Traveling wave solutions for wave equations with exponential nonlinearities S C Mancas Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114-3900, USA e-mail:
[email protected]
H C Rosu and M. P´erez-Maldonado
arXiv:1708.00542v1 [math-ph] 1 Aug 2017
1
IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Camino a la presa San Jos´e 2055, Col. Lomas 4a Secci´ on, 78216 San Luis Potos´ı, S.L.P., Mexico e-mails:
[email protected],
[email protected] We use a simple method which leads to the quadrature involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained while when that term is nonzero we give all the basic traveling wave solutions based on a detailed study of the corresponding elliptic equations of several well-known particular cases with important applications in physics. PACS numbers: 02.30.Ik, 02.30.Gp, 02.30.Hq, 02.30.Jr Keywords: Liouville equation; Tzitz´eica, Dodd-Bullough; Dodd-Bullough-Mikhailov; sine-Gordon; sinh-Gordon; Weierstrass function
I.
INTRODUCTION
In this paper, we will consider second-order differential equations with two exponential nonlinear terms ψuv = αeaψ + βebψ ,
(1)
where a and b are nonzero real constants, while α and β are real constants not simultaneously zero. This kind of equations can be turned into polynomial nonlinear equations ∂2 log h = αha + βhb , ∂u∂v
(2)
by using the change of variables ψ = log h. Particular cases of this type of equations are the Tzitz´eica equation [1–3] and its variants, such as the Dodd-Bullough equation [4] and the Dodd-Bullough-Mikhailov equation [5, 6], and also the sine-Gordon and the sinh-Gordon equations belong to this class, all of them with important applications in solid state physics, nonlinear optics, biological physics, and quantum field theory. Along the two characteristics z = u − λv, t = u + λv, λ 6= 0, Eq. (1) becomes the nonlinear wave equation ψtt − ψzz = or in the logarithmic variable
1 αeaψ + βebψ λ
h(htt − hzz ) − (ht 2 − hz 2 ) =
h2 αha + βhb . λ
(3)
(4)
We are concerned with finding closed-form solutions of Eq. (4) using the traveling wave ansatz h(z, t) = h(ξ) with ξ = kz − ωt, and k 6= ±ω which yields the following ordinary differential equation (ODE) hhξξ − hξ 2 =
h2 αha + βhb ≡ f (h), λγ
(5)
with γ = ω 2 − k 2 6= 0. This ODE will be analyzed in the next two sections of the paper, both in its full generality and simplified to the important cases mentioned above.
2 II.
THE IMPLICIT SOLUTION
du [7], to obtain A simple method to solve ODEs of type (5) is to let hξ = u(h(ξ)), and use hξξ = u dh
hu
du − u2 = f (h), dh
(6)
which can be turned into a Bernoulli equation using the substitution u2 = z dz 2 2 − z = f (h). dh h h
(7)
The solution for this equation is Z f (h) dh , z = h c0 + 2 h3 √ and using back the transformations u = ± z = dh dξ h is obtained by the quadrature 2
Z
dh
q =± h c1 + αa ha + βb hb
r
2 λγ
Z
dξ .
(8)
(9)
In general, this quadrature can be performed only if c1 = 0, when one obtains the following implicit solution involving the hypergeometric function r a β βhb b bα a−b αha −b = ∓√ + ;1 − ;− h h (ξ − ξ0 ) , (10) 2 F1 1, 1 − a b 2(a − b) 2(a − b) aβ 2λγ which also implies β 6= 0. We will show in the next section how the quadrature in (9) is solved in the important particular cases mentioned in the introduction for c1 6= 0. III.
PARTICULAR CASES
A.
Liouville equation
We start with the simplest case, which is the Liouville equation [8], one of the fundamental equations in nonlinear mathematical physics and differential geometry. It corresponds to α = 1, β = 0, a = 1, b = 0 which is ∂2 log h = h . ∂u∂v
(11)
The quadrature in Eq. (9) takes the form Z
dh √ =± h c1 + h
r
2 (ξ − ξ0 ), λγ
(12)
and the function h is obtained by solving the elliptic equation hξ 2 = a3 h3 + a2 h2 + a1 h + a0
(13)
with the coefficients given by the system a0 a1 a2 a3 For convenience, denote r =
1 λγ ,
and p =
2c1 λγ ;
=0 =0 1 = 2c λγ 2 = λγ .
(14)
then Eq. (13) becomes the reduced elliptic equation h2ξ = 2rh3 + ph2 ,
(15)
3 with soliton solution if p > 0, or periodic solution if p < 0 √ p sech2 21 p(ξ − ξ0 ) , p > 0 h(ξ) = − 2r √ p sec2 21 −p(ξ − ξ0 ) , p < 0 , h(ξ) = − 2r
(16)
which by using system (14) give the solutions
q c1 h(ξ) = −c1 sech2 (ξ − ξ ) , 0 q 2λγ −c1 , h(ξ) = −c1 sec2 2λγ (ξ − ξ0 )
c1 2λγ c1 2λγ
>0
(17)
< 0.
Since we are in the case β = 0, we cannot apply (10) when c1 = 0, but the integration of (12) corresponds to the most degenerate case of the Weierstrass elliptic equation (25) when both germs g2 and g3 are zero, which leads to the rational solution h(ξ) =
2λγ . (ξ − ξ0 )2
(18)
Plots of all these three types of Liouville solutions are given in Fig. (1). h( )=eψ ξ 3.0
2.5
2.0
c1 =-1,λ=-2,γ=1 c1 =-1,λ=2,γ=1
1.5
c1 =0,λ=2,γ=1 1.0
0.5
-6
-4
-2
0
2
4
6
FIG. 1: The soliton and periodic solutions (17) and the rational solution (18) of the Liouville equation.
B.
The Tzitz´ eica equation
Tzitz´eica’s equation, ∂2 1 log h = h − 2 , ∂u∂v h
(19)
emerged in 1907-1910 in the area of geometry, but only after eighty years it has been found to have applications in physics. For example, Euler’s equations for an ideal gas with a special equation of motion can be reduced to the Tzitz´eica equation, and a 2+1-dimensional system in magneto-hydrodynamics has been shown to be in one-to-one correspondence with it [9, 10]. Very recently, dark optical solitons and traveling waves of Tzitz´eica type have been also discussed in the literature [11, 12]. For Tzitz´eica’s equation, we identify the constants in (1) as α = 1, β = −1, a = 1, b = −2 which gives the quadrature r Z 2 dh q (ξ − ξ0 ). (20) =± λγ 1 3 2 h + c1 h + 2
4 In implicit form h satisfies Eq. (10) which simplifies to 1 1 4 ξ − ξ0 . h 2 F1 , ; ; −2h3 = ± √ 2 3 3 λγ
(21)
The solution h is obtained explicitly by solving the elliptic equation (13) with coefficients given by the system a0 a1 a2 a3
1 = λγ =0 1 = 2c λγ 2 = λγ .
(22)
which becomes hξ 2 = 2rh3 + ph2 + r .
(23)
Using the scale shift transformation h(ξ) = Eq. (23) becomes the Weierstrass equation
1 p 2℘(ξ; g2 , g3 ) − , r 6
(24)
℘ξ2 = 4℘3 − g2 ℘ − g3 .
(25)
The germs of the Weierstrass function are given by g2 = g3 =
2 a22 −3a1 a3 = p12 = 2(e21 + e22 + e23 ) 12 9a1 a2 a3 −27a0 a23 −2a32 p3 = − 41 r3 + 54 432
= 4e1 e2 e3
(26)
and together with the modular discriminant
∆ = g2 3 − 27g3 2 = −
r3 3 (p + 27r3 ) = 16(e1 − e2 )2 (e1 − e3 )2 (e2 − e3 )2 16
(27)
are used to classify the solutions of Eq. (23), where the constants ei are the roots of the cubic polynomial s3 (t) = 4t3 − g2 t − g3 = 4(t − e1 )(t − e2 )(t − e3 ) = 0.
(28)
− 23 .
Case (1). If ∆ ≡ 0 ⇒ p = −3r ⇒ c1 = This degenerate case implies that s3 (t) has repeated root of multiplicity two. Then the Weierstrass solutions can be simplified since ℘ degenerates into hyperbolic or trigonometric functions. Because of the degeneracy Eq. (20) can be factored as r Z dh 2 q (ξ − ξ0 ). (29) =± λγ (h − 1)2 h + 12 Depending on the sign of g3 we have the sub-cases: 3
Case (1a). r > 0 with g2 > 0 ⇒ g3 = − r8 < 0 ⇒ λγ > 0, so Eq. (29) has the soliton solution r 3 1 1 h(ξ) = −1 + 3tanh2 (ξ − ξ0 ) . 2 2 λγ
(30)
By letting e1 = e2 = eˆ > 0 then e3 = −2ˆ e < 0, hence
g2 = 12ˆ e2 > 0 g3 = −8ˆ e3 < 0
(31)
the Weierstrass ℘ solution to Eq. (25) reduces to √ ℘(ξ; 12ˆ e2 , −8ˆ e3) = eˆ + 3ˆ e csch2 ( 3ˆ eξ) . For eˆ =
1 4λγ
> 0, the Weierstrass solution replaced by (32) gives r 3 1 3 2 (ξ − ξ0 ) . h(ξ) = 1 + csch 2 2 λγ
(32)
(33)
5 3
Case (1b). r < 0 with g2 > 0 ⇒ g3 = − r8 > 0 ⇒ λγ < 0, so Eq. (29) has the periodic solution r 1 3 1 2 (ξ − ξ0 ) . h(ξ) = − 1 + 3tan 2 2 −λγ
(34)
By letting e2 = e3 = −˜ e < 0 with e˜ > 0, then e1 = 2˜ e > 0, hence g2 = 12˜ e2 > 0 3 g3 = 8˜ e >0
(35)
√ eξ). ℘(ξ; 12˜ e2 , 8˜ e3 ) = −˜ e + 3˜ e csc2 ( 3˜
(36)
the Weierstrass ℘ solution reduces to
1 For e˜ = − 4λγ > 0, the Weierstrass solution replaced by (36) gives
3 h(ξ) = 1 − csc2 2
r 3 1 (ξ − ξ0 ) . 2 −λγ
(37)
All the Tzitz´eica solutions corresponding to these cases are displayed in Fig. (2). h(ξ)=eψ ξ 2
1
-6
-4
-2
2
4
6
ξ
c1 =-1.5,λ=-2,γ=-1 c1 =-1.5,λ=-2,γ=1
-1
-2
-3 h(ξ)=eψ ξ
4
2
c1 =-1.5,λ=-2,γ=-1 -6
-4
-2
2
4
6
ξ
c1 =-1.5,λ=-2,γ=1
-2
-4
FIG. 2: The Tzitz´eica soliton (30) and periodic solution (34) (top). The Tzitz´eica solution (33) and periodic solution (37) (bottom).
Case (2). If ∆ 6= 0 ⇒ p 6= −3r ⇒ c1 6= − 32 we include two particular solutions which will fix the integration constant c1 as follows: the equianharmonic (g2 = 0) and lemniscatic case (g3 = 0), respectively.
6 3
27 6 i) For the equianharmonic case g2 = 0 ⇒ p = 0 ⇒ g3 = − r4 . Because ∆ = − 16 r < 0, then s3 (t) has a pair of conjugate complex roots, and since c1 = 0 the solution to Eq. (20) reduces to 1 (38) h(ξ) = 2λγ℘ ξ − ξ0 ; 0, − 3 3 . 4λ γ √ √ 3 6 ii) For the lemniscatic case g3 = 0 ⇒ p = −3 3 2r ⇒ g2 = 3 4 4 r2 . Because ∆ = 27 16 r > 0, then s3 (t) has three √ √ g2 g2 distinct real roots given by e3 = − 2 , e2 = 0, and e1 = 2 . Although the Weierstrass unbounded function has poles aligned on the real axis of the ξ − ξ0 complex plane, we can choose ξ0 in such a way to shift these poles a half of period above the real axis, so that the elliptic function simplifies using the formula [13] √ e1 − e3 (ξ − ξ0′ ); m (39) ℘(ξ; g2 , 0) = e3 + (e2 − e3 )sn2
with elliptic modulus m =
q
e2 −e3 e1 −e3 .
Using the values of the roots together with ξ0′ = 0 we obtain √ g2 2 cn ℘(ξ; g2 , 0) = − 2
√ 4 g2 ξ;
√ ! 2 . 2
(40)
3 the solutions for the lemniscatic case are reduced using the transformation (24) to periodic Because c1 = − √ 3 4 cnoidal waves, and they become " √ !# √ 4 √ 1 3 2 h(ξ) = √ 1 − 3 cn2 √ . (41) √ ξ; 3 3 2 4 2 λγ
iii) For the most general case, g2 6= 0, g3 6= 0 the general solution to Eq. (20) is c1 4c1 3 + 27 c1 2 . − h(ξ) = λγ 2℘ ξ − ξ0 ; 2 2 , − 3λ γ 108λ3 γ 3 3λγ
(42)
The equianharmonic, lemniscatic, and Weierstrass solutions of Tzitz´eica’s equation are displayed in Fig. (3). C.
The Dodd-Bullough equation
This variant of Tzitz´eica’s equation has been introduced in the first dedicated study of the polynomial conserved quantities of the sine-Gordon equation [4]. Its form in the h variable is ∂2 1 log h = −h + 2 , ∂u∂v h thus, we identify the constants to be α = −1, β = 1, a = 1, b = −2 which gives the quadrature r Z dh 2 q (ξ − ξ0 ). =± λγ 1 3 2 −h + c1 h − 2 In implicit form, the h functions satisfies Eq. (10) which simplifies to ξ − ξ0 1 1 4 . , ; ; −2h3 = ∓ √ h 2 F1 2 3 3 −λγ
Using Tzitz´eica solutions provided that r → −r ⇒ λγ → −λγ and c1 → −c1 , we have: Case (1). If ∆ ≡ 0 ⇒ c1 =
3 2
, then
(43)
(44)
(45)
7 h(ξ)=eψ ξ 2.0 1.5 1.0
c1 =0,λ=1,γ=1 0.5
c1 =-10
-5
5
10
ξ
3 3
,λ=1,γ=1
4
-0.5 -1.0 -1.5 h(ξ)=eψ ξ 3
2
1
c1 =-1,λ=-2,γ=1 -10
-5
5
10
ξ
c1 =-1,λ=2,γ=1
-1
-2
-3
FIG. 3: The equianharmonic and lemniscatic solutions (38), (41) of the Tzitz´eica equation (top). The Weierstrass solution (42) of the Tzitz´eica equation (bottom).
Case (1a). λγ < 0 so Eq. (44) has soliton solution h(ξ) =
r 1 3 1 −1 + 3tanh2 (ξ − ξ0 ) , 2 2 −λγ
while the second linearly independent solution corresponding to (33) is r 3 3 1 h(ξ) = 1 + csch2 (ξ − ξ0 ) . 2 2 −λγ
(46)
(47)
Case (1b). λγ > 0 so Eq. (44) has periodic solution h(ξ) = −
r 1 3 1 1 + 3tan2 (ξ − ξ0 ) 2 2 λγ
and the second linearly independent solution corresponding to (37) is r 3 3 1 h(ξ) = 1 − csc2 (ξ − ξ0 ) . 2 2 λγ Case (2). If ∆ 6= 0 ⇒ c1 6=
3 2
then the Weierstrass solution of Eq. (44) is c1 c1 2 4c1 3 − 27 . − h(ξ) = −λγ 2℘ ξ − ξ0 ; 2 2 , − 3λ γ 108λ3 γ 3 3λγ
(48)
(49)
(50)
8 i) For the equianharmonic case, c1 = 0, we have: h(ξ) = −2λγ℘ ξ − ξ0 ; 0, ii) For the lemniscatic case, c1 =
3 √ 3 , 4
1 4λ3 γ 3
.
(51)
we have:
" √ 1 1 − h(ξ) = √ 3 cn2 3 4
√ !# √ 4 3 2 √ . ξ; √ 3 2 2 −λγ
(52)
Respecting the rules of changing the signs of the parameters, the Dodd-Bullough solutions are identical to the Tzitz´eica solutions and consequently we will not plot them here. D.
The Tzitz´ eica-Dodd-Bullough equation
This variant equation reads ∂2 1 log h = h + 2 , ∂u∂v h thus, we identify the constants to be α = 1, β = 1, a = 1, b = −2 which gives the quadrature r Z dh 2 q (ξ − ξ0 ) . =∓ λγ h3 + c1 h2 − 12
(53)
(54)
We can use the Dodd-Bullough solutions, Eqs. (46)-(50,) with h → −h and ξ → −ξ. E.
The Dodd-Bullough-Mikhailov equation
The last Tzitz´eica variant equation reads 1 ∂2 log h = −h − 2 , ∂u∂v h thus the constants are identified as α = −1, β = −1, a = 1, b = −2 which gives the quadrature r Z dh 2 q (ξ − ξ0 ) . =∓ λγ −h3 + c1 h2 + 21
(55)
(56)
From the polynomial in the integrand, one can see that the solutions of this variant equation are obtained from Tzitz´eica solutions by h → −h and ξ → −ξ. F.
The sine-Gordon equation
According to [14], Tzitz´eica’s equation is the “nearest relative” of the well-known sine-Gordon equation which can be written as 1 i ∂2 h − h−i = sin(log h) . log h = ∂u∂v 2i
Thus, we identify the constants to be α = Z
1 2i ,
(57)
1 β = − 2i , a = i, b = −i which gives the quadrature
dh p =± h c1 − cos(log h)
r
2 (ξ − ξ0 ), λγ
(58)
9 that is equivalent to Z
dψ
p =± c1 − cos(ψ)
r
2 (ξ − ξ0 ). λγ
In implicit form, h satisfies Eq. (10) which simplifies to p 3 5 ξ − ξ0 . hi cos(log h) 2 F1 1, ; ; −h2i = ∓ √ 4 4 2 2λγ For the special case of c1 = 1, Eq. (59) simplifies to Z 2 dψ = ± √ (ξ − ξ0 ). λγ sin ψ2 for λγ > 0, and using the identity tan θ2 =
1−cos θ sin θ
dψ 2 = ±√ (ξ − ξ0 ) ψ −λγ cos 2
Z for λγ < 0, and using the identity tan
θ 2
+
π 4
=
(60)
(61)
we obtain the kink-antikink solutions
ξ−ξ0 ±√ ψ(ξ) = 4 arctan e λγ .
When c1 = −1, Eq. (59) simplifies to
(59)
1−cos θ sin θ
(62)
(63)
the shifted kink-antikink solution is obtained
ξ−ξ0 ±√ ψ(ξ) = −π + 4 arctan e −λγ .
(64)
Plots of the c1 = ±1 arctan solutions (62) and (64) are displayed in Fig. (4). When c1 6= ±1, then ψ satisfies the elliptic equation ψξ 2 =
2 (c1 − cos ψ), λγ
(65)
with solutions given by r 2 c1 − 1 (ξ − ξ0 ); ψ(ξ) = 2 am ± 2λγ 1 − c1
(66)
where the Jacobi amplitude ψ = am(ξ; m) is the inverse of the elliptic integral of the first kind F (ψ; m) = ξ given by Eq. (59) [15, 16]. For plots of the sine-Gordon Jacobi amplitude solutions, see Fig. (5). G.
The sinh-Gordon equation
The hyperbolic version of the sine-Gordon equation, the sinh-Gordon equation, is also extensively used in integrable quantum field theory [17, 18], kink dynamics [19], and hydrodynamics [20]. Here, we will use the parametrization in [17] and write the corresponding equation in the variable h in the form ∂2 1 2 log h = h − h−2 = sinh(2 log h) . ∂u∂v 2
(67)
Thus, we identify the constants to be α = 21 , β = − 21 , a = 2, b = −2 which gives the quadrature Z
dh q h c1 +
1 2
cosh(2 log h)
=±
r
2 (ξ − ξ0 ), λγ
(68)
10 ψ(ξ) 6
4
c1 =1,λ=1,γ=1 2
-10
c1 =-1,λ=-1,γ=1
-5
5
ξ
10
-2
FIG. 4: The arctan kink solutions (62) and (64) of the sine-Gordon equation. ψ() 2
1
c1 =0.5,λ=-1,γ=1 -10
-5
5
10
c1 =-0.5,λ=2,γ=-1
-1
-2 ψ(ξ) 15
10
5
c1 =-2,λ=2,γ=-1 -10
-5
5
10
ξ
c1 =-1.5,λ=2,γ=-1
-5
-10
-15
FIG. 5: The Jacobi amplitude solution (66) of the sine-Gordon equation for |m| > 1 (top) and |m| < 1 (bottom).
which is equivalent to Z
dψ q c1 +
1 2
cosh(2ψ)
=±
r
2 (ξ − ξ0 ). λγ
(69)
11 In implicit form, h satisfies Eq. (10) which simplifies to 1 1 5 ξ − ξ0 4 . h 2 F1 , , ; −h = ± √ 2 4 4 2λγ
(70)
For the special case of c1 = − 12 , Eq. (69) simplifies to Z
dψ =± sinh ψ
r
2 (ξ − ξ0 ). λγ
(71)
with the kink-antikink solutions q 2 ± λγ (ξ−ξ0 ) . ψ(ξ) = 2 arctanh e
(72)
When c1 = 21 , Eq. (69) simplifies to r 2 dψ = gd(ψ) = ± (ξ − ξ0 ), cosh ψ λγ where the Gudermannian function gd(ψ) = 2 arctan tanh ψ2 gives also the kink-antikink solutions Z
1 (ξ − ξ0 ) . ψ(ξ) = 2 arctanh tan ± √ 2λγ
(73)
(74)
The arctanh solutions are plotted in Fig. (6). When c1 6= ± 21 , then ψ satisfies the elliptic equation 1 2 c1 + cosh(2ψ) , ψξ 2 = λγ 2
(75)
with solutions given by r 2 2c1 + 1 . (ξ − ξ0 ); ψ(ξ) = i am ± −λγ 2c1 + 1
(76)
Plots of the Jacobi amplitude solution are given in Fig. (7). ψ(ξ)
2
1
c1 =0.5,λ=1,γ=1, -10
-5
5
10
ξ
c1 =-0.5,λ=1,γ=1
-1
-2
FIG. 6: The arctanh solutions (72) and (74) of the sinh-Gordon equation.
12 ψ(ξ) 0.6 0.4 0.2
-10
c1 =-1,λ=1,γ=-1
-5
5
10
ξ
c1 =-1,λ=2,γ=-2
-0.2 -0.4 -0.6 ψ(ξ)
6 4 2
-10
c1 =0.75,λ=1,γ=1
-5
5
10
ξ
c1 =0.75,λ=2,γ=2
-2 -4 -6 -8
FIG. 7: The Jacobi amplitude solution (76) of the sinh-Gordon equation for |m| > 1 (top) and |m| < 1 (bottom).
IV.
CONCLUSION
In summary, we have used a very simple method to obtain the quadrature of the general form of wave equations with two exponential nonlinearities whose particular cases correspond to celebrated equations in mathematical physics, such as Liouville, Tzitz´eica and its variants, sine-Gordon, and sinh-Gordon equations. All the basic soliton, periodic and Weierstrass solutions of these equations are obtained consistently in the traveling variable by a thorough analysis of the elliptic equation. Particular implicit solutions in terms of a generic hypergeometric function are also obtained through a direct integration. Some of these solutions, such as the Weierstrass solutions of the Tzitz´eica class of equations and the amplitude Jacobi solutions of the sine/sinh-Gordon equations cannot be obtained by other methods in the literature, for example the well-known tanh method, although they can also be obtained via the integral bifurcation method [21]. Of course, for more complicated multiple-soliton (multi-phase soliton) solutions, one should use Darboux and B¨acklund transformations [14, 22, 23].
Acknowledgements SCM would like to acknowledge partial support from the Dean of Research & Graduate Studies at Embry-Riddle Aeronautical University while on a short visit to IPICyT, San Luis Potos´ı, Mexico. MPM thanks CONACyT for a doctoral fellowship.
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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Tzitz´eica G, “Sur une nouvelle classe de surfaces” C.R. Acad. Sci. Paris 144 1257 (1907). Tzitz´eica G, “Sur une nouvelle classe de surfaces” Rend. Circ. Mat. Palermo 25 180 (1908); “ibid” 28 210 (1909). Tzitz´eica G, “Sur une nouvelle classe de surfaces” C.R. Acad. Sci. Paris 150 955 and 1227 (1910). Dodd R K and Bullough R K, “Polynomial conserved quantities for the sine-Gordon equations” Proc. R. Soc. A 352 481 (1977). Mikhailov A V, “Integrability of a two-dimensional generalization of the Toda chain” JETP Lett. 30 414 (1979). Wazwaz A M, “The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the TzitzeicaDodd-Bullough equations” Chaos, Solitons & Fractals 25 55 (2005). Mancas S C and Rosu H C, “Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations” Phys. Lett. A 377 1434 (2013). Liouville J, “Sur l’´equation aux diff´erences partielles d2 ln λ/dudv ± λ/2a2 = 0” J. Math. Pures Appl. 18 71 (1853). Gaffet B, “SU(3) symmetry of the equations of unidimensional gas flow, with arbitrary entropy distribution” J. Math. Phys. 25 245 (1984). Gaffet B, “A class of 1-d gas flows soluble by the inverse scattering transform” Physica D 26 123 (1987). Manafian J and Lakestanhi M, “Dispersive dark optical soliton with Tzitzeica type nonlinear evolution equations arising in nonlinear optics” Opt. Quant. Electron. 48 116 (2016). Hosseini K, Bekir A, and Kaplan M, “New exact traveling wave solutions of the Tzitzeica-type evolution equations arising in non-linear optics” J. Mod. Opt. 64 1688 (2017). Whittaker E T and Watson G N, A Course of Modern Analysis Chap XX (Cambridge: Cambridge University Press, 1958). Boldin A Yu, Safin S S, and Sharipov R A, “On an old article of Tzitzeica and the inverse scattering method” J. Math. Phys. 34 5801 (1993). Robertson J S, “Gudermann and the simple pendulum” The College Math. J. 28 271 (1997). Mondaini L, “The rise of solitons in sine-Gordon field theory: From Jacobi amplitude to Gudermannian function” J. Appl. Math. Phys. 2 1202 (2014) Bajnok Z, Rim C and Zamolodchikov Al, “Sinh-Gordon boundary TBA and boundary Liouville reflection amplitude” Nucl Phys. B 796 622 (2008). Perring J K and Skyrme T H, “A model unified field equation” Nucl Phys. 31 550 (1962). Fu Z, Liu S, and Liu S, Exact solutions to double and triple sinh-Gordon equations Z. Naturforsch. 59a 933 (2004). Andreev V K, Kaptsov O V, Pukhnachov V V and Rodionov A A, Applications of Group-Theoretical Methods in Hydrodynamics (Dordrecht: Kluwer, 1998). Rui W, “Exact traveling wave solutions for a nonlinear evolution equation of generalized Tzitz´eica-Dodd-BulloughMikhailov type” J. Appl. Math. 2013 395628 (2013). Brezhnev Yu M, “Darboux transformation and some multi-phase solutions of the Dodd-Bullough-Tzitzeica equation: Uxt = eU − e−2U ” Phys. Lett. A 211 94 (1996). Conte R, Musette M and Grundland A M, “B¨ acklund transformation of partial differential equations from the Painlev´eGambier classification. II Tzitz´eica equation” J. Math. Phys. 40 2092 (1999).
