Research Article | Open Access
Some Analytical Solitary Wave Solutions for the Generalized q-Deformed Sinh-Gordon Equation:
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.
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 , crystal lattice formation , and string dynamics in curved space-time . 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.
Let us first define the generalized q-deformed Sinh-Gordon equation aswhere , , and For , (1) admits a trivial constant solutionIt 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 towhere and is the Arai q-deformed function defined byand we have alsowhere For we get the standard , , , and functions. We list below some simple and useful relations for q-deformed functionswhere and are, respectively, the inverse functions of and . Since and vanish at thenIt 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 asby changing and .
Let us return to (1), by defining the following new normalized coordinates:and using(1) becomesHere is defined by the sign of and . Using the standard transformationandwith is an arbitrary constant, we obtain thenThis 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 byandwhere
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 coordinateso 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 verifiesIt 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 getso we have an implicit equation for the traveling wavewhich can be written also aswhere and are free parameters. We can derive general explicit solutions for the considered q-deformed Sinh-Gordon equation. In factand 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 thenFinally, 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 caseAfter multiplication of (45) by and integration, we havewhere 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 thenLet us first mention that is not a solution for (50). By using the transformationwe getIn order to solve this nonlinear second-order equation we definesoand (52) will be transformed toThis is a first-order inhomogeneous linear differential equation with the general solutionHere is an arbitrary constant. From definition (53) and (56) the expression of the soliton in the implicit form is thenwhere and are free parameters. We can writewhereThe expression of the solution for (52) is thenand consequently the general expression of the soliton in the moving frame iswhere and are free parameters.
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 , rational expansion method , expansion method , and auxiliary method . 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.
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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