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Abstract

In this article, dynamical analysis of fractional order Schrödinger equation governing the optical wave propagation is reported in detail. The validity criteria for the application of the semi-analytic asymptotic methods are exploited. Comparison between the solutions obtained by the two asymptotic techniques, that is, the fractional homotopy analysis transform method and the optimal homotopy analysis method is performed to select the most accurate technique for the stated problem.

Keywords

Fractional calculus fractional nonlinear Schrödinger equation optical pulse homotopy analysis method

Introduction

During the past few decades, logical devices using optical solitons have been developed and new types of optical solitons in media with Kerr-type nonlinearity and in resonant media have been explored. Multimode fiber research has attracted a lot of attention due to the growing demand of optical fiber communications and fiber lasers in science and engineering. In nonlinear science, they represent an exciting environment for complex nonlinear waves. Soliton concepts applied to the description of intense electromagnetic beams and ultrashort pulse propagation in various media have contributed much to this field. The notion of solitons has proved to be very useful in describing wave processes in hydrodynamics, plasma physics, and condensed matter physics.¹ Moreover, it is also of great importance in optics for ultrafast information transmission and storage and radiation propagation in resonant media.

The propagation of high-intensity optical waves is influenced by the nonlinear properties of the medium. Such nonlinearities of the medium lead to optical phenomena such as second harmonic generation, frequency mixing, self-refraction, self-phase modulation (SPM), and soliton which have all found interesting applications in optoelectronics and optical communications.^2,3

Saha and Sarma⁴ reported exact bright and dark solitary wave solution of the nonlinear Schrödinger equation (NLSE) in cubic-quintic non-Kerr medium adopting phase–amplitude ansatz method along with the modulational instability. They obtained the parameters along with the constraints under which bright or dark solitons exist in cubic-quintic non-Kerr medium. Herzallah and Gepreel⁵ solved the cubic nonlinear fractional Schrödinger equation with time and space fractional derivatives. They obtained the exact solution of the cubic NLSE as a special case.

Dai et al.⁶ presented self-similar rogue wave solutions (rational solutions) of the inhomogeneous NLSE using similarity transformation connected with the standard NLSE. They investigated the nonlinear tunneling effect for self-similar rogue waves. They showed that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged, or decreasing amplitudes via the modulation of the ratio of the amplitudes of rogue waves to the barrier or well height.

In the recent decades, fractional calculus has proved to be a useful branch of mathematics to understand the complex physical problems. Fractional analysis has been considered to study the electric and magnetic fields,⁷ theory of hydrologic modeling,⁸ fluid dynamics, study of visco-elasticity,⁹ modeling of heat exchanger,¹⁰ mechanical engineering problems,¹¹ solitons theory,¹² and many other engineering problems including mass transportation.

Fractional calculus is widely used in different applications related to the fields such as biological sciences, modern physics, chemistry, and medicine.^13,14 Fractional calculus is the field that generalizes the concepts of classical calculus from integer order to non-integer and complex order.¹⁵ Importantly, the time fractional derivative contains information of historical states and non-local distributive effects of the solution to understand behavior of complex dynamical system better and more clear. The fractional calculus gives the possible analytical approach on modeling of life cycles in biological sciences to span multiple scales. More specifically, the largest human organ is skin that participates in the protection and communication between the body and environment. The human skin is highly ordered multiple layered organ, so that in particular, it has been used as an appropriate model for applying fractional calculus approach.¹⁶

Different semi-analytical and numerical methods have been used in the literature to find the solution of partial differential equations governing the nonlinear dynamics, for instance, Laplace transform method (LTM),¹⁷ parameter expansion method (PEM),¹⁸ variational iteration method (VIM),¹⁹ homotopy perturbation method (HPM),²⁰ modified homotopy perturbation method (MHPM),²¹ and homotopy analysis method (HAM).^22,23 Remarkable contribution has been made by introducing the modern techniques to find solution of fractional differential equations such as Jacobi collocation technique,²⁴ implicit meshless collocation method,²⁵ generalized Legendre operational matrix method,²⁶ HAM, and HPM. LTM is a very effective tool not only for the nonlinear differential equations with constant coefficients but also for the differential equations with variable coefficients.¹⁷ Similarly, HAM is the very strong tool because it has been used to find the solution of various nonlinear partial differential equations.

In the past few decades, semi-analytical techniques such as perturbation, iteration, and variational parameter techniques received operational attention to find approximate solutions of the problems arising in the field of nano-electromechanical systems, textile engineering, and problems defined by different weak as well as strong nonlinear equations.^1,27–29 More particularly, a method of multiscale asymptotic expansion and its corresponding finite element algorithm is discussed by He et al.³⁰ Recently, He’s max-min method is applied to solve the strong nonlinear oscillators and higher order Duffing equation.³¹

The optimal homotopy analysis method (OHAM), which is an asymptotic method, is effective to obtain the series solution of nonlinear partial differential equations and is a suitable approach to control convergence of approximate solution. The OHAM is reliable not only for small parameters but shows its effectiveness and capability to solve nonlinear problems in engineering sciences and technology.^32–34

“Fractional homotopy analysis transform method” (FHATM), which is a semi-analytic technique, is based on two methods, the HAM and the LTM.³⁵ The important feature of FHATM is that two effective methods are used in harmony to obtain the series solution of partial differential equations with swift convergence.^36,37

In this article, after summarizing the current challenges and solutions to the integer and fractional order optical wave problems, we have compared the two semi-analytic techniques and have presented the approximate results. Our analysis shows that the NLSE can be best solved by the OHAM by controlling the order of accuracy.

Problem description

The mathematical description of optic-solitons is well documented by NLSE satisfied by the pulse envelope $v (x, t)$ in the presence of group velocity dispersion (GVD) and SPM.³⁸

Consider the one-dimensional equation

i v_{x} - \frac{s}{2} v_{tt} + N^{2} | v |^{2} v = 0

(1)

where $s = sgn (β_{2})$ , N represents the order of the soliton, and $β_{2}$ is defined as second-order dispersion effects for normal and anomalous GVD such that

s = sgn (β_{2}) = {\begin{matrix} 1 & + ve GVD \\ - 1 & - ve GVD \end{matrix}

(2)

The NLSE is well known in the soliton literature because it belongs to a special class of nonlinear partial differential equations that can be solved exactly with a mathematical technique known as the inverse scattering method.³⁹ The NLSE supports solitons for both normal and anomalous GVD, pulse-like solitons are found only in the case of anomalous dispersion. When $β_{2} > 0$ (i.e. in the case of normal dispersion), the solutions are demonstrated by a dip in a constant intensity background. Such solutions, referred to as dark solitons, are discussed graphically here in this article and are elaborated in detail by Agrawal.³⁸

We will first simulate equation (1) for pulse-like solitons (bright solitons) and latter for the two types of dark solitons (black and gray). Now, using the space time transformation,³⁹ we obtain

i v_{t} + \frac{c_{0}^{3}}{2} v_{xx} + c_{0} N^{2} | v |^{2} v = 0

(3)

where $c_{0}$ is the phase speed.

The Caputo fractional derivative $D_{t}^{α} w$ of order $0 < α \leq 1$ of $v : R_{+} \to R$ is defined by

D_{t}^{α} w (x, t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} {(t - s)}^{- α} w' (x, s) ds

provided the expression on right hand side is defined.

We will now introduce the fractional order derivative in time in the Caputo sense and solve the equation using OHAM

{iD}_{t}^{α} v + \frac{c_{0}^{3}}{2} v_{xx} + c_{0} N^{2} | v |^{2} v = 0

(4)

with the initial condition

v (x, 0) = N sech x

Application of OHAM

Method description

Using the OHAM technique, the following necessary steps are involved to obtain the analytic approximate solution of fractional partial differential equation. For a given nonlinear fractional partial differential equation

N {D_{t}^{α} w (x, t)} = 0

(5)

where $N$ is the nonlinear operator and $D_{t}^{α}$ is the fractional derivative. By means of HAM the zeroth-order deformation equation

(1 - q) L {ψ (x, t, q) - w_{0} (x, t)} = q ℏ H (t) N {D_{t}^{α} ψ (x, t, q)}

(6)

where $q \in [0, 1]$ is the embedding parameter of the function $ψ (x, t, q)$ , and $ℏ$ , H, L, and $w_{0}$ are the auxiliary parameter, auxiliary function, auxiliary linear operator, and initial guess, respectively

ψ (x, t, 0) = w_{0}; ψ (x, t, 1) = w (x, t)

(7)

$ψ$ can be expanded in Taylor series with respect to q

ψ (x, t, q) = w_{0} + \sum_{m = 1}^{\infty} w_{m} q^{m}

(8)

and

w_{m} = \frac{1}{m!} \frac{\partial^{m} ψ}{\partial q^{m}} |_{q = 0}

(9)

Choose $w_{0}$ , $ℏ$ , H, and L in such a way that solution $w (x, t)$ is convergent.

Now, we define

{\vec{w}}_{n} (t) = {w_{0}, w_{1}, w_{2}, \dots, w_{n}}

Differentiating m times with respect to q, the mth-order deformation equation is

L {w_{m} (x, t) - χ_{m} w_{m - 1} (x, t)} = ℏ H (t) R_{m} ({\vec{w}}_{m - 1})

(10)

where

R_{m} ({\vec{w}}_{m - 1}) = {\frac{1}{(m - 1)!} \frac{\partial^{m - 1} N {ψ (x, t, q)}}{\partial q^{m - 1}} |}_{q = 0}

(11)

and

χ_{m} = {\begin{matrix} 0 & m \leq 1 \\ 1 & m > 1 \end{matrix}

(12)

Now, applying Riemann–Liouville integral operator $I^{α}$ on both sides of equation (10), we have

\begin{array}{l} w_{m} (x, t) = χ_{m} w_{m - 1} (x, t) \\ - χ_{m} \sum_{i = 0}^{n - 1} w_{m - 1}^{i} (0) \frac{t^{2}}{i!} + ℏ H I^{α} R_{m} ({\vec{w}}_{m - 1}) \end{array}

(13)

Application and accuracy

In order to find the solution using OHAM, we again consider equation (4). To handle the term involving absolute, equation (4) can be written as

D_{t}^{α} v - i \frac{c_{0}^{3}}{2} v_{xx} - i c_{0} N^{2} v^{2} \bar{v} = 0

(14)

with the initial condition

v (x, 0) = N sech x

Choose the linear operator

L {ϕ (x, t, q)} = \frac{\partial^{α} ϕ (x, t, q)}{\partial t^{α}}

(15)

with $L {c} = 0$ where c is an arbitrary constant, and nonlinear operator is defined as

N {ϕ (x, t, q)} = \frac{\partial^{α} ϕ}{\partial t^{α}} - i \frac{c_{0}^{3}}{2} \frac{\partial^{2} ϕ}{\partial x^{2}} - i c_{0} N^{2} ϕ^{2} \bar{ϕ}

(16)

Zeroth-order deformation equation is given as

(1 - q) L {ϕ (x, t, q) - v_{0} (x, t)} = q ℏ H (t) N {ϕ (x, t, q)}

(17)

and

ϕ (x, t, 0) = v_{0}; ϕ (x, t, 1) = v (x, t)

(18)

Now, the mth-order deformation equation is given as

L {v_{m} (x, t) - χ_{m} v_{m - 1} (x, t)} = ℏ H (t) R_{m} (v_{m - 1})

(19)

where

\begin{array}{l} R_{m} (v_{m - 1}) = \frac{\partial^{α} v_{m - 1}}{\partial t^{α}} - \frac{i c_{0}^{3}}{2} \frac{\partial^{2} v_{m - 1}}{\partial x^{2}} \\ - i c_{0} N^{2} \sum_{i = 0}^{m - 1} \sum_{j = 0}^{i} v_{j} v_{i - j} \bar{v_{m - 1 - i}} \end{array}

(20)

Choose $H (t) = 1$ . Now, the solution of equation (19) becomes

v_{m} (x, t) = χ_{m} v_{m - 1} (x, t) + ℏ J^{α} R_{m} (v_{m - 1})

(21)

The analytic approximate zeroth-, first-, and second-order solutions are

v_{0} (x, t) = N sech x

(22)

v_{1} (x, t) = - ℏ N c_{0} i {\frac{c_{0}^{2}}{2} sech x + (c_{0}^{2} + N^{4}) {sech}^{3} x} \frac{t^{α}}{Γ (1 + α)}

(23)

Similarly

\begin{matrix} v_{2} (x, t) = - (1 + ℏ) ℏ N c_{0} i {\frac{c_{0}^{2}}{2} + (c_{0}^{2} + N^{4}) {sech}^{2} x} sech x \frac{t^{α}}{Γ (1 + α)} - \frac{N ℏ c_{0}^{4}}{2} \\ \times {{\frac{c_{0}^{2}}{2} + 3 (c_{0}^{2} + N^{4}) {sech}^{2} x} (1 + 2 sech x) sech x + \frac{4 N^{4}}{c_{0}^{2}} {\frac{c_{0}^{2}}{2} + (c_{0}^{2} + N^{4}) {sech}^{2} x}} \\ \times {sech}^{3} x \frac{t^{2 α} Γ (1 + α)}{Γ (1 + 2 α)} + ℏ c_{0}^{2} N {sech}^{x} {\frac{c_{0}^{2}}{2} + (c_{0}^{2} + N^{4}) {sech}^{2} x} \frac{t^{3 α} Γ (1 + 2 α)}{Γ (1 + 3 α)} \end{matrix}

(24)

The convergence can be controlled by minimizing the square residual error

Δ (ℏ) = \int_{Ω} {N \sum_{i = 0}^{M} w_{i} (τ)}^{2} d τ

(25)

as $M \to \infty$ , $Δ (ℏ) \to 0$ .

We obtained the optimal value of $ℏ$ using a nonlinear algebraic equation $d Δ ℏ / d ℏ = 0$ in a manner similar to that prescribed by Liao.³² We obtained the minimum error relative to $ℏ = - 1.25$ for third-order approximation. This result was suitable to confirm the convergence analysis. The details of such criteria can be best found in the work by Liao.⁴⁰

Application of FHATM

Method description

To understand the procedure of finding the solution of nonlinear time fractional partial differential equation using FHATM, we consider the generalized equation

\begin{array}{l} D_{t}^{α} w (x, t) + R w (x, t) + N w (x, t) = g (x, t), \\ t > 0, x \in R, n - 1 < α < n \end{array}

(26)

where $D_{t}^{α} = \partial^{α} / \partial t^{α}$ , R is the linear operator in the variable x, $N$ is the general nonlinear operator, and $g (x, t)$ is continuous function. Here, we consider the simple case, so ignoring initial and boundary conditions. The first step toward the solution is applying Laplace transform on both sides of equation (26), which can be written as

L {D_{t}^{α} w (x, t)} + L {Rw (x, t)} + L {N w (x, t)} = L {g (x, t)}

(27)

By the differentiation property of Laplace transform, we have

\begin{array}{l} ℒ {w (x, t)} - \frac{1}{s^{α}} \sum_{k = 0}^{n - 1} s^{α - k - 1} w^{k} (x, 0) \\ + \frac{1}{s^{α}} ℒ {R w (x, t) + N w (x, t) - g (x, t)} = 0 \end{array}

Here, we define the nonlinear operator

\begin{array}{l} N {ψ (x, t; r)} = ℒ {ψ (x, t; r)} - \frac{1}{s^{α}} \sum_{k = 0}^{n - 1} s^{α - k - 1} w^{k} (x, 0) \\ + \frac{1}{s^{α}} ℒ R w (x, t) + N w (x, t) - g (x, t) \end{array}

(28)

where $r \in [0, 1]$ is an embedding parameter of real function $ψ (x, t; r)$ . The zeroth-order deformation equation established by Liao is as follows

(1 - r) L {ψ (x, t; r) - w_{0} (x, t)} = r ℏ H (x, t) N {D_{t}^{α} ψ (x, t; r)}

(29)

Provided that the auxiliary parameter $ℏ \neq 0$ , the auxiliary function $H (x, t) \neq 0$ , $w_{0} (x, t)$ is an initial estimate of $w (x, t)$ , and $ψ (x, t; r)$ is an unknown real function. Clearly, when $r = 0$ and $r = 1$ , the following holds

ψ (x, t; 0) = w_{0} (x, t); ψ (x, t; 1) = w (x, t)

respectively.

As the embedding parameter r approaches from 0 to 1, the solution approaches from the initial guess $w_{0} (x, t)$ to the final solution $w (x, t)$ .

Expanding the function $ψ (x, t; r)$ in Taylor’s series expansion with respect to r, that is

ψ (x, t; r) = w_{0} (x, t) + \sum_{p = 1}^{\infty} w_{p} (x, t) r^{p}

(30)

where

w_{p} (x, t) = {\frac{1}{p!} \frac{\partial^{p} ψ (x, t; r)}{\partial r^{p}} |}_{r = 0}

Suppose the auxiliary linear operator, the initial guess, the auxiliary parameter $ℏ$ , and the function $H (x, t)$ are considered such that equation (30) is convergent at $r = 1$ , we obtain

w (x, t) = w_{0} (x, t) + \sum_{p = 1}^{\infty} w_{p} (x, t)

Now, we define the vector

\vec{w_{n}} (t) = {w_{0} (x, t), w_{1} (x, t), w_{2} (x, t), \dots, w_{n} (x, t)}

We obtain the pth-order deformation equation by differentiating the zeroth-order deformation equation (equation (29)), p times with respect to r, setting $r = 0$ and dividing by $p!$ , that is

L {w_{p} (x, t) - χ_{p} w_{p - 1} (x, t)} = ℏ H (x, t) R_{p} ({\vec{w}}_{p - 1})

(31)

Applying the inverse Laplace transform on equation (31), we obtain the following recursive relation

w_{p} (x, t) = χ_{p} w_{p - 1} + ℏ L^{- 1} {H (x, t) R_{p} ({\vec{w}}_{p - 1})}

(32)

where $R_{p} ({\vec{w}}_{p - 1})$ and $χ_{p}$ can be defined by the expression

R_{p} ({\vec{w}}_{p - 1}) = {\frac{1}{(p - 1)!} \frac{\partial^{p - 1} N {ψ (x, t; r)}}{\partial r^{p - 1}} |}_{r = 0}

(33)

and

χ_{p} = {\begin{matrix} 0; p \leq 1 \\ 1; p > 1 \end{matrix}

(34)

By this procedure, it is easy to get $w_{p} (x, t)$ for $p \geq 1$ at pth order, we have

w (x, t) = \sum_{p = 0}^{M} w_{p} (x, t)

(35)

when $M \to \infty$ , we obtain a more accurate approximate solution of the general fractional order partial differential equation (equation (26)).

Application and accuracy

In order to find the solution using FHATM, we again consider the same equation (4)

i v_{t} + \frac{c_{0}^{3}}{2} v_{xx} + c_{0} N^{2} | v |^{2} v = 0

(36)

where $c_{0}$ is the phase speed. To deal with the absolute term, we can assume that $v (x, t)$ can be decomposed into real and imaginary parts

v (x, t) = w (x, t) e^{i (kx + ω t)}

(37)

where $ω$ is the frequency and k is the wave number. Therefore, equation (36) takes the form

i w_{t} + \frac{c_{0}^{3}}{2} w_{xx} + {ic}_{0}^{3} k w_{x} - (2 ω + c_{0}^{3} k^{2}) \frac{w}{2} + c_{0} N^{2} w^{3} = 0

(38)

with the initial condition

w (x, 0) = N sech x

The Caputo fractional derivative $D_{t}^{α} w$ of order $0 < α \leq 1$ of $v : R_{+} \to R$ is defined by

D_{t}^{α} w (x, t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} {(t - s)}^{- α} w' (x, s) ds

provided the expression on right hand side is defined.

We will now introduce the fractional order derivative in time in the Caputo sense

{iD}_{t}^{α} w + \frac{c_{0}^{3}}{2} w_{xx} + {ic}_{0}^{3} k w_{x} - (2 ω + c_{0}^{3} k^{2}) \frac{w}{2} + c_{0} N^{2} w^{3} = 0

(39)

On simplifying equation (39), we have

D_{t}^{α} w - i \frac{c_{0}^{3}}{2} w_{xx} + c_{0}^{3} k w_{x} + i (2 ω + c_{0}^{3} k^{2}) \frac{w}{2} - i c_{0} N^{2} w^{3} = 0

(40)

Now, applying Laplace transform on both sides of equation (40) gives

\begin{array}{l} ℒ {D_{t}^{α} w} - \frac{i c_{0}^{3}}{2} ℒ {w_{x x}} + c_{0}^{3} k ℒ {w_{x}} \\ c + \frac{i}{2} (2 ω + c_{0}^{3} k^{2}) ℒ {w} - i c_{0} N^{2} ℒ {u^{3}} \end{array}

(41)

By the differentiation property of Laplace transform, we obtain

\begin{array}{l} s^{α} ℒ {w (x, t)} - s^{α - 1} w_{0} \\ + ℒ {\frac{- i c_{0}^{3}}{2} w_{x x} + c_{0}^{3} k w_{x} + \frac{i}{2} (2 ω + c_{0}^{3} k^{2}) w - i c_{0} N^{2} w^{3}} = 0 \end{array}

which can be written as

\begin{array}{l} ℒ {w (x, t)} - \frac{1}{s} w_{0} + \frac{1}{s^{α}} ℒ \\ {\frac{- i c_{0}^{3}}{2} w_{x x} + c_{0}^{3} k u_{x} + \frac{i}{2} (2 ω + c_{0}^{3} k^{2}) w - i c_{0} N^{2} w^{3}} = 0 \end{array}

Now, we choose the linear operator, that is

L {ψ (x, t; r)} = L {ψ (x, t; r)}

having the property $L {c} = 0$ , where c is constant. Now, we define the nonlinear operator

\begin{array}{l} N {ψ (x, t; r)} = ℒ {ψ} - \frac{1}{s} A sech x + \frac{1}{s^{α}} ℒ \\ {\frac{- i c_{0}^{3}}{2} ψ_{x x} + c_{0}^{3} k ψ_{x} + \frac{i}{2} (2 ω + c_{0}^{3} k^{2}) ψ - i c_{0} N^{2} ψ^{3}} \end{array}

(42)

Using the above definition equation (42) with assumption $H (x, t) = 1$ , we establish the zeroth-order deformation equation

(1 - r) {ψ (x, t; r) - w_{0} (x, t)} = r ℏ N {ψ (x, t; r)}

(43)

for $r = 0$ and $r = 1$ , we have

ψ (x, t; 0) = w_{0} (x, t) = w (x, 0), ψ (x, t; 1) = w (x, t)

Thus, we obtain pth-order deformation equation by differentiating zeroth-order deformation equation (equation (43)) p times with respect to r and then setting $r = 0$ and dividing by $p!$ , we have

L (w_{p} (x, t) - χ_{p} w_{p - 1} (x, t)) = ℏ R_{p} ({\vec{w}}_{p - 1}, x, t)

(44)

Taking inverse Laplace transform of equation (44), we get

w_{p} (x, t) = χ_{p} w_{p - 1} (x, t) + ℏ L^{- 1} {R_{p} ({\vec{w}}_{p - 1}, x, t)}

(45)

where

\begin{matrix} R_{p} ({\vec{w}}_{p - 1}, x, t) = L {w_{p - 1} (x, t)} - (\frac{1 - χ_{p}}{s}) A sech x \\ - \frac{{ic}_{0}^{3}}{2 s^{α}} L {\frac{\partial^{2} w_{p - 1}}{\partial x^{2}}} + \frac{c_{0}^{3} k}{s^{α}} L {\frac{\partial w_{p - 1}}{\partial x}} \\ \times {sech}^{3} x \frac{t^{2 α} Γ (1 + α)}{Γ (1 + 2 α)} + ℏ c_{0}^{2} N^{5} sech x \\ {\frac{c_{0}^{2}}{2} + (c_{0}^{2} + N^{4}) {sech}^{2} x} \frac{t^{3 α} Γ (1 + 2 α)}{Γ (1 + 3 α)} \end{matrix}

(46)

By equations (45) and (46), we end up the series of approximations

w (x, t) = w_{0} (x, t) + w_{1} (x, t) + w_{2} (x, t) + w_{3} (x, t) \dots

The expressions $w_{0} (x, t)$ and $w_{1} (x, t)$ take the form

w_{0} (x, t) = N sech x

and

\begin{matrix} w_{1} (x, t) = \frac{t^{α} N ℏ sech x}{2 Γ (1 + α)} {- i (c_{0}^{3} k^{2} + 2 ω) \\ - i c_{0} (c_{0}^{2} - 2 N^{4}) \overset{2}{\sin h} x + 2 c_{0}^{3} k \tanh x + c_{0}^{3} i \overset{2}{\tan h} x} \end{matrix}

where $w_{2} (x, t)$ and $w_{3} (x, t)$ are listed in Appendix 1. Moreover, the rest of the approximations $w_{p} (x, t)$ for all $p \geq 3$ can be completely obtained by approximating the analytic solution by the truncated series

w (x, t) = lim_{M \to \infty} \sum_{p = 0}^{M} w_{p} (x, t)

To facilitate the requirement of the accuracy, we obtained the optimal value of $ℏ$ using a nonlinear algebraic equation $d Δ ℏ / d ℏ = 0$ , where $Δ ℏ$ is obtained using equation (25). We obtained the minimum error relative to $ℏ = - 1.97$ for fifth-order approximation. This result was suitable to confirm the convergence analysis.

Graphical interpretation

Results from the asymptotic methods were compared with a robust numerical solver of Maple and it was depicted that OHAM appeared to be better in accuracy for such type of problems since it provided suitable results with better convergence rates. The parameter N represents the order of the soliton and is a dimensionless combination of the pulse and fiber parameters. An optical pulse whose parameters satisfy the condition $N = 1$ is called the fundamental soliton (left panel of Figures 1 –3). Pulses corresponding to other integer values of N are called higher order solitons (right panel of Figures 1 –3).

Figure 1.

Bright solitons.

Figure 2.

Gray solitons.

Figure 3.

Black solitons.

From Figure 1, it is obvious that input pulse acquires a phase shift $x / 2$ as it propagates inside the fiber, but its amplitude remains constant. It is this property of a fundamental soliton that makes it an ideal candidate for optical communications. Essentially, the effects of fiber dispersion are exactly balanced by the fiber nonlinearity. The NLSE describing dark (gray and black) solitons is obtained from equation (1) by changing the sign of the second term. The gray solitons and the black solitons presented in Figures 2 and 3 are obtained by considering two values (0 and $π / 4$ ) for the internal phase angle involved in the solution of the NLSE.

From the graphs, it is obvious that as the fractional order increases and approaches the integer order, the soliton dynamics changes in the three cases (Bright, Gray, and Black) uniquely. It is graphically obvious that an optical fiber can force any input pulse to evolve toward a soliton as the order $α$ approaches unity. The fractional analysis helps to reveal the dynamics of the solitons more explicitly. In case of the dark solitons (Gray and Black, as shown in Figures 2 and 3), the intensity profile of the resulting solutions displays a dip in a uniform background, and it is the dip that remains unchanged during propagation inside the fiber as conjectured by Agrawal.³⁸

Conclusion

The current discussion can prove to be helpful in understanding the unrevealed dynamics of the optical solitons. A review of the approaches available in the literature to solve integer as well as fractional order optical solitons is made. The fractional approach is considered since it gives an excellent instrument for the description of memory and hereditary properties of the medium and the wave dynamics. During this research, a comparison between the integer order and fractional order NLSE solution has been made. We have showed that the application of new asymptotic methods is useful to study the fractional order optical wave dynamics. Thus, our analysis can help to approximate the solution and to predict the upcoming state of the nonlinear problems in plasma physics, engineering, and optoelectronic studies. The future work includes modeling the solution process as a Petri net^41–43 for explicit visualization.

Footnotes

Appendix 1

The second- and third-order analytic approximations to the solution of NLSE obtained by FHATM are listed below

\begin{matrix} w_{2} (x, t) = \frac{ℏ N t^{α} sech x}{4 Γ (1 + α) Γ (1 + 2 α)} {(1 + ℏ) Γ (1 + 2 α) {3 {ic}_{0}^{3} + {ic}_{0}^{3} k^{2} - 4 i c_{0} N^{4} + 2 i ω \\ - {ic}_{0}^{3} \cosh 2 x + {ic}_{0}^{3} k^{2} \cosh 2 x + 2 i ω \cosh 2 x - 2 c_{0}^{3} k \sinh 2 x} + ℏ t^{α} Γ (1 + α) \\ \times {- 5 c_{0}^{6} + 12 c_{0}^{4} N^{4} - 12 c_{0}^{2} N^{8}} {sech}^{4} x + {{ic}_{0}^{3} k^{2} + 2 i ω - 2 c_{0}^{3} k \tanh x - {ic}_{0}^{3} \overset{2}{\tanh} x}^{2} \\ + 2 {sech}^{2} x {- 3 c_{0}^{6} k^{2} + 4 c_{0}^{4} k^{2} N^{4} - 2 c_{0}^{3} ω + 8 c_{0} N^{4} ω - 10 c_{0}^{6} ik \tanh x + 12 c_{0}^{4} ik N^{4} \tanh x \\ + 9 c_{0}^{6} \overset{2}{\tanh} x - 12 c_{0}^{4} N^{4} \overset{2}{\tanh} x}} \end{matrix}

\begin{matrix} w_{3} (x, t) = \frac{1}{8 {(Γ (1 + α))}^{2} Γ (1 + 2 α) Γ (1 + 3 α)} {ℏ N t^{α} sech x {2 (1 + ℏ)^{2} Γ (1 + α) Γ (1 + 2 α) \\ \times Γ (1 + 3 α) {sech}^{2} x {3 i c^{3} + i k^{2} c^{3} - 4 ic N^{4} + 2 i ω - i c^{3} \cosh 2 x + i k^{2} c^{3} \cosh 2 x + 2 i ω \cosh 2 x \\ - 2 c^{3} k \sinh 2 x} - \frac{3}{2} c h^{2} i N^{4} t^{2 α} {(Γ (1 + 2 α))}^{2} {sech}^{6} x {3 i c^{3} + i k^{2} c^{3} - 4 ic N^{4} + 2 iw - i c^{3} \cosh 2 x \\ + i k^{2} c^{3} \cosh 2 x + 2 iw \cosh 2 x - 2 c^{3} k \sinh 2 x}^{2} + ℏ t^{α} (Γ (1 + α))^{2} {(\frac{1}{2} (1 + ℏ) Γ (1 + 3 α) {sech}^{4} x) \\ \times {- 115 c^{6} + 20 c^{6} k^{2} - 10 c^{6} k^{2} - 3 c^{6} k^{4} + 192 c^{4} N^{4} + 32 c^{4} k^{2} N^{4} - 96 c^{2} N^{8} - 20 c^{3} w - 12 c^{3} k^{2} w \\ + 64 c N^{4} w - 12 w^{2} + (- 16 c^{6} k^{2} + 76 c^{6} - 8 c^{6} k^{2} - 4 c^{6} k^{4} - 96 c^{4} N^{4} + 32 c^{4} k^{2} N^{4} - 16 c^{3} N - 16 c^{3} ω \\ - 16 c^{3} k^{2} ω + 64 c N^{4} ω - 16 ω^{2}) \cosh 2 x + (6 c^{6} k^{2} - c^{6} - c^{6} k^{4} + 4 c^{3} ω - 4 c^{3} k^{2} ω - 4 ω^{2}) \cosh 4 x - (88 c^{6} ik \\ - 8 c^{6} i k^{3} + 96 c^{4} ik N^{4} - 16 c^{3} ik) \sinh 2 x + (4 c^{6} ik - 4 c^{6} i k^{3} - 8 c^{3} ik ω) \sinh 4 x} + ℏ t^{α} Γ (1 + 2 α) \\ \times {- 61 c^{9} i + 138 c^{7} i N^{4} - 132 c^{5} i N^{8} + 72 c^{3} i N^{1} 2} + {c^{3} i k^{2} + 2 i ω + 2 c^{3} k \tanh x - c^{3} i \overset{2}{\tanh} x}^{3} \\ + c^{2} i {sech}^{4} x {- 75 c^{7} k^{2} + 120 c^{5} k^{2} N^{4} - 60 c^{3} k^{2} N^{8} - 30 c^{4} ω + 96 c^{2} N^{4} ω - 120 N^{8} ω + (- 366 c^{7} ik \\ - 600 c^{5} ik N^{4} - 264 c^{3} ik N^{8}) \tanh x + (479 c^{7} - 864 c^{5} N^{4} + 444 c^{3} N^{8}) \overset{2}{\tanh} x} + {sech}^{2} x {- 15 c^{9} i k^{4} \\ + 14 c^{7} i k^{4} N^{4} - 36 i k^{2} ω + 56 c^{4} i k^{2} ω N - 12 c^{3} i ω^{2} + 56 ci ω^{2} N^{4} + (- 100 c^{9} k^{3} - 96 c^{7} k^{3} n^{4} + 120 c^{6} k ω \\ - 192 c^{4} k ω N^{4}) \tanh x + (270 c^{9} i k^{2} - 276 c^{9} i k^{2} N^{4} + 108 c^{6} i ω - 216 c^{4} i) \overset{2}{\tanh} x + (- 348 c^{9} k \\ - 684 c^{7} k N^{4}) \overset{3}{\tanh} x + (- 179 c^{9} i + 222 c^{7} i N^{4}) \overset{4}{\tanh} x}}}} \end{matrix}

Academic Editor: Praveen Agarwal

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the Science and Technology Development Fund, MSAR, under Grant No. 078/2015/A3.

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Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons

Abstract

Keywords

Introduction

Problem description

Application of OHAM

Method description

Application and accuracy

Application of FHATM

Method description

Application and accuracy

Graphical interpretation

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References