SOLITONS AND OTHER SOLUTIONS TO LONG-WAVE SHORT-WAVE INTERACTION EQUATION

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1 SOLITONS AND OTHER SOLUTIONS TO LONG-WAVE SHORT-WAVE INTERACTION EQUATION H. TRIKI 1, M. MIRZAZADEH 2, A. H. BHRAWY 3,4, P. RAZBOROVA 5, ANJAN BISWAS 3,5 1 Radiation Physics Laboratory, Department of Physics Faculty of Sciences, Badji Mokhtar University PO Box 12, Annaba, Algeria 2 Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C , Rudsar-Vajargah, Iran 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt 5 Department of Mathematical Sciences, Delaware State University, Dover, DE , USA Received July 29, 2014 This paper studies the long-wave short-wave interaction equation that produces soliton solutions as well as other exact solutions. Exact 1-soliton solutions are obtained for this equation, with time-dependent coefficients, by the aid of ansatz method. Subsequently, the simplest equation approach also gives soliton solutions as well as other solutions such as singular periodic solutions and plane waves. There are several constraints that naturally emerge for the solutions to exist. Key words: Solitons; integrability; conservation laws. PACS: Lm, Fg, Yv. 1. INTRODUCTION The dynamics of propagation of soliton pulses through nonlinear media has attracted much more attention in recent years [1]-[49]. Originally, the term soliton was reserved for a particular set of exact solutions existing as a result of the delicate balance between dispersion or diffraction and nonlinearity [1]. The main feature of solitons is that they propagate for a long time without visible changes [2]. The distinction between solitary wave and soliton solutions is that when any number of solitons interact they do not change form, and the only outcome of the interaction is a phase shift [3]. With the development of many powerful methods, soliton-type solutions for certain nonlinear evolution equations NLEEs were derived. The sinecosine methods [4]-[6], the subsidiary ordinary differential equation method [7]-[9], Hirota s method [10], the Petrov-Galerkin method [11], the collocation method [12]- [14], the solitary wave ansatz method [15]-[18], Exp-function method [19], and many RJP Rom. 60Nos. Journ. Phys., 1-2, Vol , Nos , P. c 72 86, 2015 Bucharest, - v.1.3a*

2 2 Solitons and other solutions to long-wave short-wave interaction equation 73 others, are examples of the methods that have been used. In this paper, the solitary wave ansatz method will be used to carry out the integration of three coupled NLEEs describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium. The study will be realized in the presence of time-dependent coefficients and power law nonlinearity. 2. GOVERNING EQUATION The evolution equations describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium [20, 21] are iφ t + αφ xx = βuφ, 1 iψ t + αψ xx = βuψ, 2 u t = β φ 2 + ψ 2, 3 x where the real-valued function ux, t characterizes the long longitudinal wave and the complex-valued functions φx, t, ψx, t are the slowly varying envelopes of the short transverse waves [21]. In 1-3, x and t represents the spatial and temporal variables respectively, while the parameters α and β are real constants. In this work, we establish the 1-soliton solutions to the basic equations in presence of timedependent coefficients and power law nonlinearity of the form: iφ t + α 1 tφ xx = β 1 tuφ, 4 iψ t + α 2 tψ xx = β 2 tuψ, 5 u t = γ 1 t φ + γ 2 t ψ, 6 x where α i t, β i t and γ i t with i = 1,2 are arbitrary time-dependent coefficients and m is the power law nonlinearity parameter. As a matter of fact, this model is much more general since nonlinear wave equations with variable coefficients can be considered as generalizations of the constant coefficient ones. We also consider the case of power law nonlinearity such that the obtained results are therefore valid for any particular nonlinearity. To our knowledge, the study of these wave equations with t-dependent coefficients and power law nonlinearity has not been widespread. RJP 60Nos. 1-2, c v.1.3a*

3 74 H. Triki et al SOLITON SOLUTIONS Here we are interested in finding the 1-soliton solutions of Eqs To solve these coupled equations, we choose the ansatze [17] where and φx,t = A 1 t sech p 1 τe iϕ 1, 7 ψx,t = A 2 t sech p 2 τe iϕ 2, 8 ux,t = A 3 t sech p 3 τ, 9 τ = Btx vtt 10 ϕ 1 = κ 1 tx + ω 1 tt + θ 1 t, 11 ϕ 2 = κ 2 tx + ω 2 tt + θ 2 t, 12 Here, in 7-12, A i with i = 1,2,3 are the amplitudes of the φ-soliton, ψ-soliton and u-soliton, respectively, B is the inverse width of the solitons while v is the solitons velocity. Also, κ 1 and κ 2 are the frequencies of the φ-soliton and ψ-soliton, respectively, while ω 1 and ω 2 are the corresponding wave numbers and θ 1 and θ 2 are the phase shifts. Since the problem is considered with time-dependent coefficients, it is therefore assumed that these parameters are also time-dependent. The values of the unknown exponent p i with i = 1,2,3 will be determined during the course of derivation of the soliton solutions. Substituting 7-12 into 4 and decomposing into real and imaginary parts, respectively, yields { A 1 sech p 1 dκ1 τ dt x dω 1 dt t ω 1 dθ } 1 + α 1 p 2 dt 1A 1 B 2 sech p 1 τ α 1 p 1 p 1 + 1A 1 B 2 sech p 1+2 τ α 1 κ 2 1A 1 sech p 1 τ β 1 A 1 A 3 sech p 1+p 3 τ = 0, 13 and da 1 dt { sech p 1 τ p 1 A 1 tanhτ sech p 1 db τ x vt B v + t dv } dt dt + 2α 1 κ 1 p 1 A 1 B tanhτ sech p 1 τ = Now substituting 7-12 into 5 and decomposing into real and imaginary parts, RJP 60Nos. 1-2, c v.1.3a*

4 4 Solitons and other solutions to long-wave short-wave interaction equation 75 respectively, yields A 2 sech p 2 dκ2 τ dt x ω 2 t dω 2 dt dθ 2 + α 2 p 2 dt 2A 2 B 2 sech p 2 τ and da 2 dt α 2 p 2 p 2 + 1A 2 B 2 sech p 2+2 τ α 2 κ 2 2A 2 sech p 2 τ β 2 A 2 A 3 sech p 2+p 3 τ = 0, 15 { sech p 2 τ p 2 A 2 tanhτ sech p 2 db τ x vt B v + t dv } dt dt while 6 reduces to da 3 dt sech p 3 τ p 3 A 3 tanhτ sech p 3 τ The balancing principle gives and + 2α 2 κ 2 p 2 A 2 B tanhτ sech p 2 τ = 0, 16 { db dt x vt B v + t dv } dt + 2γ 1 p 1 ma 1 B tanhτ sech 2p 1m τ + 2γ 1 γ 2 p 2 ma 2 B tanhτ sech 2p τ = 0, 17 p 1 = p 2 = 1 m 18 p 3 = 2 19 Now from 13, setting the coefficients of the linearly independent functions sech p 1+j τ to zero, where j = 0,2, gives after the integration with p 1 = 1/m: κ 1 t = k 1, 20 θ 1 t = θ 01, 21 ω 1 t = 1 β1 A 3 t m α 1k1 2 dt, 22 Bt = m β 1A 3 α 1 m + 1, 23 where k 1 and θ 01 are integration constants related to the initial frequency and the initial phase of the φ-soliton, respectively. We clearly see from 20 and 21 that the frequency and the phase shift of the φ-soliton remain constant when the pulse propagates in the varying system, while the wave number ω 1 in 22 is affected by the t-dependent coefficients α 1 t and β 1 t. RJP 60Nos. 1-2, c v.1.3a*

5 76 H. Triki et al. 5 Now from 14, setting the coefficients of the linearly independent functions sech p 1 τ and tanhτ sech p 1 τ to zero, gives after integration the expressions: A 1 t = λ 1 24 Bt = B 0 25 vt = 2k 1 t α 1tdt 26 where λ 1 and B 0 are integration constants related to the initial pulse amplitude of the φ-soliton and inverse width, respectively. From 15, setting the coefficients of the linearly independent functions sech p 2+j τ to zero, where j = 0,2, gives after the integration with p 2 = 1/m: κ 2 t = k 2, 27 θ 2 t = θ 02, 28 ω 2 t = 1 β2 A 3 t m α 2k2 2 dt, 29 Bt = m β 2A 3 α 2 m + 1, 30 where k 2 and θ 02 are integration constants related to the initial frequency and the initial phase of the ψ-soliton, respectively. Next, from 16, setting the coefficients of the linearly independent functions sech p 2 τ and tanhτ sech p 2 τ to zero, and integrating gives A 2 t = λ 2, 31 Bt = B 0, 32 vt = 2k 2 t α 2tdt, 33 where λ 2 is an integration constant related to the initial pulse amplitude of the ψ- soliton. Again, from 17, setting the coefficients of the linearly independent functions to zero, and integrating gives for p 1 = p 2 = 1/m and p 3 = 2 A 3 t = λ 3, 34 Bt = B 0 35 vt = 1 λ 3 t γ 1t { λ 1 + γ 2 tλ } 2 dt, 36 where λ 3 is an integration constant related to the initial amplitude of the u-soliton. RJP 60Nos. 1-2, c v.1.3a*

6 6 Solitons and other solutions to long-wave short-wave interaction equation 77 Equating the three values of vt from 26, 33 and 36 reveals the condition: k 1 α 1 t = k 2 α 2 t = γ 1t { λ 1 + γ 2 tλ } λ 3 which serves as a constraint relation between the model coefficients. Accordingly, the ratio α 1 t/α 2 t in 37 should be a constant for solitons to exist. The following restriction is obtained β 1 α 1 = β 2 α 2 38 Therefore, the ratio β 1 t/β 2 t should be a constant. Hence, the 1-soliton solution of the coupled equations 4-6 is given by φx,t = λ 1 sech 1 m [B0 x vtt]e i k 1x+ω 1 tt+θ 01, 39 ψx,t = λ 2 sech 1 m [B0 x vtt]e i k 2x+ω 2 tt+θ 02, 40 ux,t = λ 3 sech 2 [B 0 x vtt], 41 where the amplitudes λ i with i = 1,2,3 are connected by 37 and the width of the solitons is given by 32 or 35. The velocity is given by 26 or 33 or 36 and the wave numbers is shown in 20 and THE SIMPLEST EQUATION METHOD The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [22, 23] and used successfully by many authors for finding exact solutions of ordinary differential equations ODEs in mathematical physics [24 30]. The modified simplest equation method [24 30] is based on the assumptions that the exact solutions can be expressed by a polynomial in U, such that U = Uξ satisfy the equations of Bernoulli and Riccati, which are well known nonlinear ordinary differential equations and their solutions can be expressed by elementary functions. Using this method in works [24 28] exact solutions of a class of equations that generalize the reaction-diffusion and reaction-telegraph equation and Fisher equation were obtained. Also, using this method in works [29, 30] exact solutions of a class of complex equations such as perturbed nonlinear Schrödinger s equation, Davey- Stewartson equation were constructed. We now summarize the simplest equation method, established in 2012 [29], the details of which can be found in [24 30] among many others. We assume that the given NLPDE for ux,t is in the form P u, u t, u x, 2 u x 2, 2 u t x, 2 u t 2,... = 0, 42 RJP 60Nos. 1-2, c v.1.3a*

7 78 H. Triki et al. 7 where P is a polynomial. The essence of the simplest equation method can be presented in the following steps: Step-1: To find the travelling wave solutions of Eq. 42, we introduce the wave variable ux,t = Uξ, ξ = x vt. 43 Substituting Eq. 43 into Eq. 42, we obtain the following ODE Q U, du dξ, d2 U dξ 2,... = Step-2: Eq. 44 is then integrated as long as all terms contain derivatives where integration constants are considered zero. Step-3: Introduce the solution Uξ of Eq. 44 in the finite series form Uξ = N a l z l, 45 l=0 where a l are real constants with a N 0 and N is a positive integer to be determined, and z = zξ are the functions that satisfy some ordinary differential equations. These ordinary differential equations are called the simplest equations. The simplest equation is characterized by the fact that it is of a lesser order than Eq. 44 and, the general solution of this equation is known or we know the way of finding its general solution, or at least we know some particular solutions of this equation. This means that the exact solutions Uξ of Eq. 44 can be presented by a finite series 45 in the general solution z = zξ of the simplest equation. As examples of simplest equations used in the literature, we can cite the Riccati equation, the equation for the Jacobi elliptic function, and the equation for the Weierstrass elliptic function. In this paper, we use the Riccati equation as simplest equation dz dξ = ρ + Azξ + Bz2 ξ, 46 where ρ, A and B are independent on ξ. When ρ = 0 and A, B 0, we obtain the Bernoulli equation dz dξ = Azξ + Bz2 ξ, 47 We found that the use of the Bernoulli equation leads to new traveling-wave and wavefront solutions of the long wave-short wave interaction equations with t-dependent coefficients and power law nonlinearity. Eq. 47 admits the following exact solutions zξ = Aexp{Aξ + ξ 0} 1 B exp{aξ + ξ 0 }, 48 RJP 60Nos. 1-2, c v.1.3a*

8 8 Solitons and other solutions to long-wave short-wave interaction equation 79 for the case A > 0, B < 0 and zξ = Aexp{Aξ + ξ 0} 1 + B exp{aξ + ξ 0 }, 49 for the case A < 0, B > 0, where ξ 0 is a constant of integration. When ρ 0 and A = 0, B = 1 we obtain the Riccati equation Equation 50 admits the following exact solutions when ρ < 0, and dz dξ = ρ + z2 ξ, 50 zξ = ρ tanh ρξ + ξ 0, zξ = ρ coth ρξ + ξ 0, 51 zξ = ρ tan ρξ + ξ 0, zξ = ρ cot ρξ + ξ 0, 52 when ρ > 0. Step-4: Determining N, can be accomplished by balancing the linear term of highest order derivatives with the highest order nonlinear term in Eq. 44. Step-5: Substituting Eq. 45 into Eq. 47 with Eq. 44, then the left hand side of Eq. 44 is converted into a polynomial in zξ; equating each coefficient of the polynomial to zero yields a set of algebraic equations for a l, A, B, v. Step-6: Solving the algebraic equations obtained in Step-5, and substituting the results into Eq. 45, then we obtain the exact travelling wave solutions for Eq. 42. Remark: In Eq. 47, when B = 1 we obtain the Bernoulli equation dz dξ = Azξ z2 ξ. 53 Equation 53 admits the following exact solutions zξ = A { [ ]} A 1 + tanh 2 2 ξ + ξ 0, 54 when A > 0, and for A < 0. zξ = A 2 { [ ]} A 1 tanh 2 ξ + ξ 0, APPLICATION OF SIMPLEST EQUATION METHOD In this section, we apply the simplest equation method to solve the long waveshort wave interaction equations with t-dependent coefficients and power law non- RJP 60Nos. 1-2, c v.1.3a*

9 80 H. Triki et al. 9 linearity iφ t + αtφ xx = βtuφ, 56 iψ t + αtψ xx = βtuψ, 57 u t = βt φ + ψ, 58 x In order to solve Eqs by the simplest equation method, we assume that Eqs admit a solution of the form φx,t = uξe i{ κx+ωtt+θ}, ψx,t = vξe i{ κx+ωtt+θ}, ux,t = wξ, ξ = x ctt 59 On substituting these into Eqs yields { φ t = i t dω dt + ω u t dc } u dt + c e i{ κx+ωtt+θ}, 60 φ xx = u 2iκu κ 2 u e i{ κx+ωtt+θ}, 61 { ψ t = i t dω dt + ω v t dc } v dt + c e i{ κx+ωtt+θ}, 62 ψ xx = v 2iκv κ 2 v e i{ κx+ωtt+θ}, 63 and u t = t dc dt + c w. 64 Substituting Eqs into Eqs and then decomposing into real and imaginary parts yields a pair of relations. The imaginary part gives ct = 2κ αtdt. 65 t while the real part gives αtu κ 2 αt + t dω αtv dt + ω κ 2 αt + t dω dt + ω u βtwu = 0, 66 v βtwv = 0, 67 2καtw = βt u + v. 68 RJP 60Nos. 1-2, c v.1.3a*

10 10 Solitons and other solutions to long-wave short-wave interaction equation 81 Integrating Eq. 68 once with respect to ξ, then we have w = βt u + v, 69 2καt where integration constant is taken to zero. Inserting Eq. 69 into Eqs yields αtu κ 2 αt + t dω dt + ω u β2 t u +1 + v u = 0, 70 2καt αtv κ 2 αt + t dω dt + ω v β2 t 2καt Balancing u with v u and v with u v in Eqs gives N + 2 = M + N M + 2 = N + M N = M = 1 m. u v + v +1 = To obtain an analytic solution, we use the transformations u = U 1 and v = V 1 in Eqs to find κα 2 t1 U 2 + κα 2 tuu 4m 2 καt κ 2 αt + t dω dt + ω U 2 2 β 2 tu 3 2 β 2 tu 2 V = 0, 72 κα 2 t1 V 2 + κα 2 tv V 4m 2 καt κ 2 αt + t dω dt + ω V 2 2 β 2 tv 3 2 β 2 tv 2 U = For the solutions of Eqs , we make the following ansatz Uξ = V ξ = N a l z l, a N 0, 74 l=0 M b l z l, b M 0, 75 l=0 where a l and b l are some functions of t to be determined and z satisfies Eq. 47. Balancing UU with U 3 and V V with V 3 in Eqs gives 2N + 2 = 3N 2M + 2 = 3M N = M = 2. RJP 60Nos. 1-2, c v.1.3a*

11 82 H. Triki et al. 11 This suggests the choice of Uξ and V ξ in Eqs as Uξ = a 0 + a 1 zξ + a 2 z 2 ξ, 76 V ξ = b 0 + b 1 zξ + b 2 z 2 ξ. 77 Substituting Eqs along with Eq. 47 into Eqs , collecting the coefficients of z, and solving the resulting system we find a 0 = b 0 = 0 a 1 = b 1 = κα2 tabm + 1 m 2 β 2 t a 2 = b 2 = κα2 tb 2 m + 1 m 2 β 2 t ωt = A2 4m 2 κ 2 4m 2 αtdt, t 78 where κ, A and B are arbitrary constants. Consequently, we obtain the exact solutions to Eqs and therefore the new exact solutions for the long wave-short wave interaction equations with t- dependent coefficients and power law nonlinearity can be written as φx,t = ψx,t = [ [ ± κα2 ta 2 Bm + 1 m 2 β 2 t ± κα2 ta 2 Bm + 1 m 2 β 2 t ux,t = ± αta2 Bm + 1 m 2 βt exp A x + 2κ ] 1 αtdt + ξ ± B exp A x + 2κ αtdt + ξ0 e i κx+ A2 4m 2 κ 2 4m 2 αtdt+θ, exp A x + 2κ ] 1 αtdt + ξ ± B exp A x + 2κ αtdt + ξ0 e i κx+ A2 4m 2 κ 2 4m 2 αtdt+θ, exp A x + 2κ αtdt + ξ ± B exp A x + 2κ αtdt + ξ0 Substituting Eqs along with Eq. 50 into Eqs , collecting the RJP 60Nos. 1-2, c v.1.3a*

12 12 Solitons and other solutions to long-wave short-wave interaction equation 83 coefficients of z, and solving the resulting system we obtain a 0 = b 0 = κα2 tρm + 1 m 2 β 2 t a 1 = b 1 = 0 a 2 = b 2 = κα2 tm + 1 m 2 β 2 t ωt = ρ + m2 κ 2 m 2 αtdt, t where κ and ρ are arbitrary constants. Thus we obtain the exact solutions to Eqs and then exact solutions for the long wave-short wave interaction equations with t-dependent coefficients and power law nonlinearity can be written as: 1 Soliton solutions [ κα 2 tρm + 1 φx,t = m 2 β 2 t [ κα 2 tρm + 1 ψx,t = m 2 β 2 t ux,t = αtρm + 1 m 2 βt 2 Singular soliton solutions φx,t = [ κα2 tρm + 1 m 2 β 2 t [ ψx,t = κα2 tρm + 1 m 2 β 2 t αtρm + 1 ux,t = m 2 βt ρ sech x 2 + 2κ sech 2 ρ x + 2κ sech 2 ρ x + 2κ αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2, ] 1 αtdt + ξ 0 ρ csch x 2 + 2κ ρ csch x 2 + 2κ csch 2 ρ x + 2κ RJP 60Nos. 1-2, c v.1.3a* e i κx ρ+m2 κ 2 αtdt+θ m 2, αtdt + ξ 0, αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2, αtdt + ξ 0 ] e i κx ρ+m2 κ 2 αtdt+θ m 2, 82 αtdt + ξ 0.

13 84 H. Triki et al Singular periodic solutions First form [ κα 2 tρm + 1 ρ φx,t = m 2 β 2 sec x 2 + 2κ t [ κα 2 tρm + 1 ψx,t = m 2 β 2 t ux,t = αtρm + 1 m 2 βt ρ sec x 2 + 2κ sec 2 ρ x + 2κ 4 Singular periodic solutions Second form [ κα 2 tρm + 1 ρ φx,t = m 2 β 2 csc x 2 + 2κ t [ κα 2 tρm + 1 ψx,t = m 2 β 2 t ux,t = 5 Plane waves φx,t = αtρm + 1 m 2 βt [ κα 2 tm + 1 m 2 β 2 t ρ csc x 2 + 2κ csc 2 ρ x + 2κ αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2 αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2, αtdt + ξ 0, αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2 αtdt + ξ 0 ] 1 e i κx ρ+m2 κ 2 αtdt+θ m 2, αtdt + ξ 0. ] x + 2κ αtdt + ξ0,, ψx,t = [ κα 2 tm + 1 m 2 β 2 t ux,t = e i κx κ2 αtdt+θ, ] x + 2κ αtdt + ξ0 αtm + 1 m 2 βt 1 x + 2κ αtdt + ξ0 2. RJP 60Nos. 1-2, c v.1.3a* e i κx κ2 αtdt+θ, 85

14 14 Solitons and other solutions to long-wave short-wave interaction equation CONCLUSIONS This paper studied in detail the integrability aspects of the long-wave shortwave vector coupled equation. This coupled system has given rise to soliton solutions. The ansatz approach revealed solitary wave solution. The second integration algorithm lead to several forms of solutions. These are solitary wave solutions, singular solitons, singular periodic solutions, and plane waves. The constraint conditions for the existence of these solutions are given out. The results of this paper stand on a strong footing for further development in this area. In future, it is possible to consider the extraction of shock wave solutions and singular soliton solutions by the aid of ansatz method. Additionally, it is possible to seek travelling wave solutions as well as apply the semi-inverse variational principle to obtain solitary waves. The results of these research activities will be reported elsewhere. REFERENCES 1. C. G. L. Tiofack, Alidou Mohamadou, Timoléon C. Kofané, and Alain B. Moubissi, Phys. Rev. E 80, S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. Lett. 88, G. Assanto, T. R. Marchant, A. A. Minzoni, and N. F. Smyth, Phys. Rev. E 84, A.-M. Wazwaz, Phys. Lett. A 360, A.-M. Wazwaz, Communications in Nonlinear Science and Numerical Simulation 12, Y. Yang, Z. L. Tao, and F. R. Austin, Applied Mathematics and Computation 216, X. Li and M. Wang, Phys. Lett. A 361, H. Triki and A. M. Wazwaz, Applied Mathematics and Computation 214, H. Triki and T. R. Taha, Chaos, Solitons and Fractals 42, K. Nakkeeran, Phys. Rev. E 64, M. S. Ismail, Applied Mathematics and Computation 202, E. H. Doha, A. H. Bhrawy, D. Baleanu, and M. A. Abdelkawy, Rom. J. Phys. 59, E. H. Doha, D. Baleanu, A. H. Bhrawy, and R. M. Hafez, Proc. Romanian Acad. A 15, A. H. Bhrawy, Applied Mathematics and Computation 222, A. Biswas, Phys. Lett. A 372, A. Biswas and H. Triki, Applied Mathematics and Computation 217, H. Triki, A. Yildirim, T. Hayat, O. M. Aldossary, and A. Biswas, Advanced Science Letters 16, H. Triki and A. M. Wazwaz, Phys. Lett. A 373, S. Zhang, Phys. Lett. A 365, S. Erbay, Chaos, Solitons and Fractals 11, H. Borluka, G. M. Muslub, and H. A. Erbay, Mathematics and Computers in Simulation 74, N. A. Kudryashov, Phys. Lett. A 342, N. A. Kudryashov, Chaos Soliton and Fractals 24, RJP 60Nos. 1-2, c v.1.3a*

15 86 H. Triki et al N. K. Vitanov and Z. I. Dimitrova, Communications in Nonlinear Science and Numerical Simulation 15, N. K. Vitanov, Z. I. Dimitrova, and H. Kantz, Applied Mathematics and Computation 216, N. K. Vitanov, Communications in Nonlinear Science and Numerical Simulation 16, N. K. Vitanov, Communications in Nonlinear Science and Numerical Simulation 15, N. Taghizadeh and M. Mirzazadeh, Communications in Nonlinear Science and Numerical Simulation 17, A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaeid, and A. Biswas, Nonlinear Analysis: Modelling and Control 17, M. Labidi, H. Triki, E. V. Krishnan, and A. Biswas, Journal of Applied Nonlinear Dynamics 1, G. Ebadi, A. Mohavir, S. Kumar, and A. Biswas, To appear in: International Journal of Numerical Methods for Heat and Fluid Flow. 32. H. Leblond, H. Triki, and D. Mihalache, Rom. Rep. Phys. 65, H. Leblond and D. Mihalache, Phys. Rep. 523, D. Mihalache, Rom. J. Phys. 57, D. Mihalache, Rom. J. Phys. 59, D. Mihalache, D. Mazilu, L.-C. Crasovan, and L. Torner, Opt. Commun. 137, D. Mihalache, D. Mazilu, J. Dörring, and L. Torner, Opt. Commun. 159, D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, Opt. Lett. 32, H. Triki, Rom. J. Phys. 59, B. S. Ahmed and A. Biswas, Proc. Romanian Acad. A 14, B. S. Ahmed, E. Zerrad, and A. Biswas, Proc. Romanian Acad. A 14, B. S. Ahmed, A. Biswas, E. V. Krishnan, and S. Kumar, Rom. Rep. Phys. 65, A. H. Bhrawy, M. A. Abdelkawy, S. Kumar, and A. Biswas, Rom. J. Phys. 58, A. Biswas, A. H. Kara, L. Moraru, A. H. Bokhari, and F. D. Zaman, Proc. Romanian Acad. A 15, A. Biswas, A. H. Bhrawy, M. A. Abdelkawy, A. A. Alshaery, and E. M. Hilal, Rom. J. Phys. 59, L. Girgis, D. Milovic, S. Konar, A. Yildirim, H. Jafari and A. Biswas, Rom. Rep. Phys. 64, M. Savescu, A. H. Bhrawy, E. M. Hilal, A. A. Alshaery, and A. Biswas, Rom. J. Phys. 59, H. Triki, Z. Jovanoski, and A. Biswas, Rom. Rep. Phys. 66, G. W. Wang, T. Zhou Xu, and A. Biswas, Rom. Rep. Phys. 66, RJP 60Nos. 1-2, c v.1.3a*