- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 537930, 24 pages
New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Ito Equations
1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
Received 20 March 2012; Revised 14 September 2012; Accepted 14 September 2012
Academic Editor: Massimo Scalia
Copyright © 2012 A. H. Bhrawy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using the extended F-expansion method based on computerized symbolic computation technique, we find several new solutions of (1+1)-dimensional and (2+1)-dimensional Ito equations. These solutions contain hyperbolic and triangular solutions. It is shown that the power of the extended F-expansion method is its ease of use to determine shock or solitary type of solutions. In addition, as an illustrative sample, the properties for the extended F-expansion solutions of the Ito equations are shown with some figures.
The nonlinear wave phenomena can be observed in various scientific fields, such as plasma physics, optical fibers, fluid dynamics, and chemical physics. The nonlinear wave phenomena can be obtained in solutions of nonlinear evolution equations (NEEs). The study of NLEEs appear everywhere in applied mathematics and theoretical physics including engineering sciences and biological sciences. These NLEEEs play a key role in describing key scientific phenomena. For example, the nonlinear Schrödinger’s equation describes the dynamics of propagation of solitons through optical fibers. The Korteweg-de Vries equation models the shallow water wave dynamics near ocean shore and beaches. Additionally, the Schrödinger-Hirota equation describes the dispersive soliton propagation through optical fibers. These are just a few examples in the whole wide world of NLEEs and their applications, (see, for instance, [1–4]). While the above mentioned NLEEs are scalar NLEEs, there is a large number of NLEEs that are coupled. Some of them are two-coupled NLEEs such as the Gear-Grimshaw equation , while there are several others that are three-coupled NLEEs. An example of a three-coupled NLEE is the Wu-Zhang equation . These coupled NLEEs are also studied in various areas of theoretical physics as well.
The exact solutions of these NEEs play an important role in the understanding of nonlinear phenomena. In the past decades, many methods were developed for finding exact solutions of NEEs such as the inverse scattering method [5, 6], improved projective Riccati equations method [7, 8], Cole-Hopf transformation method , exp-function method [10–16], bifurcation theory method , -expansion method [18, 19], homotopy perturbation method , tanh function method [20–24], and Jacobi and Weierstrass elliptic function method [25, 26]. Although Porubov et al. [27–29] have obtained some exact periodic solutions to some nonlinear wave equations, they use the Weierstrass elliptic function and involve complicated deducing. A Jacobi elliptic function (JEF) expansion method, which is straightforward and effective, was proposed for constructing periodic wave solutions for some nonlinear evolution equations. The essential idea of this method is similar to the tanh method by replacing the tanh function with some JEFs such as , , and . For example, the Jacobi periodic solution in terms of may be obtained by applying the -function expansion. Many similarl repetitious calculations have to be done to search for the Jacobi doubly periodic wave solutions in terms of and .
Recently, F-expansion method [31–34] was proposed to obtain periodic wave solutions of NLEEs, which can be thought of as a concentration of JEF expansion since F here stands for every function of JEFs. The objectives of this work are twofold. First, we seek to extend others works to establish new exact solutions of distinct physical structures for the nonlinear equations (1.1) and (1.2). The extended F-expansion (EFE) method will be used to achieve the first goal. The second goal is to show that the power of the EFE method is its ease of use to determine shock or solitary type of solutions. In this paper, we study two well-known PDEs, namely, generalized (1+1)-dimensional and generalized (2+1)-dimensional Ito equations. Many studies are concerning the (1+1)-dimensional Ito equation and the (2+1)-dimensional Ito equation [35–42].
The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870, and, finally, Korteweg and de Vries in 1895 . The KdV equation was not studied much after this until Zabusky and Kruskal (1965)  discovered numerically that its solutions seemed to decompose at large times into a collection of “solitons”: well-separated solitary waves. Ito [41, 42] obtained the well-known generalized (1+1)-dimensional and generalized (2+1)-dimensional Ito equations by generalization of the bilinear KdV equation as Also Sawada-Kotera-Ito (SK-Ito) seventh-order equation is the special case of the generalized seventh-order KdV equation as SK-Ito equation is characterized by the presence of three dispersive terms , , and , respectively. SK-Ito seventh-order equation is completely integrable and admits of conservation laws . Moreover, the Ito-type coupled KdV (ItcKdV) equation , written in the following form: if we take the special values , , and . Equation (1.4) describes the interaction process of two internal long waves which has infinitely many conserved quantities [45, 46].
In this paper, we extend the EFE method with symbolic computation to (1.1) and (1.2) for constructing their interesting Jacobi doubly periodic wave solutions. It is shown that soliton solutions and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions. In addition, the algorithm that we use here is also a computerized method, in which we are generating an algebraic system.
2. Extended F-Expansion Method
In this section, we introduce a simple description of the EFE method, for a given partial differential equation as We like to know whether travelling waves (or stationary waves) are solutions of (2.1). The first step is to unite the independent variables , , and into one particular variable through the new variable as where is wave speed, and reduce (2.1) to an ordinary differential equation (ODE) as Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE. For this purpose, let us simply use as the expansion in the form where the highest degree of , is taken as where , , and are constants, and in (2.3) is a positive integer that can be determined by balancing the nonlinear term(s) and the highest order derivatives. Normally is a positive integer, so that an analytic solution in closed form may be obtained. Substituting (2.1)–(2.5) into (2.3) and comparing the coefficients of each power of in both sides, we will get an overdetermined system of nonlinear algebraic equations with respect to . We will solve the over-determined system of nonlinear algebraic equations by use of Mathematica. The relations between values of , , , and corresponding JEF solution of (2.4) are given in Table 1. Substituting the values of , , , and the corresponding JEF solution chosen from Table 1 into the general form of solution, then an ideal periodic wave solution expressed by JEF can be obtained.
, cn, and dn are the JE sine function, JE cosine function, and the JEF of the third kind, respectively. And with the modulus .
When . the Jacobi functions degenerate to the hyperbolic functions, that is, when , the Jacobi functions degenerate to the triangular functions, that is,
3. Generalized (1+1)-Dimensional Ito Equation
We first consider the generalized (1+1)-dimensional Ito equation (1.1) as follows: if we use the transformation , it carries (3.1) into if we use transforms (3.2) into the ODE, we have where by integrating once we obtain, upon setting the constant of integration to zero, if we use the transformation , then (3.4) can be written as follows: Balancing the term with the term we obtain then
Substituting (3.6) into (3.5) and comparing the coefficients of each power of in both sides, we will get an over-determined system of nonlinear algebraic equations with respect to , . Solving the over-determined system of nonlinear algebraic equations by use of Mathematica, we obtain three groups of constants
(3) The solutions of (3.1) are
3.1. Soliton Solutions
3.2. Triangular Periodic Solutions
4. Generalized (2+1)-Dimensional Ito Equation
In this section we consider the generalized (2+1)-dimensional Ito equation (1.2) as follows: if we use the transformation , it carries (4.1) into if we use carries (4.2) into the ODE, we have where by integrating twice we obtain, upon setting the constant of integration to zero, if we use the transformation , it carries (4.4) into Balancing the term with the term , we obtain , then
Proceeding as in the previous case, we obtain
(3) The solutions of (4.1) are
4.1. Soliton Solutions
Some solitary wave solutions can be obtained, if the modulus approaches to 1 in (4.10) as follows:
4.2. Triangular Periodic Solutions
Some trigonometric function solutions can be obtained, if the modulus approaches to zero in (4.10) as follows:
If we take , in the two Sections 3 and 4, we obtain the solutions degenerated by the hyperbolic extended hyperbolic functions methods (tanh, coth, sinh, sech,, etc.) (see, for example ). Moreover, when , the solutions obtained by triangular and extended triangular functions methods (tan, sine, cosine, sec,, etc.) are found as disused in Sections 3.1, 3.2, 4.1, and 4.2.
By introducing appropriate transformations and using extended F-expansion method, we have been able to obtain, in a unified way with the aid of symbolic computation system-mathematica, a series of solutions including single and the combined Jacobi elliptic function. Also, extended F-expansion method showed that soliton solutions and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions. When , the Jacobi functions degenerate to the hyperbolic functions and give the solutions by the extended hyperbolic functions methods. When , the Jacobi functions degenerate to the triangular functions and give the solutions by extended triangular functions methods. In fact, the disadvantage of extended F-expansion method is the existence of complex solutions which are listed here just as solutions.
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.
- A. Biswas, E. Zerrad, J. Gwanmesia, and R. Khouri, “1-Soliton solution of the generalized Zakharov equation in plasmas by He's variational principle,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4462–4466, 2010.
- A. Biswas and M. S. Ismail, “1-Soliton solution of the coupled KdV equation and Gear-Grimshaw model,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3662–3670, 2010.
- P. Suarez and A. Biswas, “Exact 1-soliton solution of the Zakharov equation in plasmas with power law nonlinearity,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7372–7375, 2011.
- X. Zheng, Y. Chen, and H. Zhang, “Generalized extended tanh-function method and its application to (1+1)-dimensional dispersive long wave equation,” Physics Letters A, vol. 311, no. 2-3, pp. 145–157, 2003.
- R. A. Baldock, B. A. Robson, and R. F. Barrett, “Application of an inverse-scattering method to determinethe effective interaction between composite particles,” Nuclear Physics A, vol. 366, no. 2, pp. 270–280, 1981.
- S. Ghosh and S. Nandy, “Inverse scattering method and vector higher order non-linear Schrödinger equation,” Nuclear Physics. B, vol. 561, no. 3, pp. 451–466, 1999.
- A. H. Salas S and C. A. Gómez S, “Exact solutions for a third-order KdV equation with variable coefficients and forcing term,” Mathematical Problems in Engineering, vol. 2009, Article ID 737928, 13 pages, 2009.
- C. A. Gomez and A. H. Salas, “Exact solutions to KdV equation by using a new approach of the projective riccati equation method,” Mathematical Problems in Engineering, vol. 2010, Article ID 797084, 10 pages, 2010.
- A. H. Salas and C. A. Gomez, “Application of the ColeHopf transformation for finding exact solutions forseveral forms of the seventh-order KdV equation (KdV7),” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010.
- H. Naher, F. A. Abdullah, and M. Ali Akbar, “New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method,” Journal of Applied Mathematics, vol. 2012, Article ID 575387, 14 pages, 2012.
- R. Sakthivel and C. Chun, “New soliton solutions of Chaffee-Infante equations,” Z. Naturforsch, vol. 65, pp. 197–202, 2010.
- R. Sakthivel, C. Chun, and J. Lee, “New travelling wave solutions of Burgers equation with finitetransport memory,” Z.Naturforsch A., vol. 65, pp. 633–640, 2010.
- J. Lee and R. Sakthivel, “Exact travelling wave solutions of the Schamel—Korteweg—de Vries equation,” Reports on Mathematical Physics, vol. 68, no. 2, pp. 153–161, 2011.
- S. T. Mohyud-Din, M. A. Noor, and A. Waheed, “Exp-function method for generalized travelling solutionsof calogero-degasperis-fokas equation,” Z. Naturforsch A, vol. 65, pp. 78–85, 2010.
- A. H. Bhrawy, A. Biswas, M. Javidi, W. X. Ma, Z. Pinar, and A. Yildirim, “New Solutions for (1 + 1)-Dimensional and (2 + 1)-Dimensional Kaup-Kupershmidt Equations,” Results in Mathematics. In press.
- C. A. Gómez and A. H. Salas, “Exact solutions for the generalized BBM equation with variable coefficients,” Mathematical Problems in Engineering, vol. 2010, Article ID 498249, 10 pages, 2010.
- S. Tang, C. Li, and K. Zhang, “Bifurcations of travelling wave solutions in the -dimensional sine-cosine-Gordon equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3358–3366, 2010.
- G. Ebadi and A. Biswas, “The method and 1-soliton solution of the Davey-Stewartson equation,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 694–698, 2011.
- H. Kim and R. Sakthivel, “Travelling wave solutions for time-delayed nonlinear evolution equations,” Applied Mathematics Letters, vol. 23, no. 5, pp. 527–532, 2010.
- R. Sakthivel, C. Chun, and A.-R. Bae, “A general approach to hyperbolic partial differential equations by homotopy perturbation method,” International Journal of Computer Mathematics, vol. 87, no. 11, pp. 2601–2606, 2010.
- A. H. Khater, D. K. Callebaut, and M. A. Abdelkawy, “Two-dimensional force-free magnetic fields describedby some nonlinear equations,” Physics of Plasmas, vol. 17, no. 12, Article ID 122902, 10 pages, 2010.
- J. Lee and R. Sakthivel, “New exact travelling wave solutions of bidirectional wave equations,” Pramana Journal of Physics, vol. 76, pp. 819–829, 2011.
- E. G. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
- A.-M. Wazwaz, “The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 2, pp. 148–160, 2006.
- S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001.
- A. H. Salas, “Solving nonlinear partial differential equations by the sn-ns method,” Abstract and Applied Analysis, vol. 2012, Article ID 340824, 25 pages, 2012.
- A. V. Porubov, “Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer,” Physics Letters A, vol. 221, no. 6, pp. 391–394, 1996.
- A. V. Porubov and M. G. Velarde, “Exact periodic solutions of the complex Ginzburg-Landau equation,” Journal of Mathematical Physics, vol. 40, no. 2, pp. 884–896, 1999.
- A. V. Porubov and D. F. Parker, “Some general periodic solutions to coupled nonlinear Schrödinger equations,” Wave Motion, vol. 29, no. 2, pp. 97–109, 1999.
- E. Fan, “Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method,” Journal of Physics A, vol. 35, no. 32, pp. 6853–6872, 2002.
- S. Zhang and T. Xia, “An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 7, pp. 1294–1301, 2008.
- J. Zhang, C. Q. Dai, Q. Yang, and J. M. Zhu, “Variable-coefficient F-expansion method and its applicationto nonlinear Schrödinger equation,” Optics Communications, vol. 252, no. 4–6, pp. 408–421, 2005.
- Y. Shen and N. Cao, “The double F-expansion approach and novel nonlinear wave solutions of soliton equation in (2 + 1)-dimension,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 683–690, 2008.
- M. Wang and X. Li, “Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations,” Physics Letters A, vol. 343, no. 1–3, pp. 48–54, 2005.
- X. B. Hu and Y. Li, “Nonlinear superposition formulae of the Ito equation and a model equation for shallow water waves,” Journal of Physics A, vol. 24, no. 9, pp. 1979–1986, 1991.
- Y. Zhang and D. Y. Chen, “N-soliton-like solution of Ito equation,” Communications in Theoretical Physics, vol. 42, no. 5, pp. 641–644, 2004.
- C. A. Gomez S, “New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 241–250, 2010.
- F. Khani, “Analytic study on the higher order Ito equations: new solitary wave solutions using the Exp-function method,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 2128–2134, 2009.
- D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of stationary waves,” Philosophical Magazine, vol. 39, p. 422, 1895.
- N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the recurrence ofinitial states,” Physical Review Letters, vol. 15, no. 6, pp. 240–243, 1965.
- M. Ito, “An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders,” Journal of the Physical Society of Japan, vol. 49, no. 2, pp. 771–778, 1980.
- A.-M. Wazwaz, “Multiple-soliton solutions for the generalized (1 + 1)-dimensional and the generalized (2 + 1)-dimensional Itô equations,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 840–849, 2008.
- A.-M. Wazwaz, “The Hirota's direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 133–138, 2008.
- Y. Chen, S. Song, and H. Zhu, “Multi-symplectic methods for the Ito-type coupled KdV equation,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5552–5561, 2012.
- M. Ito, “Symmetries and conservation laws of a coupled nonlinear wave equation,” Physics Letters A, vol. 91, pp. 405–420, 1982.
- C. Guha-Roy, “Solutions of coupled KdV-type equations,” International Journal of Theoretical Physics, vol. 29, no. 8, pp. 863–866, 1990.