In this work, a general (2+1)-dimensional breaking soliton equation is investigated. The Hereman’s simplified method is applied to derive multiple soliton solutions, hence to confirm the model integrability.Publisher's Versio

Wazwaz, Abdul-Majid

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. Eng. Math. V.1, N.1, 2011, pp. 69-74GENERALIZED (2+1)-DIMENSIONAL BREAKING SOLITONEQUATIONA. M. WAZWAZ1 §Abstract. In this work, a general (2+1)-dimensional breaking soliton equation is inves-tigated. The Hereman’s simplified method is applied to derive multiple soliton solutions,hence to confirm the model integrability.Keywords: Breaking soliton equation; multiple soliton solutions; multiple singular solitonsolutions.AMS Subject Classification: 35Q51, 35Q53, 37K101. IntroductionMany reliable methods are used in the literature to investigate completely integrableequations that admit multiple soliton solutions [1–6]. The algebraic-geometric method[2–4], the inverse scattering method, the Bäcklund transformation method, the Darbouxtransformation method, the Hirota bilinear method [7–14], and other methods are used tomake progress and new developments in this filed. The Hirota’s bilinear method is ratherheuristic and possesses significant features that make it practical for the determinationof multiple soliton solutions, and for multiple singular soliton solutions [15–23] for a wideclass of nonlinear evolution equations in a direct method. Hereman et. al [10] developeda modified form of the Hirota’s method that facilitates the computational work. Thecomputer symbolic systems such as Maple, Mathematica can be used to overcome thetedious calculations.In this work, we will study a generalized (2+1)-dimensional breaking soliton equation[1](uxt − β(4uxyux + 2uxxuy − uxxxy)− γ(6uxuxx − uxxxx))x = −α2(βuyyy + 3γuxyy). (1)For α = 0, we obtain the (2+1)-dimensional equationuxt − β(4uxyux + 2uxxuy − uxxxy)− γ(6uxuxx − uxxxx) = 0, (2)1 Department of Mathematics, Saint Xavier University, Chicago, IL 60655,e-mail: wazwaz@sxu.edu§ Manuscript received 04 May 2011.TWMS Journal of Applied and Engineering Mathematics Vol.1 No.1 c© Işık University, Departmentof Mathematics 2011; all rights reserved.6970 TWMS J. APP. ENG. MATH. V.1, N.1, 2011which describes the interaction between the Riemann wave propagation along the y-axisand the long wave propagating along the x-axis [1–3]. For α = 0 and γ = 0, Eq. (1)reduces to the (2+1)-dimensional breaking soliton equationuxt − β(4uxyux + 2uxxuy − uxxxy) = 0. (3)Eq. (3) was proved in [2] to be completely integrable equation.Our aim from this work is to derive multiple regular soliton solutions and multiplesingular soliton solutions for the (2+1)-dimensional equation (1). The modified form ofthe Hirota’s bilinear method, that was established by Hereman et. al. [11] will be usedto achieve the goal set for this work. The Hereman’s method is now well-known in theliterature, for more details see [2,10–23].2. Multiple soliton solutionsIn this section we will apply the Hereman’s method which is a simplified form of the Hi-rota’s bilinear method to study a generalized (2+1)-dimensional breaking soliton equation[1](uxt − β(4uxyux + 2uxxuy − uxxxy)− γ(6uxuxx − uxxxx))x = −α2(βuyyy + 3γuxyy). (4)To determine the dispersion relation we substituteu(x, y, t) = eθi , θi = kix + riy − ωi t, (5)into the linear terms of (4), and solving the resulting equation for ωi, we find the dispersionrelation is defined byωi =k4i (βri + γki) + α2r2i (βri + 3γki)k2i, i = 1, 2, · · ·N, (6)and hence θi becomesθi(x, y, t) = kix + riy − k4i (βri + γki) + α2r2i (βri + 3γki)k2it, i = 1, 2, · · ·N. (7)We next substituteu(x, y, t) = R∂ ln f(x, y, t)∂x= Rfx(x, y, t)f(x, y, t), (8)where R is a constant that should be determined, and the auxiliary function f(x, y, t)readsf(x, y, t) = 1 + ek1x+r1y− k41(βr1+γk1)+α2r21(βr1+3γk1)k21t, (9)into Eq. (4) and solve to find thatR = −2. (10)Substituting (9) and (10) into (8) gives the single soliton solutionu(x, y, t) = − 2k1ek1x+r1y− k41(βr1+γk1)+α2r21(βr1+3γk1)k21t1 + ek1x+r1y− k41(βr1+γk1)+α2r21(βr1+3γk1)k21t. (11)For the two soliton solutions, we substituteu(x, y, t) = −2∂ ln f(x, y, t)∂x, (12)A. M. WAZWAZ: GENERALIZED (2+1)-DIMENSIONAL BREAKING SOLITON EQUATION 71where the auxiliary function f(x, y, t) for the two soliton solutions readsf(x, y, t) = 1 + eθ1 + eθ2 + a12eθ1+θ2 , (13)into Eq. (4), where θ1 and θ2 are given above in (7) to obtain the phase shift a12 bya12 =k21k22(k1 − k2)2 − α2(k1r2 − k2r1)2k21k22(k1 + k2)2 − α2(k1r2 − k2r1)2, (14)and this can be generalized for the phase shifts byaij =k2i k2j (ki − kj)2 − α2(kirj − kjri)2k2i k2j (ki + kj)2 − α2(kirj − kjri)2, 1 ≤ i < j ≤ 3. (15)It is interesting to point out that the phase shifts do not depend on the parameters β andγ. Only the parameter α affects these phase shifts. Moreover, for α = 0, the phase shiftsreduce toaij =(ki − kj)2(ki + kj)2, 1 ≤ i < j ≤ 3, (16)that is consistent with the result obtained in [2]. The last result depends only on thecoefficients of the space variable x only.The auxiliary function f(x, y, t) for the two soliton solutions is given byf(x, y, t) = 1 + ek1x+r1y− k41(βr1+γk1)+α2r21(βr1+3γk1)k21t+ ek2x+r2y−k42(βr2+γk2)+α2r22(βr2+3γk2)k22t+ k21k22(k1−k2)2−α2(k1r2−k2r1)2k21k22(k1+k2)2−α2(k1r2−k2r1)2× e(k1+k2)x+(k1+k2)y−(k41(βr1+γk1)+α2r21(βr1+3γk1)k21+k42(βr2+γk2)+α2r22(βr2+3γk2)k22) t.(17)To determine the two soliton solutions explicitly, we substitute (17) into (8).To determine the three soliton solutions, we use the auxiliary functionf(x, y, t) = 1 + exp(θ1) + exp(θ2) + exp(θ3)+ a12exp(θ1 + θ2) + a23exp(θ2 + θ3) + a13exp(θ1 + θ3)+ b123exp (θ1 + θ2 + θ3),(18)and proceed as before to find thatb123 = a12a13a23, (19)where the phase shifts aij are defined in (15). To determine the three soliton solutionsexplicitly, we substitute the last result for f(x, y, t) into (8). The higher level solitonsolutions, for N ≥ 4 can be obtained in a parallel manner. This confirms the fact that the(2+1)-dimensional breaking soliton equation (4) is completely integrable and gives rise tomultiple soliton solutions of any order.3. Multiple singular soliton solutionsIn this section we will proceed as before study the multiple singular soliton solutionsfor a generalized (2+1)-dimensional breaking soliton equation(uxt − β(4uxyux + 2uxxuy − uxxxy)− γ(6uxuxx − uxxxx))x = −α2(βuyyy + 3γuxyy). (20)72 TWMS J. APP. ENG. MATH. V.1, N.1, 2011The dispersion relation is the same as derived before, hence we setωi =k4i (βri + γki) + α2r2i (βri + 3γki)k2i, i = 1, 2, · · ·N, (21)and θi becomesθi(x, y, t) = kix + riy − k4i (βri + γki) + α2r2i (βri + 3γki)k2it, i = 1, 2, · · ·N. (22)We next substituteu(x, y, t) = R∂ ln f(x, y, t)∂x= Rfx(x, y, t)f(x, y, t), (23)where R is a constant that should be determined, and the auxiliary function f(x, y, t) forthe singular case readsf(x, y, t) = 1− ek1x+r1y−k41(βr1+γk1)+α2r21(βr1+3γk1)k21t, (24)into Eq. (20) and solve to find thatR = −2. (25)Substituting (24) and (25) into (23) gives the single singular soliton solutionu(x, y, t) =2k1ek1x+r1y− k41(βr1+γk1)+α2r21(βr1+3γk1)k21t1− ek1x+r1y−k41(βr1+γk1)+α2r21(βr1+3γk1)k21t. (26)For the two soliton solutions, we substituteu(x, y, t) = −2∂ ln f(x, y, t)∂x, (27)where the auxiliary function f(x, y, t) for the two soliton solutions is given byf(x, y, t) = 1− eθ1 − eθ2 + a12eθ1+θ2 , (28)into Eq. (20), where θ1 and θ2 are given above in (22) to obtain the phase shift a12 bya12 =k21k22(k1 − k2)2 − α2(k1r2 − k2r1)2k21k22(k1 + k2)2 − α2(k1r2 − k2r1)2, (29)and this can be generalized for the phase shifts byaij =k2i k2j (ki − kj)2 − α2(kirj − kjri)2k2i k2j (ki + kj)2 − α2(kirj − kjri)2, 1 ≤ i < j ≤ 3. (30)The phase shifts do not depend on the parameters β and γ, but depend on α only.The auxiliary function f(x, y, t) for the two soliton solutions is given byf(x, y, t) = 1− ek1x+r1y−k41(βr1+γk1)+α2r21(βr1+3γk1)k21t − ek2x+r2y−k42(βr2+γk2)+α2r22(βr2+3γk2)k22t+ k21k22(k1−k2)2−α2(k1r2−k2r1)2k21k22(k1+k2)2−α2(k1r2−k2r1)2× e(k1+k2)x+(k1+k2)y−(k41(βr1+γk1)+α2r21(βr1+3γk1)k21+k42(βr2+γk2)+α2r22(βr2+3γk2)k22) t.(31)To determine the two singular soliton solutions explicitly, we substitute (31) into (23).A. M. WAZWAZ: GENERALIZED (2+1)-DIMENSIONAL BREAKING SOLITON EQUATION 73To determine the three soliton solutions, we use the auxiliary functionf(x, y, t) = 1− exp(θ1)− exp(θ2)− exp(θ3)+ a12exp(θ1 + θ2) + a23exp(θ2 + θ3) + a13exp(θ1 + θ3)+ b123exp (θ1 + θ2 + θ3),(32)and proceed as before to find thatb123 = −a12a13a23, (33)where the phase shifts aij are defined in (30). To determine the three singular solitonsolutions explicitly, we substitute the last result for f(x, y, t) into (23). The higher levelsingular soliton solutions, for N ≥ 4 can be obtained in a parallel manner.4. DiscussionIn this work we proved the integrability of a generalized (2+1)-dimensional breakingsoliton equation. Multiple soliton solutions were formally derived for this equation. More-over, multiple singular soliton solutions were derived as well. The Hereman’s methodshows effectiveness and reliability in handling nonlinear evolution equations.References[1] Bogoyavlenskii, O.I., (1990), Breaking solitons in 2+1-dimensional integrable equations, Uspekhi Mat.Nauk., 45(4), 17-27.[2] Wazwaz, A.M., (2010), Integrable (2 + 1)-dimensional and (3 + 1)-dimensional breaking solitonequations, Phys. Scr., 81, 035005.[3] Ma, S.-H., Peng, J. and Zhang, C., (2009), New exact solutions of the (2+1)-dimensional breakingsoliton system via an extended mapping method, Chaos, Solitons and Fractals, 46, 210-214.[4] Gao, Y.-T. and Tian, B., (1995), New family of overturning soliton solutions for a typical breakingsoliton equation, Comput. Math. Applic., 12, 97-100.[5] Hirota, R., (1974), A new form of Bäcklund transformations and its relation to the inverse scatteringproblem, Progress of Theoretical Physics, 52(5), 1498-1512.[6] Hirota, R., (2004), The Direct Method in Soliton Theory, Cambridge University Press, Cambridge.[7] Hirota, R., (1971), Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons,Physical Review Letters, 27(18), 1192-1194.[8] Hietarinta, J., (1987), A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations, J. Math. Phys., 28(8), 1732-1742.[9] Hietarinta, J., (1987), A search for bilinear equations passing Hirota’s three-soliton condition. II.mKdV-type bilinear equations, J. Math. Phys., 28(9), 2094-2101.[10] Hereman, W. and Nuseir, A., (1997), Symbolic methods to construct exact solutions of nonlinearpartial differential equations, Mathematics and Computers in Simulation, 43, 13-27.[11] Wazwaz, A.M., (2007), Multiple-soliton solutions for the KP equation by Hirota’s bilinear methodand by the tanh-coth method, Appl. Math. Comput., 190, 633-640.[12] Wazwaz, A.M., (2007), Multiple-front solutions for the Burgers equation and the coupled Burgersequations, Appl. Math. Comput., 190, 1198-1206.[13] Wazwaz, A.M., (2007), New solitons and kink solutions for the Gardner equation, Communicationsin Nonlinear Science and Numerical Simulation, 12(8), 1395-1404.[14] Wazwaz, A.M., (2007), Multiple-soliton solutions for the Boussinesq equation, Appl. Math. Comput.,192, 479-486.[15] Wazwaz, A.M., (2008), The Hirota’s direct method and the tanh-coth method for multiple-solitonsolutions of the Sawada-Kotera-Ito seventh-order equation, Appl. Math. Comput., 199(1), 133-138.[16] Wazwaz, A.M., (2008), Multiple-front solutions for the Burgers-Kadomtsev-Petvisahvili equation,Appl. Math. Comput., 200, 437-443.74 TWMS J. APP. ENG. MATH. V.1, N.1, 2011[17] Wazwaz, A.M., (2008), Multiple-soliton solutions for the Lax-Kadomtsev-Petvisahvili (Lax-KP) equa-tion, Appl. Math. Comput., 201(1/2), 168-174.[18] Wazwaz, A.M., (2008), The Hirota’s direct method for multiple-soliton solutions for three modelequations of shallow water waves, Appl. Math. Comput., 201(1/2), 489-503.[19] Wazwaz, A.M., (2008), Multiple-soliton solutions of two extended model equations for shallow waterwaves, Appl. Math. Comput., 201(1/2), 790-799.[20] Wazwaz, A.M., (2008), Single and multiple-soliton solutions for the (2+1)-dimensional KdV equation,Appl. Math. Comput., 204, 20-26.[21] Wazwaz, A.M., (2008), Solitons and singular solitons for the Gardner-KP equation, Appl. Math.Comput., 204, 162-169.[22] Wazwaz, A.M., (2008), Regular soliton solutions and singular soliton solutions for the modifiedKadomtsev-Petviashvili equations , Appl. Math. Comput., 204, 817-823.[23] Wazwaz, A.M., (2008), Multiple kink solutions and multiple singular kink solutions for the (2+1)-dimensional Burgers equations, Appl. Math. Comput., 204, 529-541.Abdul-Majid Wazwaz is a Professor of Mathematics at Saint Xavier Uni-versity in Chicago, Illinois, U.S.A. He received his Ph.D in Applied Mathe-matics from University of Illinois at Chicago in 1981. He has both authoredand co-authored more than 300 papers in applied mathematics, mathemat-ical physics, numerical analysis, and integral equations. He is the author offive books on the subjects of discrete mathematics, integral equations andpartial differential equations. Furthermore he has contributed extensively totheoretical advances in the Adomian decomposition method, the tanh-cothmethod and solitary waves theory.

Generalized (2+1)−dimensional breaking soliton equation

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Generalized (2+1)−dimensional breaking soliton equation

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