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**Authors :**O.P. Swami, Vijendra Kumar, A.K. Nagar -
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**Volume & Issue :**AMRP – 2013 (Volume 1 – Issue 01) -
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#### Stability of Parametrically Driven Dark Lattice Solitons in Nanoscale Systems

O.P. Swami*, Vijendra Kumar, A.K. Nagar

Department of Physics, Govt. Dungar College, Bikaner, Rajasthan 334001, India

*omg1789@gmail.com

wave function, X n satisfies the stationary equation

Abstract

C2 X n X X X 0

3

n

n

n

(2)

In this paper, we consider a parametrically driven nanoscale device modelled by discrete nonlinear Schrodinger equation. To determine the stability of fundamental dark solitons, analytical and numerical calculations are performed. We show that a parametric driving can change the stability of dark solitons. Stability windows of fundamental dark solitons are presented and stability approximations are derived using perturbation

To examine the stability of X n , we introduce the linearization ansatz

n X n Yn

where 1, and substitute this into Eq.(1), it yield the following linearized equation at O( ) :

n 2 n n n n n n n

iY C Y 2 X 2 Y X 2 Y Y Y (3) writing Yn An iBn , and linearizing in , we find

theory, with numerical results.

A A

(4)

n N n

where

N

Bn Bn

0 M (C)

(5)

1. Introduction

M (C) 0

n

We consider a nanoelectromechanical system (NEMS)

and

M (C) C2

( X 2 )

governed by a parametrically driven discrete nonlinear Schrodinger (PDNLS) equation

M (C) C (3X 2 )

2

n

Let the eigenvalues of N be denoted by id, which

i

C

2

(1)

implies that

X is stable if Im(d)=0. Since Eq. (5) is linear,

n

n

n 2 n n n n n

where n n (t)

is a real-valued wave function and n is

we can eliminate one of the eigenvectors, for instance B ,

the lattice site index. The overdot and the overline denote then we obtain the following eigenvalue problem

the time derivative and complex conjugation, respectively.

M (C)M (C) d2 A

A

(6)

n n

C is the coupling constant between two adjacent sites,

2n n1 n1 2n

is the one-dimensional (1-D)

discrete Laplacian, is the parametric driving coefficient with frequency . Parametrically driven electromechanical resonators have been discussed in Ref. [1,2]. Discrete bright

3. Analytical calculation

soliton type systems have been discussed before e.g.in Ref.

In the uncoupled limit C 0 , we denote the exact

[3-7]. In undriven case, Eq. (1) reduces to the standardsolutions of (2) by X X (0) , in which each

X (0)

must take

discrete nonlinear Schrodinger (DNLS) equation, which

n n n

appears in many applications [8]. The same equation also applies to the study of discrete modulation instability in parametrically driven optical lattice [9]. The long bosonic Josephson junctions and Bose-Einstein condensates trapped Optical lattices are also studied using same equation [10,11].

In this paper we examine the condition for stability of fundamental dark soliton in defocusing PDNLSE.

For small C, the perturbation theory is used, followed by numerical computations in MATLAB.

2. Analytical setup and Perturbative results

Stationary solution of system (1) in the form of

one of the three values given by 0,

.

n X n , where X n is a time independent and real-valued

Following Ref. [1], using a perturbative expansion, the dark soliton solutions are obtained as

6. Figures

2

1 C / , n 1 1

X n 0 ,

n 0

(7)

1 C /

2

, n 1

0.8

0.6

and its eigenvalues for small C are given by

2 2 4C O(C2 )

(8)

0.4

stable

unstable

The instability of discrete dark soliton is due to the collision of the smallest eigenvalue (8) with an eigenvalue bifurcating from lower and upper edge of continuous

0.2

stable

spectrum, for small and large , respectively. Equating these quantities we find the critical value of as a function of the coupling constant C i.e.

0

0 0.2 0.4 0.6 0.8 1

C

Figure. 1

The stability-instability region in the two parameter space

cr

1 0.4 1.6C 0.2 92 28C 16C2

cr

2 2 4C

(9)

(10)

C . The solid red and black lines are the analytical approximations of Eqs. (9) and (10). Red and black dash- dotted lines are their respective numerical approximations.

cr

Both 1

and

2 give approximate boundaries of the

cr

instability region in the (C, ) plane.

1

Comparison with numarical calculations

0

Xn

-1

-25 -20 -15 -10 -5 0 5 10 15 20 25

n

Using NewtonRaphson method, we have numerically solved the static equation (2), and analyzed the stability of the numerical solution by solving the eigenvalue problem (4). We consider 0.9 in the model.

Figure 1 provides a full description of the dynamics of the parametrically driven DNLS model regarding the intervals of stability/instability of the model. Analytical prediction for the stability range as obtained by the conditions of collision of the phase mode eigenfrequency with the continuous spectrum from Eqs. (9)-(10).

Figure 2,3 illustrate the typical instability scenario for different values of parametric drives and coupling constant C . Where the left panels present the structure of just before the collision (stable) whereas the right panels represent just after the collision (unstable).

Conclusions

In this paper, we have considered a parametrically driven nanoelectromechanical system which is modeled by DNLS. The stability of fundamental dark solitons is determined using perturbative analysis, which is followed by numerical computations in MATLAB. We have shown that the presence of parametric driving can change the stability of governed model. It destabilizes the dark soliton. We have

1

Im(d)

0

-1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Re(d)

(a)

1

Xn

0

-1

-25 -20 -15 -10 -5 0 5 10 15 20 25

n

1

Im(d)

0

-1

-1.5 -1 -0.5 0 0.5 1 1.5

Re(d)

(b)

Figure. 2

The eigenvalue structure of dark soliton for 0.03 and

C 0.008 (a), as well as C 0.03 (b).

considered the frequency of parametric drive as 0.9 ,

which is smaller than that in Ref. [1]. The result is a downward shift in C graph (Figure 1), which is expected from analysis.

2

Xn

0

-2

-25 -20 -15 -10 -5 0 5 10 15 20 25

n

1

Im(d)

0

-1

-1.5 -1 -0.5 0 0.5 1 1.5

Re(d)

(a)

2

Xn

0

-2

-25 -20 -15 -10 -5 0 5 10 15 20 25

n

-3

x 10

5

Im(d)

0

-5

-1 -0.5 0 0.5 1

Re(d)

(b)

Figure. 3

The eigenvalue structure of dark soliton for 0.4 and

C 0.01 (a), as well as C 0.028 (b).

7. References

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M. Syafwan, "The existence and stability of solitons in discrete nonlinear Schrodinger equations", Ph.D. Thesis, Nottingham University, 2012.

O. P. Swami, V. Kumar, and A. K. Nagar, Int. J. Mod. Phys.

#### 22, 570-575 (2013).

H. Susanto, Q. E. Hoq, and P. G. Kevrekidis, Phys. Rev. E4, 067601-4 (2006).

D. Hennig and G. Tsironis, Phys. Rep. 307, 333- 432 (1999).

G. L. Alfimov, V. A. Brazhnyi, and V. V. Konotop, Phys. Rev.

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D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis,

Physica. D212,1-19 (2005).

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O. P. Swami, A. Sharma, and A. K. Nagar, Discrete modulational instability in parametrically driven optical lattices, AIP Conference Proceedings 1536, American Institute of Physics, Melville, NY, 2013, pp. 757-758. D. Hennig and G. Tsironis, Phys. Rep. 307, 333- 432 (1999).

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