Solitons solutions of nonlinear Schrödinger equation in the left-handed metamaterials by three different techniques

This paper, derives the exact traveling-wave solution and soliton solutions of nonlinear Schrodinger equation (NLSE) with higher-order nonlinear terms of Left-handed metamaterials (LHMs), the authors apply three different methods, namely: csch function method, the exp ( − ϕ ( ξ ) ) -Expansion method and the simplest equation method. The results obtained are Dark, Bright solitons and other solutions, which are well known in optics metamaterials and LHMs.


Introduction
It is well known that the partial differential equation (PDEs) of the non-linear Schrodinger equation with hightorder nonlinear terms are near the complex physics phenomena which are concerned many fields from physics to biology etc [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, some effective methods for getting solitons solutions in LHMs and optics has attracted many researchers attention because of soliton theory which is a very important and fascinating area of research in nonlinear left-handed metamaterials and optics. Houwe Alphonse et al [4] studied optical soliton in left-handed metamaterials. M Mirzazadeh et al [5] reported solitons to generalized resonant dispersive nonlinear Schrödinger's equation with power law nonlinearity. I. V. Shadrivov et al [6] studied Spatial solitons in left-handed metamaterials. Ekici Mehmet et al [7] investigated optical solitons in birefringent fibers with Kerr nonlinearity. Biswas et al [8] obtained bright and dark solitons for MMs. Anjan Biwas et al [9] demonstrated the existence of singular solitons in optical metamaterials by ansatz method and simplest equation approach. Alphonse HOUWE et al obtained solitons of the perturbed nonlinear Schrödingers equation in the nonlinear left-handed transmission lines [18]. In this perspective, many methods for obtaining exact solutions of NLSE was investigated, such as Tan-sech method [10,11], Exponential rational function method [12], the sine-cosine method [13,14], the modified simple equation method [15], and so on.
In [16], N Taghizadeh et al used the first integral method to find exact soliton solution of the nonlinear Schrodinger equation and Ma and Chen [19] is used Direct search method to obtain exact solutions of the same nonlinear Schrodinger equation. This cubic nonlinear Schrodinger equation [16,19], which is similar to that obtained in a left-handed transmission lines loaded with a varactor is in the following form: Where u=u (x, t) is the complex-valued function of two real variables x , t. a is the group velocity dispersion and the term c is the nonlinearity coefficient. The index m>0,is the full nonlinearity parameter. For a=p=1, c=q=μ, and m=1 correspond to the non-linear Schrodinger equation and have been discussed [19]. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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Solitary waves of nonlinear schrödingers equation in the left-handed metamaterials can paved the way for relevant studies, e.g., modulational instability. In the present paper, to befall many exact solutions and solitons solutions of this model of equation (1), the authors has used three integrations schemes. They are csch method, the exp(−f(ξ))-Expansion method and the simplest equation method that will uncover solitons solutions to the model. The beginning hypothesis is the traveling-wave transformation. The elaboration are all recorded in the upcoming section.

Traveling wave assumption
The solution of equation (1) is supposed to be where U(ξ) is the amplitude component of the wave and x =x vt, while v is its speed. Here θ (x, t)=−kx+ω t+θ 0 represents the phase component of the soliton. The parameters ω, k and θ 0 are respectively the inverse pulse width, the frequency and the phase constant.
After changing the variables, and substituting equation (2) into equation (1), and separating the real and imaginary parts it is obtained: from equation (3) leads to the speed wave of the soliton : Now, multiplying equation (4) by ¢ U and integrating once with zero constant gives w ¢ - can be written as follows w ¢ -

Application
In this section, three different integrations tools will be applied to befall exact solution and soliton solutions

csch function method
The solutions of many nonlinear equation can be expressed in form [20] x and admits the following derivative where A, τ and μ are parameters to be determined, μ is the wave number Substituting equations (10) and (9) into the reduced equation equation To Balance the terms of the csch functions to find τ. 2τ+2=(m+2)τ, and t =  Solving the system of equations equation (12) and equation (13) result is: then, if m>2, and therefore The key step is to suppose that the solution of equation (8) can be expressed by a rational polynomial as the following : the parameter N, it obtained by balancing the highest-order linear term with the nonlinear term, where i=(0, 1, K.., N) and f (ξ) satisfies the following ordinary differential we balance The constraint condition is m 2 −4>0.

The simplest equation method
The demarche is to suppose that V(ξ) satisfies the Bernoulli and Riccati equations method [22,23]. The step is to introduce the solution V(ξ) of equation (8) in the following finite series form where a i are real constants with ¹ a 0 N , and N is a positive integer to be determined. f(ξ) satisfies the following ordinary differential equation Where ρ, A and B are independent on ξ, and will be determined later To obtain different exact solution and other solutions dependent of the parameters ρ, A and B two cases will be present If we surmise m=2, equation (8) becomes: By balancing the linear term of highest order derivatives with the highest order nonlinear term in equation (32), leads to (N−1) 2 =0, and N=1 Then equation (30) becomes  (2), is obtained  (2), the result is  (2), the following solutions of equation (1) is obtained For A>0, where ξ 0 is the integration constant.

Some graphical representations
In this part of the paper, the application of the results obtained above are illustrated. Figures 1-5 are the graphical representation of equation (41). By varying the parameters k, a 1 , a, c, v, ω,one arrives at graphic representations well known in LHMs and optical fiber from the different graphical representations above, the solitons solutions (dark, bright) and other solutions obtained by the simple equation method.The results obtained are comparable to those well known in [18,24].

summary
In this study, the authors apply successfully three different methods namely: csch function method, the f x -( ( )) exp -Expansion method and simplest equation method to construct soliton solutions and other  solutions to the nonlinear Schrodinger equation (1). The results obtained are dark, bright and singular 1-soliton solutions. Note that the first two integrations failed to find known solitons. In the future, this model can be studied from a different perspectives. Subsequently, the model will be consider perturbation terms and spatiotemporal dispersion. Certainly abundant 1-solitons solutions and other solutions will be obtained. These results will be later disposable.