Optical soliton solutions of the Biswas-Arshed model by the tan(⊝/2) expansion approach
OOptical soliton solutions of the Biswas-Arshed modelby the tan( (cid:127) / expansion approach Md Fazlul Hoque and Harun-Or-RoshidDepartment of Mathematics, Pabna University of Science and Technology, Pabna-6600,BangladeshE-mail: f azlul _ math @ yahoo.co.in ; harunorroshidmd @ gmail.com Abstract
In this paper, we consider the Biswas-Arshed model (BAM) with nonlinearKerr and power law. We integrate these nonlinear structures of the BAM toobtain optical exact solitons that passing through the optical fibers. To retrievethe solutions, we apply the tan( (cid:127) /
2) expansion integral scheme to the structuresof the BAM nonlinearity. The novel solutions present optical shock wave, doubleperiodic optical solitons, interaction between optical periodic wave and opticalsolitons, and optical periodic and rogue waves for both structures of the model.It is shown that the amplitude of the periodic double solitons waves graduallyincreases and reached the highest peak at the moment of interaction, and it goesto diminish for a much larger time. In fact, we show that the amplitude of thewave for the interaction between periodic and optical solitons, gradually increaseswith beat phenomena. To the purpose, all these types of optical solitons can befrequently used to amplify or reduce waves for a certain hight. Moreover, wedescribe the physical phenomena of the solitons in graphically.
Keywords : The Biswas-Arshed model, the extension tan( (cid:127) /
2) approach, shock wavedouble solitons, optical double solitons, rogue wave solitons.
Optical transmission of solitons motion plays a central role in a variety of branchesin communication sciences. The general concepts of transmission optical solitons innonlinear optical fiber systems are fundamentally important in controlling optical con-tinuum and transferring informations for very long distances. The general nonlinearSchrodinger equation can be successfully addressed the picosecond pulses of the wavedynamics involving the coefficient of the group velocity dispersion (GVD) and the1 a r X i v : . [ n li n . PS ] J u l elf-phase modulation (SPM) [1, 2]. If ultrashort pulses durations less than 100 fem-toseconds, it would be motivated, attractive and desirable to increase the power ofhigh-bit-rate transmission systems, higher-order Kerr dispersion, the slow nonlinearactivity, and the third-order dispersion play a vital role in optical fiber media [1, 3].Due to valuable applications in optical communications and optical signal transmittingsystems, the theoretically predicted solitons as well as experimentally study of the dy-namical temporal optical solitons were observed in fiber lasers [4]. The study of solitonscovers as well other areas of physics including plasma physics, fluid dynamics, biolog-ical systems and so on. With such effects, there are many classes of nonlinear mod-els such as Lakshmanan-Porsezian-Daniel model, complex Ginzburg-Landau equation,Kundu-Eckhaus equation, Fokas-Ienells equation, Kaup-Newll model, Radhkrishnan-Kundu-Lakshmanan equation, resonant nonlinear Schrodinger’s equation, Kadomtsev-Petviashvili-Benjamin -Bona-Mahony, Boiti-Leon-Manna-Pempinelli equation and thereferences therein [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. These models havebeen successfully addressed dynamical transmission solitons solutions.In recent years, Biswas and Arshed newly introduced one governing nonlinear modeland studied the wave profile of optical solitons in existence of super order dispersionswith negligibly SPM [19]. They also pointed out the bright, singular and combo-solitonsfor the two integration structures of the model. It is fundamentally effective in inves-tigating the dynamical solitons in optical fibers and metameterials in case of both lowGVD and nonlinearity. The model known as Biswas-Arshed model (BAM) has beenused by several authors to obtain various type optical solitons via the trial solution tech-nique [20], modified simple equation technique [21], Kerr and power law nonlinearity[22, 23], mapping method [24], the extended trial function method [25] and parameterrestriction approach [26]. All these approaches have previously been restricted mainlyto obtain optical single solitons of the BAM. However, the generalization of the interac-tion of double optical solitons and rogue type solitons is still an unexplored subject viaother existing methods. The interaction of optical nonlinearity solitons often lead toperiodic double solitons and rogue waves, and the nature of these solitons is unknown.The purpose of this paper is to show how we can present optical shock wave, opticaldouble solitons, interaction between optical periodic wave and optical solitons, andoptical periodic and rogue waves for both structures of the model. We also show thathow they play to increase or decrease the amplitude of the waves for the solitons ingraphically. It is based on the tan( (cid:127) /
2) expansion approach [27, 28, 29]. To ourknowledge these type of investigations for the BAM is a first step in the study of thetan( (cid:127) /
2) expansion method. 2
Review of the tan( (cid:127) / expansion approach In the section, we review the tan( (cid:127) /
2) expansion approach for the nonlinear evolutionequations (NLEEs). These methods are highly effective and algebraic schemes to drivethe general optical solitons, periodic solitons and rogue waves solutions, and moredeeply understand the properties of the model [27, 28]. tan( (cid:127) / expansion approach This method has been extensively used to a class of NLEEs [27, 28] to obtain novelexact solutions. The main algorithm of the approach is as follows:
Step 1
Let us consider the nonlinear evolution equation and the travelling variabletransformation in the following form:Γ( u xx , u xt , u tt , u t , u n , . . . . ) = 0 , (2.1)where u ( x, t ) is an unknown function, Γ is a polynomial of u ( x, t ) and its derivatives.The NLEE (2.1) can be converted to the ordinary differential equation by the trans-formation relation ζ = kx − ωt as follows: H (Ψ , Ψ , Ψ , . . . ) = 0 . (2.2)The constant k is the wave number and ω is the frequency of the wave. Step 2
One considers the general form of the trial solution [27, 28],Ψ( (cid:127) ) = n X r =0 L r tan r ( (cid:127) , (2.3)where L r are free constants to be later calculated, such that L n = 0 and the (cid:127) is afunction of ζ , which satisfies the condition, d (cid:127) dζ = λ sin( (cid:127) ( ζ )) + µ cos( (cid:127) ( ζ )) + ν. (2.4)The condition (2.4) leads to the following exact solutions:Set 1: If Λ = λ + µ − ν <
0, and µ − ν = 0, then (cid:127) ( ζ ) = 2 tan − h λµ − ν − √− Λ µ − ν tan( √− Λ2 ¯ ζ ) i .Set 2: If Λ = λ + µ − ν >
0, and µ − ν = 0, then (cid:127) ( ζ ) = 2 tan − h λµ − ν + √ Λ µ − ν tanh( √ Λ2 ¯ ζ ) i .3et 3: If Λ = λ + µ − ν >
0, and µ = 0 and ν = 0, then (cid:127) ( ζ ) = 2 tan − (cid:20) λµ + √ λ + µ µ tanh( √ λ + µ ¯ ζ ) (cid:21) .Set 4: If Λ = λ + µ − ν <
0, and µ = 0 and ν = 0, then (cid:127) ( ζ ) = 2 tan − h − λν + √ ν − λ ν tan( √ ν − λ ¯ ζ ) i .Set 5: If Λ = λ + µ − ν >
0, and µ − ν = 0 and λ = 0, then (cid:127) ( ζ ) = 2 tan − (cid:20)q µ + νµ − ν tanh( √ µ − ν ¯ ζ ) (cid:21) .Set 6: If λ = 0, and ν = 0, then (cid:127) ( ζ ) = tan − h e µ ¯ ζ − e µ ¯ ζ +1 , e µ ¯ ζ e µ ¯ ζ +1 i .Set 7: If λ = 0, and ν = 0, then (cid:127) ( ζ ) = tan − h e µ ¯ ζ e µ ¯ ζ +1 , e µ ¯ ζ − e µ ¯ ζ +1 i .Set 8: If λ + µ = ν , then (cid:127) ( ζ ) = 2 tan − h λ ¯ ζ +2( µ − ν )¯ ζ i .Set 9: If λ = µ = ν = kλ , then (cid:127) ( ζ ) = 2 tan − h e kλ ¯ ζ − i .Set 10: If λ = ν = kλ and µ = − λk , then (cid:127) ( ζ ) = − − h e kλ ¯ ζ e kλ ¯ ζ − i .Set 11: If ν = λ , then (cid:127) ( ζ ) = − − (cid:20) ( λ + µ ) e µ ¯ ζ − λ − µ ) e µ ¯ ζ − (cid:21) .Set 12: If λ = ν , then (cid:127) ( ζ ) = 2 tan − (cid:20) ( µ + ν ) e µ ¯ ζ +1( µ − ν ) e µ ¯ ζ − (cid:21) .Set 13: If ν = − λ , then (cid:127) ( ζ ) = 2 tan − (cid:20) e µ ¯ ζ + µ − λe µ ¯ ζ − µ − λ (cid:21) .Set 14: If µ = − ν , then (cid:127) ( ζ ) = 2 tan − h λe λ ¯ ζ − νe λ ¯ ζ i .Set 15: If µ = 0 and λ = ν , then (cid:127) ( ζ ) = − − h ν ¯ ζ +2 ν ¯ ζ i .Set 16: If λ = 0 and µ = ν , then (cid:127) ( ζ ) = 2 tan − h ν ¯ ζ i .Set 17: If λ = 0 and µ = − ν , then (cid:127) ( ζ ) = − − h ν ¯ ζ i .Set 18: If λ = 0 and µ = 0, then (cid:127) ( ζ ) = νζ + K .Set 19: If µ = ν , then (cid:127) ( ζ ) = 2 tan − h e λ ¯ ζ − νλ i ,4here ¯ ζ = ζ + K , where L , L r are constants to be later calculated. Step 3.
Inserting the above solutions into (2.3) together with (2.4) and substitutinginto (2.2), and then solving the algebraic equations for the coefficients L , L r and ω .Finally, putting the results into (2.3) with the above set of solutions, they providethe new exact solutions of the NLEEs (2.1). In the following sections, we apply thisapproach to the first and second structures of the Biswas-Arshed model. In this section, we consider the first structure of the Biswas-Arshed model (BAM) [19]to obtain the optical double solitons and the rogue waves via the tan( (cid:127) /
2) expansionapproach. The first structure of the BAM is [19, 22], i V t + a V xx + a V xt + i ( b V xxx + b V xxt ) = i [ ε ( |V| V ) x + σ ( |V| ) x V + ϑ |V| V x ] . (3.1)The function V ( x, t ) in the model represents the wave profile of soliton, a is the coeffi-cient of the GVD, a is the spatio-temporal dispersion, b is the coefficient of third-orderdispersion, b is the coefficient of spatio-temporal third-order dispersion, and ε is theeffect of self-steepening, and σ, ϑ are the effect of dispersions. This model was firstintroduced by Biswas-Arshed [19], known as the so-called BAM, and it was shown theoptical solitons in the presence of higher-order dispersions with negligibly self-phasemodulation. Such a model with optical transmission wave solutions has also found inthe context of Kerr and non-Kerr law media [22]. This model considered here could havewider applicability and other aspects as optical double solitons, periodic solitons androgue type wave solutions could be investigated via the tan( (cid:127) /
2) expansion method.This method makes the Biswas-Arshed model to be highly interesting. In this section,we apply the transformation variable to (2.1)
Let us consider the transformation variable [19], V ( x, t ) = U ( ζ ) e iφ ( x,t ) , (3.2)where U ( ζ ) is the amplitude portion with ζ = x − δt , the phase component φ ( x, t ) = − kx + wt + ρ , and the constants δ, ρ, k, w are respectively, the soliton velocity, the phase5onstant, the wave number and the frequency of the soliton. After a long computationand an integration, one can present the differential equation [19, 22] of (3.1), AU − BU − CU = 0 , (3.3)where A = a − a δ + 3 b k − b δk − b w , B = a k − a wk + b k − b wk + w and C = k ( ε + ϑ ). In the following section, we apply the extension tan( (cid:127) /
2) method to(3.3) and obtain the exact soliton solutions of (3.1). tan( (cid:127) / method to the first structureof BAM We now compute the balance number of (3.3) from power balance of the highest deriva-tive with the highest-order nonlinear terms [19], that leads to n = 1. Then the trialsolution (2.3) in the tan( (cid:127) /
2) expansion method takes the form,Ψ( (cid:127) ) = L + L tan( (cid:127) . (3.4)Putting (3.4) into (3.3) along with (2.4), we obtain a polynomial of sin( ζ ) and cos( ζ )functions, whose equating coefficients lead to a system of equations, and the solutionsof the system of equations via symbolic computation, yield the following constraints: w = ( λ + µ − ν )( a − δa − δkb + 3 kb ) + 2 b k + 2 a k k b + λ b + µ b − ν b + 2 ka − , L = ± λ r k b b + 3 k a b + ka a − kb − a − δ (2 k b + 3 k a b + ka − kb − a ) k ( ε + ϑ )(2 k b + λ b + µ b − ν b + 2 ka − , L = ± ( ν − µ ) r δ (2 k b + 3 k a b + ka − kb − a ) − k b b − k a b − ka a + 3 kb + a k ( ε + ϑ )(2 k b + λ b + µ b − ν b + 2 ka − . (3.5) Now if we combine the above constraints with (3.4), and substituting into (3.2), andthe solutions Set [1-19] of (2.4), we obtain the following seventeen valid exact solitonsolutions of (3.1), V , ( x, t ) = n L + L h λµ − ν − √− Λ µ − ν tan( √− Λ2 ¯ ζ ) io e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h λµ − ν + √ Λ µ − ν tanh( √ Λ2 ¯ ζ ) io e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20) λµ + √ λ + µ µ tanh( √ λ + µ ¯ ζ ) (cid:21)(cid:27) e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h − λν + √ ν − λ ν tan( √ ν − λ ¯ ζ ) io e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20)q µ + νµ − ν tanh( √ µ − ν ¯ ζ ) (cid:21)(cid:27) e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) tan − h e µ ¯ ζ − e µ ¯ ζ +1 , e µ ¯ ζ e µ ¯ ζ +1 i(cid:17)o e i ( − kx + wt + ε ) ,6 , ( x, t ) = n L + L tan (cid:16) tan − h e µ ¯ ζ e µ ¯ ζ +1 , e µ ¯ ζ − e µ ¯ ζ +1 i(cid:17)o e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h λ ¯ ζ +2( µ − ν )¯ ζ io e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L tan (cid:18) − tan − (cid:20) ( λ + µ ) e µ ¯ ζ − λ − µ ) e µ ¯ ζ − (cid:21)(cid:19)(cid:27) e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20) ( µ + ν ) e µ ¯ ζ +1( µ − ν ) e µ ¯ ζ − (cid:21)(cid:27) e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20) e µ ¯ ζ + µ − λe µ ¯ ζ − µ − λ (cid:21)(cid:27) e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h λe λ ¯ ζ − νe λ ¯ ζ io e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) − tan − h ν ¯ ζ +2 ν ¯ ζ i(cid:17)o e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h ν ¯ ζ io e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) − tan − h ν ¯ ζ i(cid:17)o e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan h νζ + K io e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h e λ ¯ ζ − νλ io e i ( − kx + wt + ε ) ,where w , L and L come from (3.5). There is a note that the solutions V , ( x, t ) and V , ( x, t ) are invalid for the constraints.To the graphical description, it is shown that V , ( x, t ), V , ( x, t ), V , ( x, t ) solutionsprovide double periodic optical solitons. In particular, we present the 2 D and 3D graphsin Figure 1 for the solution V , ( x, t ); and the solutions V , ( x, t ) and V , ( x, t ) havesimilar dynamical characteristics to the Figure 1.The solutions V , ( x, t ), V , ( x, t ) and V , ( x, t ) represent interaction between periodicwaves and solitons, which produce an optical double solitons. We present graphs inFigure 2 for a particular solution V , ( x, t ). It is shown that the amplitude of thesolitons increases after the interaction ( t = 0). These types of optical solitons can beextensively used to amplify waves for a certain hight.The solutions V , ( x, t ), V , ( x, t ), V , ( x, t ), V , ( x, t ), V , ( x, t ), V , ( x, t ) and V , ( x, t )provide to similar interaction between periodic waves and optical solitons. We givegraphs in Figure 3 for the solution V , ( x, t ). The graphs show that the amplitude ofthe optical solitons gradually increases with beat phenomena.The solutions V , ( x, t ), V , ( x, t ), V , ( x, t ), and V , ( x, t ) provide to similar interac-tion between periodic waves and optical solitons. We present a periodic double solitonsfor the solution V , ( x, t ) in Figure 4. The graphs show that the amplitude of the wavegradually increases and reaches highest peak at the moment of interaction, and thenthe amplitude goes to diminish for a larger time. As it is presented various interactionsof optical solitons, and such interactions are stable localized wave packets that cantravel large distance in optical fibers remaining their structures [4]. Thus these types7f interactions solitons would be interested in optical fiber communications. Let us consider the second structure of Biswas-Arshed model [19], i V t + a V xx + a V xt + i ( b V xxx + b V xxt ) = i [ ε ( |V| q V ) x + σ ( |V| q ) x V + ϑ |V| q V x ] , (4.1)where q leads to the nonlinearity of the model. In this section, we apply the tan( (cid:127) / q = 1, the model (4.1) becomes the firststructure of BAM (3.1). Using the transformation (3.2) into the model (4.1) leads to the following ODE [19], AU − BU − CU q +1 = 0 , (4.2)where A = a − a δ + 3 b k − b δk − b w , B = a k − a wk + b k − b wk + w and C = k ( ε + ϑ ). After scaling the ODE (4.2) by U = T q yields, A{ (1 − q )( T ) + 2 q T T } − q BT − q CT = 0 . (4.3) tan( (cid:127) / approach to thesecond structure of BAM The balance number of (4.3) comes from power balance of the highest derivative withthe highest-order nonlinear terms [19] that leads to n = 2, and hence the trial solution(2.3) of the tan( (cid:127) /
2) expansion method takes the form,Ψ( (cid:127) ) = L + L tan( (cid:127) L tan ( (cid:127) . (4.4)Putting (4.4) into (4.3) along with (2.4), we obtain a polynomial of sinh( ζ ), and cosh( ζ )functions, whose coefficients after equating lead to a system of equations, and theseequations enable to give us the following set of constraints:8 = (4 k q b + 2 δkλ b + 2 δkµ b − δkν b + 4 k q a + δλ a + δµ a − δν a − kλ b − kµ b + 3 kν b − λ a − µ a + ν a ) / (4 k q b + 4 kq a − λ b − µ b + ν b − q ), L = (2 δk µ qb − δk ν qb + 2 δk µ b − δk ν b + 3 δk µ qa b − δk ν qa b − k µ qb b + 2 k ν qb b + 3 δk µ a b − δk ν a b + δkµ qa − δkν qa − k µ b b +2 k ν b b − k µ qa b +3 k ν qa b − δkµ qb + δkµ a +2 δkν qb − δkν a − k µ a b +3 k ν a b − kµ qa a + kν qa a − δkµ b + 2 δkν b − δµ qa + δν qa + 3 kµ qb − kµ a a − kν qb + kν a a − δµ a + δν a + 3 kµ b − kν b + µ qa − ν qa + µ a − ν a ) / (( ε + ϑ )(4 k q b + 4 kq a − λ b − µ b + ν b − q ) k ), L = (2 λ (2 δk µqb − δk νqb + 2 δk µb − δk νb + 3 δk µqa b − δk νqa b − k µqb b +2 k νqb b +3 δk µa b − δk νa b + δkµqa − δkνqa − k µb b +2 k νb b − k µqa b + 3 k νqa b − δkµqb + δkµa + 2 δkνqb − δkνa − k µa b + 3 k νa b − kµqa a + kνqa a − δkµb + 2 δkνb − δµqa + δνqa + 3 kµqb − kµa a − kνqb + kνa a − δµa + δνa +3 kµb − kνb + µqa − νqa + µa − νa )) / ( k (4 k q εb +4 k q ϑb +4 kq εa + 4 kq ϑa − λ εb − λ ϑb − µ εb − µ ϑb + ν εb + ν ϑb − − q ε − q ϑ )), L = − (2 δk µ qb − δk µνqb +2 δk ν qb +2 δk µ b − δk µνb +2 δk ν b +3 δk µ qa b − δk µνqa b + 3 δk ν qa b − k µ qb b + 4 k µνqb b − k ν qb b + 3 δk µ a b − δk µνa b + 3 δk ν a b + δkµ qa − δkµνqa + δkν qa − k µ b b + 4 k µνb b − k ν b b − k µ qa b + 6 k µνqa b − k ν qa b − δkµ q + δkµ a + 4 δkµνqb − δkµνa − δkν qb + δkν a − k µ a b +6 k µνa b − k ν a b − kµ qa a +2 kµνqa a − kν qa a − − δkµ b + 4 δkµνb − δkν b − δµ qa + 2 δµνqa − δν qa + 3 kµ qb − kµ a a − kµνqb + 2 kµνa a + 3 kν qb − kν a a − δµ a + 2 δµνa − δν a + 3 kµ b − kµνb +3 kν b + µ qa − µνqa + ν qa + µ a − µνa + ν a ) / ( k (4 k q εb +4 k q ϑb +4 kq εa + 4 kq ϑa − λ εb − λ ϑb − µ εb − µ ϑb + ν εb + ν ϑb − q ε − q ϑ )).Now using the above constraints into (4.4), and combining this with U = T q to sub-stitute into (3.2), then the solutions Set [1-19] of (2.4), provides us the following exactsoliton solutions of (4.1): V , ( x, t ) = n L + L h λµ − ν − √− Λ µ − ν tan( √− Λ2 ¯ ζ ) i + L h λµ − ν − √− Λ µ − ν tan( √− Λ2 ¯ ζ ) i (cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h λµ − ν + √ Λ µ − ν tanh( √ Λ2 ¯ ζ ) i + L h λµ − ν + √ Λ µ − ν tanh( √ Λ2 ¯ ζ ) i (cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20) λµ + √ λ + µ µ tanh( √ λ + µ ¯ ζ ) (cid:21) L (cid:20) λµ + √ λ + µ µ tanh( √ λ + µ ¯ ζ ) (cid:21) ) q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L h − λν + √ ν − λ ν tan( √ ν − λ ¯ ζ ) i + L h − λν + √ ν − λ ν tan( √ ν − λ ¯ ζ ) i (cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L (cid:20)q µ + νµ − ν tanh( √ µ − ν ¯ ζ ) (cid:21) + L (cid:20)q µ + νµ − ν tanh( √ µ − ν ¯ ζ ) (cid:21) ) q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) tan − h e µ ¯ ζ − e µ ¯ ζ +1 , e µ ¯ ζ e µ ¯ ζ +1 i(cid:17) + L tan (cid:16) tan − h e µ ¯ ζ − e µ ¯ ζ +1 , e µ ¯ ζ e µ ¯ ζ +1 i(cid:17)o q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) tan − h e µ ¯ ζ e µ ¯ ζ +1 , e µ ¯ ζ − e µ ¯ ζ +1 i(cid:17) + L tan (cid:16) tan − h e µ ¯ ζ e µ ¯ ζ +1 , e µ ¯ ζ − e µ ¯ ζ +1 i(cid:17)o q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L h λ ¯ ζ +2( µ − ν )¯ ζ i + L h λ ¯ ζ +2( µ − ν )¯ ζ i (cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) − tan − h e kλ ¯ ζ e kλ ¯ ζ − i(cid:17) + L tan (cid:16) − tan − h e kλ ¯ ζ e kλ ¯ ζ − i(cid:17)o q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L tan (cid:18) − tan − (cid:20) ( λ + µ ) e µ ¯ ζ − λ − µ ) e µ ¯ ζ − (cid:21)(cid:19) + L tan (cid:18) − tan − (cid:20) ( λ + µ ) e µ ¯ ζ − λ − µ ) e µ ¯ ζ − (cid:21)(cid:19)(cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = ( L + L (cid:20) ( µ + ν ) e µ ¯ ζ +1( µ − ν ) e µ ¯ ζ − (cid:21) + L (cid:20) ( µ + ν ) e µ ¯ ζ +1( µ − ν ) e µ ¯ ζ − (cid:21) ) q e i ( − kx + wt + ε ) , V , ( x, t ) = ( L + L (cid:20) e µ ¯ ζ + µ − λe µ ¯ ζ − µ − λ (cid:21) + L (cid:20) e µ ¯ ζ + µ − λe µ ¯ ζ − µ − λ (cid:21) ) q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L h λe λ ¯ ζ − νe λ ¯ ζ i + L h λe λ ¯ ζ − νe λ ¯ ζ i (cid:27) q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) − tan − h ν ¯ ζ +2 ν ¯ ζ i(cid:17) + L tan (cid:16) − tan − h ν ¯ ζ +2 ν ¯ ζ i(cid:17)o q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan (cid:16) − tan − h ν ¯ ζ i(cid:17) + L tan (cid:16) − tan − h ν ¯ ζ i(cid:17)o q e i ( − kx + wt + ε ) , V , ( x, t ) = n L + L tan h νζ + K i + L tan h νζ + K io q e i ( − kx + wt + ε ) , V , ( x, t ) = (cid:26) L + L h e λ ¯ ζ − νλ i + L h e λ ¯ ζ − νλ i (cid:27) q e i ( − kx + wt + ε ) ,where L , L and L come from the above set of constraints. There is a note that thesolutions V , ( x, t ) and V , ( x, t ) are invalid for the constraints.10o the graphical representations, it is shown that V , ( x, t ), V , ( x, t ), V , ( x, t ) so-lutions provide double period optical solitons. In particular, we present the 2 D and 3Dgraphs in Figure 5 for the solution V , ( x, t ); and the solutions V , ( x, t ) and V , ( x, t )have similar dynamical characteristics to the Figure 5.The solutions V , ( x, t ), V , ( x, t ) and V , ( x, t ) represent interaction between periodicwaves and solitons which produce an optical double solitons. We present graphs inFigure 6 for a particular solution V , ( x, t ). It is shown that the solution presents roguewaves whose amplitude increased two or three times higher than the surrounding waveswithin a tiny time. These types of rogue waves can be predicted to control the ampli-tude of the waves.The solutions V , ( x, t ), V , ( x, t ) V , ( x, t ), V , ( x, t ), V , ( x, t ), V , ( x, t ), V , ( x, t )and V , ( x, t ) provide to similar interaction of periodic wave with optical solitons. Wegive graphs in Figure 7 for the solution V , ( x, t ). The graphs show that the amplitudeof the optical solitons gradually decreases.The solutions V , ( x, t ), V , ( x, t ), and V , ( x, t ) provide to similar interaction of peri-odic wave with optical solitons. We present a periodic double solitons for the solution V , ( x, t ) in Figure 8. The graphs show that the amplitude of the wave graduallyincreases and reaches the highest peak at the moment of interaction, and then theamplitude goes to diminish for a larger time. Remark.
When L = 0 and q = 1 / q , it is shown that the amplitude of the wavesincreases more, and the waves can be transferred to a very large distance remainingsame nonlinear dynamical shapes. The main results of this paper is the determination of exact solitons to the Biswas-Arshed model (BAM) with nonlinear Kerr and power law via the tan( (cid:127) /
2) expansionintegral scheme. We retrieve optical shock waves, double periodic optical solitons,interaction between optical periodic waves and optical solitons, and optical periodicwaves and rogue waves of the model. We explain how to display the nonlinear feathersof the waves at the moment of interactions, that is, how to the amplitude of the opticalperiodic double solitons gradually increases, reaches to the highest peak, and goes todisappear for a much longer time. In fact, all these types of optical solitons can befrequently used to amplify or reduce waves on account of a certain hight. The physical11henomena of the solitons are graphically presented in the 2D and 3D plots. To ourknowledge these types of solitons for the Biswas-Arshed model have not been exploredbefore [21, 22, 23, 24]. In fact, the model could be investigated to construct multi-and rogue waves solitons by the other existing methods, in particular, Hirota bilinearapproach [30, 31] and Darboux transformation [32].
References [1] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press San Diego (1989).[2] A. Hasegawa, and Y. Kodama, Solitons in Optical Communications, Oxford Uni-versity Press, New York (1995).[3] L.C. Zhao, S.C. Li and L.M. Ling, W-shaped solitons generated from a weak mod-ulation in the Sasa-Satsuma equation, Phys. Rev. E , 021313 (2019).[5] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Lakshmanan-Porsezian-Daniel model by modified simple equation method, Optik (2), 983-994 (2019).[13] W. Liu, Y. Zhang, Z. Luan, Q. Zhou, M. Mirzazadeh, M. Ekici and A. Biswas,Dromion-like soliton interactions for nonlinear Schrödinger equation with variablecoefficients in inhomogeneous optical fibers, Nonlinear Dyn. , 729-736 (2019).[14] X. Liu, W. Liu, H. Triki, Q. Zhou and A. Biswas, Periodic attenuating oscil-lation between soliton interactions for higher-order variable coefficient nonlinearSchrödinger equation, Nonlinear Dyn. , 801-809 (2019).[15] T. A. Sulaiman, Three-component coupled nonlinear Schrödinger equation: opticalsoliton and modulation instability analysis, Phys. Scr. , 065220(2020).[17] J. Manafian, O. A. Ilhan and A. Alizadeh, Periodic wave solutions and stabilityanalysis for the KP-BBM equation with abundant novel interaction solutions, Phys.Scr. , 065208 (2020).[19] A. Biswas, and S. Arshed, Optical solitons in presence of higher order dispersionsand absence of self-phase modulation, Optik, , 452-459 (2018).[20] Y. Yildirim, Optical solitons of Biswas-Arshed equation by trial equation tech-nique, Optik, , 876-883 (2019). 1321] Y. Yildirim, Optical solitons of Biswas-Arshed equation by modified simple equa-tion technique, Optik, , 986-994 (2019).[22] M. Tahir, and A.U. Awan, Optical travelling wave solutions of the Biswas-Arshedmodel in Kerr and non-Kerr law media, Pramana-J. Phys., , 29 (2020).[23] M. Tahir, A.U. Awan, and H.U.Rehman, Dark and singular optical solitons to theBiswas-Arshed model with Kerr and power law nonlinearity, Optik, , 777-783(2019).[24] H.U. Rehman, M.S. Saleem, M. Zubair, S. Jafar and I. Latif, Optical solitons withBiswas-Arshed equation using mapping method, Optik, , 163091 (2019).[25] M. Ekici, and A. Sonmezoglu , Optical solitons with Biswas-Arshed equation bythe extended trial function method, Optik, , 13-20 (2019).[26] S. Aouadi, A. Bouzida, A.K. Daoui, H. Triki, Q. Zhou and S. Liu, W-shaped,bright and dark solitons of Biswas-Arshed equation, Optik, , 272-232 (2019).[27] O.A. Ilhan, J. Manafian, A. Alizadeh and H.M. Baskonus, New exact solutionsfor nematicons in liquid crystals by the tan( φ/ , 113 (2020).[28] J. Manafian, Optical soliton solutions for Schrodinger type nonlinear evolutionequations by the tan( φ/ , 4222-4245 (2016).[29] J. Manafian and M. Lakestani, Optical soliton solutions for the Gerdjikov-Ivanovmodel via the tan( φ/ , 9603-9620 (2016).[30] H. O. Roshid and W. X. Ma, Dynamics of mixed lump-solitary waves of an ex-tended (2+1)-dimensional shallow water wave model, Physics Letters A, , 45,3262-3268 (2018).[31] M. B. Hossen, H. O. Roshid and M Z. Ali, Characteristics of the solitary waves androgue waves with interaction phenomena in a (2+ 1)-dimensional Breaking Solitonequation, Phys. Lett. A, , 45, 1268-1274 (2018).[32] V.G. Bagrov and B. F. Samsonov, Darboux transformation and elementary exactsolutions of the Schrodinger equation, Pramana J. Phys., , 563-580 (1997).14igure 1: The 2D and 3D double periodic solitons of V , ( x, t ), for the real part of V , ( x, t ) with a = b = b = δ = λ = µ = k = 1, a = 2, and ν = 3, ε = 0.Figure 2: The 3D periodic double solitons of V , ( x, t ), for the imaginary part of V , ( x, t ) with a = b = b = δ = λ = k = 1, a = 2, µ = 3 and ν = 1, ε = 0.15igure 3: The 2D and 3D periodic double solitons of V , ( x, t ), for the real part of V , ( x, t ) with a = 0 . a = b = b = λ = ν = 1, δ = √− µ = 3, k = 20, ε = 0 and ϑ = 0 . V , ( x, t ), for the real part of V , ( x, y ) with a = a = b = b = λ = µ = 1, a = δ = 2, k = 5, ϑ = 2, ε = 0 and ν = 2. 16igure 5: The 2D and 3D double periodic solitons of V , ( x, t ), for the real part of V , ( x, t ) with a = b = b = δ = k = ϑ = λ = 1, a = q = 2, µ = 1, ε = 0 and ν = 3.Figure 6: The 2D and 3D rogue wave solitons of V , ( x, t ), for the real part of V , ( x, t )with a = b = b = δ = k = ϑ = λ = 1, a = q = 2, µ = 3, ε = 0 and ν = 1.17igure 7: The 2D and 3D periodic double solitons of V , ( x, t ), for the real part of V , ( x, t ) with a = 0 . a = b = b = q = 1, δ = 1 + √− k = 25, ε = 0, ϑ = µ = 3and λ = ν = 0.Figure 8: The 2D ( x = 0) and 3D periodic double solitons of V , ( x, t ), for the imaginarypart of V , ( x, t ) with a = a = b = b = δ = k = ϑ = µ = 1, ε = λ = 0, q = 2 and ν = −−