# Some Analytical Solitary Wave Solutions for the Generalized q-Deformed Sinh-Gordon Equation:∂2θ/∂z∂ξ=αsinhq⁡(βθγ)p-δ

Authors: H. Eleuch
Journal: Advances in Mathematical Physics
 (2018)

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## 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.


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## Body

## 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(1)∂2θ∂z∂ξ=αsinhq⁡βθγp-δ,where α,γ,p∈R∗, δ∈R, and β∈R+. For δ/α⩾0, (1) admits a trivial constant solution(2)θz,ξ=1βln⁡δα1/p+δα2/p+q1/γ.It is useful to mention that any first-order polynome (Po1(s)=a1s+a0; with a1≠0) in z or in ξ or any combination of two arbitrary first-order polynomes ( α0Po1(z)+α1po1(ξ)) can not be a solution to the generalized q-deformed Sinh-Gordon equation (1). Equation (1) is also equivalent to(3)∂2θ1γ1∂z∂ξ=αsinhq⁡βθ1p-δ,where θ1=θγ and γ1=1/γ.  sinhq⁡ is the Arai q-deformed function defined by(4)sinhq⁡x=ex-qe-x2,and we have also(5)coshq⁡x=ex+qe-x2,(6)sec⁡hqx=1coshq⁡x,(7)tanhq⁡x=ex-qe-xex+qe-x=sinhq⁡xcoshq⁡x,where 0<q≤1. For q=1 we get the standard sinh, cosh, sech, and tanh functions. We list below some simple and useful relations for q-deformed functions(8)coshq⁡x2-sinhq⁡x2=q,(9)sinhq⁡-x=-qsinh1/q⁡x,(10)coshq⁡-x=qcosh1/q⁡x,(11)ddxsinhq⁡x=coshq⁡x,(12)ddxcoshqx=sinhq⁡x,(13)ddxtanhq⁡x=qcoshq⁡x2=1-tanhq2⁡x,(14)ddxsec⁡hqx=-tanhq⁡xsec⁡hqx,(15)ddxarcsin⁡hqx=1q+x2,(16)ddxarctan⁡hqx=11-x2,where arctan⁡hq and arcsinhq are, respectively, the inverse functions of tanhq and sinhq. Since tanhq(x) and sinhq(x) vanish at x=1/2ln⁡(q) then(17)arcsinhq⁡x=∫0xdx′q+x′2+12ln⁡q=ln⁡x+x2+q,(18)arctanhq⁡x=∫0xdx′1-x′2+12ln⁡q=12ln⁡q1+x1-x;x<1.It is worth mentioning that the generalized q-deformed Sinh-Gordon equation (1) with δ=0(19)∂2θ∂z∂ξ=αsinhq⁡βθγpcan be easily transformed to the generalized q-deformed Sin-Gordon equation defined as(20)∂2θ∂z∂ξ=αsinq⁡βθγp,by changing θ→(i)1/γθ and z→(i)(1/γ-p)z.Let us return to (1), by defining the following new normalized coordinates:(21)z1=β1/2γαz,(22)ξ1=β1/2γαξ,and using(23)u=β1/γθ,(1) becomes(24)∂2u∂z1∂ξ1=±sinhq⁡uγp-δ1.Here ± is defined by the sign of α and δ1=δ/α. Using the standard transformation(25)x=az1+ξ1aand(26)t=az1-ξ1a,with a is an arbitrary constant, we obtain then(27)∂2u∂x2-∂2u∂t2=sinhq⁡uγp-δ1.This equation is obtained for positive α, and for negative α we obtain the same equation as (27) by interchanging x→t and t→x. Equation (27) has two conservative quantities, namely, the total energy E and the momentum P [77, 78] defined by(28)E=∫-∞∞∂2u∂x2-∂2u∂t2-Fudx,and(29)P=12∫-∞∞∂u∂t∂u∂xdx,where(30)Fu=∫sinhq⁡uγpdu-δ1.


## 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(31)η=11-c2x-ct,so(32)∂∂t=-c1-c2∂∂η,∂∂x=11-c2∂∂η.c can be interpreted as the speed of the traveling wave in the space-time (x-t). Note that by defining η1=x±t we get the trivial constant solution for δ1⩾0(33)uη1=ln⁡δ11/p+δ12/p+q1/γ.Here we take c≠±1; the traveling wave in the moving frame η verifies(34)d2uηdη2=sinhq⁡uγηp-δ1.It is worth mentioning that, for p=1, γ=1, δ1=0, and q=1, (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 (p=1, γ=1, and δ1=0)

For=1, δ1=0, and γ=1 (in this case (27) can be identified as a deformed Sinh-Gordon equation). We can multiply both side of (34) by du(η)/dη and after the integration we get(35)12duηdη2=coshq⁡uη+c1′,so we have an implicit equation for the traveling wave(36)∫udu′2coshq⁡u′+c1=±η+c2′,which can be written also as(37)Hu±η=C,where(38)Hu=∫udu′2coshq⁡u′+c1.C and c1 are free parameters. We can derive general explicit solutions for the considered q-deformed Sinh-Gordon equation. In fact(39)Hu=∫udu′eu′+qe-u′+c1=∫eudyy3+qy+c1y2,and let us first remind the definition of the Weierstrass Elliptic function ϱ(x):(40)x=∫ϱx;g2;g3∞dw4w3-g2w-g3.so we need to transform the expression of H(u) and in particular the polynomial P3(y)=y3+c1y2+qy. By defining y1=y+c1/3 the polynomial P3(y) will be transformed to 1/4(4y13+4(3q-c12/3)y1+4(2c13/27-qc1/3)), where g2=-4(3q-c12/3) and g3=-4(2c13/27-qc1/3), and we have then(41)Hu=∫eu+c1/32dy14y13-g2y1-g3.Finally, we obtain the general expression of the soliton in the moving frame η(42)uη=ln⁡ϱ±12η+c2;g2;g3-c13,with(43)g2=4c12-3q3,(44)g3=4qc13-2c1327.c1 and c2 are free parameters. Let us now study another case.


### 3.2. Special Case (2): (p=-2, γ=1, and δ1=0)

In this section we shall derive analytical soliton solutions for the case(45)d2uηdη2=1sinhq⁡uη2.After multiplication of (45) by du(η)/dη and integration, we have(46)duηdη=±4q-e2uη+c1,where c1 is a free parameter.Equation (46) can be integrated and we obtain then the following general implicit solutions:(47)0=q4+c1qln⁡c1q2+4q+c1q2+4qe2uη-qc1e2uη-c1q-4-2+c1qe2uη+1c1ln⁡2+c1q-c1e2x-c1e2uη-qc1e2uη-c1q-4-4q4+c1quη±2η+c2.It is worth mentioning that this equation is valid only for c1≠-4/q and c1≠0.By choosingc1=0 or c1=-4/q we can generate from direct integration of (46) two other sets of implicit solutions, namely,(48)q-e2uη-qarctanh⁡q-e2uηq±2η+c2=0,and(49)qe-2uη-1+arctan⁡1qe-2uη-1±2qη+c2=0.


### 3.3. Special Case (3): (p=2, γ=1, and δ1=-q/2)

Another interesting case to study is for (p=2, γ=1, and δ1=-q/2), the traveling wave equation is then(50)d2uηdη2=sinhq⁡uη2+q2=12coshq2⁡2uη.Let us first mention that u=cst is not a solution for (50). By using the transformation(51)vη=e2uη,we get(52)2vηv′′η-2v′2-vη3-q2vη=0.In order to solve this nonlinear second-order equation we define(53)wv=dvηdη2,so(54)d2vηdη2=12dwdvv,and (52) will be transformed to(55)vdwdvv-2wv=v3+q2v.This is a first-order inhomogeneous linear differential equation with the general solution(56)wv=v3+c1′v2-q2v.Here c1′ is an arbitrary constant. From definition (53) and (56) the expression of the soliton in the implicit form is then(57)∫vηdvv3+c1v2-q2v=±η+C,where C and c1 are free parameters. We can write(58)∫vηdvv3+c1v2-q2v=-∫vη+c1/32dv14v13-g4v1-g5,where(59)g4=4c12+3q23,(60)g5=-4q2c13+2c1327,The expression of the solution for (52) is then(61)vη=ϱ±12η+c2;g4;g5-c13,and consequently the general expression of the soliton in the moving frame η=1/1-c2(x-ct) is(62)uη=ln⁡ϱ±12η+c2;g4;g5-c13,where c1 and c2 are free parameters.


## 3.1. Special Case (1): The Deformed Sinh-Gordon Equation (p=1, γ=1, and δ1=0)

For=1, δ1=0, and γ=1 (in this case (27) can be identified as a deformed Sinh-Gordon equation). We can multiply both side of (34) by du(η)/dη and after the integration we get(35)12duηdη2=coshq⁡uη+c1′,so we have an implicit equation for the traveling wave(36)∫udu′2coshq⁡u′+c1=±η+c2′,which can be written also as(37)Hu±η=C,where(38)Hu=∫udu′2coshq⁡u′+c1.C and c1 are free parameters. We can derive general explicit solutions for the considered q-deformed Sinh-Gordon equation. In fact(39)Hu=∫udu′eu′+qe-u′+c1=∫eudyy3+qy+c1y2,and let us first remind the definition of the Weierstrass Elliptic function ϱ(x):(40)x=∫ϱx;g2;g3∞dw4w3-g2w-g3.so we need to transform the expression of H(u) and in particular the polynomial P3(y)=y3+c1y2+qy. By defining y1=y+c1/3 the polynomial P3(y) will be transformed to 1/4(4y13+4(3q-c12/3)y1+4(2c13/27-qc1/3)), where g2=-4(3q-c12/3) and g3=-4(2c13/27-qc1/3), and we have then(41)Hu=∫eu+c1/32dy14y13-g2y1-g3.Finally, we obtain the general expression of the soliton in the moving frame η(42)uη=ln⁡ϱ±12η+c2;g2;g3-c13,with(43)g2=4c12-3q3,(44)g3=4qc13-2c1327.c1 and c2 are free parameters. Let us now study another case.


## 3.2. Special Case (2): (p=-2, γ=1, and δ1=0)

In this section we shall derive analytical soliton solutions for the case(45)d2uηdη2=1sinhq⁡uη2.After multiplication of (45) by du(η)/dη and integration, we have(46)duηdη=±4q-e2uη+c1,where c1 is a free parameter.Equation (46) can be integrated and we obtain then the following general implicit solutions:(47)0=q4+c1qln⁡c1q2+4q+c1q2+4qe2uη-qc1e2uη-c1q-4-2+c1qe2uη+1c1ln⁡2+c1q-c1e2x-c1e2uη-qc1e2uη-c1q-4-4q4+c1quη±2η+c2.It is worth mentioning that this equation is valid only for c1≠-4/q and c1≠0.By choosingc1=0 or c1=-4/q we can generate from direct integration of (46) two other sets of implicit solutions, namely,(48)q-e2uη-qarctanh⁡q-e2uηq±2η+c2=0,and(49)qe-2uη-1+arctan⁡1qe-2uη-1±2qη+c2=0.


## 3.3. Special Case (3): (p=2, γ=1, and δ1=-q/2)

Another interesting case to study is for (p=2, γ=1, and δ1=-q/2), the traveling wave equation is then(50)d2uηdη2=sinhq⁡uη2+q2=12coshq2⁡2uη.Let us first mention that u=cst is not a solution for (50). By using the transformation(51)vη=e2uη,we get(52)2vηv′′η-2v′2-vη3-q2vη=0.In order to solve this nonlinear second-order equation we define(53)wv=dvηdη2,so(54)d2vηdη2=12dwdvv,and (52) will be transformed to(55)vdwdvv-2wv=v3+q2v.This is a first-order inhomogeneous linear differential equation with the general solution(56)wv=v3+c1′v2-q2v.Here c1′ is an arbitrary constant. From definition (53) and (56) the expression of the soliton in the implicit form is then(57)∫vηdvv3+c1v2-q2v=±η+C,where C and c1 are free parameters. We can write(58)∫vηdvv3+c1v2-q2v=-∫vη+c1/32dv14v13-g4v1-g5,where(59)g4=4c12+3q23,(60)g5=-4q2c13+2c1327,The expression of the solution for (52) is then(61)vη=ϱ±12η+c2;g4;g5-c13,and consequently the general expression of the soliton in the moving frame η=1/1-c2(x-ct) is(62)uη=ln⁡ϱ±12η+c2;g4;g5-c13,where c1 and c2 are free parameters.


## 4. Conclusion

We have introduced a generalized q-deformed Sinh-Gordon equation defined by (1) (∂2θ/∂z∂ξ=αsinhq(βθγ)p-δ). We have derived general explicit analytical soliton solutions for two sets of parameters, namely, (p=1, γ=1, and δ=0) and (p=2, γ=1, and δ=-q/2α). Furthermore, we have given general implicit soliton solutions for the case: p=-2, γ=1, and δ=0. Future investigations will be focusing on generating more soliton solutions using algebraic methods such as Tanh-method [79], rational expansion method [80], F-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.



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