Solitons and conservation laws to the resonance nonlinear Shrödinger's equation with both spatio-temporal and inter-modal dispersions

doi:10.1016/j.ijleo.2017.06.010

Optik

Volume 142, August 2017, Pages 509-522

https://doi.org/10.1016/j.ijleo.2017.06.010 Get rights and content

Abstract

The resonant nonlinear Shrödinger's equation (RNLSE) with both spatio-temporal (STD) and inter-modal (IMD) dispersions which describes the modelling of fluids and propagation dynamics of optical solitons is studied using three analytical schemes. These are generalized projective-Riccati equation method (GPRE), Bernoulli sub-ODE method and the Riccati-Bernoulli sub-ODE. The presented problem is studied with Kerr law nonlinearity. Dark optical, singular, and combined formal solitons are acquired. The constraint conditions that naturally fall out of the solution structure guarantee the existence of these solitons. We derive the Lie point symmetry generators of a system of partial differential equations (PDEs) obtained by decomposing the underlying equation into real and imaginary components. Then we used these symmetries to construct a set of nonlocal conservation laws (Cls) using the technique introduced by Ibragimov.

Introduction

Optical solitons is one of the fast growing fields in the field of optoelectronics and nano electronics. In recent years, several results have been presented in many journals and books. A lot of progress are made in the field of nonlinear optics that deals with meta materials, optical fibers, birefringent fibers, and many more. Several models and techniques describe the dynamics of soliton [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. This paper addresses the kerr law nonlinearity to a NLSE that appears with STD and IMD dispersions [22]. It should be noted that the governing equation for the propagation of optical solitons in nonlinear media is well-posed only when the additional STD is considered [5].

Section snippets

Governing equation

In the presence of IMD, the well-posed nonlinear dynamical model that will be investigated in this work is given by the following RNLSE [22]: ${iq}_{t} + α q_{xx} + β q_{xt} + dF (| q |^{2}) q + γ (\frac{| q |_{xx}}{| q |}) q - i δ q_{x} = 0, i = \sqrt{- 1},$ where x represents the non-dimensional distance along the fiber and t is the temporal variable. The term q(x, t) representing the dependent variable is a complex valued wave profile, α and β represent the coefficients of STD and group-velocity dispersion, d and γ are the coefficients of non-Kerr law

Generalized projective Riccati equations method

The proposed method can be summarized in the following steps. Suppose that a NLPDE in two independent variables x and t is represented by $\begin{matrix} P (q, q_{t}, q_{x}, q_{tt}, q_{xx}, q_{xt}, \dots) = 0, \end{matrix}$ where q(x, t) is an unknown function and P is a polynomial of q(x, t) and its partial derivatives.

Step 1:
We use the travelling wave transformation $q (x, t) = u (ξ), ξ = (x \pm vt),$ to get the following ordinary differential equation (ODE) $\begin{matrix} P (u, u^{'}, u^{″}, \dots) = 0, \end{matrix}$ where $u^{'} (ξ) = \frac{du}{d ξ}$ .
Step 2:
Suppose that Eq. (12) has a solution of the form $u (ξ) = A_{0} + \sum_{i = 1}^{N} σ {(ξ)}^{i - 1} [A_{i} σ (ξ) + B_{i} τ (ξ)],$

Lie symmetry analysis

In this section, we study the Lie symmetry analysis [49] of the kerr law nonlinearity which will be used to construct the Cls. In order to study the Lie symmetries, we express the complex envelope as $q (x, t) = u (x, t) e^{i υ (x, t)},$ with high-frequency waves u(x, t) and υ(x, t). Substituting Eq. (81) into Eq. (8) and separating the real and imaginary parts, we get the following system $\begin{matrix} u υ_{t} = d u^{3} + δ u υ_{x} - β u υ_{t} υ_{x} - α u {(υ_{x})}^{2} + β u_{xt} + α u_{xx} + γ u_{xx}, \\ u_{t} = δ u_{x} - β υ_{t} u_{x} - β u_{t} υ_{x} - 2 α u_{x} υ_{x} - β u υ_{xt} - α u υ_{xx} . \end{matrix}$

The Lie point symmetries of system Eq. (82)

Nonlocal conservation Laws

In this section, we construct the nonlocal conservation laws for system Eq. (82). We recall the following theorems and definition from [44], [45], [46], [47], [48].

Theorem 5.1

The system of $\bar{m}$ differential equations $F_{\bar{α}} (\bar{x}, u, u_{(1)}, \dots, u_{(s)}) = 0, \bar{α} = 1, \dots, m,$

with m dependent variables u = (u¹, …, u^m) has formal Lagrangian $L = z^{\bar{β}} F_{\bar{β}} (\bar{x}, u, u_{(1)}, \dots, u_{(s)}),$

and adjoint equation $F_{\bar{α}}^{*} (\bar{x}, u, z, u_{(1)}, z_{(1)}, \dots, u_{(s)}, z_{(s)}) = \frac{δ (z^{\bar{β}} F_{\bar{β}})}{δ u^{\bar{α}}}, \bar{α} = 1, \dots, m,$

where $\frac{δ}{δ u^{\bar{α}}} = \frac{\partial}{\partial u^{\bar{α}}} - D_{i} \frac{\partial}{\partial u_{i}^{\bar{α}}} + {(- 1)}^{s} D_{i_{1}} D_{i_{2}}, \dots, D_{i_{s}} \frac{\partial}{\partial u_{i_{1}, \dots, i_{s}}^{\bar{α}}} + \dots$

where $z^{\bar{β}} = z^{\bar{β}} (\bar{x}, t)$ is a new dependent

Physical properties of acquired solutions

In this section, we describe the physical interpretation of the acquired solutions. Solitons are solitary waves with stretchy dispersion possessions. They describe many physical phenomena in soliton physics and optical fiber. Soliton preserve their shapes and speed after colliding with each other and also give ascend to particle-like structures, such as magnetic monopoles, etc. Eqs. (32), (34) with surface view Fig. 1 illustrate the combined formal soliton like solutions of Eqs. (8), (36) and

Conclusion

This paper studies the dynamics of optical solitons to the RNLSE with both STD and IMD dispersions. Exact dark optical, periodic singular, dark singular and combined formal soliton solutions for the Kerr law nonlinearity are obtained using three integration schemes. There are constraint conditions that must hold in order for the solitons to exist. Using the point symmetry generators of a system of PDEs obtained by decomposing the underlying equation into real and imaginary components, the

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Original research articleSolitons and conservation laws to the resonance nonlinear Shrödinger's equation with both spatio-temporal and inter-modal dispersions

Abstract

Introduction

Section snippets

Governing equation

Generalized projective Riccati equations method

Lie symmetry analysis

Nonlocal conservation Laws

Physical properties of acquired solutions

Conclusion

Optik

Optik

J. Math. Anal. Appl.

Chaos Solitons Fract.

Appl. Math. Comput.

Optik

Adv. Differ. Equ.

Chaos, Solitons Fract.

Adv. Differ. Equ.

Optik

Optik

Optik

J. Math. Anal. Appl.

Commun. Nonlinear Sci. Numer. Simul.

J. Math. Anal. Appl.

Darboux transformation for an integrable generalization of the nonlinear Schrodinger equation

Optik

Optical and other solitons for the fourth-order dispersive nonlinear Shrödinger's equation with dual-power law nonlinearity

Superlattices Microstruct.

Dark and singular optical solitons with spatiotemporal dispersion using modified simple equation method

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Original research article
Solitons and conservation laws to the resonance nonlinear Shrödinger's equation with both spatio-temporal and inter-modal dispersions