A few years ago many authors found the different types of exact traveling and solitary wave solutions of both nonlinear models GZK-BBM and GCH-equations. These both equations are well-known nonlinear evaluation equations and play important role in many scientific fields. The GZK-BBM equation used in the studies of acoustic waves, acoustic-gravity waves, surface waves with long wavelength, hydromagnetic waves, these all mentioned waves have source harmonic crystals, compressible fluids, cold plasma and liquids, respectively. The GCH equation play important role in shallow water waves. In 2005 Wazwaz  studied the GZK-BBM equation for the first time and found some complex solutions, kink type solutions, periodic wave solutions and solitons solutions with the help of the sine-cosine method. Wazwaz  found the two types of compactons and solitary patterns wave solutions of ZK-BBM equation by applying the extended tanh method. Abdou  found the set of exact solutions of ZK-BBM equation with the help of extended F-Expansion method. Mahmoudi et al.  investigated the periodic solitary wave solutions of ZK-BBM equation by applying the exp-function method. Wang and Tang  studied the existence property of smoothness of traveling wave solutions of ZK-BBM equation by apply the bifurcation theory of planner. Song and Yang  with the help of bifurcation technique found the travelingwave, solitary wave and kink type solutions of ZK-BBM equation.
Camassa and Holm  derived a Camassa-Holm equation (CH-equation) by using the Hamiltonian methods, which is a completely integrable dispersive water waves equation by holding two terms, which are neglected in the limit of shallow water waves, having small amplitude. After that many authors started to investigate the different types of travelling solitary wave solutions of CH equation by using various methods. Cooper and Shepard  found the solitary wave solutions of GCH-equation by using the variational function. Liu et al.  improved? CH equation and found traveling wave solutions. Zhang and Bi  studied the bifurcation technique of CH-equation. Liu and Tang  investigated the bifurcation phenomena and found the periodic solutions of GCH-equation with the help of integrated scheme. Deng et al.  found the compacton, kink and anti-kink, periodic solitary wave and solitons solutions of GCH-Degaspersi-Procesi-equation. Kalla and Klein  found the multidimensional theta functions independent derivation solutions of GCH equation with the help of technique that is related to Fay’s identity.
Recently, Liu and Song  found the smooth periodic and blow-up periodic solutions of GZK-BBM equation by applying the bifurcation method. Khadijo Adem and Masood Khalique  investigated the traveling waves solutions and conservation laws of GZK-BBM equation with the help of (G′ / G)− expansion method. Harun-Or-Roshid et al.  found the families of solitary waves solutions of GZK-BBM and RLW equations by using the modified simple equation method. Seadawy et al.  found the families of exact travelling and solitary wave solutions of GZK-BBM equation with the help of exp (−φ(ξ))-expansion method. Many other authors have investigated the travelling solitary wave solutions of GZK-BBM equation and GCH-equation see Ref. [18, 19, 20, 26, 27, 28, 29, 30,].
The nonlinear system of partial differential equations is very useful to study the physical nature in many different scientific fields, such as engineering, physics, geophysics, optics, chemistry, biology, material science, computer science, mechanics, electricity, ultrasound, thermodynamics and so on. The solitary and travelling wave solutions of NPDEs have many applications to understanding the process and physical phenomena in many areas of applied science. In the last five decades a lot of new methods have been developed by many groups of mathematicianas and engineers to investigate the (NPDEs). For example some important methods such as, exp-function method; modified Extended tanh-expansion method; modified simple equation method; homotopy perturbation method; novel (G′ /G)− expansion method; extended modified direct algebraic method; generalized kudryashov method; modified extended Kudryashov method; exp (−φ(ξ))-expansion method; extended Jacobian method; extended trial equation method and so on [31, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,].
The main aim of this research is to investigate the exact traveling and solitary wave solutions of GZK-BBM and simplified modified form of CH-equations. These new solutions are obtained with the help of new method, which is modified extended auxiliary equation mapping method. The arrangement of this article is organized as fellows. Description of the ,modified extended auxiliary equation mapping method is given in Section 2. Section 3 deals with the investigation of the solitary wave solutions of GZK-BBM-equation and simplified modified CH-equation by using the described method. Finally, the conclusion are presented in Section 4.
2 Modified extended auxiliary equation mapping method
Consider the general form of (2+1)-dimensional NPDEs as(1)
here F denotes the polynomial function of U(x, y, t) and its all derivatives which contained highest order nonlinear terms and highest order partial derivatives. Herewe explain the important steps of the new method as:
Step1. We apply the traveling wave transformations as(2)
where l and m are the wave numbers and ω is the frequency of the wave. We obtained the ODE of Eq.1 as(3)
here P is the polynomial function in U(ξ) and its derivatives.
Step2. We consider the general solution of Eq.(2), in the following form(4)
where a0, a1, ...an , b1, b2, ...bn , c2, c3, ...cn , d1, d2, ...dn are constants parameter to be find later, the values of Ψ(ξ) and its derivative Ψ′ (ξ) satisfy to the given auxiliary equation(5)
Where βis are real constants, which determine later such that βn ≠0.
Step4. Substituting Equation 5 into Equation 4 and combining each coefficients of Ψi(ξ)(i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a families of algebraic equations, solving this system of equations with the help of Mathematica, the constants a0, a1, ...an , b1, b2, ...bn , c2, c3, ...cn , d1, d2, ...dn can be determined.
3 Application of the proposed method
Now we applying the modified extended auxiliary equation mapping method to investigate the families of new solitary wave solutions for the (2+1)-dimensional GZK-BBM-equation and simplified modified form of CH-equation.
3.1 Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation
We consider a (2+1)-dim GZK-BBM-equation as(6)
where γ and α are non zero constants. Consider the travel-ingwave transformation U(x, y, t) = U(ξ), ξ = lx+my+ωt, by this transformation we obtained ordinary differential equation of Eq.6 as(7)
we integrate the Eq.7 once time according to ξ and integration constant equal to zero, then we obtained as(8)
Balance the highest order nonlinear term and highest order partial derivative in Eq.8 obtained the value of n =
1. The general solution of Eq.8 takes form of(9)
Substituting Equation 9 into Equation 8 and combining each coefficients of Ψ′ j(ξ)Ψi(ξ)(j = 0,1;i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a set of algebraic equations. We solve this system of equations with the aid of Mathematica. The parameters a0, a1, b1, d1 can be determined as
3.2 Simplified modified form of Camassa-Holm equation
We consider a simplified modified form of CH-equation as(38)
where β and δ are non zero constants. Consider the wave transformation as; U(x, y, t) = U(ξ), ξ = kx + ωt. By this transformation we obtained ordinary differential equation of Eq.38 as(39)
we integrate Eq.39 once time according to the ξ and constant of integration equal to zero, then we obtained as(40)(41)
Substituting Eq.41 into Eq.40 and combining each coefficients of Ψ′ j(ξ)Ψi(ξ)(j = 0,1; i = 1, 2, 3, ...n), then making a every coefficient equal to zero and obtaining a set of algebraic equations. We solve this system of equations with the aid of Mathematica. The parameters a0, a1, b1, d1 can be determined as
We have successfully applied a new method on two nonlinear evaluation equations. We have obtained a new exact traveling and solitary wave solutions of GZK-BBM-equation and simplified modified form of CH-equation by applying the Modified extended auxiliary equation mapping method. As a results, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitions, bright and dark solitons, periodic solitary wave and travelling wave solutions and also show two and three dimensional graphs with the help of Mathematica. These new families of solutions show the power, effectiveness, capability, realizabilities and fruitfulness of this new method. We can solve other nonlinear physical phenomena, which are related to nonlinear evaluation equations with the help of this new method.
Wazwaz A.M., Compact and noncompact physical structures for the ZK-BBM equation, Appl. Math. Comput. 2005, 169, 713-725. Google Scholar
Abdou M.A., Exact periodic wave solutions to some nonlinear evolution equations, Int. J. Nonlinear Sci., 2008, 6, 145-153. Google Scholar
Song M., Yang C.X., Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, Appl. Math. Comput., 2010, 216, 3234-3243. Google Scholar
Liu Z.R., Qian. T.F., Peakons and their bifurcation in a generalized Camassa–Holm equation, International Journal of Bifurcation and Chaos, 2001, 11, 781-792. Google Scholar
Zhang Z.D., Bi Q.S., Bifurcations of a generalized Camassa-Holm equation, International Journal of Nonlinear Sciences and Numerical Simulation, 2005, 6, 81-86. Google Scholar
Liu Z.R., Tang H., Explicit periodic wave solutions and their bifurcations for generalized Camassa-Holm equation, International Journal of BifurcationI and Chaos, 2010, 20, 2507-2519. CrossrefGoogle Scholar
Kalla C., Klein C. New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation, Proceedings of the Royal Society A- Mathematical Physical and Engineering Sciences, 2012, 468, 1371-1390. CrossrefGoogle Scholar
Song M., Liu Z., Periodic wave solutions and their limits for the ZK–BBM equation[J], Applied Mathematics & Computation, 2012, 2012, 9-26. Google Scholar
Adem K. R., Khalique C.M., Conservation Laws and Traveling Wave Solutions of a Generalized Nonlinear ZK-BBM Equation[J], Abstract & Applied Analysis, 2014, 2014(2), 1-5. Google Scholar
Roshid H.O., Roshid M.M., Rahman N., et al. New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation[J], Propulsion & Power Research, 2017, 6(1), 49-57. CrossrefGoogle Scholar
Khater A.H., Helal M.A., Seadawy A.R., General soliton solutions of n-dimensional nonlinear Schr?dinger equation, IL Nuovo Cimento, 2000, 115B, 1303-1312. Google Scholar
Seadawy A.R., Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations, Appl. Math. Inf. Sci., 2016, 10, 1, 209-214.Google Scholar
Khater A.H., Callebaut D.K., Helal M.A. and Seadawy A.R., Variational Method for the Nonlinear Dynamics of an Elliptic Magnetic Stagnation Line, The European Physical Journal D, 2006, 39, 237-245.CrossrefGoogle Scholar
Seadawy A.R., The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrodinger equation and its solutions, Optik - International Journal for Light and Electron Optics, 2017, 139, 31-43.CrossrefGoogle Scholar
Seadawy A.R. Ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma, Mathematical methods and applied Sciences, 2017, 40, (5), 1598-1607.CrossrefGoogle Scholar
Lenells. Traveling wave solutions of the Camassa-Holm equation[J]. Journal of Differential Equations, 2005, 217(2), 393-430. Google Scholar
Baleanu1 D., Inc M., Yusuf A., Aliyu I.A., Optical solitons, nonlinear self-adjointness and conservation laws for Kundu-Eckhaus equation, Chinese Journal of Physics, 2017, 55, 2341-2355. CrossrefGoogle Scholar
Inc M., Yusuf A., Aliyu I.A., Baleanu D., Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics, Opt Quant Electron, 2018, 50, 190. CrossrefGoogle Scholar
Inc M., Yusuf A., Aliyu I.A., Baleanu D., Soliton structures to some time-fractional nonlinear differential equations with conformable derivative, Opt Quant Electron, 2018, 50, 20. CrossrefGoogle Scholar
Inc M., Yusuf A., Aliyu I.A., Baleanu D., Dark and singular optical solitons for the conformable space-time nonlinear Schr?dinger equation with Kerr and power law nonlinearity, Optik, 2018, 162, 65-75.CrossrefGoogle Scholar
Lu D., Seadawy A.R., Ali A., Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications, Results in Physics, 2018, 9.Google Scholar
Helal M.A. and Seadawy A.R., Exact soliton solutions of an D-dimensional nonlinear Schrödinger equation with damping and diffusive terms, Z. Angew. Math. Phys. (ZAMP) 2011, 62, 839-847. CrossrefGoogle Scholar
Lu D., Seadawy A.R., Khater M.A., Structure of solitary wave solutions of the nonlinear complex fractional generalized Zakharov dynamical system, Advances in Difference Equations, 2018, 2018, (1), 266. Google Scholar
Ali A., Seadawy A.R., Lu D., New solitary wave solutions of some nonlinear models and their applications, Advances in Difference Equations, 2018, 2018,(1), 232. Google Scholar
Seadawy A.R., Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves, The European Physical Journal Plus, 2017, 132, 29, 1-13. Google Scholar
Seadawy A.R., Fractional travelling wave solutions of the higher order extended KdV equations in a stratified shear flow, part I, Computers and Mathematics with Applications, 2015, 70, 345-352. CrossrefGoogle Scholar
Lu D., Seadawy A.R., Arshad M., Bright-dark solitary wave and elliptic function solutions of unstable nonlinear Schrödinger equation and their applications[J]. Optical & Quantum Electronics, 2018, 50(1):23. CrossrefGoogle Scholar
Iqbal M., Seadawy A.R. and Lu D., Construction of solitary wave solutions to the nonlinear modified Kortewege-de Vries dynamical equation in unmagnetized plasma via mathematical methods, Modern Physics Letters A, 2018, 33, 1850183, 1-13. Google Scholar
Arshad M., Seadawy A.R., Lu D., Exact bright–dark solitary wave solutions of the higher-order cubic-quintic nonlinear SchrOdinger equation and its stability, Optik 2017, 138, 40-49. CrossrefGoogle Scholar
Lu D., Seadawy A.R., Arshad M., Wang J., New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications, Results Phys, 2017, 7, 899-909. CrossrefGoogle Scholar
Tariq K., Seadawy A.R., Bistable Bright-Dark solitary wave solutions of the (3 +1)-dimensional Breaking soliton, Boussinesq equation with dual dispersion and modified Kortewegde Vries Kadomtsev Petviashvili equations and their applications, Results Phys 2017, 7, 1143-1149. CrossrefGoogle Scholar
Seadawy A.R., Traveling wave solutions of the Boussinesq and generalized fifth-order KdV equations by using the direct algebraic method, Appl Math Sci 2012, 6, (82), 4081-4090. Google Scholar
Seadawy A.R., Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries Zakharov Kuznetsov equation in a magnetized electron-positron plasma, Phys A 2016, 455, 44. CrossrefGoogle Scholar
Seadawy A.R., Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method, The European Physical Journal Plus, 2015, 130, 182, 1-10. Google Scholar
Iqbal M., Seadawy A.R. and Lu D., Dispersive solitary wave solutions of nonlinear further modified Kortewege-de Vries dynamical equation in a unmagnetized dusty plasma via mathematical methods, Modern Physics Letters A, 2018, 33, 1850217, 1-19. Google Scholar
Lu D., Seadawy A.R., Iqbal M., Mathematical physics via construction of traveling and solitary wave solutions of three coupled system of nonlinear partial differential equations and their applications, Results in Physics, 2018, 11, 1161-1171.CrossrefGoogle Scholar
About the article
Published Online: 2018-12-31
Citation Information: Open Physics, Volume 16, Issue 1, Pages 896–909, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0111.
© 2018 Dianchen Lu et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0