = 0.0157 from a repulsive point defect with ε (a, a')

and from a comparable bond-defect with µ = ε/2 (b, b'). ε = 0.0050 for (a), (b) and ε = 0.0051 for (a'), (b'). All other parameters are the same as in Fig. 1.


382

Acknowledgments. This work is supported in part by the National Science Fund of Bulgaria under Grant No. DO 02-264.

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(2009). 10. M. T. Primatarowa, R. S. Kamburova, and K. T. Stoychev, J. Optoel. Adv. Mat., 11, 1388 (2009). 11. R. F. Wallis, D. L. Mills, and A. D. Boardman, Phys. Rev. B, 52, R3828 (1995). 12. S. Rakhmanova and D. L. Mills, Phys. Rev. B, 58, 11458 (1998). 13. M. T. Primatarowa and R. S. Kamburova, J. Phys.: Conf. Ser., 253, 012022 (2010). 14. M. T. Primatarowa, K. T. Stoychev, and R. S. Kamburova, Phys. Rev. B, 52, 15291 (1995). 15. M. T. Primatarowa, K. T. Stoychev, and R. S. Kamburova, Phys. Rev. E, 77, 066604 (2008).

interaction of narrow dark solitons with impurities … · 2 interaction of narrow dark solitons...

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