The genesis of two-hump, W-shaped and M-shaped soliton propagations of the coupled Schrödinger-Boussinesq equations with conformable derivative
aa r X i v : . [ n li n . S I] F e b EPJ manuscript No. (will be inserted by the editor)
The genesis of two-hump, W-shaped and M-shaped solitonpropagations of the coupled Schr ¨ odinger-Boussinesq equationswith conformable derivative Prakash Kumar Das a Department of Mathematics, Trivenidevi Bhalotia College, Raniganj, Paschim Bardhaman, West Bengal, India-713347Received: date / Revised version: date
Abstract.
This work oversees with the coupled Schr¨odinger-Boussinesq equations with conformable deriva-tive, which have lots of applications in laser and plasma. The said equations are reduced to a coupledstationary form using complex travelling wave transformation. Next Painlev´e test applied to derived theintegrable cases of the reduced equation, after that using RCAM derived the solution of reduced equa-tions integrable and nonintegrable cases. Few theorems have been presented and proved to ensure theirboundedness. All presented boundedness cases have been checked and explained by plotting the solutionsfor particulars values of parameters satisfying them. The obtained solutions of stationary form utilizedto derive solutions of the coupled Schr¨odinger-Boussinesq equations with conformable derivative. The de-rived solutions have been plotted and explained. From this, it appears that these solutions propagate bymaintaining their two-hump, W-shaped, M-shaped solutions shapes.
Key words.
Coupled Schr¨odinger-Boussinesq equations – conformable derivative – Exact solution –Boundedness – Painlev´e test – W-shaped, M-shaped solitons – Rapidly convergent approximation method
PACS.
XX.XX.XX No PACS code given
Diverse wave propagations observed in abundant fields of the environment are modelled by non-linear ordinary orpartial differential equations (ODEs/PDEs) involving integer or fractional order derivative. Their chaotic featurescan be explained by the solutions of the equations which describes them. Due to the existence of interaction termsin such equations, it is not always an easy task to derive an exact solution to these equations. Despite that, a vastamount of literature exists for constructing exact travelling wave and soliton solutions [39,63,64,44,47,30,52,54] andapproximation solutions [42,1,60,3,24,4,5,6,7,56,57,55,58,59] of ODEs and PDEs involving integer derivative. Alsothere exists few direct methods [37,46,25,53,34] to derive exact solutions and and analytical methods [62,9,12,45,8,38,32,48] to deal with approximate solutions of nonlinear fractional differential equations. These direct methods oftenneed to guess the syntax of solutions and solve a system of nonlinear algebraic equations. Thus these schemes fail whenthe choice of a form of solution is not suitable or unable to solve the system of equations. And the above mentionedapproximate methods are often observed to have slow convergent rate and always unable to provide the close form ofthe series solution. To overcome the above-mentioned limitations of those methods we adopt the rapidly convergentapproximation method (RCAM) [18,17,16,23,19,22,15,21,20]. This current work deals with this scheme to obtainedsome new solutions for the coupled Schr¨odinger-Boussinesq systems (SBS) with conformable derivative.To attained the goal, the paper is organised as follows: in sections 2 and 3 we present the basic properties ofconformal derivative and methodology of RCAM to solve a system of ODEs respectively. In section 4 we reduced SBSto coupled ODEs by employing a complex wave transformation. The Painlev´e test has been employed to the reducedcoupled ODEs and identified its integrable cases in section 5. In section 6, the integrable and nonintegrable cases of thereduced ODEs have been solved by RCAM. Few theorems have been presented to study their boundedness and utilisedto plot the stationary form of the solutions. These solutions used to derive explicit solutions of SBS. Furthermore, the
Send offprint requests to : a e-mail: [email protected] P.K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative main characteristics of these derived solutions are graphically discussed. We summarize our outlook on the presentwork in section 7. This portion deals with few definitions and properties of conformable derivative [37,2]:
Definition 1
The conformable derivative of a function f : [0 , ∞ ) → R of order α is defined by T α ( h )( t ) = lim ǫ → h (cid:0) t + ǫ t − α (cid:1) − h ( t ) ǫ , (1) for all t > , α ∈ (0 , . If f is α -differentiable in some (0 , a ) , a > , and lim t → f ( α ) ( t ) exists, then define f ( α ) (0) =lim t → f ( α ) ( t ) . In the corresponding sections of this paper we replace the notation T α ( f )( t ) by f ( α ) ( t ), to prevail the conformablederivatives of f of order α . Some notable features of conformable derivative are listed below:If α ∈ (0 , , and f, g be α -differentiable at a point t > T α ( a f + b g ) = a T α ( f ) + b T α ( g ) , for all a, b ∈ R . T α ( t p ) = pt p − α for all p ∈ R . T α ( λ ) = 0, for all constant functions f ( t ) = λ. T α ( f g ) = f T α ( g ) + g T α ( f ) . T α (cid:16) fg (cid:17) = g T α ( f )+ f T α ( g ) g .6. if f is differentiable, then T α ( f )( t ) = t − α dfdt ( t ) . Theorem 1 (Chain Rule [2, 28, 27, 13])
Let f, g : (0 , ∞ ) → R be two differentiable functions and also f is α -differentiable, then, one has the following rule: T α ( f og )( t ) = t − α g ′ ( t ) f ′ ( g ( t )) . Take into account a system of ODEs X ′′ − A . X = N , (2)where X , A and N are the matrix of dependent variables, constant coefficients and interaction terms respectively,having form X = U ( x ) U ( x )... U k ( x ) , A = λ · · · λ · · · · · · λ k , and N = N ( U ( x ) , · · · , U k ( x )) N ( U ( x ) , · · · , U k ( x ))... N k ( U ( x ) , · · · , U k ( x )) . To solve (2), we remodel it in an exponential matrix operator conformation O [ X ]( x ) = N , (3)where linear exponential matrix operator can be recast in the formˆ O [ · ]( x ) = e A .x ddx (cid:18) e − A .x ddx (cid:0) e A .x [ · ] (cid:1)(cid:19) . (4)The inverse operator ˆ O − of O []( x ) is conferred byˆ O − [ · ]( x ) = e −A .x Z x e A .x ′ Z x ′ e −A .x ′′ [ · ] dx ′′ dx ′ . (5) .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 3 Operating O − on O [ X ]( x ) and employing integration by parts yieldsˆ O − (cid:16) X ′′ − A . X (cid:17) = X − C .e A .x − D .e −A .x , (6)where C = c c ... c k and D = d d ... d k are matrices of integration constants. Operating O − on (3) and utilising (6),provides X = C .e A .x + D .e −A .x + ˆ O − [ N ]( x ) , (7)where C , and D are three arbitrary constants matrices. To derive the unknown X terms in the R.H.S of (7), we recastthem in the syntax X ∼ = U ( x ) U ( x )... U k ( x ) = ∞ X m =0 U ,m ( x ) U ,m ( x )... U k,m ( x ) (8)and N = P ∞ m =0 ∆ m ( x ) , with ∆ m ∼ = ∆ ,m ( x ) ∆ ,m ( x )... ∆ k,m ( x ) = 1 m ! d m dǫ m N ( x ) N ( x )... N k ( x ) ǫ =0 (9)and N j ( x ) , j = 1 , , · · · , k are given by N j ( x ) = ∞ X k =0 U ,k ǫ k , ∞ X k =0 U ,k ǫ k , · · · , ∞ X k =0 U k,k ǫ k ! . The terms ∆ i,m ( x ) = ∆ i,m ( U , ( x ) , U , ( x ) , . . . .., U ,m ( x ) , · · · , U k, ( x ) , U k, ( x ) , . . . .., U k,m ( x )) , i = 1 , , · · · , k areAdomian polynomials [3,24,4,5,6,7] outturn from the formula (9). Use of (9) in (11) provides X = c e λ x + d e − λ x c e λ x + d e − λ x ... c k e λ k x + d k e − λ k x + ˆ O − ∞ X m =0 ∆ ,m ( x ) ∆ ,m ( x )... ∆ k,m ( x ) . (10)We obey the footsteps of [16], to get the higher order iteration terms as X ∼ = U , ( x ) U , ( x )... U k, ( x ) = c e λ x + d e − λ x c e λ x + d e − λ x ... c k e λ k x + d k e − λ k x , (11) X n +1 ∼ = U ,n +1 ( x ) U ,n +1 ( x )... U k,n +1 ( x ) = ˆ O − ∆ ,n ( x ) ∆ ,n ( x )... ∆ k,n ( x ) , (12) n ≥
0. In case λ i >
0, treatment of the vanishing boundary condition U i ( ∞ ) = 0 in (10) for the localized solutionleads us to c i = 0 , i = 1 , , · · · , k . Therefore the leading and higher order iteration terms of the series solution are P.K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative produced by (12) and X ∼ = U , ( x ) U , ( x )... U k, ( x ) = d e − λ x d e − λ x ... d k e − λ k x . (13)Further to obtain the localized solution in case λ i <
0, for the boundary condition U i ( −∞ ) = 0 , we go ahead withrestraining the term involving e λ i x . In this case, the commanding and subsequent terms of the solution are yields by(12) with X ∼ = U , ( x ) U , ( x )... U k, ( x ) = c e λ x c e λ x ... c k e λ k x . (14)One can easily obtain the iterative terms and the general term of the series (or generating function) by taking advantageof symbolic software. That leads to the exact solution of the discussed system of ODEs. Consider the generalized Schr¨ o dinger-Boussinesq system (SBS) [40,11,14] in the form i (cid:18) ∂E∂t + δ ∂E∂x (cid:19) + δ ∂ E∂x = δ N E,∂ N∂t + µ ∂ N∂x + µ ∂ N∂x + µ ∂ N ∂x = µ ∂ | E | ∂x , (15)where E ( x, t ) is complex wave field, N ( x, t ) is real wave field, δ i , i = 1 , , µ j , j = 1 , , , i (cid:16) E ( β ) t + δ E ( α ) x (cid:17) + δ E ( α ) xx = δ N E,N ( β ) tt + µ N ( α ) xx + µ N ( α ) xxxx + µ (cid:0) N (cid:1) ( α ) xx = µ (cid:0) | E | (cid:1) ( α ) xx , (16)where f ( α ) ∗ and f ( β ) ∗ represents α and β order conformal derivative of f with respect to suffix variables respectively.Here our aim is to study its integrability and derive its new stationary different shaped exact solutions by employingRCAM. To attain the target we impose the travelling wave transformation [35,27,13] E ( x, t ) = u ( ξ ) e iη , η = k x α α + k t β β + c ,N ( x, t ) = v ( ξ ) , ξ = k x α α + c t β β + ξ , (17)to (16) and equating real and imaginary parts we get c + k δ + 2 k k δ = 0 ,k δ u ′′ − ( k + k δ + k δ ) u − δ u v = 0 , (18) k µ v (4) + ( c + µ k ) v ′′ − k µ ( uu ′ ) ′ + 2 k µ ( vv ′ ) ′ = 0 . Integrating the last equation of the system twice and taking integration constant zero, reduces system (18) to .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 5 u ′′ ( ξ ) − λ u ( ξ ) = α u ( ξ ) v ( ξ ) ,v ′′ ( ξ ) − λ v ( ξ ) = β u ( ξ ) + γ v ( ξ ) , (19)for the values of the parameters c = − k ( δ + 2 k δ ) , α = δ δ k , β = µ µ k , γ = − µ µ k ,λ = k ( δ + δ k ) + k δ k , λ = − δ + 4 δ δ k + 4 k δ + µ µ k . (20)Here u ( ξ ) and v ( ξ ) are real fields, ξ is the (real) independent variable and all the other remaining quantities arefree parameters. It is important to note here that the equations are invariant under the transformations (i) ξ → − ξ (ii) u → − u , and (iii) ξ → ξ + C , where C is a constant. ´ e test of Eq.(19) The existence of solutions is a necessary condition of integrability, but it is not sufficient. To confirm the integrabilityother tests such as the Lax pair or the Painlev´e test should be used for the proposed model. The Painlev´e analysis isa powerful scheme to check the integrability of a system. Here we apply this important tools Mathematica packagePainleveTest.m [10] to examine the integrability of equation (19). Application of the package yields the resonances ofthe considered equation as −
1; 6; − p − α (23 α − γ ) − α α ; p − α (23 α − γ ) + 5 α α . It can be checked that for resonances −
1; 6, this model (19) fails the Painlev´e test, and the remaining two resonancesare symbolic so we can not proceed further. To move forward we assume that these symbolic resonances are equal topositive integer k (say), which yields ± p − α (23 α − γ ) ± α α = k or γ = 112 (cid:0)
12 + k − k (cid:1) α . Subsequent, setting different positive integer values for k and using the same Mathematica package, we get the followingtwo integrable cases;1. For k = 8 compatibility condition is γ = 3 α .
2. For k = 2 or 3 compatibility condition is γ = α , λ = λ . In this section we solve above presented integrable and one nonintegrable cases of (19) (or (16)) by RCAM. γ = 3 α In this case, applying RCAM we get following correction terms ( u ( ξ ) = u − e λ ξ ,v ( ξ ) = v − e λ ξ u ( ξ ) = α u − v − e ( λ λ ξ λ (2 λ + λ ) ,v ( ξ ) = β u − e λ ξ λ − λ + α v − e λ ξ λ u ( ξ ) = α u − e λ ξ ( β λ u − e λ ξ − α λ ( λ − λ ) v − e λ ξ ) λ λ − λ λ ,v ( ξ ) = α v − e λ ξ ( α λ ( λ − λ ) v − e λ ξ − β λ u − e λ ξ ) λ λ ( λ − λ ) , P.K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative u ( ξ ) = α u − v − e ( λ λ ξ ( β λ (6 λ + λ ) u − e λ ξ − α λ ( λ − λ )(2 λ + λ ) v − e λ ξ ) λ (2 λ − λ ) λ (2 λ + λ ) ,v ( ξ ) = α h β λ u − (2 λ + λ ) e λ ξ − α β λ ( λ − λ ) ( λ + λ ) λ u − × v − e λ + λ ) ξ + 2 α λ (2 λ + λ ) (cid:0) λ − λ (cid:1) v − e λ ξ i / h λ λ (2 λ + λ ) (cid:0) λ − λ (cid:1) i ...therein u − and v − are integration constants. Likewise, other higher order correction terms can be calculated usingsymbolic computations available in software packages Mathematica and Maple. Moreover, after calculating ten orhigher order iteration terms the said software packages can easily provide the generating functions (or general term)of the iteration terms. Hither we have acquired the subsequent generating functions u ( ξ, ǫ ) = n λ (2 λ − λ ) (2 λ + λ ) u − e λ ξ (cid:0) λ + λ ) λ + α ( λ − λ ) × v − ǫe λ ξ (cid:1)(cid:9) / Q ( ξ, ǫ ) ,v ( ξ, ǫ ) =4 (2 λ − λ ) (2 λ + λ ) ǫ n α β (2 λ − λ ) λ u − v − ǫ e (4 λ + λ ) ξ +16 β λ u − ǫ (2 λ + λ ) e λ ξ (cid:0) λ (2 λ + λ ) λ + α λ v − ǫ × ( λ − λ ) e λ ξ − α (cid:0) λ − λ λ (cid:1) v − ǫe λ ξ (cid:17) − λ ( λ − λ ) × (2 λ + λ ) λ v − e λ ξ o / Q ( ξ, ǫ ) (21)where Q ( ξ, ǫ ) = α β u − ǫ e λ ξ n α ( λ − λ ) v − ǫe λ ξ − λ (2 λ + λ ) o + 8 λ (2 λ − λ ) (2 λ + λ ) (cid:0) λ − α v − ǫe λ ξ (cid:1) . (22)Ergo, the exact solution can be derived by setting ǫ = 1 in (21)-(22) as u ( ξ ) = n λ (2 λ − λ ) (2 λ + λ ) u − e λ ξ (cid:0) λ + λ ) λ + α ( λ − λ ) × v − e λ ξ (cid:1)(cid:9) / Q ( ξ ) ,v ( ξ ) =4 (2 λ − λ ) (2 λ + λ ) n α β (2 λ − λ ) λ u − v − e (4 λ + λ ) ξ + 16 β λ × u − (2 λ + λ ) e λ ξ (cid:16) λ (2 λ + λ ) λ + α λ ( λ − λ ) v − e λ ξ − α (cid:0) λ − λ λ (cid:1) v − e λ ξ (cid:17) − λ ( λ − λ ) (2 λ + λ ) λ v − e λ ξ o / Q ( ξ ) (23)where Q ( ξ ) = α β u − e λ ξ n α ( λ − λ ) v − e λ ξ − λ (2 λ + λ ) o + 8 λ × (2 λ − λ ) (2 λ + λ ) (cid:0) λ − α v − e λ ξ (cid:1) . (24)Due to the existence of many free parameters in the solution (23)-(24), it is always not bounded ( for all values ofthese free parameters). To study physical relevant properties modelled by these equations, one always need to derivea bounded solution. To ensure the boundedness of this derived solution, below a theorem, have been presented. Theorem 2
The solution (23)-(24) will be bounded if parameters λ , λ , α , β involved in the equation and inte-gration constants u − , v − satisfy any one of the conditions given in the table 1.Proof Solution (23)-(24) have a common factor in the denominator, which under transformation e ξ = Z , reduces to ageneralized Dirichlet polynomial [36] P ol ( Z ) = a Z λ + λ + a Z λ + a Z λ + a , .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 7 Table 1: Boundedness conditions of theorem 2.
Case Con. No. λ α β u − v − λ < λ < − + − − − − − + − +1(c) − + − + − − − + + + λ > − λ > − + − − − − − + − +2(c) − + − + − − − + + + λ < − λ < − − − − + − +3(c) + + − + − − + + + λ > λ > − − − − + − +4(c) + + − + − − + + + λ < λ < − + + − − − − − − +5(c) − + + + − − − − + +0 < λ < − λ − + + − − − − − − +6(c) − + + + − − − − + + − λ < λ < − − − − − +7(c) + + + + − − − + +0 < λ < λ − − − − − +8(c) + + + + − − − + + where a = α β ( λ − λ ) u − v − , a = − α β λ (2 λ + λ ) u − ,a = − α λ (2 λ − λ ) (2 λ + λ ) v − , a = 16 λ (2 λ − λ ) λ (2 λ + λ ) . Solution (23)-(24) is unbounded if there exists at least one real positive root. So the boundedness of the solutionis ensured by the conditions that the polynomial never have positive real root. Such conditions can be provided byDescartes’ rule of signs [36]. Which states that the polynomial
P ol ( Z ) does not contain any positive real root ifits coefficients never changes their signs. That leads us to the condition that either all a i > i = 1 , , ,
4) or all a i < i = 1 , , , a i > i = 1 , , ,
4) yields the condition 1.(a)-4.(d) of the theorem, whereasremaining conditions provided by a i < i = 1 , , , Solution-I
P.K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative
Table 2: Particular values of parameters satisfying conditions presented in Theorem 2 used in Fig. 1. λ λ α β u − v − Figure − . − . . − . − . − . − . − . − . . − . . − . − . − . . − . − . − −
19 5 5 6 1(d)2 . − − . − − . − − . − . − . . − − . − . − .
15 5 6 2(d) − . . − . − − − . . −
19 15 − − . . . − . − . . − . . . . − . − − . . − . − . . − .
15 5 − . . . − . − . − . . . − . − . − . − . − . − . − . . − . − . . . . − . − . − . − . − . . . − . . . − . − . . − . − . − . − . .
01 6(b) . − . . . . − .
01 6(c) . − . − . − . . .
01 6(d) − . . . . − . − . − . . − . − . − . . − . . . . . − − . . − − . . . − . − . . − . − − . . . . − . . − − . So in this case solution of (16) can be obtained from (23)-(24) with (17) and (20) in the form E ( x, t ) = n λ (2 λ − λ ) (2 λ + λ ) u − e λ ξ (cid:0) λ + λ ) λ + α ( λ − λ ) × v − e λ ξ (cid:1)(cid:9) / Q ( ξ ) e i ( k xαα + k tββ + c ) ,N ( x, t ) = 4 (2 λ − λ ) (2 λ + λ ) n α β (2 λ − λ ) λ u − v − e (4 λ + λ ) ξ +16 β λ u − (2 λ + λ ) e λ ξ (cid:16) λ (2 λ + λ ) λ + α λ ( λ − λ ) × v − e λ ξ − α (cid:0) λ − λ λ (cid:1) v − e λ ξ (cid:17) − λ ( λ − λ ) (2 λ + λ ) × λ v − e λ ξ (cid:9) / Q ( ξ ) (25)where Q ( ξ ) = α β u − e λ ξ n α ( λ − λ ) v − e λ ξ − λ (2 λ + λ ) o + 8 λ × (2 λ − λ ) (2 λ + λ ) (cid:0) λ − α v − e λ ξ (cid:1) , ξ = k x α α + c t β β + ξ . (26) .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 9 This solution exists provided δ = − δ µ µ , k ( δ + δ k ) + k δ k > , and − δ + 4 δ δ k + 4 δ k + µ µ k > , (27)which are yield from the integrability condition γ = 3 α and the requirement that λ , λ are real. The boundednessof solution (25)-(27) can be easily ensured by theorem 2 with the conditions (20). Utilising one of such conditionsthe solution (25)-(27) has been plotted in figure 2. From the Fig. 1, on the (x,t)-plane, it is clear that absolute valueof solution E represents W-shaped soliton wave, real and imaginary parts of E of the wave solution represent theAkhmediev breather (AB) wave, which can evolve periodically along a certain angle with the t axis and component N propagate with W-shaped soliton wave. α = 2 γ , λ = λ = λ (say) In this case, RCAM gives the following correction terms ( u ( ξ ) = u − e − λ ξ ,v ( ξ ) = v − e − λ ξ , ( u ( ξ ) = γ u − v − e − λ ξ λ ,v ( ξ ) = e − λ ξ ( β u − + γ v − ) λ , u ( ξ ) = γ u − e − λξ ( β u − +3 γ v − ) λ ,v ( ξ ) = γ v − e − λξ ( β u − + γ v − ) λ , u ( ξ ) = γ u − v − e − λξ ( β u − + γ v − ) λ ,v ( ξ ) = γ e − λξ ( β u − +6 β γ u − v − + γ v − ) λ , ... u m ( ξ ) = ( m +1) e − λξ m +1 m √ β (cid:20)(cid:0) √ β u − − √ γ v − (cid:1) (cid:20) e − λξ ( γ v − −√ β √ γ u − ) λ (cid:21) m + (cid:0) √ β u − + √ γ v − (cid:1) (cid:20) e − λξ ( √ β √ γ u − + γ v − ) λ (cid:21) m (cid:21) ,v m ( ξ ) = ( m +1) e − λξ m +1 m √ γ (cid:20)(cid:0) √ γ v − − √ β u − (cid:1) (cid:20) e − λξ ( γ v − −√ β √ γ u − ) λ (cid:21) m + (cid:0) √ β u − + √ γ v − (cid:1) (cid:20) e − λξ ( √ β √ γ u − + γ v − ) λ (cid:21) m (cid:21) , ...where u − and v − are integration constants. Summing the above series terms one can obtain close form solution of (19)in the form u ( ξ ) = λ u − e λξ ( λ e λξ + β γ u − − γ v − )( λ e λξ − β γ u − + γ v − − γ λ v − e λξ ) ,v ( ξ ) = λ e λξ ( β λ u − e λξ − β γ u − v − + γ v − − γ λ v − e λξ +36 λ v − e λξ )( λ e λξ − β γ u − + γ v − − γ λ v − e λξ ) . (28)In the following, a theorem has been proposed and proved to ensure the boundedness of this derived solution. Theorem 3
The solution (28) will be bounded if real parameters λ, β , γ present in the equation and integrationconstants v − , u − involved in solution satisfies any one of the following conditionsIa. β > γ < Ib. β < γ > IIa. β > , γ > , v − < − q γ v − β < u − < q γ v − β IIb. β < , γ < , v − > − q γ v − β < u − < q γ v − β Proof
The solution (23) components have common denominator given by the cubic polynomial
P ol ( Z ) = − β γ u − + γ v − − γ λ v − Z + 36 λ Z , where Z = exp( λ ξ ). Solution (28) is bounded if the roots of the polynomial are either negative real or complex. Inrenaming cases they are unbounded. So boundedness of solution demands that roots of P ol ( Z ) have to be negativereal, complex, or both. Roots of the above polynomial are given by Z ± = γ v − ± √ β √ γ u − λ Roots Z ± will be complex if β γ <
0, which provides boundedness conditions Ia & Ib of the theorem. Remainingboundedness conditions can be obtained when Z ± are negative real. Now Z − will be negative real if parameters satisfythe relationsI. β > , γ > γ v − − √ β √ γ u − ) < β < , γ < γ v − − √ β √ γ u − ) < . Which further can be put in the formsI.a. β > , γ > , v − ≤ , & u − > − q γ v − β I.b. β > , γ > , v − > , & u − > q γ v − β II.a. β < , γ < , v − > , & u − > − q γ v − β II.b. β < , γ < , v − ≤ , & u − > q γ v − β . Now Z + will be negative real if parameters satisfy the relations1. β > , γ > γ v − + √ β √ γ u − ) < β < , γ < γ v − + √ β √ γ u − ) < β > , γ > , v − > u − < − q γ v − β β > , γ > , v − ≤ u − < q γ v − β β < , γ < , v − ≤ , & u − < − q γ v − β β < , γ < , v − > , & u − < q γ v − β . Now combining negative real Z ± conditions, we get the valid two common regions summarised in conditions IIa. &IIb. of the theorem. Hence the theorem is proved.Next to establishing the conditions presented in the above theorem and study the features of the solution (28), wehave taken particular values of different parameters satisfying the conditions of the theorem in table 3 and utilisingthem to plot the solution in figure 3. From the 2D plots, it is clear that the solution is enriched with several two-hump,W-shape, M-shape soliton-like profile. Solution-II
So in this case solution of (16) can be obtained from (28) with (17) and (20) in the form E ( x, t ) = 36 λ u − e λξ (cid:0) λ e λξ + β γ u − − γ v − (cid:1)(cid:0) λ e λξ − β γ u − + γ v − − γ λ v − e λξ (cid:1) e i ( k xαα + k tββ + c ) ,N ( x, t ) =36 λ e λξ (cid:2) β λ u − e λξ − β γ u − v − + γ v − − γ λ v − e λξ +36 λ v − e λξ (cid:3) / (cid:2) λ e λξ − β γ u − + γ v − − γ λ v − e λξ (cid:3) , (29) .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 11 Table 3: Particular values of parameters satisfying conditions presented in Theorem 3 & 4, used in Fig. 3. λ α β γ u − v − Fig. . −
20 2 −
15 Ia . −
30 20 − . .
44 -3 IIa . − −
20 18 . . . − . − . − − − . . − − − where ξ = k x α α + c t β β + ξ . This solution exist provided δ = − δ µ µ , k = − δ δ + δ k (cid:0) δ + µ (cid:1) + δ (cid:0) k (cid:0) δ + µ (cid:1) + µ (cid:1) µ , and − δ + 4 δ δ k + 4 δ k + µ µ k > , (30)which are derived using the integrability condition α = 2 γ , λ = λ = λ and demand that λ is real.The boundedness of solution (29)-(30) can be easily ensured by theorem 3 with the conditions (20). Using oneamong those conditions solution (29)-(30) is plotted in figure 3. Fig. 3 shows that the absolute value of solution E have M-shaped soliton profile, whereas real and imaginary parts of E of the wave solution constitute the Akhmedievbreather (AB) wave, the wave is not the space-periodic breather but the time-periodic breather and component N propagate with keeping M-shaped form soliton wave. In the previous cases, we have derived exact solutions of SBSby eliminating one or more free parameters using compatibility conditions of the Painlev´e test. Then it is of naturalcuriosity to explore whether it is possible to obtain an exact solution of the equations without using compatibilityconditions (nonintegrable case) containing all parameters involved in the equation? To answer that here we considera nonintegrable case λ = λ . To solve the considered system of equations rather eliminating any other additionalparameters, we eliminate one integration constant. v − = √ β u − √ α − γ , λ = λ = λ (say) In this case, RCAM gives the following correction terms ( u ( ξ ) = u − e − λ ξ ,v ( ξ ) = √ β u − e − λ ξ √ α − γ , u ( ξ ) = α √ β u − e − λξ λ √ α − γ ,v ( ξ ) = α β u − e − λ ξ λ ( α − γ ) , u ( ξ ) = α β u − e − λ ξ λ ( α − γ ) ,v ( ξ ) = α β u − e − λ ξ λ ( α − γ ) , u ( ξ ) = α β u − e − λ ξ λ ( α − γ ) ,v ( ξ ) = α β u − e − λ ξ λ ( α − γ ) , ... u m ( ξ ) = ( m +1) u − e − λξ m h α √ β u − e − λξ λ √ α − γ i m ,v m ( ξ ) = λ ( m +1)6 m α h α √ β u − e − λξ λ √ α − γ i m +1 , ...where u − and v − are integration constants. Summing the above series terms one can derive the close form solution of(19) in the form u ( ξ ) = u − λ ( α − γ ) e λξ ( λ ( α − γ ) e λξ − u − α √ β √ α − γ ) ,v ( ξ ) = u − λ √ β ( α − γ ) e λξ ( λ ( α − γ ) e λξ − u − α √ β √ α − γ ) . (31)To make the solution (31) physically relevant below we present a theorem to derive its bounded cases. Theorem 4
The solution (31) will be bounded if parameters α , β , γ involved in the equation and integrationconstant u − involve in solution satisfies any one of the following conditionsIIIa α > , β > , u − < α > γ IIIb α > , β < , u − < α < γ IIIc α < , β > , u − > α > γ IIId α < , β < , u − > α < γ Proof
The components u ( ξ ) and v ( ξ ) of the solution (31) have a common denominator given by the quartic polynomial (cid:16) λ ( α − γ ) e λξ − u − α p β √ α − γ (cid:17) . The repeated roots of the polynomial given by ξ = 1 λ log (cid:20) α u − √ β λ √ α − γ (cid:21) . The positive real values of these roots make denominator zero that leads to an unbounded solution. So the requirementof boundedness of solution suggests that the argument of the log has to be complex or negative. The complex case isnot admissible here because it makes solutions complex, so the real bounded solutions given by the conditions amongparameters β ( α − γ ) > α u − < . These restrictions can be split to derive the conditions of the theorem.Few particular values of the free parameters which satisfy the conditions of the theorem presented in table 3 and usingthem solution have been plotted in figure 3. The 2D plots ensure that solution always has a one-hump soliton-likeprofile.
Solution-III
So the solution of (16) can be obtained from (31) with (17) and (20) in the form E ( x, t ) = 36 u − λ ( α − γ ) e λξ (cid:0) λ ( α − γ ) e λξ − u − α √ β √ α − γ (cid:1) e i ( k xαα + k tββ + c ) ,N ( x, t ) = 36 u − λ √ β ( α − γ ) e λξ (cid:0) λ ( α − γ ) e λξ − u − α √ β √ α − γ (cid:1) , ξ = k x α α + c t β β + ξ , (32)provided k = − δ δ + δ k (cid:0) δ + µ (cid:1) + δ (cid:0) k (cid:0) δ + µ (cid:1) + µ (cid:1) µ , and − δ + 4 δ δ k + 4 δ k + µ µ k > , (33)which are given by condition λ = λ = λ and assertion that λ is real. One can easily derive the boundednessconditions of solution (32)-(33) by using theorem 4 with the conditions (20). Using one among those conditionsgraphical representations of the solution (32)-(33) are presented in figure 3. The figure shows that the absolute valueof solution E has always soliton profile, whereas real and imaginary parts of E of the wave solution constitute aspace-periodic breather and component N propagate with maintaining soliton shaped form. .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 13 In this work, a few new exact solutions of three integrable/nonintegrable cases of SBS with space-time conformablehave been derived. Here we have applied a complex travelling wave transformation and a modified RCAM to solve theSBS. Painlev´e test is used to identify the integrable cases of the stationary form of SBS. In general Painlev´e analysiscannot handle the case when considered equation contains parameter coefficients. In this case, the arbitrariness ofparameters leads to the symbolic resonances and one can not execute the remaining steps of Painlev´e test. But herewe have proposed an alternative technique to handle this limitation and derived integrable cases. We have classifiedall the bounded physically relevant cases of the solutions and presented them in three theorems. General theories ofalgebra have been utilized to prove the theorems. In addition to that, all bounded cases have been checked and usedin plots to establish our claims. The presented 2D and 3D plots of solutions of SBS reflect the appearance of a fewnew two-hump, W-shaped, M-shaped of solution propagation state. We believe these findings are new, not available inthe literature for the considered equation. The results can be helpful for analyzing the dynamics of nonlinear localizedwaves in the generalized coupled SBS and other coupled systems.
Acknowledgements
The author express his sincere thanks to all the faculty members of department of mathematics, T.D.B. CollegeRaniganj, for their valuable comments.
References
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Fig. 3: Plot of solutions (28) and (31) for the conditions presented in Theorem 3 & 4, using values of parameters givenin Table 3. E ) Re( E )Im( E ) N Fig. 4: Plot of the solution (29)-(30) for values of parameters values k = 0 . , k = 1 . , δ = 0 . , δ = 0 . , µ =0 . , µ = − , µ = 0 . , µ = 0 . , u − = 0 . , v − = 0 . , α = . , β = . , c = 1 , ξ = − . .K.Das: Two-hump soliton propagations of the coupled SB equations with conformable derivative 19Abs( E ) Re( E )Im( E ) N Fig. 5: Plot of solution (32) for values of parameters values k = 0 . , k = 0 . , δ = − , δ = 0 . , δ = 0 . , µ =0 . , µ = − , µ = − . , µ = − . , α = . , β = . , c = 9 , ξ = 3 , u − = − .4