## Abstract

In this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find *n*-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions.

## 1 Introduction

Nonlinear phenomena appear in a wide variety of scientific applications such as plasma physics, solid-state physics, fluid dynamics, and so on. Partial differential equations (PDEs) have been the focus of many studies due to their frequent appearance in various applications in many fields, such as physics, biology, engineering, signal processing, control theory, and so on. Recently, a large amount of literature has been provided to construct the solutions of the PDEs. Several powerful methods have been proposed to obtain approximate and exact solutions of these equations, such as the inverse scattering transform [1], the Bäcklund transformation method [2], the Hirota bilinear method [3], the Adomian decomposition method [4, 5], the variational iteration method [6–8], the homotopy analysis method [9–12], the homotopy perturbation method [13–15], the Lagrange characteristic method [16], the fractional sub-equation method [17], the (*G*′/*G*)-expansion method [18, 19], the transformed rational function method [20], the multiple exp-function method [21, 22], the generalised Riccati equation method [23], the Frobenius decomposition technique [24], the local fractional differential equations method [25, 26], the local fractional variation iteration method [27], the multiple (*G*′/*G*)-expansion method [28], the cantor-type cylindrical coordinate method [29], the Riccati equation method combined with the (*G*′/*G*)-expansion method [30], the fractional complex transform method [31], the modified simple equation method [32–35], the first integral method [36–38], the linear superposition principle [39], and so on.

The objective of this article is to apply two interesting methods, namely, the multiple exp-function method and the linear superposition principle to construct the exact solutions for the following (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation [40–42]:

where *u*=*u*(*x*, *z*, *t*). The CBS equation was first constructed by Bogoyavlenskii and Schiff in different ways [41, 42]. Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same equation by reducing the self-dual Yang–Mills equation. Equation (1) has been discussed in [43] by using the Hirota bilinear method.

## 2 Description of the Multiple Exp-Function Method

Consider the following nonlinear PDE:

We describe the main steps [21, 22] of the multiple exp-function method to solve (2) as follows:

**Step 1:** We introduce *n*-wave variables *η*_{i}=*η*_{i} (*x*, *t*), (*i*=1, 2, …, *n*), such that

where *k*_{i} are the angular wave number and *λ*_{i} are the wave frequencies, while *c*_{i} are any constants, positive or negative.

**Step 2:** Assume that (2) has the rational solution

where *p* and *q* are two polynomials of degree “*n*” and of the new variables *η*_{i}=*η*_{i} (*x*, *t*), (*i*=1, 2, …, *n*).

**Step 3:** Substituting (4) along with (3) into (2) yields a rational equation that can be written in the following form:

where *R*(*η*_{1}, *η*_{2}, …, *η*_{n}) ≠ 0.

**Step 4:** Setting the numerator of (5) to be zero yields a system of algebraic equations, which can be solved using the Maple to determine the two polynomials *p* and *q* and the exponents *ξ*_{i}, (*i*=1, 2, …, *n*). Consequently, we can get the exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions of (2).

## 3 One-Wave, Two-Wave, and Three-Wave Solutions to the Nonlinear CBS Equation

Let us apply the multiple exp-function method to the (2+1)-dimensional CBS equation (1). We analyze three cases of the two polynomials *p* and *q* for (1) to construct their multiple-wave solutions.

### 3.1 Case 1: One-Wave Solutions

With reference to [21, 22], let us try a pair of two polynomials *p*(*η*_{1}) and *q*(*η*_{1}) of degree one, such that

where *ξ*_{1}=*k*_{1}*x* + *m*_{1}*z* − *λ*_{1}*t*, while *a*_{0}, *a*_{1}, *b*_{0}, *b*_{1}, *c*_{1}, *k*_{1}, *m*_{1}, and *λ*_{1} are constants to be determined later. By the multiple exp-function method, we assume that (1) has the rational solution

Substituting (6) into (1), we have the polynomial equation

where

On solving the above algebraic equations using the Maple, we get the following result:

Now, the corresponding one-wave solution of (1) can be written in the new form

where *c*_{1}, *a*_{1}, *m*_{1}, *b*_{0}, *b*_{1}, and *k*_{1} are arbitrary constants. On comparing our new result (9) with the well-known solution (25) obtained in [43] using the Hirota method, we deduce that they are equivalent in the special case *b*_{0}=*b*_{1}=*c*_{1}=1, *a*_{1}=2*k*_{1}.

### 3.2 Case 2: Two-Wave Solutions

With reference to [21, 22], let us try a particular pair of two polynomials *p*(*η*_{1}, *η*_{2}) and *q*(*η*_{1}, *η*_{2}) of degree two, such that

where *c*_{i}, *k*_{i}, *m*_{i}, *λ*_{i} (*i*= 1, 2), and *a*_{12} are constants to be determined. By the multiple exp-function method, we assume that (1) has the rational solution

Substituting (10) into (1), we have the polynomial equation

where the coefficients *c*_{i} (*i*=1, 2, …, 12) of the polynomial equation (11) have been determined in terms of *η*_{1}, *η*_{2}, which are omitted here for simplicity. On setting these coefficients to be zero, we have a system of algebraic equations, which can be solved by the Maple, to get the following result:

Now, the corresponding two-wave solution of (1) has the new form

where *a*_{12}=(*k*_{1} − *k*_{2})^{2}/(*k*_{1} + *k*_{2})^{2}, *i*=1, 2).

One specific solution of two-wave solution is plotted in Figure 1.

### 3.3 Case 3: Three-Wave Solutions

With reference to [21, 22], let us try a particular pair of two polynomials *p*(*η*_{1}, *η*_{2}, *η*_{3}) and *q*(*η*_{1}, *η*_{2}, *η*_{3}) of degree three, such that

where *i*=1, 2, 3), while *c*_{i}, *k*_{i}, *m*_{i}, *λ*_{i} and *a*_{12}, *a*_{13}, *a*_{23} are constants to be determined. By the multiple exp-function method, we assume that (1) has the rational solution

Substituting (14) into (1), we have the equation

where the coefficients *c*_{i} (*i*=1 − 57) of the polynomial equation (15) have been determined in terms of *η*_{1}, *η*_{2}, *η*_{3}, which are omitted here for simplicity. On setting these coefficients to zero, we get a system of algebraic equations, which can be solved using the Maple, to get the following result:

Now, the corresponding three-wave solution of (1) can be written in the new form

where

and *a*_{ij}=(*k*_{i} − *k*_{j})^{2}/(*k*_{i} + *k*_{j})^{2}, 1 ≤ *i* ≤ *j* ≤ 3.

The specific solution of three-wave solution is plotted in Figure 2.

## 4 The Construction of the *n*-Wave Solution of (1)

In this section, we apply the linear superposition principle [39] to find the *n*-wave solution of (1).

Through the dependent variable transformation *u*(*x*, *z*, *t*)=2 [ln (1 + *f*(*x*, *z*, *t*)]_{x}, (1) can be written in the Hirota bilinear form

which is equivalent to

Let us introduce the *n*-wave variables

and *n*-exponential wave functions

where *k*_{i}, *m*_{i}, and *λ*_{i} (*i* =1, 2, …, *n*) are constants to be determined.

Then, the *n*-wave solution condition (2.8) of [39] becomes

By inspection, a solution to this equation is given by

where *k* is an arbitrary constant.

Therefore, by the linear superposition principle in theorem 2.1 of [39], the nonlinear CBS equation (1) has the following *n*-wave solution:

where *ε*_{i} are arbitrary constants.

## 5 Conclusions

The multiple exp-function method and the linear superposition principle are applied successfully for solving the (2+1)-dimensional CBS equation (1). The first method gives one-wave, two-wave, and three-wave solutions including one-soliton-, two-soliton-, and three-soliton-type solutions of (1). It is our guess that higher-wave solutions to the (2+1)-dimensional CBS equation (1) could be presented in a parallel manner. But the required computation is pretty complicated, even in the case of four-wave solutions. The second method gives a specific subclass of *n*-soliton solutions formed by linear combinations of exponential traveling waves. Finally to our knowledge, the obtained solutions (9), (13), and (17) of (1) using the multiple exp-function method and the obtained solution (24) of (1) using the linear superposition principle are all new and not published elsewhere.

## Acknowledgments

The authors wish to thank the referees for their comments on this article.

## References

[1] M. J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York 1991.10.1017/CBO9780511623998Search in Google Scholar

[2] M. R. Miurs, Bäcklund Transformation, Springer, Berlin 1978.Search in Google Scholar

[3] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge 2004.10.1017/CBO9780511543043Search in Google Scholar

[4] A. M. A. El-sayed, S. Z. Rida, and A. A. M. Arafa, Commu. Theor. Phys. **52**, 992 (2009).Search in Google Scholar

[5] M. Safari, D. D. Ganji, and M. Moslemi, Comput. Math. Appl. **58**, 2091 (2009).Search in Google Scholar

[6] M. Inc, J. Math. Anal. Appl. **345**, 476 (2008).Search in Google Scholar

[7] G. C. Wu and E. W. M. Lee, Phys. Lett. A **374**, 2506 (2010).10.1016/j.physleta.2010.04.034Search in Google Scholar

[8] F. Fouladi, E. Hosseinzad, and A. Barari, Heat Transfer Res. **41**, 155 (2010).Search in Google Scholar

[9] L. N. Song and H. Q. Zhang, Chaos Soliton. Fract. **40**, 1616 (2009).Search in Google Scholar

[10] S. Abbasbandy and A. Shirzadi, Numer. Algorithms **54**, 521 (2010).10.1007/s11075-009-9351-7Search in Google Scholar

[11] H. Bararnla, G. Domariy, and M. Gorji, Numer. Meth. Part. D. E. **26**, 1 (2010).Search in Google Scholar

[12] M. M. Rashidi, G. Domairry, A. Doosthosseini, and S. Dinarvand, Int. J. Math. Anal. **12**, 581 (2008).Search in Google Scholar

[13] Z. Ganji, D. Ganji, A.D. Ganji, and M. Rostamain, Numer. Meth. Part. D. E. **26**, 117 (2010).Search in Google Scholar

[14] K. A. Gepreel, Appl. Math. Lett. **24**, 1428 (2011).Search in Google Scholar

[15] P. K. Gupta and M. Singh, Comput. Math. Appl. **61**, 250 (2011).Search in Google Scholar

[16] G. Jumarie, Appl. Math. Lett. **19**, 873 (2006).Search in Google Scholar

[17] S. Zhang and H. Q. Zhang, Phys. Lett. A **375**, 1069 (2011).10.1016/j.physleta.2011.01.029Search in Google Scholar

[18] B. Zheng, Commu. Theor. Phys. **58**, 623 (2012).Search in Google Scholar

[19] B. Zheng, Adv. Differ. Eqs. **199**, 1 (2013).Search in Google Scholar

[20] W. X. Ma and J. H. Lee, Chaos Soliton. Fract. **42**, 1356 (2009).Search in Google Scholar

[21] W. X. Ma, T. Huang, and Y. Zhang, Phys. Script. **82**, 065003 (2010).Search in Google Scholar

[22] W. X. Ma and Z. Zhu, Appl. Math. Comput. **218**, 11871 (2012).Search in Google Scholar

[23] W. X. Ma and B. Fuchssteliner, Int. J. Nonlinear Mech. **31**, 329 (1966).Search in Google Scholar

[24] W. X. Ma, H. Y. Wu, and J. S. He, Phys. Lett. A **364**, 29 (2007).10.1016/j.physleta.2006.11.048Search in Google Scholar

[25] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong Kong 2011.Search in Google Scholar

[26] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York 2012.Search in Google Scholar

[27] X. J. Yang and D. Baleanu, Therm. Sci. **17**, 2 (2012).Search in Google Scholar

[28] J. Chen and B. Li, Pramana J. Phys. **78**, 3 (2012).10.1007/s12043-011-0237-6Search in Google Scholar

[29] X. J. Yang, H. M. Srivastava, J. H. He, and D. Baleanu, Phys. Lett. A **377**, 1996 (2013).10.1016/j.physleta.2013.04.012Search in Google Scholar

[30] E. M. E. Zayed and Y. A. Amer. Sci. Res. Ess. **10**, 86 (2015).Search in Google Scholar

[31] W. H. Su, X. J. Yang, H. Jafari, and D. Baleanu, Adv. Diff. Eqs. **1**, 97 (2013).Search in Google Scholar

[32] A. J. M. Jawad, M. D. Petkovic, and A, Biswas, Appl. Math. Comput. **217**, 869 (2010).Search in Google Scholar

[33] E. M. E. Zayed, Appl. Math. Comput. **218**, 3962 (2011).Search in Google Scholar

[34] E. M. E. Zayed and H. Ibrahim, Chin. Phys. Lett. **29**, 060201 (2012).Search in Google Scholar

[35] E. M. E. Zayed and A. H. Arnous, Appl. Appl. Math. **8**, 553 (2013).Search in Google Scholar

[36] Z. S. Feng, J. Phys. A **35**, 343 (2002).10.1088/0305-4470/35/2/312Search in Google Scholar

[37] G. M. Moatimid, R. M. El-Shiekh, and A.-G. A. A. H. Al-Nowehy, Nonlinear Sci. Lett. A **4**, 1 (2013).Search in Google Scholar

[38] R. M. El-Shiekh and A.-G. Al-Nowehy, Z. Naturforsch. **68a**, 255 (2013).10.5560/ZNA.2012-0108Search in Google Scholar

[39] W. X. Ma and E. Fan, Comput. Math. Appl. **61**, 950 (2011).Search in Google Scholar

[40] G. M. Moatimid, R. M. El-Shiekh, and A.-G. A. A.H. Al-Nowehy, Appl. Math. Comput. **220**, 455 (2013).Search in Google Scholar

[41] M. S. Bruzon, M. L. Gandarias, C. Muriel, J. Ramierez, S. Saez, et al., Theor. Math. Phys. **137**, 1367 (2003).Search in Google Scholar

[42] Y. Peng, Int. J. Theor. Phys. **45**, 1779 (2006).Search in Google Scholar

[43] A. M. Wazwaz, Appl. Math. Comput. **203**, 592 (2008).Search in Google Scholar

**Received:**2015-3-31

**Accepted:**2015-7-7

**Published Online:**2015-8-1

**Published in Print:**2015-9-1

©2015 by De Gruyter