^{1}

^{*}

^{2}

^{3}

^{4}

^{4}

^{1}

^{2}

^{3}

^{4}

Edited by: Devendra Kumar, University of Rajasthan, India

Reviewed by: Haci Mehmet Baskonus, Harran University, Turkey; K. S. Nisar, Prince Sattam Bin Abdulaziz University, Saudi Arabia; Jagdev Singh, JECRC University, India

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The research paper aims to investigate the space-time fractional cubic-quartic non-linear Schrödinger equation in the appearance of the third, and fourth-order dispersion impacts without both group velocity dispersion, and disturbance with parabolic law media by utilizing the extended sinh-Gordon expansion method. This method is one of the strongest methods to find the exact solutions to the non-linear partial differential equations. In order to confirm the existing solutions, the constraint conditions are used. We successfully construct various exact solitary wave solutions to the governing equation, for example, singular, and dark-bright solutions. Moreover, the 2D, 3D, and contour surfaces of all obtained solutions are also plotted. The finding solutions have justified the efficiency of the proposed method.

Non-linear partial differential equations have different types of equations, one of them is the non-linear Schrödinger equation (NLSE) that relevant to the classical and quantum mechanics. The non-linear Schrödinger equation is a generalized (1 + 1)-dimensional version of the Ginzburg-Landau equation presented in 1950 in their study on supraconductivity and has been specifically reported by Chiao et al. [

where ^{α} of order α, where α ∈ (0, 1] is the fractional derivative, the parameters γ and β are real constants, a real-valued algebraic function ^{2}) is

By using the relation of

on Equation (1), we obtain the fractional non-linear Schrödinger equations with Parabolic law as follows:

The extended sinh-Gordon expansion method is intended to a generalization of the sine-Gordon expansion equation because it is based on an auxiliary equation namely the sine-Gordon equation (see previous studies [

The outline of paper are organize the paper as follows: A short review of the conformable derivative is presented in section 2. Section 3 deals with the analysis of the ShGEM. In section 4, the method is applied to solve the non-linear Schrödinger equation involving the fractional derivatives with the parabolic law. Eventually, in section 5, we presented our conclusion of this paper.

The basic definitions of the conformable derivative of order α are given as follows [

if the Riemann improper integral exists.

In the current section, we presented the main steps of the e ShGEM (see previous study [

Consider the following fractional non-linear PDE:

where

Consider the wave transformation

by substitute relation Equation (8) into Equation (7), we obtain the following non-linear ODE:

Consider the trial solution of Equation (9) of the form

The parameters _{j}, _{j}, for (_{0} are real constants, and θ is a function of η that hold the following ODE:

The homogeneous balance principle is applied on Equation (9) to find the value of

The exact solutions of Equation (12) may be given as

and

Letting solutions of Equation (10) along with Equations (13) and (14) as the form

Finding the value of

we gather a group of over-defined non-linear algebraic equations in _{0}, A_{j}, _{j}, putting the coefficients of ^{i}(θ) ^{j}(θ) to zero, and finding the solutions of acquired system, we gain the values of _{0}, A_{j}, _{j}, _{1}, _{2}, κ, and ω. Putting the values of _{0}, A_{j}, _{j}, _{1}, _{2}, κ, and ω into Equations (15) and (16), we can find the solutions of Equation (7).

The implementation of the extended ShGEM to the cubic-quartic non-linear Schrödinger equation with conformable derivative is provided in this section.

Consider the wave transformation

In Equation (18), θ(

Multiply both sides of Equation (19) by

From Equation (20), we get constraint conditions ν = 4γκ^{3} − 3βκ^{2} and β = 4γκ. Balancing the terms ^{‴}^{6} yields κ = 1. With κ = 1, Equations (10), (16), and (17) change to

and

respectively.

Inserting Equation (22) along with Equation (12) into Equation (21), and using constraint conditions provides a non-linear algebraic system. Equaling each coefficient of ^{i}(θ) ^{j}(θ) with the same power to zero, and finding the obtained system of algebraic equations, we gain the values of the parameters. Putting the obtained values of the parameters into Equations (23) and (24), give the solutions of Equation (3).

we get

we get

we get

we get

we obtain

we get

we get

In this article, we have successfully used the extended sinh-Gordon expansion method to solve the problem for the non-linear cubic-quartic Schrödinger equations involving fractional derivatives with the Parabolic law. A traveling wave transforms in the sense of the comfortable derivative has been used to convert the governing equation into a NODE. The various optical solutions of the studied model have been constructed, for example, the singular soliton solutions as shown in

3D, 2D, and contour surfaces of Equation (26) where ω = 0.1, _{2} = 0.1, κ = 2, α = 0.8.

3D, 2D, and contour surfaces of Equation (28) where γ = 0.5, _{2} = 0.2, κ = 0.4, α = 0.7.

3D, 2D, and contour surfaces of Equation (30) where γ = 0.5, _{1} = 0.7, α = 0.9.

3D, 2D, and contour surfaces of Equation (32) where γ = 5, _{1} = 7, α = 0.7.

3D, 2D, and contour surfaces of Equation (34) where _{1} = 0.2, κ = 0.4, ω = 6, α = 0.4.

3D, 2D, and contour surfaces of Equation (36) where _{2} = 0.2, _{1} = 0.3, α = 0.7.

3D, 2D, and contour surfaces of Equation (38) where α = 0.3, _{1} = 0.5, _{2} = 0.2, κ = 0.4.

All datasets generated for this study are included in the article/supplementary material.

HD contributed in developing the proofs and edited the article for possible improvement. HG, KA, and RY contributed in developing the main results and proofs. All authors read the final version and approved it.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.