Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential

Wei- Ling Zhu; Ying-Ji He

doi:10.1364/OE.18.017053

1. Introduction

Optical solitons are self-trapped light beams or pulses supported by the balance between diffraction and/or dispersion (in the spatial and/or temporal domain) and nonlinearities of various types [1–4]. Spatial solitons have a potential for the use in all-optical data-processing schemes designed for switching, pattern recognition, parallel information storage and other applications to all-optical data processing schemes [5,6]. Motivated by the previous works in which it was studied a similar effect, viz., the drift of cavity solitons induced by a phase gradient in the holding laser beam [7–11], in this paper, we derived the analytical and numerical results for stable moving dissipative solitons based on the cubic-quintic (CQ) complex Ginzburg-Landau (CGL) equations by utilizing a linear potential. It is relevant to mention that CGL equations represent a broad class of models with applications to superconductivity and superfluidity, nonlinear optics, hydrodynamics and plasmas, and quantum field theories [12–23]. The stability criterion for stationary dissipative soliton solutions in CQ CGL equations without external potential has been well studied [24]. In this work, we study stable moving solitons in the 1D, 2D, and 3D equations of the CQ CGL type, which include a linear potential (alias linear refractive index modulation, LRIM, in terms of optics). The constant force induced by the linear potential leads to solitons drift. Analytical considerations by using the generalized variational approximation (VA) and direct simulations reveal the conditions of stable moving dissipative solitons and solitons trajectory. The largest slope which admits the existence of stable solitons is found too.

2. The model

We consider the CQ CGL equation of the general form which describes the propagation of an electromagnetic field with local amplitude u in an optical medium [19,23,24]:

i u_{z} + (1 / 2) Δ u + {| u |}^{2} u + ν {| u |}^{4} u = i R [u] - α x u,

where

Δ = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2} + \partial^{2} / \partial t^{2}

is the transverse Laplacian (x and y are the transverse coordinates, and t is the temporal variable, while z is the propagation distance). Further, ν accounts for the quintic self-defocusing, and

R [u] = δ u + β Δ u + ε {| u |}^{2} u + μ {| u |}^{4} u,

where δ is the coefficient of the linear loss (δ <0) or gain (δ >0), μ is the quintic-loss parameter, ε is the cubic-gain coefficient, and β denotes effective diffusion (viscosity). The last term in Eq. (1) is the LRIM, i.e., an effective linear potential that may be implemented via the local modulation of the refractive index in the transverse plane. The LRIM has been proposed as a tool for the controls of soliton dynamics in Kerr medium [25] and soliton interaction in dissipative system [26].

3. The stability of dissipative solitons within the framework of the VA

To analyze the stability of the moving dissipative solitons with a linear potential, we use the following isotropic ansatz in the general 3D case (cf. Refs [24,26].):

u = A (z) \exp {- \frac{{[x - q (z)]}^{2} + y^{2} + t^{2}}{2 w^{2} (z)} + i c (z) [{(x - q (z))}^{2} + y^{2} + t^{2}] + i L (z) [x - q (z)] + i ψ (z)},

and its straightforward reductions in the 2D and 1D cases. Here, A, w, c, and ψ represent the amplitude, width, wave front curvature (alias chirp), and overall phase, respectively. Peak position q(z) and conjugate momentum L(z) account for the motion of the soliton along the direction of x, which results from the combination of the LRIM potential and friction force induced by the transverse-viscosity term in Eq. (1).

Firstly, we use the generalized VA to solve the CGL equation, as proposed in Refs [24,27]. Using ansatz (2), after a straightforward algebra the following system of evolution equations for its parameters is derived in the D-dimensional case (D = 1, 2, 3):

d A / d z = [δ + 2^{- (2 + D / 2)} (4 + D) ε A^{2} + 3^{- (1 + D / 2)} (3 + D) μ A^{4} - D β w^{- 2} - D c - β L^{2}] A,

d w / d z = [- 2^{- (1 + D / 2)} ε A^{2} - 2 \cdot 3^{- (1 + D / 2)} μ A^{4} + β w^{- 2} - 4 β c^{2} w^{2} + 2 c] w,

d c / d z = {(2 w^{4})}^{- 1} - 2^{- (1 + D / 2)} A^{2} w^{- 2} - 2 \cdot 3^{- (1 + D / 2)} ν A^{4} w^{- 2} - 4 β c w^{- 2} - 2 c^{2},

d L / d z = α - 2 β L w^{- 2} (1 + 4 c^{2} w^{4}),

d ψ / d z = 2 D β c - D {(2 w^{2})}^{- 1} - 2^{- (2 + D / 2)} (4 + D) A^{2} + 3^{- (1 + D / 2)} (3 + D) ν A^{4} + α q + L^{2} / 2 - 4 β c L^{2} w^{2},

d q / d z = L - 4 β c L w^{2} .

Introducing small perturbations about a fixed point of Eqs. (3), A = A ₀ + ∆α, w = w ₀ + ∆w, c = c₀ + ∆c, L = L₀ + ∆L, we derive the respective linearized equations:

d (Δ a) / d z = b_{1} Δ a + b_{2} Δ w + b_{3} Δ c + b_{4} Δ L,

d (Δ w) / d z = b_{5} Δ a + b_{6} Δ r + b_{7} Δ c,

d (Δ c) / d z = b_{8} Δ a + b_{9} Δ r + b_{10} Δ c,

d (Δ L) / d z = b_{11} Δ w + b_{12} Δ c + b_{13} Δ L,

where we define

b_{1} = 2^{- (1 + D / 2)} (4 + D) ε A_{0}^{2} + 4 \cdot 3^{- (1 + D / 2)} (3 + D) μ A_{0}^{4},

b_{2} = 2 A_{0} D β w_{0}^{- 3},

b_{3} = - D A_{0},

b_{4} = - 2 β A_{0} L_{0},

b_{5} = - 2^{- D / 2} ε A_{0} w_{0} - 8 \cdot 3^{- (1 + D / 2)} μ A_{0}^{3} w_{0},

b_{6} = - (2 β w_{0}^{- 2} + 8 β c_{0}^{2} w_{0}^{2}),

b_{7} = 2 w_{0} - 8 c_{0} w_{0}^{3} β,

b_{8} = - A_{0} {(2^{D / 2} w_{0}^{2})}^{- 1} - 8 v A_{0}^{3} {(3^{1 + D / 2} w_{0}^{2})}^{- 1},

b_{9} = - 2 w_{0}^{- 5} + 8 β c_{0} w_{0}^{- 3} + A_{0}^{2} {(2^{D / 2} w_{0}^{3})}^{- 1} + 4 ν A_{0}^{4} {(3^{(1 + D / 2} w_{0}^{3})}^{- 1},

b_{10} = - (4 c_{0} + 4 β / w_{0}^{2}),

b_{11} = 4 β L_{0} w_{0}^{- 3} - 16 β c_{0}^{2} w_{0} L_{0},

b_{12} = - 16 β c_{0} w_{0}^{2} L_{0},

b_{13} = - (2 β w_{0}^{- 2} + 8 β c_{0}^{2} w_{0}^{2}) .

Solutions to Eqs. (4) are looked for with the z-dependence in the form of exp(λz), where the eigenvalues are determined by the fourth-order equation with real coefficients,

| \begin{matrix} b_{1} - λ & b_{2} & b_{3} & b_{4} \\ b_{5} & b_{6} - λ & b_{7} & 0 \\ b_{8} & b_{9} & b_{10} - λ & 0 \\ 0 & b_{11} & b_{12} & b_{13} - λ \end{matrix} | = 0.

The fixed-point solution to Eqs. (4) is stable when the roots of Eq. (5) have negative or zero real parts. Rewriting the above equation in the form of λ⁴ + a ₁λ³ + a ₂λ² + a ₃λ + a ₄ = 0, the stability condition amounts to the Routh-Hurwitz criterion, i.e., the coefficients a _1,2,3 must satisfy four inequalities:

a_{1} > 0, | \begin{matrix} a_{1} & 1 \\ a_{3} & a_{2} \end{matrix} | > 0, | \begin{matrix} a_{1} & 1 & 0 \\ a_{3} & a_{2} & a_{1} \\ 0 & a_{4} & a_{3} \end{matrix} | > 0, | \begin{matrix} a_{1} & 1 & 0 & 0 \\ a_{3} & a_{2} & a_{1} & 1 \\ 0 & a_{4} & a_{3} & a_{2} \\ 0 & 0 & 0 & a_{4} \end{matrix} | > 0,

where

a_{1} = - (b_{1} + b_{6} + b_{10} + b_{13}),

a_{2} = - b_{2} b_{5} - b_{3} b_{8} - b_{7} b_{9} + b_{6} b_{10} + b_{6} b_{13} + b_{10} b_{13} + b_{1} (b_{6} + b_{10} + b_{13}),

\begin{array}{l} a_{3} = b_{1} (b_{7} b_{9} - b_{6} b_{10}) - b_{4} (b_{5} b_{11} + b_{8} b_{12}) - b_{13} [b_{1} b_{6} - b_{7} b_{9} + b_{10} (b_{1} + b_{6})] \\ + b_{3} [b_{8} (b_{6} + b_{13}) - b_{5} b_{9}] + b_{2} [b_{5} (b_{1}_{0} + b_{13}) - b_{7} b_{8}], \end{array}

\begin{array}{l} a_{4} = b_{4} [- b_{7} b_{8} b_{11} + b_{6} b_{8} b_{12} + b_{5} (b_{10} b_{11} - b_{9} b_{12})] \\ + b_{13} [b_{3} (b_{5} b_{9} - b_{6} b_{8}) + b_{2} (b_{7} b_{8} - b_{5} b_{10}) + b_{1} (b_{6} b_{10} - b_{7} b_{9})] . \end{array}

Equation (3)d) relates the linear momentum L to the slope of the LRIM α, diffusivity β and parameters of the soliton,

L_{0} = α w_{0}^{2} / [2 β (1 + 4 c_{0}^{2} w_{0}^{4})] .

Substituting

L_{0} = α w_{0}^{2} / [2 β (1 + 4 c_{0}^{2} w_{0}^{4})]

into Eq. (3)f), one gets the expression of the peak position of soliton q(z):

q (z) = α w_{0}^{2} (1 - 4 β c_{0} w_{0}^{2}) / [2 β (1 + 4 c_{0}^{2} w_{0}^{4})] z .

From Eq. (7), the peak position q(z) linearly changes with the propagation distance z when fixing the slope of the LRIM α and the parameters of soliton, which is verified by the numerical simulations below, see Figs. 2(a) , 3(a) , and 4(a) . This is the consequence that the transverse push forces from the linear potential and the friction force induced by the effective diffusion (with β term in Eq. (1)) reach to balance. If without the effective diffusion, β = 0, from Eqs. (3d) and (3c), obviously, dL/dz = α and dq/dz = L, and thus q(z) ~αz², which exhibits quadratic increase with z, similar to the moving soliton in the conservative system [25]. Using Eqs. (3a)–(3d), we find A₀, w₀, c₀, and L₀ for given α, and substitute the results into inequalities (6). Solving them in a numerical form, we thus identify the largest value of the slope, α = α_cr, up to which (at α≤α_cr) the solitons are stable.

Fig. 2 Numerical simulation for moving 1D dissipative soliton. (a) Stable propagation of soliton with α = 0.1 and (b) soliton decay with α = 0.3.

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Fig. 3 Numerical simulation for moving 2D dissipative soliton in (a) Stable propagation of soliton with α = 0.2 and (b) soliton decay with α = 0.45.

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Fig. 4 Numerical simulation for moving 3D dissipative soliton in (a) Stable propagation of soliton with α = 0.04 and (b) soliton decay with α = 0.1. In (a) and (b), the isosurfaces $| u | = 0.8.$

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4. Results and analysis

The generic case can be adequately represented by the choice of parameters δ = −0.5, β = 0.5, μ = −1, and for the 1D case: ε = 1.85 and ν = −0.115; for the 2D case: ε = 2.4 and ν = −0.01; and for the 3D case: ε = 2.5 and ν = −0.115. For all cases, the stable soliton solutions were obtained in the numerical form, see Figs. 1(a) , 1(b), and 1(c). The robustness of the stable solitons is additionally tested in direct simulations by multiplying Eq. (3) with [1 + ρ(x,y,t)], where ρ(x,y,t) is a Gaussian random function, whose maximum is 10% of the soliton’s amplitude.

Fig. 1 Profiles of solitons for 1D, 2D, and 3D cases, respectively, in (a), (b), and (c).

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In 1D case, the largest slope of the linear potential at conditions (6), which was derived by means of the VA, admits a stable single-soliton solution at α_cr ≈0.24. On the other hand, direct simulations of Eq. (1) yield α_cr ≈0.23, also indicated in Ref [26]. By using the same method, in 2D case, the largest slope of the linear potential is α_cr ≈0.4, and direct simulations of Eq. (1) obtain α_cr ≈0.41 [26], while in 3D case, the largest slope of the linear potential is α_cr ≈0.08, and direct simulations of Eq. (1) find α_cr ≈0.082 [26]. All these cases, the largest slope of the linear potential induced by numerical simulation demonstrates the accuracy provided by the VA. The numerical examples are shown in Figs. 2(a) and 2(b), Figs. 3(a) and 3(b), Figs. 4(a) and 4(b), respectively, for the 1D, 2D, and 3D cases. It is clearly demonstrated that soliton can maintain stable propagation when the slope of linear potential is smaller than the critical value [Figs. 2(a), 3(a), and 4(a)]. On contrary, for the slope of linear potential is larger than the critical value, soliton will disappear due to loss large energy [Figs. 2(b), 3(b), and 4(b)], because that stronger linear modulation pushes soliton to move faster, and the effective diffusion of model (1) leads to larger friction force when the soliton is drifted faster. In addition, the stable moving solitons exhibit linearly transverse drift, as above analytical result given by Eq. (7).

5. Conclusions

In this work, we have studied the stability of isolated dissipative solitons in the 1D, 2D, and 3D versions of the complex Ginzburg-Landau equation with cubic-quintic nonlinearity. A key element of the model is the linear potential term, which represents the linear refractive-index modulation effect in laser-cavity models. The existence and stability of single solitons was studied using the generalized version of the variational approximation based on the Gaussian ansatz. In particular, the largest value of the slope of the linear potential was identified, below which the solitons are stable. Predictions of the variational approximation are well corroborated by direct numerical results. The results suggest a simple method allowing one to stabilize and control one- and multidimensional moving dissipative solitons, with potential applications to the all-optical data processing systems.

Acknowledgment

This work was supported by Guangdong Province Natural Science Foundation of China (Grant No. 9451063301003516).

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Stability conditions for moving dissipative solitons in one- and multidimensional systems with a linear potential

Abstract

1. Introduction

2. The model

3. The stability of dissipative solitons within the framework of the VA

4. Results and analysis

5. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (16)

Optics Express