This work proposes the construction of a prototype of pulse-kink hybrid solitary waves with a strong Kink dosage of the Sasa-Satsuma equation which describes the dynamics of the wave propagating in an optical fiber where the stimulated Raman scattering effect is bethinking during modeling. The ultimate goal of this work is to propose a plateful of solutions likely to serve as signals during studies on computer or laboratory propagation studies. The resolution of such an equation is not always the easiest thing, and we used the Bogning-Djeumen Tchaho-Kofané method extended to the implicit functions of Bogning to obtain the results. The flexibility of the iB-functions made it possible to deduce the trigonometric solutions, from the obtained solitary wave solutions with a hyperbolic analytical sequence of the studied Sasa-Satsuma equation. A better appreciation of the nature of the solutions obtained is made through the profiles of some solutions obtained during the different analyses.
Nowadays, nonlinear optics is a field that increasingly instigates the curiosity of many researchers around the world. This curiosity can be justified by the fact that many telecommunication industries use optical fibers more as a medium for transmitting or transporting large quantities of data over long distances, especially transoceanic, transcontinental and many other distances. This understanding can also be extended to the fact of the wide application of which optical solitons constitute the basic element of data transmission technology. It is undoubtedly for this reason that a good number of works encountered in the literature track down and reveal the optical solitons which coexist with mathematical models such as the nonlinear Schrödinger equation [
The aim of this work is to construct new prototypes of optical solitary waves of the Sasa-Satsuma dynamical equation in order to further enrich the literature with new varieties of more robust solitary wave solutions. And at the same time, allow new investigations in laboratories during the propagation tests which will lead to the understanding of new phenomena that occur in the studied model. Owing to all of this, our manuscript is organized as follows: Section 2 gives a brief description of the chosen model while; Section 3 explains the outline of the Bogning-Djeumen Tchaho-Kofané method (BDKm) [
The Sasa-Satsuma equation which is the chosen model is written as being [
i ∂ Φ ∂ t + λ ∂ 2 Φ ∂ x 2 + γ | Φ | 2 Φ + i ( α ∂ 3 Φ ∂ x 3 + β | Φ | 2 ∂ Φ ∂ x + θ ∂ | Φ | 2 ∂ x Φ ) = 0. (1)
The first term describes the temporal evolution of the optical soliton molecules while the second term provided the group velocity dispersion (GVD). Then, the third term is the well-known Kerr-law fiber nonlinearity, Φ ( x , t ) the optical soliton profile, the factor of the imaginary i through the coefficients β , θ , α sequentially provides the self-steepening, stimulated Raman scattering in additionally third-order dispersion. Two mathematical techniques namely, improved F-expansion and improved auxiliary methods was used in [
The Bogning-Djeumen Tchaho-Kofané method [
X ( Φ , Φ t , Φ x y , Φ x z t , Φ t y , Φ y z , Φ t t z , Φ x x x , ⋯ , | Φ | 2 , ( Φ | Φ | 2 ) t , ⋯ ) = 0, (2)
where Φ ( x , y , z , t ) is an unknown function to be determined, X is some function of Φ and its derivatives with respect to x , y , z , t and X includes the highest order derivatives and the nonlinear terms. Most often, we use the change of variables Φ ( x , t ) = Ω ( ξ ) , ξ = ∑ k = 0 p α k x k . In the case where Φ is a function of x , y , z and t, ξ becomes ξ = x + y + z − ν t , where ν is the wave speed. In this context, Equation (2) gives rise to the ordinary differential equation (ODE) below:
X O D E ( Ω , Ω ′ , Ω ″ , ⋯ , Ω ′ | Ω | 2 , ⋯ ) = 0, (3)
where Ω ′ , Ω ″ represent respectively the first and second derivatives of the envelope Ω with respect to ξ . Then, the solution we are looking for can be expressed under contracted form
Ω ( ξ ) = ∑ i j μ i j J i , j ( η ξ ) , (4)
where η is a real constant, μ i j are the unknown constants to be determined and J n , m ( α x ) are the iB-functions whose explicit hyperbolic form is written as:
J n , m ( ∑ i = 0 p α i x i ) = sinh m ( ∑ i = 0 p α i x i ) cosh n ( ∑ i = 0 p α i x i ) . (5)
where α i , ( i = 0 ; 1 ; 2 ; ⋯ ; p ) are the parameters associated to the independent variables x i , ( i = 0 ; 1 ; 2 ; ⋯ ; p ) , m and n are powers of both terms of Equation (5). For more details, see [
∑ i j n A n ( μ i j , η , ν ) J n , 0 ( η ξ ) + ∑ i j m B m ( μ i j , η , ν ) J m , 1 ( η ξ ) + ∑ i j k C k ( μ i j , η , ν ) J − k , 0 ( η ξ ) + ∑ i j l D l ( μ i j , η , ν ) J − l , 1 ( η ξ ) + ∑ i j E ( μ i j , η , ν ) J 0 , 0 ( η ξ ) = 0 , (6)
where i , j , k , l are positive natural integers and n , m the real numbers [
Now, we address the resolution of Equation (1) by applying the BDKm with certain transformations specific to traveling waves.
The traveling wave solutions that we seek to construct can be considered in the form below
Φ ( x , t ) = Ψ ( ξ ) e i ϕ ( x , t ) , (7)
where ϕ ( x , t ) = − k x + ω t ; ξ = x − ν t ; ω is the angular frequence; k the wave number and ν the wave speed. Thus, the insertion of Equation (7) into Equation (1) yields the following equation
m 1 Ψ + m 2 ∂ 2 Ψ ∂ ξ 2 + m 3 | Ψ | 2 Ψ + i ( m 4 ∂ Ψ ∂ ξ + α ∂ 3 Ψ ∂ ξ 3 + β | Ψ | 2 ∂ Ψ ∂ ξ + θ ∂ | Ψ | 2 ∂ ξ Ψ ) = 0, (8)
where m 1 = α k 3 − λ k 2 − ω ; m 2 = λ + 3 α k ; m 3 = γ + β k ; m 4 = − ( 2 λ k + 3 α k 2 + ν ) . Equation (8) describes the dynamics of the amplitude Ψ . We decided to look for Ψ ( ξ ) verifying Equation (8) such that
Ψ ( ξ ) = a J 1,0 ( η ξ ) + i b J 1,1 ( η ξ ) + c J 2,1 ( η ξ ) , (9)
where J 1,0 ( η ξ ) , J 1,1 ( η ξ ) and J 2,1 ( η ξ ) are hyperbolic iB-functions; a , b , c are constant real coefficients, η the inverse of the width at half height of each of the solitons contained in Equation (9), and i an imaginary such that i = − 1 . Equation (9) is a complex multi-soliton whose basic form is represented by the first two terms of coefficient a and b, respectively. This basic shape is disturbed by the addition of a hybrid soliton of amplitude c. We believe that it is this disturbance term that is at the origin of the emergence of new hybrid structures in the propagation media. In this context, inserting of Equation (9) into Equation (8) leads to gamuts main equation below
∑ l = 1 6 P l ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J l , 0 ( η ξ ) + ∑ n = 2 6 Q n ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J n , 1 ( η ξ ) + i [ ∑ j R j ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J j , 0 ( η ξ ) + ∑ p = 1 6 S p ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J p , 1 ( η ξ ) ] = 0 , (10)
where j ∈ { 1 ; 2 ; 3 ; 4 ; 5 ; 7 } . Equation (10) is made up of four gamuts of equations, two of which are preponderant in the order of resolution in the case where we want to approach or force the solutions, in particular the ranges of equations resulting from the global coefficients P l and R j . In the case c ≠ 0 , we obtain the approximate solution while the case c = 0 produces exact solutions. Thus, we obtain two families of solutions.
According to the theory briefly presented in Section 3, only range equations which resulting from the coefficients of the terms, in J l ,0 ( η ξ ) and in J j ,0 ( η ξ ) allow to best approach the solutions given by Equation (9) in the case of this Family I. The cape being fixed, one obtains from the identification at zero of the ranges of equations offered by the coefficients P l and R j :
- From the real part
the term in J 6,0 ( η ξ ) ,
η ( β + 4 θ ) b c 2 = 0 , (11)
the term in J 5,0 ( η ξ ) ,
− 3 m 3 a c 2 = 0 , (12)
the term in J 4,0 ( η ξ ) ,
[ η ( β + 2 θ ) ( b 2 − a 2 ) − η ( β + 6 θ ) c 2 + 6 α η 3 ] b = 0, (13)
the term in J 3,0 ( η ξ ) ,
[ m 3 ( a 2 − b 2 + 3 c 2 ) − 2 η 2 m 2 ] a = 0, (14)
the term in J 2,0 ( η ξ ) ,
[ 2 η θ a 2 − η ( β + 2 θ ) b 2 + 2 η θ c 2 − m 4 η − 4 α η 3 ] b = 0, (15)
the term in J 1,0 ( η ξ ) ,
( m 3 b 2 + m 2 η 2 + m 1 ) a = 0, (16)
- From the imaginary part
the term in J 7,0 ( η ξ ) ,
− 2 η ( β + 2 θ ) c 3 = 0, (17)
the term in J 5,0 ( η ξ ) ,
[ 4 η ( β + 2 θ ) a 2 − 2 η ( β + θ ) b 2 + 3 η ( β − 2 θ ) c 2 − 24 α η 3 ] c = 0, (18)
the term in J 4,0 ( η ξ ) ,
− 2 m 3 a b c = 0 , (19)
the term in J 3,0 ( η ξ ) ,
[ η ( 3 β + 2 θ ) b 2 − 3 η ( β + 2 θ ) a 2 − η ( β + 2 θ ) c 2 + 2 m 4 η + 20 α η 3 ] c = 0, (20)
the term in J 2,0 ( η ξ ) ,
2 m 3 a b c = 0 , (21)
the term in J 1,0 ( η ξ ) ,
− ( η β b 2 + α η 3 + m 4 η ) c = 0, (22)
From the structuring of the above equations, it appears from Equations (11) and (17) that we can have β = − 4 θ or b = 0 or c = 0 and β = − 2 θ or c = 0 respectively, while Equations (12), (19), (21) enforce to choose m 3 = 0 ⇔ γ = − β k or a = 0 or c = 0 or b = 0 . Given all these conditions, only the case m 3 = 0 ⇔ γ = − β k ; β = − 2 θ ; a ≠ 0 ; b ≠ 0 ; c ≠ 0 allows to obtain non-trivial approximate solutions. Under these conditions, Equations (12), (17), (19) and (21) are verified. On the other hand, Equation (11) proposes β = − 4 θ and b = c = 0 , which means in this case that, the contribution of this equation to this order of power of the constitutive hyperbolic functions of the ansatz given by Equation (9) is not significant, and therefore negligible. Continuing our analysis, Equations (14) and (16) lead successively to m 2 = 0 ⇔ λ = − 3 α k and m 1 = 0 ⇔ ω = α k 3 − λ k 2 while the combination of Equations (13) and (15) gives, respectively:
c = ± η 3 2 α θ , (23)
and
a = ± 1 2 ( η 2 − 3 k 2 ) α θ − k λ θ − 1 2 ν θ , (24)
with α θ ≻ 0 ; λ θ ≺ 0 ; ν ≺ 0 and η ≥ k 3 . Then, Equations (18) and (20) give, respectively
θ b 2 − 6 θ c 2 − 12 α η 2 = 0 (25)
and
b = ± 1 2 ( 10 η 2 − 3 k 2 ) α θ − k 10 λ θ − ν θ . (26)
Given Equation (23), the successive combinations of Equations (22), (24) and (25) provides the two respective constraints
ν = 9 α η 2 + 9 5 λ k (27)
and
α = − 19 λ θ k ( 120 η 2 + 15 k 2 ) θ + 120 η 2 . (28)
Finally, we obtain the approximate expression of Family I of solutions of Equation (1) in the form
Φ ( x , t ) = [ ± 1 2 ( η 2 − 3 k 2 ) α θ − k λ θ − 1 2 ν θ J 1,0 ( η ξ ) ± i 1 2 ( 10 η 2 − 3 k 2 ) α θ − k 10 λ θ − ν θ J 1,1 ( η ξ ) ± η 3 2 α θ J 2 , 1 ( η ξ ) ] e i ( − k x + 4 α k 3 t ) ; (29)
with α θ ≻ 0 ; λ θ ≺ 0 ; ν θ ≺ 0 ; η ≥ k 3 ; β = − 2 θ ; γ = 2 k θ ; λ = − 3 α k ; ω = 4 α k 3 (these expressions of ω , λ , γ , β , are obtained from equations m 1 = 0 , m 2 = 0 , m 3 = 0 and (17) whose are constraints imposed by the studied system.) and the constraints given by Equations (27) and (28) respectively. It should be emphasized here that the third term of Equation (29) is a hybrid soliton of the bright-dark or dark-bright type. By carefully observing this Equation (29), we realize that it is a complex multi-soliton prototype whose third term is at the origin of the appearance in the propagation medium of multiform solitary wave structures, this in relation with the values taken by the parameters of the wave. This can be justified in the sense that the magnitude of the amplitude Ψ stages the squares of the first and the second term, which, in the absence of the third term, generates either a bright soliton or a classic dark soliton. It is necessary to note in this case that, the dynamics of the amplitude given by the Equation (9) is only described by the traveling wave equation provided by the imaginary part of the Equation (8) and which is under the form:
m 4 ∂ Ψ ∂ ξ + α ∂ 3 Ψ ∂ ξ 3 + β | Ψ | 2 ∂ Ψ ∂ ξ + θ ∂ | Ψ | 2 ∂ ξ Ψ = 0. (30)
Equation (30) compared with Equation (8) suggests that the dynamics of the amplitude given by Equation (29) is in a marginal mode with respect to the GVD and of the Kerr law nonlinearity. Stimulated Raman scattering keeps a great influence on the dynamics of Ψ . This influence is illustrated with the coefficients β , α and θ , respectively.
For c = 0 , all the equations resulting from the coefficients Q n and R j (see Equations (17)-(22)) are verified. Then, only some equations provided by the coefficients P l (see Equations (11) and (12)) and S p are verified while Equation (10) reduces to
∑ l = 1 4 P ′ l ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J l , 0 ( η ξ ) + i [ ∑ p = 1 4 S ′ p ( a , b , c , m 1 , m 2 , m 3 , m 4 , β , η , θ , α ) J p , 1 ( η ξ ) ] = 0. (31)
Thus, the identification to zero of the equations resulting from the coefficients P ′ l and S ′ p leads to the following two series of equations.
- From the real part
the term in J 4,0 ( η ξ ) ,
[ ( β η + 2 η θ ) ( b 2 − a 2 ) + 6 α η 3 ] b = 0, (32)
the term in J 3,0 ( η ξ ) ,
[ m 3 ( a 2 − b 2 ) − 2 η 2 m 2 ] a = 0, (33)
the term in J 2,0 ( η ξ ) ,
[ 2 η θ a 2 − ( η β + 2 η θ ) b 2 − η m 4 − 4 α η 3 ] b = 0, (34)
the term in J 1,0 ( η ξ ) ,
( m 3 b 2 + η 2 m 2 + m 1 ) a = 0, (35)
- From the imaginary part
the term in J 4,1 ( η ξ ) ,
[ ( η β + 2 η θ ) ( b 2 − a 2 ) + 6 α η 3 ] a = 0 , (36)
the term in J 3,1 ( η ξ ) ,
[ m 3 ( a 2 − b 2 ) − 2 η 2 m 2 ] b = 0 , (37)
the term in J 2,1 ( η ξ ) ,
( η β b 2 + η m 4 + α η 3 ) a = 0 , (38)
the term in J 1,1 ( η ξ ) ,
( m 3 b 2 + m 1 ) b = 0. (39)
A thorough observation of the two ranges of equations above shows that, apart from a factor, Equations (32) and (36) are identical. The same is true for Equations (33) and (37). Under this observation including the structuring of the rest of the above equations, and taking into account the fact that we are looking for the non-trivial solutions of Equation (1), this Family II of solutions reveals four subfamilies: Case a ≠ 0 , b ≠ 0 , c = 0 , β ≠ − 2 θ , γ ≠ − 2 θ k ; case m 1 = m 2 = m 3 = 0 , a ≠ 0 , b ≠ 0 , c = 0 ; case a ≠ 0 , b = c = 0 ; case a = 0 , b ≠ 0 , c = 0 .
1) Subfamily I of the Family II of solutions: case: a ≠ 0 , b ≠ 0 , c = 0 , β ≠ − 2 θ , γ ≠ 2 θ k
For this first case, the combination of Equations (32) and (33) or of Equations (36) and (37) results in the same constraint which gives the expression of α as a function of the other parameters as being
α = ( β + 2 θ ) λ 3 ( γ − 2 θ k ) (40)
with β ≠ − 2 θ ; γ ≠ 2 θ k . Thereafter, the combination of Equations (34) and (35) gives, respectively
b = ± ( η 2 − k 2 ) λ + ( k 3 + 3 η 2 k ) α − ω γ + β k (41)
and
a = ± [ 2 ( k 2 − η 2 ) θ − ( k 2 + η 2 ) β − 2 γ k ] λ + [ ( η 2 − 3 k 2 ) γ + ( 2 θ − 2 β ) ( k 3 + η 2 k ) ] α − ( γ + β k ) ν 2 ( γ + β k ) θ
(42)
with { [ 2 ( k 2 − η 2 ) θ − ( k 2 + η 2 ) β − 2 γ k ] λ + [ ( η 2 − 3 k 2 ) γ + ( 2 θ − 2 β ) ( k 3 + η 2 k ) ] α − ( γ + β k ) ν } ( γ + β k ) θ ≻ 0 and
[ ( η 2 − k 2 ) λ + ( k 3 + 3 η 2 k ) α − ω ] ( γ + β k ) ≻ 0 . On the other side, the combination of Equations (38) and (39) offers the constraint
ω = [ ( 2 k 2 − η 2 ) α + λ k + ν ] k + [ ( 3 k 2 − η 2 ) α + 2 λ k + ν ] γ β (43)
where β ≠ 0 . So, we obtain Subfamily I of Family II of the solutions of Equation (1) and which is represented by the exact solution below
Φ ( x , t ) = [ ± [ 2 ( k 2 − η 2 ) θ − ( k 2 + η 2 ) β − 2 γ k ] λ + [ ( η 2 − 3 k 2 ) γ + ( 2 θ − 2 β ) ( k 3 + η 2 k ) ] α − ( γ + β k ) ν 2 ( γ + β k ) θ J 1 , 0 ( η ξ ) ± i ( η 2 − k 2 ) λ + ( k 3 + 3 η 2 k ) α − ω γ + β k J 1 , 1 ( η ξ ) ] e i ( − k x + { [ ( 2 k 2 − η 2 ) α + λ k + ν ] k + [ ( 3 k 2 − η 2 ) α + 2 λ k + ν ] γ β } t ) ,
(44)
with constraints obtained for the cause. Equation (44) is also a kind of complex multi-soliton, which due to the squares existing in the modulus of the amplitude Ψ , makes appear in the propagation medium either a bright-soliton, or a dark-soliton, and this according to the values taken by the wave parameters. This gives freedom of choice of the structure that one would like to use during laboratory propagation tests.
2) Subfamily II of the Family II of solutions: case: ω = α k 3 − λ k 2 , a ≠ 0 , b ≠ 0 , c = 0 , β = 2 θ , γ = − 2 θ k , λ = − 3 α k
For m 1 = 0 ⇔ ω = α k 3 − λ k 2 , Equations (39) and (33), (35), (37) impose successively taking m 3 = 0 ⇔ γ = − β k and m 2 = 0 ⇔ λ = − 3 α k while Equations (32), (34), (36) and (38) remain unchanged. Herein, we also notice that Equations (32) and (36) up to a factor, are identical. Under this ascertainment, the combination of Equations (32), (34) and (38); or the Equations (36), (34) and (38) gives
β = 2 θ , (45)
a = ± ( 2 η 2 + 3 k 2 + 2 k ) α β + ν β (46)
b = ± ( 3 k 2 + 2 k − η 2 ) α β + ν β . (47)
So, the Subfamily II of Family II of the solutions of Equation (1) is the following exact solution
Φ ( x , t ) = [ ± ( 2 η 2 + 3 k 2 + 2 k ) α β + ν β J 1 , 0 ( η ξ ) ± i ( 3 k 2 + 2 k − η 2 ) α β + ν β J 1 , 1 ( η ξ ) ] e i [ − k x + ( α k 3 − λ k 2 ) t ] , (48)
with α β ≻ 0 ; ν β ≻ 0 and 3 k 2 + 2 k ≥ η 2 . Equation (48) presents the same characteristics noted in the case of Equation (44) with the only difference that its dynamics is described by the reduced traveling wave equation given by Equation (30).
3) Subfamily III of the Family II of solutions: case: a ≠ 0 , b = 0 , c = 0 , β ≠ − 2 θ ; γ ≠ 2 θ k
For b = 0 ; c = 0 , Equations (32), (34), (37) and (39) are verified while the combination of Equations (35) and (38) leads to the explicite expression of ω as being
ω = 1 2 [ ( 10 α 2 k + 8 λ α ) k 2 + ( 2 λ 2 + 3 α ν ) k + λ ν ] . (49)
Then, the combined resolution of Equations (33) and (36) gives rise to, respectively
α = ( β + 2 θ ) λ 3 ( γ − 2 θ k ) (50)
and
a = ± 2 λ γ − 2 θ k (51)
with ( γ − 2 θ k ) λ ≻ 0 . Thus, the Subfamily III of Family II of solutions of Equation (1) is reduced to the exact solution below
Φ ( x , t ) = [ ± 2 λ γ − 2 θ k J 1 , 0 ( η ξ ) ] e i { − k x + 1 2 [ ( 10 α 2 k + 8 λ α ) k 2 + ( 2 λ 2 + 3 α ν ) k + λ ν ] t } (52)
with ( γ − 2 θ k ) λ ≻ 0 . The solution given Equation (52) shows that the Sasa-Satsuma dynamical equation admits a bright-soliton as an exact pure real solution.
4) Subfamily IV of the Family II of solutions: case: a = 0 , b ≠ 0 , c = 0 ; β ≠ − 2 θ ; γ ≠ − β k
When equating a = c = 0 , Equations (33), (35), (36) and (38) are verified while the real part gives, from Equations (32) and (34), respectively
b = ± η − 6 α β + 2 θ (53)
and
b = ± 2 λ k + ( 3 k 2 − 4 η 2 ) α + ν β + 2 θ (54)
with α ( β + 2 θ ) ≺ 0 ; η ≥ k 2 3 ; λ ( β + 2 θ ) ≻ 0 and ν ( β + 2 θ ) ≻ 0 . Then, the imaginary part leads, from Equations (37) and (39), successively
b = ± η − 2 λ + 6 α k γ + β k (55)
and
b = ± λ k 2 + ω − α k 3 γ + β k (56)
with ( γ + β k ) ( 2 λ + 6 α k ) ≺ 0 and ( γ + β k ) ( λ k 2 + ω − α k 3 ) ≻ 0 . Since the coefficient b must be unique; on the one hand, Equations (53) and (54) give the equality b 2 = b 2 , which leads to the constraint
α = − 2 λ k + ν 2 η 2 + 3 k 2 (57)
with k ≠ 0 ; η ≠ 0 . On the other hand, Equations (55) and (56) also give the equality b 2 = b 2 , leading in turn to the constraint
α = ω + ( k 2 + 2 η 2 ) λ k ( k 2 − 6 η 2 ) (58)
with k ≠ 0 ; η ≠ k 6 6 . If the coefficient b is unique, then the parameter α must also be unique. In this context, the combination of Equations (57) and (58) reveals
ω = [ 4 ( k 2 − 1 ) η 2 − 5 k 4 ] λ + k ( 6 η 2 − k 2 ) ν 2 η 2 + 3 k 2 (59)
with ( γ + β k ) [ ( 3 k 2 − 2 ) λ + 3 k ν ] ≻ 0 . So, the Subfamily IV of Family II of solutions of Equation (1) can be expressed as
Φ ( x , t ) = [ ± i η 2 ( 3 k 2 − 2 ) λ + 6 k ν ( γ + β k ) ( 2 η 2 + 3 k 2 ) J 1 , 1 ( η ξ ) ] e i { − k x + [ 4 ( k 2 − 1 ) η 2 − 5 k 4 ] λ + k ( 6 η 2 − k 2 ) ν 2 η 2 + 3 k 2 t } (60)
with the constraints given by Equations (57); (58); (59) and ( γ + β k ) [ ( 3 k 2 − 2 ) λ + 3 k ν ] ≻ 0 . This last subfamily IV of the Family II of solutions shows that Equation (1) likewise admits a kink-soliton as an exact pure imaginary solution.
The transition from hyperbolic forms to trigonometric forms of iB-functions is done by means of the relation [
J n , m ( η x ) = i m T n , m ( η x ) (61)
with
T n , m ( η x ) = sin m ( η x ) cos n ( η x ) (62)
such that T n , m ( η x ) is the secondary form of iB-functions related to trigonometric functions. Thus, when we come back to our work, by making the correspondences η ← i η ( i = − 1 ) , x ← ξ , we obtain from Equations (29), (44), (48), (52) and (60), the respective trigonometric solutions.
Φ ( x , t ) = { ± 1 2 ( η 2 − 3 k 2 ) α θ − k λ θ − 1 2 ν θ cosec ( η ξ ) ± 1 2 ( 10 η 2 − 3 k 2 ) α θ − k 10 λ θ − ν θ tan ( η ξ ) ± i η 3 2 α θ cosec 2 ( η ξ ) sin ( η ξ ) } e i ( − k x + 4 α k 3 t ) , (63)
Φ ( x , t ) = [ ± a cosec ( η ξ ) ± b tan ( η ξ ) ] e i ( − k x + { [ ( 2 k 2 − η 2 ) α + λ k + ν ] k + [ ( 3 k 2 − η 2 ) α + 2 λ k + ν ] γ β } t ) , (64)
Φ ( x , t ) = [ ± a cosec ( η ξ ) ± b tan ( η ξ ) ] e i [ − k x + ( α k 3 − λ k 2 ) t ] , (65)
Φ ( x , t ) = ± a cosec ( η ξ ) e i [ − k x + ( α k 3 − λ k 2 ) t ] , (66)
and
Φ ( x , t ) = ± b tan ( η ξ ) e i [ − k x + ( α k 3 − λ k 2 ) t ] , (67)
where a, b and c are expressions obtained in Subsection 4.1, including the accompanying constraints. Equations (63)-(67) constitute the trigonometric versions of the solutions obtained in Subsection 4.1.
This subsection is dedicated to the display of the profiles of certain traveling wave solutions given by Equations (29) and (44), respectively. The graphical tool which made it possible to achieve this result is MAPLE. Thus, we have, respectively.
Note here that, the choice of values is linked to constraints given by the obtained analytical expressions. At this level, several values of variables can be defined. This is for example, the case of
In this section, it is important to note that the obtained analytical or graphical results corroborate with the theoretical predictions about the multi-soliton characters which consist of the proposed ansatz, and this with a more or less good accuracy. An illustration of this corroboration may be observed through:
- the different profiles of
- the different profiles of
the third term of Equation (29)), are equals tendencies of bright-dark or dark-bright soliton. On the other hand,
In summary, we note that the values assigned to each of the parameters of the wave in particular, then extended to the parameters of the system studied in general, are fundamental in the formation of the resulting structure. In addition,
In our previous work [
This work is carried out with the support provided by both ministries of higher education of Cameroon and Gabon through their research assistance programs. The authors would also like to thank Madame Nchong Glory Arrey who, despite her multiple occupations, accomplished the reading in order to improve the quality of the English used in this paper.
The authors declare no conflicts of interest regarding the publication of this paper.
Tchaho, C.T.D., Ngantcha, J.P., Omanda, H.M., Um, B.R.M., Ekogo, T.B. and Bogning, J.R. (2022) Sasa-Satsuma’s Dynamical Equation and Optical Solitary Wave Solutions. Optics and Photonics Journal, 12, 128-145. https://doi.org/10.4236/opj.2022.125010