Solitons in nonlocal nonlinear kerr media with exponential response function

doi:10.1364/OE.20.007469

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Solitons in nonlocal nonlinear kerr media with exponential response function

Jian Jia and Ji Lin »View Author Affiliations

Optics Express, Vol. 20, Issue 7, pp. 7469-7479 (2012)
http://dx.doi.org/10.1364/OE.20.007469

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▸ Article Outline
– Abstract
1. Introduction
2. The classical Lie-group reduction
3. Analytic solutions and numerical solution
4. Conclusions
– References and links

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Abstract

In this paper, we find some exact analytical solutions including bright soliton solution, dipole-mode soliton solution, double soliton solution and periodic solution when a slit laser beam propagates in Kerr-type nonlinear, nonlocal media with exponential response function. Furthermore, we address the energy flow is a monotonically growing function of d₂ and the Hamiltonian decreases while the energy flow increases. And we also obtain an Airy-like soliton by numerical method.

1. Introduction

It is well known that solitons are self-guided wave packets propagating in nonlinear media that keep their self-trapped shape. And the balance between the material nonlinearity and diffraction in the spatial domain or dispersion in the temporal domain lead to the existence of the optical solitons [1

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).

]. During the last two decades spatial solitons have become a subject of intense investigation because of their unique physical features [2

M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998). [CrossRef]

]. Properties of solitons supported by media with local nonlinear response are now well established. However, the recent interest in the study of nonlocal optical solitons was stronger because of experimental observations and theoretical treatments of self-trapping effects and spatial solitary waves in different types of nonlocal nonlinear media. The nonlocality which has been shown to profoundly affect the properties and interactions of solitons is a characteristic feature of nonlocal nonlinear media. Spatial nonlocality means that the response of the medium at a particular point is not only determined by the wave intensity at that point, but also depends on the wave intensity in its vicinity [3

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004). [CrossRef]

]. Principally new effects of nonlocality have been studied in photorefractive crystals [4

M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998). [CrossRef]

, 5

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, 1110–1113 (1997). [CrossRef]

], nematic liquid crystals [6

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002). [CrossRef]

, 7

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005). [CrossRef] [PubMed]

], plasmas [8

A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).

], thermo-optical materials [9

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]

], and Bose-Einstein condensates with long-range interparticle interactions [10

L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000). [CrossRef] [PubMed]

,11

V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000). [CrossRef]

]. For example, the nonlocal nonlinear response suppresses the modulation instability of the plane waves in focusing media [12

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001). [CrossRef]

,13

M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003). [CrossRef]

]; it can arrest catastrophic collapse of multidimensional beams [14

S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985). [CrossRef]

–16

D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001). [CrossRef]

] and stabilizes complex soliton structures, such as vortex solitons [17

W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004). [CrossRef]

,18

J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]

]; it also can make the colliding solitons merge into a standing wave [19

Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009). [CrossRef]

] and the dark solitons form bound states [20

N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004). [CrossRef] [PubMed]

] which was observed in Ref. [21

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006). [CrossRef] [PubMed]

]. Furthermore, stable dipole solitons in a medium with a Gaussian response function also have been predicted recently [22

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]

]. Besides, the quadratic nonlinear materials also display a nonlocal nonlinearity [23

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003). [CrossRef]

] which can explain (1+2) -dimensional X waves based on a Bessel type nonlocal response function [24

P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006). [CrossRef]

]. And the group-velocity mismatch (GVM) induces asymmetric nonlocal Raman responses that accurately explain the stationary and nonstationary regimes in cascaded quadratic soliton compressors [25

M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007). [CrossRef] [PubMed]

]. At the same time, the compression limit was already studied in Ref. [26

M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008). [CrossRef] [PubMed]

]. Thus, nonlocality has become important in nonlinear optics recently.

As far as we know, the relationship between the width of the response function and the width of the intensity profile divides the degree of nonlocality into four types, namely the local, weakly nonlocal, general and strongly response [12

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001). [CrossRef]

]. However, in this paper, our aim is to study the nonlocal focusing Kerr-type medium with exponential response function. And the features of modulational instability in nonlocal Kerr media with exponential response function have been discussed in Ref. [27

J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002). [CrossRef]

In this Letter, our theoretical model is based on two coupled phenomenological equations for dimensionless complex light field amplitude q and nonlinear correction to the refractive index n describing the propagation of a slit laser beam along the ξ axis of a nonlocal focusing Kerr-type medium:

\begin{array}{l} A & = i \frac{\partial q}{\partial ξ} + \frac{1}{2} \frac{\partial^{2} q}{\partial η^{2}} + q n = 0, \\ B & = n - d \frac{\partial^{2} n}{\partial η^{2}} - {| q |}^{2} = 0, \end{array}

(1)

where the two variables, namely, η and ξ stand for the transverse and longitudinal coordinates scaled to the the beam width and the diffraction length, respectively, and the parameter d stands for the degree of nonlocality of the nonlinear response. The model (1) for diffusive non-linerities becomes identical to the model for quadratic nonlinearity which is also determined by an exponential response function [23

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003). [CrossRef]

]. When d → 0, Eq. (1) describes a local nonlinear response. Contrarily, the case d → ∞ corresponds to the strongly nonlocal response. The nonlinear contribution to refractive index is given by

n = - \int_{- \infty}^{\infty} G (η - λ) {| q (λ, ξ) |}^{2} d λ

where G(η) = (1/2d^1/2)×exp(− |η|/d^1/2) is the response function of the nonlocal medium. As far as we know, such equations have been already studied by the numerical method and the bright, dark, and gray solitons were obtained. It had been proved that these solitons could exist under certain values of the degree of nonlocality of the nonlinear response. Besides, multiple-mode solitons also are found and bound states are stable if they contain fewer than five solitons [28

Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. 30, 3171–3173 (2005). [CrossRef] [PubMed]

]. However, it is worth noting that all the solitons above are found by numerical method, not by analytic method. In order to known more about the characters of this system, looking for the analytic solution is particularly important.

It is well known that Eq. (1) is non-integrable system, so it is difficult to find the exact solutions analytically. However, in this paper, a few analytically exact solutions of Eq. (1) have been obtained by the classical Lie-group method. This method is effective for searching analytically the exact solution by reducing the evolution equation to some similarity equations firstly. Then we search some exact solutions of these similarity equations in different cases by test solution method. At the same time, we also use numerical method to get numerical solution.

The paper is organized as follows. In Sec.II, the process of reducing the evolution equation to some similarity equations by the classical Lie-group method is described in detail. Sec.III is devoted to find solutions of similarity equations. We obtain some exact solutions and numeric soliton solutions from these similarity equations and also figure out their energy flow and the Hamiltonian. Sec.IV is conclusion.

2. The classical Lie-group reduction

In order to reduce Eq. (1) to ordinary differential equations and obtain analytical solutions, we apply the classical Lie-group method [29

P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986). [CrossRef]

, 30

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973). [CrossRef]

] to Eq. (1). First we introduce the Lie group of transformations of independent and dependent variables (η,ξ,q,q^*,n)

{η, ξ, q, q^{*}, n} \to {η, ξ, q, q^{*}, n} + ε {χ, τ, ζ, ζ^{*}, δ}

(2)

where ε is an infinite parameter. The corresponding of infinitesimal symmetries is the set of vector fields of the form

V = χ \partial_{x} + τ \partial_{z} + ζ \partial_{u} + ζ^{*} \partial_{u^{*}} + δ \partial_{n} .

(3)

According to the infinitesimal invariance criterion for symmetry, Eq. (1) is invariant under the transformations (2) if and only if

\begin{array}{l} \Pr^{(2)} V \cdot A = 0, \\ \Pr^{(2)} V \cdot B = 0. \end{array}

(4)

Pr⁽²⁾V is the second prolongation of V

\Pr^{(2)} V = V + ζ^{ξ} \partial_{q_{ξ}} + ζ^{η η} \partial_{q_{η η}} + δ^{η η} \partial_{q}_{η η},

(5)

where ζ^ξ, ζ^ηη and δ^ηη are given explicitly in terms of invariants χ, τ, ζ and n and their derivatives,

\begin{array}{l} ζ^{ξ} = D_{ξ} (ζ - χ q_{η} - τ q_{ξ}) + χ q_{η ξ} + τ q_{ξ ξ}, \\ ζ^{η η} = D_{η η} (ζ - χ q_{η} - τ q_{ξ}) + χ q_{η η η} + τ q_{η η ξ}, \\ δ^{η η} = D_{η η} (δ - χ n_{η} - τ n_{ξ}) + χ n_{η η η} + τ n_{η η ξ} . \end{array}

(6)

Solving Eq. (4) by substituting of Eqs. (5) and (6) into it and collecting together the coefficients of like-derivative terms of q and n, then setting all of them to zero, we get a system of linear partial differential equations from which we can find χ, τ, ζ, ζ^* and δ,

χ = b ξ + c_{0}, τ = c_{1}, ζ = i b q η + i a q, ζ^{*} = - i b q^{*} η + i a q^{*}, δ = 0,

(7)

where a, b, c₀ and c₁ are constants.

We can have a similarity variable ψ, similarity solutions q and n by integrating the following characteristic equations:

\frac{d η}{χ} = \frac{d ξ}{τ} = \frac{d q}{ζ} = \frac{d q^{*}}{ζ^{*}} = \frac{d n}{δ} .

(8)

Taking the constant b is zero or not in Eq. (7), we can obtain two different types of similarity reductions of Eq. (8). When b is zero, Eq. (7) becomes

χ = c_{0}, τ = c_{1}, ζ = i a q, ζ^{*} = i a q^{*}, δ = 0.

(9)

By solving the characteristic equations Eq. (8), We have

\begin{matrix} ψ = c_{1} η - c_{0} ξ, \\ q = u (ψ) exp (\frac{i a ξ}{c_{1}}) exp (i \int (\frac{c_{2}}{u {(ψ)}^{2}} + \frac{c_{0}}{c_{1}^{2}}) d ψ), \\ n = Q, \end{matrix}

(10)

where the integral invariant u(ψ) stands for a normalized real function. The quantity

\frac{c_{0}}{c_{1}}

which is associated with the angle between the central wave vector and the propagation axis, represents a transverse velocity. Substitution of Eq. (10) into Eq. (1) yields the similarity reduced equations

- \frac{2 a}{c_{1}} u + \frac{c_{0}^{2}}{c_{1}^{2}} u - \frac{c_{2}^{2} c_{1}^{2}}{u^{3}} + u_{ψ ψ} c_{1}^{2} + 2 u Q = 0,

(11)

Q - d Q_{ψ ψ} c_{1}^{2} - u^{2} = 0,

(12)

where u_ψ = du/dψ. In the other case, we will study Eq. (7) with b ≠ 0. Now the substitution of Eq. (7) into Eq. (8) arrives at

\begin{matrix} ψ = c_{1} η - \frac{b ξ^{2}}{2} - c_{0} ξ, \\ q = u (ψ) exp (\frac{i b}{c_{1}^{2}} (c_{1} η ξ - \frac{b ξ^{3}}{3} - \frac{c_{0} ξ^{2}}{2} + \frac{c_{1} a ξ}{b})) exp (i \int (\frac{c_{3}}{u {(ψ)}^{2}} + \frac{c_{0}}{c_{1}^{2}}) d ψ), \\ n = Q . \end{matrix}

(13)

where c₃ is an integral constant. Making use of Eqs. (1) and (13), we obtain

u_{ψ ψ} c_{1}^{2} - (\frac{2 b ψ}{c_{1}^{2}} + \frac{2 a}{c_{1}} - \frac{c_{0}^{2}}{c_{1}^{2}} - 2 Q) u - \frac{c_{3}^{2} c_{1}^{2}}{u^{3}} = 0,

(14)

Q - d Q_{ψ ψ} c_{1}^{2} - u^{2} = 0.

(15)

3. Analytic solutions and numerical solution

It is noted that Eqs. (11) and (12) are complicated nonlinear ordinary differential equations(ODEs) which are difficult to obtain some exact solutions directly. But we can use other method to solve it, for instance test solution method. Such method is always applied to find the exact solutions of nonlinear wave equations in the nonlinear problems. It is shown that the periodic solutions obtained by this method include some shock wave solutions and solitary wave solutions. So we use test solution method to solve Eqs. (11) and (12). There are eight types of solutions for the functions u and Q.

3.1. Soliton solutions

3.1.1. Bright soliton

After using test solution method, we get the corresponding soliton solution:

\begin{matrix} q = \sqrt{\frac{3 d_{2}}{2}} sech {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} exp (\frac{i ξ (3 c_{0}^{2} + 4 c_{1}^{2} d_{2})}{6 c_{1}^{2}}) exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = d_{2} sech {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2}, \end{matrix}

(16)

with

d = \frac{3}{4 d_{2}}

and

a = \frac{3 c_{0}^{2} + 4 c_{1}^{2} d_{2}}{6 c_{1}}

. It is noted that c₀, c₁ are arbitrary constants and c₂ is zero. As Figs. 1(a) and 1(b) show, it is evident that we can find out that u and Q are ground-state bright solitons. We can recall the properties of ground-state solitons, namely, the width of a ground-state soliton increases while its peak amplitude decreases with increasing degree of nonlocality d according to Ref. [31

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000). [CrossRef]

]. Further, it is easy to obtain the energy flow

U = \int_{- \infty}^{\infty} {| q |}^{2} d η = 2 \sqrt{3 d_{2}},

(17)

and the Hamiltonian

H = \frac{1}{2} \int_{- \infty}^{\infty} ({| \frac{\partial q}{\partial η} |}^{2} - n {| q |}^{2}) d η = - \frac{8 {(d_{2})}^{\frac{3}{2}}}{5 \sqrt{3}} = - \frac{U^{3}}{45} .

(18)

Fig. 1 (a) Evolution of bright soliton u and (b) evolution of refractive index n in Eq. (16) for d₂ = 3, c₀ = 3, c₁ = 3. (c) relationship between U and d₂. (d) relationship between H and U.

Accordingly, Fig. 1(c) depicts that the energy flow U is a monotonically growing function of d₂. As d₂ → 0 the soliton broadens drastically while its energy flow vanishes. The soliton is stable in the entire domain of its existence and achieve the absolute minimum of Hamiltonian H for a fixed energy flow U[Fig. 1(d)]. And, the analytical solution we get is different from the steady-state analytical solution because the soliton will evolve along line

η = \frac{c_{0} ξ}{c_{1}}

. Therefor, if c₀ = 0, the solution is reduced to

\begin{matrix} q = \sqrt{\frac{3 d_{2}}{2}} sech {(\frac{\sqrt{3 d_{2}} η}{3})}^{2} exp (i ξ \frac{2 d_{2}}{3}), \\ n = d_{2} sech {(\frac{\sqrt{3 d_{2}} η}{3})}^{2} . \end{matrix}

(19)

which is equal to the steady-state analytical solution.

3.1.2. Dipole-mode soliton

\begin{matrix} q = \sqrt{3 d_{2}} sech (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}}) tanh (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}}) exp (\frac{i ξ (3 c_{0}^{2} + c_{1}^{2} d_{2})}{6 c_{1}^{2}}) \\ exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = d_{2} sech {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} . \end{matrix}

(20)

As is shown in Fig. 2(a), the solution is a dipole-mode soliton which can be viewed as nonlinear combinations (bound states) of fundamental solitons with alternating phases. Such bound states can not exist in a local Kerr-type medium because a π phase difference between solitons leads to a local decrease of refractive index in the overlap region and results in repulsion. By comparison, the whole intensity distribution in the transverse direction decides the refractive-index change in the overlap region in nonlocal media. And under appropriate conditions the nonlocality can cause an increase in refractive index and attraction between solitons. Thus, the proper choice of separation between solitons forms bound state. In fact, we can find the bright and dipole-mode solitons we obtain are similar to the approximate analytical solutions in quadratic nonlinear materials with exponential response function [23

N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003). [CrossRef]

Fig. 2 (a) Profiles of dipole-mode soliton u when ξ = 0(black line), 2(red line), 4(blue line) with d₂ = 2, c₀ = 2, c₁ = 3. (b) refractive index n when ξ = 0(black line), 2(red line), 4(blue line) from left to right with d₂ = 2, c₀ = 2, c₁ = 3

3.1.3. Double soliton

\begin{matrix} q = \sqrt{3 d_{2} tanh {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} - 3 d_{2} tanh {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{4}} exp (\frac{i ξ (3 c_{0}^{2} + 4 c_{1}^{2} d_{2})}{6 c_{1}^{2}}) \\ exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = d_{2} sech {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} . \end{matrix}

(21)

where c₂ = 0. Obviously, as the Figs. 3(a), 3(b) and 3(c) show, the solution we obtain is double soliton. The center of double soliton is located in η = 0 when the light is incident on the media, namely ξ = 0. With the light propagates along the ξ axis, although the center of the solition moves right, the peak amplitude is still invariant.

Fig. 3 (a), (b) and (c) Profiles of the double soliton u when ξ = 0, 4, 8 with d₂ = 1, c₀ = 2, c₁ = 3. (d) refractive index when ξ = 0(black line), 2(red line), 4(blue line) from left to right with d₂ = 1, c₀ = 2, c₁ = 3

3.1.4. Divergent solution

\begin{matrix} q = \sqrt{\frac{3 d_{2}}{2}} csch {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} exp [\frac{i ξ (3 c_{0}^{2} + 4 c_{1}^{2} d_{2})}{6 c_{1}^{2}}] exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = d_{2} csch {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 c_{1}})}^{2} . \end{matrix}

(22)

Here, we do not show the profile of the solution because it is evident that (22) is a divergent solution.

3.2. Period solutions

3.2.1. sn type period solution

Solving with Eqs. (11) and (12), we find the corresponding solution:

\begin{array}{l} u = \frac{d_{2} (m^{2} + 1 + γ - 3 m^{2} s n {(\frac{\sqrt{3 d_{2}} ψ}{3 m c_{1}}, m)}^{2})}{m \sqrt{6 d_{2} γ}}, \\ Q = \frac{d_{2} (m^{2} + 1 + γ - 3 m^{2} s n {(\frac{\sqrt{3 d_{2}} ψ}{3 m c_{1}}, m)}^{2})}{3 m^{2}}, \end{array}

(23)

with

\begin{matrix} d_{2} = \frac{3 m^{2} ((m^{2} + 1) γ + (m^{4} + m^{2} + 1))}{2 d ((2 + 3 m^{6}) + α γ)}, \\ a = \frac{3 m^{2} c_{0}^{2} (α + 2 γ (m^{2} + 1)) + 4 c_{1}^{2} d_{2} (2 (1 + m^{6}) + γ α)}{6 c_{1} m^{2} (α + 2 γ + 2 m^{2} γ)} . \end{matrix}

(24)

where α = 2m⁴ + m² + 2 and

γ = \sqrt{m^{4} - m^{2} + 1}

. Obviously, it is well known that sn is the usual Jacobi elliptic sine function and m is the modular of the function sn. And it is a periodic solution when 0 < m < 1 [see Fig. 4(a)]. The substitution of Eq. (23) into Eq.(10), then arrives at

\begin{matrix} q = \frac{d_{2} (m^{2} 1 + γ - 3 m^{2} s n {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m)}^{2})}{m \sqrt{6 d_{2} γ}} exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)) \\ exp (\frac{i ξ (3 m^{2} c_{0}^{2} (α + 2 γ (m^{2} + 1)) + 4 c_{1}^{2} d_{2} (2 (1 + m^{6}) + γ α))}{6 c_{1} m^{2} (α + 2 γ + 2 m^{2} γ)}), \\ n = \frac{d_{2} (m^{2} + 1 γ - 3 m^{2} s n {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m)}^{2})}{3 m^{2}} . \end{matrix}

(25)

The Fig. 4(a) illustrates the period of the solution decreases while its peak amplitude increases with increasing d₂.

Fig. 4 (a) Profiles of period solution u where d₂ = 1(black line), 2(red line) and 3(blue line) with c₀ = 3, c₁ = 3 and m = 0.5. (b) refractive index n with d₂ = 2, c₀ = 3, c₁ = 3 and m = 0.5.

3.2.2. sn⁻¹ type divergent solution

\begin{matrix} q = (\frac{- d_{2} (m^{2} + 1 + γ)}{m \sqrt{6 d_{2} γ}} + \frac{\sqrt{6 d_{2} γ}}{2 γ s n {(k ψ, m)}^{2}}) exp (\frac{i a ξ}{c_{1}}) exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = \frac{d_{2} (m^{2} + 1 + γ)}{3 m^{2}} - d_{2} \frac{1}{s n {(k ψ, m)}^{2}}, \end{matrix}

(26)

where the parameters are

\begin{matrix} d_{2} = \frac{3 (m^{4} + m^{2} + 1) + 3 (m^{2} + 1) γ}{2 d (α γ - 2 (m^{6} + 1))}, \\ a = \frac{4 d_{2}^{2} c_{1}^{2} (2 (1 + m^{2}) + α γ))}{12 c_{1} d_{2} ((2 m^{2} + 1) γ + (1 + m^{2} + m^{4}))} \\ + \frac{- 3 d_{0} c_{0}^{2} (α + 2 (1 + m^{2}) γ)}{12 c_{1} d_{2} ((2 m^{2} + 1) γ + (1 + m^{2} + m^{4}))}, \end{matrix}

(27)

with c₀, c₁ are arbitrary constants and c₂ = 0.

3.2.3. cn−dn type period solution

\begin{matrix} q = \frac{\sqrt{d_{2} (3 m^{2} + 3)} c n (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m) d n (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m)}{1 + m^{2}} exp (\frac{i a ξ}{c_{1}}) \\ exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = \frac{d_{2} (2 - (m^{2} + 1) s n {(\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m)}^{2})}{1 + m^{2}}, \end{matrix}

(28)

when

d_{2} = \frac{3 m^{2}}{2 d (1 + m^{2})}, a = \frac{- c_{1}^{2} d_{1} (1 - 10 m^{2} + m^{4}) + 3 m^{2} c_{0}^{2} (1 + m^{2})}{6 m^{2} c_{1} (1 + m^{2})} .

(29)

Apparently, the parameters c₀, c₁ are arbitrary constants and c₂ is zero.

3.2.4. sn−cn type period solution

\begin{matrix} q = m \sqrt{\frac{3 d_{2}}{m^{2} - 2}} s n (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}) c n (\frac{\sqrt{3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}) exp (\frac{i a ξ}{c_{1}}) \\ exp (i \int \frac{c_{0}}{c_{1}^{2}} d (c_{1} η - c_{0} ξ)), \\ n = \frac{d_{2} (- 1 - (m^{2} - 2) s n {(\frac{\sqrt{- 3 d_{2}} (c_{1} η - c_{0} ξ)}{3 m c_{1}}, m)}^{2})}{m^{2} - 2} . \end{matrix}

(30)

with

\begin{matrix} a = \frac{- c_{1}^{2} d_{1} (m^{2} - 1) (8 - 8 m^{2} - m^{4}) + 3 m^{2} c_{0}^{2} (4 m^{2} - 4 - m^{4})}{6 m^{4} c_{1} (4 + m^{2} - 4 m^{2})}, \\ d_{2} = \frac{3 m^{2}}{2 d (m^{2} - 2)} . \end{matrix}

(31)

It is evident that this is a periodic solution.

3.3. Airy-like solution

Because of

- \frac{2 b ψ}{c_{1}^{2}} u

in Eq. (14), it is difficult to use test solution method to find solutions of Eqs. (14) and (15) directly. But when c₃ = 0, Eq. (14) can be given

u_{ψ ψ} - ρ ψ u + κ u + λ u Q = 0.

(32)

Here, we can search for stationary soliton solutions of Eqs. (15) and (32) numerically. We get an Airy-like solution when ρ = −8, κ = −0.01, λ = 3.1 and d = 0.1. As Fig. 5(a) displays, the amplitude of the light field has a strong Airy tail. It sharply increases when ψ < 0, while it possesses the maximum at ψ = 0. Then it oscillates around u = 0 and decays. However, the amplitude of the refractive index oscillates above u = 0 and decays. At last, it tends to a constant which is greater than zero.

Fig. 5 (a) and (b) are profiles of a Airy-like solution u and refractive index n at ρ = −8, κ = −0.01, λ = 3.1 and d = 0.1.

4. Conclusions

In this paper, we pay attention to the Kerr-type nonlinear, nonlocal media with exponential response function (as in liquid crystals). We study the evolution equation, namely Eq. (1) by using the classical Lie-group method. Firstly we reduce it to some similarity equations and study these similarity equations in detail. It is difficult to obtain some exact solutions directly because the similarity equations are complicated ODEs. Thus, by the test solution method, we study these similarity equations and find bright soliton solution, periodic solution, dipole-mode soliton solution and double soliton solution. We also investigate the energy flow and the Hamiltonian, find that the energy flow is related with the parameter d₂. In addition, we obtain a Airy-like solution from the similarity equations by numerical method. However, in this paper, we neglect to find the solution for another similarity equations in case c₂ ≠ 0. In the further work ,we would like to consider the case with c₂ is not zero.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 10875106 and No. 11175158, by program for Innovative Research Team in Zhejiang Normal University .

References and links

1.	Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).
2.	M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998). [CrossRef]
3.	W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004). [CrossRef]
4.	M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998). [CrossRef]
5.	A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, 1110–1113 (1997). [CrossRef]
6.	M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002). [CrossRef]
7.	M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005). [CrossRef] [PubMed]
8.	A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).
9.	J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]
10.	L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000). [CrossRef] [PubMed]
11.	V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000). [CrossRef]
12.	W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001). [CrossRef]
13.	M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003). [CrossRef]
14.	S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985). [CrossRef]
15.	O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002). [CrossRef]
16.	D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001). [CrossRef]
17.	W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004). [CrossRef]
18.	J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]
19.	Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009). [CrossRef]
20.	N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004). [CrossRef] [PubMed]
21.	A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006). [CrossRef] [PubMed]
22.	S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]
23.	N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003). [CrossRef]
24.	P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006). [CrossRef]
25.	M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007). [CrossRef] [PubMed]
26.	M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008). [CrossRef] [PubMed]
27.	J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002). [CrossRef]
28.	Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. 30, 3171–3173 (2005). [CrossRef] [PubMed]
29.	P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986). [CrossRef]
30.	A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973). [CrossRef]
31.	W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 7, 2011
Revised Manuscript: February 16, 2012
Manuscript Accepted: February 27, 2012
Published: March 19, 2012

Citation
Jian Jia and Ji Lin, "Solitons in nonlocal nonlinear kerr media with exponential response function," Opt. Express 20, 7469-7479 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7469

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References

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).
M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today51(8), 42–45 (1998). [CrossRef]
W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B.6, 288–294 (2004). [CrossRef]
M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett.80, 4657–4600 (1998). [CrossRef]
A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A56, 1110–1113 (1997). [CrossRef]
M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett.27, 1460–1462 (2002). [CrossRef]
M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett.30, 415–417 (2005). [CrossRef] [PubMed]
A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys.1, 31–33 (1975).
J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys.36, 3–8 (1965). [CrossRef]
L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett.85, 1791–1794 (2000). [CrossRef] [PubMed]
V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E62, 4300–4308 (2000). [CrossRef]
W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E64, 016612–016619 (2001). [CrossRef]
M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E68, 025602–025605 (2003). [CrossRef]
S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys.64, 797–801 (1985). [CrossRef]
O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E66, 046619–046623 (2002). [CrossRef]
D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett.26, 1185–1187 (2001). [CrossRef]
W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E70, 036617–036621 (2004). [CrossRef]
J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E71, 065603 (2005). [CrossRef]
Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A80, 013838–013844 (2009). [CrossRef]
N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett.29, 286–288 (2004). [CrossRef] [PubMed]
A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett.96, 043901 (2006). [CrossRef] [PubMed]
S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett.31, 1100–1102 (2006). [CrossRef] [PubMed]
N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E68, 036614–036618 (2003). [CrossRef]
P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E73, 036614–036623 (2006). [CrossRef]
M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett.32, 2490–2492 (2007). [CrossRef] [PubMed]
M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express16, 3273–3287 (2008). [CrossRef] [PubMed]
J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E66, 066615–066627 (2002). [CrossRef]
Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett.30, 3171–3173 (2005). [CrossRef] [PubMed]
P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986). [CrossRef]
A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett.23, 171–172 (1973). [CrossRef]
W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E63, 016610–016615 (2000). [CrossRef]

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Optics Express

Solitons in nonlocal nonlinear kerr media with exponential response function

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Abstract

1. Introduction

2. The classical Lie-group reduction

3. Analytic solutions and numerical solution

3.1. Soliton solutions

3.1.1. Bright soliton

3.1.2. Dipole-mode soliton

3.1.3. Double soliton

3.1.4. Divergent solution

3.2. Period solutions

3.2.1. sn type period solution

3.2.2. sn⁻¹ type divergent solution

3.2.3. cn−dn type period solution

3.2.4. sn−cn type period solution

3.3. Airy-like solution

4. Conclusions

Acknowledgments

References and links

References

Appl. Phys. Lett.

J. Appl. Phys.

J. Opt. B.

Opt. Express

Opt. Lett.

Phys. Rev. A

Phys. Rev. E

Phys. Rev. Lett.

Phys. Today

Sov. J. Plasma Phys.

Theor. Math. Phys.

Other

Cited By

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OpticsInfoBase

Optics Express

Solitons in nonlocal nonlinear kerr media with exponential response function

Article Tools

Article Navigation

Abstract

1. Introduction

2. The classical Lie-group reduction

3. Analytic solutions and numerical solution

3.1. Soliton solutions

3.1.1. Bright soliton

3.1.2. Dipole-mode soliton

3.1.3. Double soliton

3.1.4. Divergent solution

3.2. Period solutions

3.2.1. sn type period solution

3.2.2. sn−1 type divergent solution

3.2.3. cn−dn type period solution

3.2.4. sn−cn type period solution

3.3. Airy-like solution

4. Conclusions

Acknowledgments

References and links

References

Appl. Phys. Lett.

J. Appl. Phys.

J. Opt. B.

Opt. Express

Opt. Lett.

Phys. Rev. A

Phys. Rev. E

Phys. Rev. Lett.

Phys. Today

Sov. J. Plasma Phys.

Theor. Math. Phys.

Other

Cited By

Figures

3.2.2. sn⁻¹ type divergent solution