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Journal of Applied Mathematics
Volume 2012, Article ID 265348, 15 pages
http://dx.doi.org/10.1155/2012/265348
Research Article

Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

1Mathematics Department, Faculty of Science, Qassim University, Buraydah 51452, Saudi Arabia
2Department of Science and Mathematics, Faculty of Education, Assiut University, New Valley Branch, El-Kharja 72511, Egypt

Received 10 October 2011; Revised 4 January 2012; Accepted 17 January 2012

Academic Editor: Shan Zhao

Copyright © 2012 Zeid I. A. Al-Muhiameed and Emad A.-B. Abdel-Salam. 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.

Abstract

With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameters p and q in the obtained solutions by using the computer simulation.

1. Introduction

The nonlinear Schrödinger (NLS) equation is a ubiquitous and significant model that naturally arises in many fields of physics, such as nonlinear optical systems, plasmas, fluid dynamics, and Bose-Einstein condensation. In the last three decades, great progress has been made on the construction of exact solutions of NLS equation. Many significant methods have been established by mathematicians and physicists to obtain special solutions of nonlinear partial differential equations (NLPDEs), including the inverse scattering method, Darboux transformation, Hirota’s bilinear method, homogeneous balance method, Jacobi elliptic function method, variational iteration method, the sine-cosine method, the expansion method, tanh-function method, F-expansion method, Lucas Riccati method, auxiliary equation method, algebraic method, and others [136]. The last five methods mentioned above belong to a class of method called subsidiary ordinary differential equation (sub-ODE) method. The key ideas of the sub-ODE method are that the travelling wave solutions of the complicated NLPDE can be expressed as a polynomial, the variable of which is one of the solutions of simple and solvable ODE that called the sub-ODE. The sub-ODEs which were often used are the Riccati equation, Jacobi elliptic equation, projective Riccati equations, and so forth. With the development of computer science, recently, the sub-ODEs with nonlinear terms of high order have attracted much attention [34, 35]. This is due to the availability of symbolic computation systems like Mathematica or Maple which enable us to perform the complex and tedious computation on computers. Recently, Sirendaoreji [2123] introduced the auxiliary equation method by using the six degree first-order nonlinear differential equation, Wang et al. [24] proposed new sub-ODE equation involving an arbitrary positive power to construct exact travelling wave solutions of NLPDEs in a unified way. In this study, we improve the method presented by Wang et al. [24] and introduce some solutions for the sub-ODE equation in terms of the generalized hyperbolic functions (GHFs) [36].

The rest of this paper is organized as follows: in the following section, we introduce the improved sub-ODE method to construct exact solutions of some NLPDEs in terms of the GHFs. In Section 3, we apply this method to the generalized Zakharov system, the Rangwala-Rao equation and the Chen-Lee-Liu equation. Finally, we conclude the paper and give some futures and comments.

2. The Improved Subsidiary Ordinary Differential Equation Method

The main idea of this method is to express the solutions of NLPDEs as polynomials in the solution of sub-ODE involving an arbitrary positive power of dependent variable that the GHFs satisfy (see the appendix). Consider a given NLPDE The improved sub-ODE method for solving NLPDE (2.1) proceeds in the following four steps [36].

Step 1. We seek its traveling wave solution of (2.1) in the form where and are constants to be determined later. Substituting (2.2) into (2.1) yields an ordinary differential equation where is a polynomial of and its various derivatives. If is not a polynomial of and its various derivatives, then we may use new variables which makes become polynomial of and its various derivatives.

Step 2. Suppose that can be expressed by a finite power series of where is a positive integer which can be determined by balancing the highest derivative term with the highest nonlinear term in (2.3) and are some parameters to be determined. The function satisfies the sub-ODE with an arbitrary positive power where , and are parameters to be determined.

Step 3. Substituting (2.4) and (2.5) into the ODE (2.3), then the left-hand side of (2.3) can be converted into a polynomial in . Setting all coefficients of the polynomial to zero yields system of algebraic equations for , and .

Step 4. Solving this system obtained in Step 3, then , , and can be expressed by , and . Substituting these results into (2.4), then the general formulae of the travelling wave solution of (2.1) can be obtained. Selecting the values of , and the corresponding GHF solution of (2.5) given bellow to obtain the exact solutions of (2.1). The definition and proprieties of the GHFs are given in the appendix.
We list various types of exact solutions of (2.5) as follows.

Case 1. If and , then (2.5) admits the following positive solution:

Case 2. If and , then (2.5) admits the following positive solution:

Case 3. If and , then (2.5) admits the following positive solution:

Case 4. If and , then (2.5) admits the following positive solution:

3. Applications

In the following we use the improved sub-ODE method to seek exact traveling wave solutions of the class of nonlinear Schrödinger-type equations which are of interest in plasma physics, wave propagation in nonlinear optical fibers, Ginzburg-Landau theory of superconductivity, and so forth.

3.1. Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role. This system has wide interest and attention for many scientists. Let us consider the generalized Zakharov system [18]: When , the generalized Zakharov system reduce to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function is the fluctuation in the ion density about its equilibrium value, and the complex unknown function is the slowly varying envelope of highly oscillatory electron field. The parameters andare real numbers, whereis proportional to the ion acoustic speed (or electron sound speed). Here we seek its traveling wave solution in the form where , and are constants and is real function. Therefore, system (3.1) is reduced to Integrating the first equation of (3.3) with respect to and taking the integration constants to zero yields Substituting (3.4) into (3.3) results in Now, by balancing the higher order derivative term to the higher power nonlinear term in (3.5), we have which gives. Since must be a positive integer, thus this gives . For simplicity, we take and . We suppose that Substituting (3.6) with (2.5) into (3.5) and equating each of the coefficients of to zero, we obtain system of algebraic equations. Solving this system by using Maple, yields Now based on the solutions of (2.5), one can obtain new types of solitary wave solution of the generalized Zakharov system. The general formulae take the form By selecting the special values of the and the corresponding function, we have the following solutions of the generalized Zakharov system (3.1): In order to understand the significance of these solutions expressed by (3.9)–(3.13), the main features of them are investigated by using direct computer simulations with the accuracy as high as . We study the effect of the parameters and of the GHF in the solution given by the first equation of (3.9) by choosing different values of the parameters and , for , , and , by means of Figures 1, 2, and 3 over same region . From Figures 1, 2 and 3, we see that the solution describes dark soliton solution (increase, decrease, and have singularity with the change of the two parameters and ). Also, we discuss given by the first equation of (3.10), by taking the same value of the parameters by means of Figures 4, 5, and 6 over the region . From Figures 4, 5, and 6, we see that the solution describes singular wave solution changed with different value of the two parameters and . Of course, we can plot the other figures of the exact solutions of (3.1), we omit them here for convenience. As a result, we find that the parameters and affect the solution structure.

fig1
Figure 1: Evolutional behaviour of describes dark soliton: (a) (b) (singular wave solution); (c) (d) (e) .
fig2
Figure 2: Evolutional behaviour of describes dark soliton: (a) (singular wave solution); (b) ; (c) .
fig3
Figure 3: Evolutional behaviour of describes dark soliton: (a) (singular wave solution); (b) ; (c) .
fig4
Figure 4: Evolutional behaviour of describes singular wave solution: (a) (b) (c) (d) (e) .
fig5
Figure 5: Evolutional behaviour of describes singular wave solution: (a) (b) (c) .
fig6
Figure 6: Evolutional behaviour of describes singular wave solution: (a) (b) ; (c) .
3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [19] where and are real constants. Rangwala and Rao introduced (3.14) as the integrability condition when they studied the mixed derivative nonlinear Schrödinger equations and looked for the Bäckland transformation and solitary wave solutions.

Suppose the exact solutions of (3.14) is of the form where , are constants determined later and are undetermined real functions with one variable only. Set the relation of as substituting (3.15) with (3.16) into (3.14) simultaneously yields Applying the homogeneous balance principle, to (3.17) we have this gives . For simplicity, we suppose Substituting (3.18) with (2.5) into (3.17) and equating each of the coefficients of to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution: The general formulae of the solution of Rangwala-Rao equation with . By selecting the special values of the and the corresponding functionyields We omitted the reminder solutions for simplicity. Now, we study the affect of the parameters and of the GHF in the intensity given by the (3.21) by choosing different values of the parameters and , for , and , by means of Figure 7 over the region . We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters and . This means that the parameters and affect in the solution structure.

fig7
Figure 7: Evolutional behaviour of the intensity describes the bright soliton: (a) (b) (singular wave solution); (c) .
3.3. Chen-Lee-Liu Equation

The Chen-Lee-Liu equation [19] were is a real constant. Similar as before, we suppose that the exact solution of (3.23)is of the form Set the relation of as substituting (3.25) with (3.24) into (3.23) simultaneously yields According the homogeneous balance principle, we have and , we let Substituting (3.27) with (2.5) into (3.26) and equating each of the coefficients of to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution: The general formulae of the solution of Chen-Lee-Liu equation with . By selecting the special values of the and the corresponding function yields We omitted the reminder solutions for simplicity. Now, we show the effect of the parameters and of the GHF in the intensity given by the (3.30) by choosing different values of the parameters and , for , and , by means of Figure 8 over the region . We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters and . This means that the parameters and affect in the solution structure.

fig8
Figure 8: Evolutional behaviour of the intensity describes the bright soliton: (a) ; (b) (singular wave solution); (c) .

Besides the solutions obtained above, the ODE equation (2.5), albeit with different parameters, has been studied in the different context [2024]. It has been shown that this equation possesses abundant solutions, including Weierstrass function solutions, kink solutions and periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.

Notice that the GHFs are generalization of the hyperbolic functions as stated in the appendix. Also, the two parameters and describe the degree of the wave energy localization in the obtained solutions. Referring to [18, 19], it is easy to see that the obtained results in this paper are new and the improved method is valuable.

4. Summary and Discussion

In this paper, a set of sub-ODEs are introduced by using GHFs. Also, we have obtained many families of exact traveling wave solutions of a class of nonlinear Schrödinger equations including the generalized Zakharov system, the Rangwala-Rao equation, and the chen-Lee-Liu equation. We study and analyze the properties of these solutions by taking different parameter values of the generalized hyperbolic functions and . As a result, we find that these parameter values affect the solution structure. These solutions include the GHF solution, hyperbolic function solution, deformed hyperbolic function solution, and others. The obtained solutions which depend on and may be of important significance for the explanation of some practical physical problems. We believe that one can apply this method to many other nonlinear differential equations in mathematical physics.

Appendix

The generalized hyperbolic sine, cosine, and tangent functions are where is an independent variable, and are arbitrary constants greater than zero and called deformation parameters. The generalized hyperbolic cotangent function is , the generalized hyperbolic secant function is , the generalized hyperbolic cosecant function is , the above six kinds of functions are said GHFs. These functions satisfy the following relations [9, 12]:

Note that if then is not odd and is not even:

Acknowledgment

The authors would like to express their sincere thanks to the referees for their valuable suggestions. This work has been supported by the Deanship of Scientific Research at Qassim University.

References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,” Zeitschrift fur Naturforschung A, vol. 57, no. 8, pp. 692–700, 2002. View at Google Scholar · View at Scopus
  3. A. H. Bokhari, G. Mohammad, M. T. Mustafa, and F. D. Zaman, “Adomian decomposition method for a nonlinear heat equation with temperature dependent thermal properties,” Mathematical Problems in Engineering, vol. 2009, Article ID 926086, 12 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A.-M. Wazwaz, “The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1196–1210, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. F. El-Sabbagh and A. T. Ali, “New exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 151–162, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. S. Liu, Z. Fu, S. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. Dai and J. Zhang, “Jacobian elliptic function method for nonlinear differential-difference equations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 1042–1047, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. M. F. El-Sabbagh, M. M. Hassan, and E. A.-B. Abdel-Salam, “Quasiperiodic waves and their interactions in the 2+1dimensional modified dispersive water-wave system,” Physica Scripta, vol. 80, no. 1, article 15006, 2009. View at Publisher · View at Google Scholar
  10. E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. I. A. Hassanien, R. A. Zait, and E. A.-B. Abdel-Salam, “Multicnoidal and multitravelling wave solutions for some nonlinear equations of mathematical physics,” Physica Scripta, vol. 67, no. 6, pp. 457–463, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Ren and H. Zhang, “New generalized hyperbolic functions and auto-Bäcklund transformation to find new exact solutions of the -dimensional NNV equation,” Physics Letters A, vol. 357, no. 6, pp. 438–448, 2006. View at Publisher · View at Google Scholar
  13. E. A. B. Abdel-Salam, “Periodic structures based on the symmetrical lucas function of the (2+1)-dimensional dispersive long-wave system,” Zeitschrift fur Naturforschung A, vol. 63, no. 10-11, pp. 671–678, 2008. View at Google Scholar · View at Scopus
  14. E. A.-B. Abdel-Salam, “Quasi-periodic structures based on symmetrical Lucas function of (2+1)-dimensional modified dispersive water-wave system,” Communications in Theoretical Physics, vol. 52, no. 6, pp. 1004–1012, 2009. View at Publisher · View at Google Scholar
  15. E. A. B. Abdel-Salam and D. Kaya, “Application of new triangular functions to nonlinear partial differential equations,” Zeitschrift fur Naturforschung A, vol. 64, no. 1-2, pp. 1–7, 2009. View at Google Scholar · View at Scopus
  16. E. A. B. Abdel-Salam, “Quasi-periodic, periodic waves, and soliton solutions for the combined KdV-mKdV equation,” Zeitschrift fur Naturforschung A, vol. 64, no. 9-10, pp. 639–645, 2009. View at Google Scholar · View at Scopus
  17. E. A.-B. Abdel-Salam and Z. I. A. Al-Muhiameed, “Generalized Jacobi elliptic function method and non-travelling wave solutions,” Nonlinear Science Letters A, vol. 1, pp. 363–372, 2010. View at Google Scholar
  18. A. Borhanifar, M. M. Kabir, and L. M. Vahdat, “New periodic and soliton wave solutions for the generalized Zakharov system and (2+1)-dimensional Nizhnik-Novikov-Veselov system,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1646–1654, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. J. L. Zhang and M. L. Wang, “Exact solutions to a class of nonlinear Schrödinger-type equations,” Pramana, vol. 67, no. 6, pp. 1011–1022, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. H. W. Schürmann, “Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation,” Physical Review E, vol. 54, no. 4, pp. 4312–4320, 1996. View at Publisher · View at Google Scholar · View at Scopus
  21. Sirendaoreji and S. Jiong, “A direct method for solving sine-Gordon type equations,” Physics Letters A, vol. 298, no. 2-3, pp. 133–139, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,” Physics Letters A, vol. 309, no. 5-6, pp. 387–396, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,” Physics Letters A, vol. 356, no. 2, pp. 124–130, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96–101, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. Z. I. A. Al-Muhiameed and E. A.-B. Abdel-Salam, “Generalized Jacobi elliptic function solution to a class of nonlinear Schrödinger-type equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 575679, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. E. A.-B. Abdel-Salam and Z. I. A. Al-Muhiameed, “Exotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system,” Turkish Journal of Physics, vol. 35, pp. 1–16, 2011. View at Google Scholar
  27. J. L. Zhang, M. L. Wang, and X. Z. Li, “The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation,” Physics Letters A, vol. 357, no. 3, pp. 188–195, 2006. View at Publisher · View at Google Scholar · View at Scopus
  28. Q. Wang and M. Tang, “New exact solutions for two nonlinear equations,” Physics Letters A, vol. 372, no. 17, pp. 2995–3000, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96–101, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. E. J. Parkes, “Observations on the basic (G/G) -expansion method for finding solutions to nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1759–1763, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. W.-X. Ma and M. Chen, “Direct search for exact solutions to the nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2835–2842, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. M. Y. Moghaddam, A. Asgari, and H. Yazdani, “Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by extended tanh-coth, sine-cosine and exp-function methods,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 422–435, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  33. R. Yang, R. Hao, L. Li, Z. Li, and G. Zhou, “Dark soliton solution for higher-order nonlinear Schrödinger equation with variable coefficients,” Optics Communications, vol. 242, no. 1-3, pp. 285–293, 2004. View at Publisher · View at Google Scholar · View at Scopus
  34. X. Li and M. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms,” Physics Letters A, vol. 361, no. 1-2, pp. 115–118, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. S. Zhang, W. Wang, and J.-L. Tong, “The improved sub-ODE method for a generalized KdV-mKdV equation with nonlinear terms of any order,” Physics Letters A, vol. 372, no. 21, pp. 3808–3813, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. M. M. Hassan and E. A.-B. Abdel-Salam, “New exact solutions of a class of higher-order nonlinear Schrödinger equations,” Journal of the Egyptian Mathematical Society, vol. 18, pp. 315–329, 2010. View at Google Scholar