Some Analytical Solitary Wave Solutions for the Generalized q-Deformed Sinh-Gordon Equation : ∂ 2 θ / ∂z∂ξ

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, KadomtsevPetviashvili 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 BoseEinstein 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.

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.

Preliminaries
Let us first define the generalized q-deformed Sinh-Gordon equation as where , ,  ∈ R * ,  ∈ R, and  ∈ R + .For / ⩾ 0, (1) admits a trivial constant solution It is useful to mention that any first-order polynome ( 1 () =  1  +  0 ; with  1 ̸ = 0) in  or in  or any combination of two arbitrary first-order polynomes (  0  1 () +  1  1 ()) can not be a solution to the generalized q-deformed Sinh-Gordon equation (1).Equation ( 1) is also equivalent to where  1 =   and  1 = 1/.sinh  is the Arai q-deformed function defined by and we have also tanh where 0 <  ≤ 1.For  = 1 we get the standard sinh, cosh, sech, and tanh functions.We list below some simple and useful relations for q-deformed functions where arctan ℎ  and arcsinh  are, respectively, the inverse functions of tanh  and sinh  .Since tanh  () and sinh  () vanish at  = (1/2) ln() then arctanh It is worth mentioning that the generalized q-deformed Sinh-Gordon equation ( 1) with  = 0 can be easily transformed to the generalized q-deformed Sin-Gordon equation defined as by changing  → () 1/  and  → () (1/−) .
Let us return to (1), by defining the following new normalized coordinates: and using (1) becomes Here ± is defined by the sign of  and  1 = /||.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

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  1 =  ±  we get the trivial constant solution for  1 ⩾ 0  ( 1 ) = [ln ( Here we take  ̸ = ±1; the traveling wave in the moving frame  verifies It is worth mentioning that, for  = 1,  = 1,  1 = 0, and  = 1, (27) describes the standard traveling wave of Sinh-Gordon equation.Let us consider now the following particular cases.
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  1 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  3 () =  3 +  1  2 + .By defining  1 =  +  1 /3 the polynomial  3 () will be transformed to (1/4)(4 3 1 + 4((3 −  2 1 )/3) 1 + 4(2 3 1 /27 −  1 /3)), where  2 = −4((3 −  2 1 )/3) and  3 = −4(2 3 1 /27 −  1 /3), and we have then Finally, we obtain the general expression of the soliton in the moving frame with 1 and  2 are free parameters.Let us now study another case.
Advances in Mathematical Physics 3.2.Special Case ( 2): ( = −2,  = 1, and  1 = 0).In this section we shall derive analytical soliton solutions for the case After multiplication of (45) by ()/ and integration, we have where  1 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  1 ̸ = −4/ and  1 ̸ = 0.By choosing  1 = 0 or  1 = −4/ we can generate from direct integration of (46) two other sets of implicit solutions, namely, and 3.3.Special Case (3): ( = 2,  = 1, and  1 = −/2).Another interesting case to study is for ( = 2,  = 1, and  1 = −/2), 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   1 is an arbitrary constant.From definition ( 53) and ( 56) the expression of the soliton in the implicit form is then where  and  1 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 where  1 and  2 are free parameters.

Conclusion
We have introduced a generalized q-deformed Sinh-Gordon equation defined by (1) ( 2 / = [sinh  (  )]  − ).We have derived general explicit analytical soliton solutions for two sets of parameters, namely, ( = 1,  = 1, and  = 0) and ( = 2,  = 1, and  = −(/2)||).Furthermore, we have given general implicit soliton solutions for the case:  = −2,  = 1, and  = 0. 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.