Optical solitons of model with integrable equation for wave packet envelope
Introduction
In recent years, there has been great interest in the study of nonlinear mathematical models which describe the pulse propagation in optical fibers. Let us mention here the only widespread non-linear mathematical models: the Radhakrishnan-Kundu-Laksmanan equation [1], [2], [3], the Triki-Biswas equation [4], [5], [6], the Schrödinger equation with anti-cubic nonlinearity [7], [8], [9], [10], the Kundu-Mukherjee-Naskar model [11], [12], [13], the perturbed Chen-Lee-Liu equation [14], [15], [16], the Fokas-Lenells equation [17], [18], [19], the Biswas-Arshed equation [20], [21], [22], the Biswas-Milovic equation [23], [24], [25], the Kudryashov equation [26], [27], [28], [29]. However, most of the above nonlinear mathematical models are described by non-integrable nonlinear differential equations. This circumstance reduces the possibility for description of propagation pulses in the optical fiber and leads to the need of finding nonlinear integrable differential equations for nonlinear optics.
In this paper we consider the partial differential equation in the formwhere q(x, t) is a complex-valued function and α, β, δ, σ, ν and ε are parameters of Eq. (1). Eq. (1) is the generalization of some well-known equation for description of solitary wave solutions in the optical fiber. In case Eq. (1) is transformed to the famous nonlinear Schrödinger equation [30], [31], [32], [33]. If take into account Eq. (1) is also reduced to the equation which is used in the nonlinear optics [34], [35], [36], [37], [38], [39], [40], [41], [42]. Therefore, Eq. (1) can be considered as a generalization of the nonlinear Schrodinger equation, which also can be used for describing nonlinear processes in optics.
In this paper we demonstrate that the nonlinear ordinary differential equation for the wave packet envelope corresponding to Eq. (1) will be integrable under some constraints on the parameters of Eq. (1). We present the Lax pair for this nonlinear fourth-order differential equation. We find the first integrals and the general solution for the wave packet envelope. We also obtain the special solutions of the fourth-order ordinary differential equation expressed via the Jacobi elliptic sine.
Section snippets
System of equations corresponds to Eq. (1)
We look for solutions of Eq. (1) using the traveling wave reduction in the formSubstituting solution (2) into Eq. (1) and equating the imaginary and real parts to zero, we obtain the following system of equationsandFrom Eq. (3) we have conditions for parameters of Eq. (1) in the formUsing conditions (5) in Eq. (4) we obtain the following equation
The Lax pair for Eq. (6) at some constraints
Without loss of generality we can assume and and . In this case Eq. (6) can be written as the followingWe will look for the Lax pair for Eq. (6) taking into account the well-known AKNS scheme [44], [45], [46], [47].
Let us suppose that there is the Lax pairs to Eq. (6) in the formwhere ψ(z), A and B are matrices in the formWe look for the Lax pair for Eq. (6) using the equation
First integrals for Eq. (21)
We can use the Lax pair (8) for finding the first integrals of Eq. (21). We know that if the matrix A satisfies Eq. (10), the first integrals corresponding to Eq. (21) can be obtained by means of calculating of traces trAk. Taking into account the condition for the matrix elements we get [46], [47]. So, we need to calculate detA with the matrix elements (17)–(20).
Equating expressions in detA at various values λ to zero, we find the first integrals for Eq. (21) in the form
Special solutions of Eq. (6)
Eq. (21) at conditions (22) is known to have the special solutions which are expressed via the elliptic functions. Taking this circumstance into account, let us look for special solutions of Eq. (6). We will search for solutions of Eq. (6) using the idea of the simplest equation by means of the following algorithm. We assume that there is a solution of Eq. (6) that is determined by solution of the following first-order differential equationwhere a, b, c, d and e are
General solution of Eq. (33)
Now let us present the general solution of Eq. (33) which is the followingUsing the new variable we can present Eq. (33) in the formwhere y1, y2, y3 and y4 are roots of the algebraic equation
We are going to look for solution of Eq. (42) in the formwhere Y(x) is a new function and E is constant which will be looked for.
Taking into account (45), we calculate the following relations
Conclusion
Let us briefly formulate the results of this work. In this paper, we have considered a mathematical model that is a generalization of the nonlinear Schrödinger equation. Unlike the nonlinear Schrödinger equation, the Cauchy problem for this equation under consideration is not solved in the general case by the inverse scattering transform. However, taking into account the traveling wave reduction, we transform this equation into a nonlinear fourth-order ordinary differential equation with the
Declaration of Competing Interest
Authors declare that they have no conflict of interest.
Acknowledgments
This work was supported by the Ministry of Science and Higher Education of the Russian Federation (state task project no. 0723-2020-0036) and was also supported by Russian Foundation for Basic Research according to the research project no. 18-29-10025.
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