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.
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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 . During the last two decades spatial solitons have become a subject of intense investigation because of their unique physical features . 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 . Principally new effects of nonlocality have been studied in photorefractive crystals [4, 5], nematic liquid crystals [6, 7], plasmas , thermo-optical materials , and Bose-Einstein condensates with long-range interparticle interactions [10,11]. For example, the nonlocal nonlinear response suppresses the modulation instability of the plane waves in focusing media [12,13]; it can arrest catastrophic collapse of multidimensional beams [14–16] and stabilizes complex soliton structures, such as vortex solitons [17,18]; it also can make the colliding solitons merge into a standing wave  and the dark solitons form bound states  which was observed in Ref. . Furthermore, stable dipole solitons in a medium with a Gaussian response function also have been predicted recently . Besides, the quadratic nonlinear materials also display a nonlocal nonlinearity  which can explain (1+2) -dimensional X waves based on a Bessel type nonlocal response function . And the group-velocity mismatch (GVM) induces asymmetric nonlocal Raman responses that accurately explain the stationary and nonstationary regimes in cascaded quadratic soliton compressors . At the same time, the compression limit was already studied in Ref. . 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 . 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. .
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:23]. 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 where G(η) = (1/2d1/2)×exp(− |η|/d1/2) 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 . 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, 30] to Eq. (1). First we introduce the Lie group of transformations of independent and dependent variables (η,ξ,q,q*,n)Eq. (1) is invariant under the transformations (2) if and only if 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,
We can have a similarity variable ψ, similarity solutions q and n by integrating the following characteristic equations:Eq. (7), we can obtain two different types of similarity reductions of Eq. (8). When b is zero, Eq. (7) becomes Eq. (8), We have Eq. (10) into Eq. (1) yields the similarity reduced equations Eq. (7) with b ≠ 0. Now the substitution of Eq. (7) into Eq. (8) arrives at Eqs. (1) and (13), we obtain
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: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. . Further, it is easy to obtain the energy flow
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 . Therefor, if c0 = 0, the solution is reduced to
3.1.2. Dipole-mode soliton
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 .
3.1.3. Double soliton
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.
3.1.4. Divergent solution
3.2. Period solutions
3.2.1. sn type period solutionFig. 4(a)]. The substitution of Eq. (23) into Eq.(10), then arrives at Fig. 4(a) illustrates the period of the solution decreases while its peak amplitude increases with increasing d2.
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 solutionEqs. (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.
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.
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 .
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