Advances in Mathematical Physics
Volume 2018 (2018), Article ID 5242757, 7 pages
https://doi.org/10.1155/2018/5242757

Research Article

Some Analytical Solitary Wave Solutions for the Generalized q-Deformed Sinh-Gordon Equation:

H. Eleuch1,2

¹Institute for Quantum Science and Engineering, Texas A&M University, College Station, TX 77843, USA
²Department of Applied Sciences and Mathematics, College of Arts and Sciences, Abu Dhabi University, Abu Dhabi, UAE

Correspondence should be addressed to H. Eleuch; hichemeleuch@yahoo.fr

Received 28 June 2018; Accepted 9 September 2018; Published 25 September 2018

Academic Editor: Boris G. Konopelchenko

Copyright © 2018 H. Eleuch. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce the generalized q-deformed Sinh-Gordon equation and derive analytical soliton solutions for some sets of parameters. This new defined equation could be useful for modeling physical systems with violated symmetries.

1. Introduction

Dynamical models are the cornerstone of several fields of science and uneven covered in the literature [1–9]. They represent an important research area in mathematics and theoretical physics. Such models are usually based on nonlinear ordinary or partial differential equations. Nonlinearity has generated crucial effects and phenomena not only in the macroscopical systems but also in the microscopical systems governed by quantum physics. We can cite as examples Rogue waves [10–12], bifurcation [13, 14], bistability [15–18], chaos [19–22], phase transition [23–26], lasing [27–31], superradiance [32–34], correlations [35–37], squeezing [38–41], and solitons [42–48]. A particular interest is focused on solitons due to their potential new application perspectives in several fields of physics and engineering [49–56]. Soliton is a propagating wave solution that conserves its shape and has a particle-like behavior. This propriety could be explained in general by a balance between dispersion and nonlinearity. Several famous nonlinear partial differential equations generate solitons [43, 57–61] such as Korteweg-de Vries equation, nonlinear Schrödinger equation, Kadomtsev-Petviashvili equation, Sin-Gordon, and Sinh-Gordon equations. The Sinh-Gordon equation has several applications such as in surface theory [62], crystal lattice formation [63], and string dynamics in curved space-time [64]. When the q-deformed hyperbolic function, proposed in the ninetieth of the last century by Arai [65, 66], is included in the dynamical system, the symmetry of the system is destroyed and consequently the symmetry of the solution. Recently, several solutions for Schrödinger equation and Dirac equation with q-deformed hyperbolic potential are generated [67–72]. q-deformed functions are very promising for modeling atom-trapping potentials or statistical distributions in Bose-Einstein condensates [73, 74] as well as for exploring vibrational spectra of diatomic molecules [75, 76].

In this work, we propose to analyze the propagating wave solutions for a more general form of the Sinh-Gordon equation, the generalized q-deformed Sinh-Gordon equation. As far as we know, this equation is introduced for the first time. We derive for some sets of parameters analytical soliton solutions. The generalized q-deformed Sinh-Gordon equation will open the door for conceiving models of physical systems where the symmetry is absent or violated.

2. Preliminaries

Let us first define the generalized q-deformed Sinh-Gordon equation as

where , , and For , (1) admits a trivial constant solution

It is useful to mention that any first-order polynome (; with ) in or in or any combination of two arbitrary first-order polynomes ( ) can not be a solution to the generalized q-deformed Sinh-Gordon equation (1). Equation (1) is also equivalent to

where and    is the Arai q-deformed function defined by

and we have also

where For we get the standard , , , and functions. We list below some simple and useful relations for q-deformed functions

where and are, respectively, the inverse functions of and . Since and vanish at then

It is worth mentioning that the generalized q-deformed Sinh-Gordon equation (1) with

can be easily transformed to the generalized q-deformed Sin-Gordon equation defined as

by changing and .

Let us return to (1), by defining the following new normalized coordinates:

and using

(1) becomes

Here is defined by the sign of and . Using the standard transformation

and

with is an arbitrary constant, we obtain then

This equation is obtained for positive , and for negative we obtain the same equation as (27) by interchanging and . Equation (27) has two conservative quantities, namely, the total energy and the momentum [77, 78] defined by

and

where

3. Traveling Wave Solution

In this section we explore the traveling wave solutions of the generalized q-deformed Sinh-Gordon equation (27). We first define a new moving coordinate

so

can be interpreted as the speed of the traveling wave in the space-time . Note that by defining we get the trivial constant solution for

Here we take ; the traveling wave in the moving frame verifies

It is worth mentioning that, for , , , and , (27) describes the standard traveling wave of Sinh-Gordon equation. Let us consider now the following particular cases.

3.1. Special Case (1): The Deformed Sinh-Gordon Equation (, , and )

For , , and (in this case (27) can be identified as a deformed Sinh-Gordon equation). We can multiply both side of (34) by and after the integration we get

so we have an implicit equation for the traveling wave

which can be written also as

where

and are free parameters. We can derive general explicit solutions for the considered q-deformed Sinh-Gordon equation. In fact

and let us first remind the definition of the Weierstrass Elliptic function

so we need to transform the expression of and in particular the polynomial . By defining the polynomial will be transformed to , where and , and we have then

Finally, we obtain the general expression of the soliton in the moving frame

with

and are free parameters. Let us now study another case.

3.2. Special Case (2): (, , and )

In this section we shall derive analytical soliton solutions for the case

After multiplication of (45) by and integration, we have

where is a free parameter.

Equation (46) can be integrated and we obtain then the following general implicit solutions:

It is worth mentioning that this equation is valid only for and .

By choosing or we can generate from direct integration of (46) two other sets of implicit solutions, namely,

and

3.3. Special Case (3): (, , and )

Another interesting case to study is for (, , and , the traveling wave equation is then

Let us first mention that is not a solution for (50). By using the transformation

we get

In order to solve this nonlinear second-order equation we define

so

and (52) will be transformed to

This is a first-order inhomogeneous linear differential equation with the general solution

Here is an arbitrary constant. From definition (53) and (56) the expression of the soliton in the implicit form is then

where and are free parameters. We can write

where

The expression of the solution for (52) is then

and consequently the general expression of the soliton in the moving frame is

where and are free parameters.

4. Conclusion

We have introduced a generalized q-deformed Sinh-Gordon equation defined by (1) (). We have derived general explicit analytical soliton solutions for two sets of parameters, namely, (, , and ) and (, , and ). Furthermore, we have given general implicit soliton solutions for the case: , , and . Future investigations will be focusing on generating more soliton solutions using algebraic methods such as Tanh-method [79], rational expansion method [80], expansion method [81], and auxiliary method [82]. The general study of the traveling wave solutions for the generalized q-deformed Sinh-Gordon equation or its asymptotic behavior as well as the analysis of some other proprieties will be for sure very interesting. The usefulness of the generalized q-deformed Sinh-Gordon equation will not be limited to the fields of applied mathematics and mathematical physics but also will open the door for applications in physics where the symmetry of the studied system is absent or violated.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

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