## Abstract and Applied Analysis

### Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation

#### Abstract

Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of $(2+1)$-dimensional nonlinear partial differential equation. The resulting theory is applied to the $(2+1)$-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 534017, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278619

Digital Object Identifier
doi:10.1155/2014/534017

Mathematical Reviews number (MathSciNet)
MR3176752

Zentralblatt MATH identifier
07022567

#### Citation

Zhao, Zhonglong; Zhang, Yufeng; Xia, Tiecheng. Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 534017, 6 pages. doi:10.1155/2014/534017. https://projecteuclid.org/euclid.aaa/1412278619

#### References

• E. Fan, “Extended tanh-function method and its applications tononlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000.
• M. L. Wang, “Exact solutions for a compound KdV-Burgers eq-uation,” Physics Letters A, vol. 213, no. 5-6, pp. 279–287, 1996.
• G. Bluman and S. Kumei, Symmetries and Differential Equations, vol. 81 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1989.
• W.-X. Ma, C.-X. Li, and J. He, “A second Wronskian formulation of the Boussinesq equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4245–4258, 2009.
• Y. Zhang, T.-F. Cheng, D.-J. Ding, and X.-L. Dang, “Wronskian and Grammian solutions for $(2+1)$-dimensional soliton equation,” Communications in Theoretical Physics, vol. 55, no. 1, pp. 20–24, 2011.
• V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991.
• R. Hirota, “Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.
• R. Hirota and J. Satsuma, “$N$-soliton solutions of model equations for shallow water waves,” Journal of the Physical Society of Japan, vol. 40, no. 2, pp. 611–612, 1976.
• X.-B. Hu and W.-X. Ma, “Application of Hirota's bilinear for-malism to the Toeplitz lattice–-some special soliton-like solutions,” Physics Letters A, vol. 293, no. 3-4, pp. 161–165, 2002.
• X.-B. Hu and W.-M. Xue, “A bilinear Bäcklund transformation and nonlinear superposition formula for the negative Volterra hierarchy,” Journal of the Physical Society of Japan, vol. 72, no. 12, pp. 3075–3078, 2003.
• S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation,” Functional Analysis and Its Applications, vol. 8, no. 3, pp. 236–246, 1974.
• A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution,” Journal of the Physical Society of Japan, vol. 47, no. 5, pp. 1701–1705, 1979.
• A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. II. Exact one-periodic and two-periodic wave solution of the coupled bilinear equations,” Journal of the Physical Society of Japan, vol. 48, no. 4, pp. 1365–1370, 1980.
• Y. C. Hon, E. Fan, and Z. Qin, “A kind of explicit quasi-periodic solution and its limit for the Toda lattice equation,” Modern Physics Letters B, vol. 22, no. 8, pp. 547–553, 2008.
• E. Fan and K. W. Chow, “On the periodic solutions for bothnonlinear differential and difference equations: a unified appr-oach,” Physics Letters A, vol. 374, no. 35, pp. 3629–3634, 2010.
• E. Fan, “Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions,” Physics Letters A, vol. 374, no. 5, pp. 744–749, 2010.
• W.-X. Ma, R. Zhou, and L. Gao, “Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in $(2+1)$ dimensions,” Modern Physics Letters A, vol. 24, no. 21, pp. 1677–1688, 2009.
• S.-F. Tian and H.-Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations,” Journal of Mathematical Analysis and Applications, vol. 371, no. 2, pp. 585–608, 2010.
• S.-F. Tian and H.-Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the $(1+1)$-dim-ensional and $(2+1)$-dimensional Ito equation,” Chaos, Solitons & Fractals, vol. 47, pp. 27–41, 2013.
• C.-W. Cao and X. Yang, “Algebraic-geometric solution to $(2+1)$-dimensional Sawada-Kotera equation,” Communications in Theoretical Physics, vol. 49, no. 1, pp. 31–36, 2008.
• A.-M. Wazwaz, “Multiple soliton solutions for $(2+1)$-dime-nsional Sawada-Kotera and Caudrey-Dodd-Gibbon equations,” Mathematical Methods in the Applied Sciences, vol. 34, no. 13, pp. 1580–1586, 2011.
• V. G. Dubrovsky and Y. V. Lisitsyn, “The construction of exactsolutions of two-dimensional integrable generalizations of Ka-up-Kuperschmidt and Sawada-Kotera equations via $\overline{\partial }$-dressing method,” Physics Letters A, vol. 295, no. 4, pp. 198–207, 2002.
• R. A. Zait, “Bäcklund transformations, cnoidal wave and travelling wave solutions of the SK and KK equations,” Chaos, Solitons & Fractals, vol. 15, no. 4, pp. 673–678, 2003.
• R. Hirota, The Direct Method in Soliton Theory, Cambridge Uni-versity Press, Cambridge, UK, 2004.
• X. Lü, T. Geng, C. Zhang, H.-W. Zhu, X.-H. Meng, and B. Tian,“Multi-soliton solutions and their interactions for the $(2+1)$-dimensional Sawada-Kotera model with truncated Painlevé expansion, Hirota bilinear method and symbolic computation,” International Journal of Modern Physics B, vol. 23, no. 25, pp. 5003–5015, 2009.
• X. Lü, B. Tian, K. Sun, and P. Wang, “Bell-polynomial manipula-tions on the Bäcklund transformations and Lax pairs for somesoliton equations with one Tau-function,” Journal of Mathematical Physics, vol. 51, no. 11, Article ID 113506, 2010. \endinput