Doubly and Triply Periodic Waves Solutions for the KdV Equation

ABSTRACT

Based on the arbitrary constant solution, a series of explicit doubly periodic solutions and triply periodic solutions for the Korteweg-de Vries (KdV) equation are first constructed with the aid of the Darboux transformation method.

Cite this paper

Huang, Y. and Xu, D. (2013) Doubly and Triply Periodic Waves Solutions for the KdV Equation.*Applied Mathematics*, **4**, 1599-1062. doi: 10.4236/am.2013.412216.

Huang, Y. and Xu, D. (2013) Doubly and Triply Periodic Waves Solutions for the KdV Equation.

References

[1] M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolutions and Inverse Scattering,” Cambridge University Press, Cambridge, 1991, pp. 23-99.

http://dx.doi.org/10.1017/CBO9780511623998

[2] J. álvarez and A. Durán, “Error Propagation When Approximating Multi-Solitons: The KdV Equation with as a Case Study,” Applied Mathematics and Computation, Vol. 217, No. 4, 2010, pp. 1522-1539.

http://dx.doi.org/10.1016/j.amc.2009.06.033

[3] J. L. Yin and L. X. Tian, “Classification of the Traveling Waves in the Nonlinear Dispersive KdV Equation,” Nonlinear Analysis, Vol. 73, No. 2, 2010, pp. 465-470.

http://dx.doi.org/10.1016/j.na.2010.03.039

[4] A. Biswas, M. D. Petkovic and D. Milovic, “Topological and Non-Topological Exact Soliton Solution of the KdV Equation,” Nonlinear Science and Numerical Simulation, Vol. 15, No. 11, 2010, pp. 3263-3269.

http://dx.doi.org/10.1016/j.na.2010.03.039

[5] M. Nivala and B. Deconinck, “Periodic Finite-Genus Solutions of the KdV Equation Are Orbitally Stable,” Physica D: Nonlinear Phenomena, Vol. 239, No. 13, 2011, pp. 1147-1158.

http://dx.doi.org/10.1016/j.physd.2010.03.005

[6] Y. Yamamoto, T. Nagase and M. Ohmiya, “Appell’s Lemma and Conservation Laws of KdV Equation,” Journal of Computational and Applied Mathematics, Vol. 233, No. 6, 2010, pp. 1612-1618.

http://dx.doi.org/10.1016/j.cam.2009.02.076

[7] N. K. Ameine and M. A. Ramadau, “A Small Time Solutions for the KdV Equation Using Bubnov-Galerkin Finite Element Method,” Journal of the Egyptian Mathematical Society, Vol. 19, No. 3, 2011, pp. 118-125.

http://dx.doi.org/10.1016/j.joems.2011.10.005

[8] X. M. Li and A. H. Chen, “Darboux Transformation and Multi-Soliton Solutions of Boussinesq-Burgers Equation,” Physics Letters A, Vol. 342, No. 5-6, 2005, pp. 413-420.

http://dx.doi.org/10.1016/j.physleta.2005.05.083

[9] Y. Wang, L. J. Shen and D. L. Du, “Darboux Transformation and Explicit Solutions for Some (2 + 1)-dimensional Equation,” Physics Letters A, Vol. 366, No. 3, 2007, pp. 230-240.

http://dx.doi.org/10.1016/j.physleta.2007.02.043

[10] H. X. Wu, Y. B. Zeng and T. Y. Fan, “Complexitons of the Modified KdV Equation by Darboux Transformation,” Applied Mathematics and Computation, Vol. 196, No. 2, 2008, pp. 501-510.

http://dx.doi.org/10.1016/j.amc.2007.06.011

[11] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformation in Soliton Theory and Its Applications on Geometry,” Shanghai Scientific and Technical Publishers, Shanghai, 2005.

[1] M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolutions and Inverse Scattering,” Cambridge University Press, Cambridge, 1991, pp. 23-99.

http://dx.doi.org/10.1017/CBO9780511623998

[2] J. álvarez and A. Durán, “Error Propagation When Approximating Multi-Solitons: The KdV Equation with as a Case Study,” Applied Mathematics and Computation, Vol. 217, No. 4, 2010, pp. 1522-1539.

http://dx.doi.org/10.1016/j.amc.2009.06.033

[3] J. L. Yin and L. X. Tian, “Classification of the Traveling Waves in the Nonlinear Dispersive KdV Equation,” Nonlinear Analysis, Vol. 73, No. 2, 2010, pp. 465-470.

http://dx.doi.org/10.1016/j.na.2010.03.039

[4] A. Biswas, M. D. Petkovic and D. Milovic, “Topological and Non-Topological Exact Soliton Solution of the KdV Equation,” Nonlinear Science and Numerical Simulation, Vol. 15, No. 11, 2010, pp. 3263-3269.

http://dx.doi.org/10.1016/j.na.2010.03.039

[5] M. Nivala and B. Deconinck, “Periodic Finite-Genus Solutions of the KdV Equation Are Orbitally Stable,” Physica D: Nonlinear Phenomena, Vol. 239, No. 13, 2011, pp. 1147-1158.

http://dx.doi.org/10.1016/j.physd.2010.03.005

[6] Y. Yamamoto, T. Nagase and M. Ohmiya, “Appell’s Lemma and Conservation Laws of KdV Equation,” Journal of Computational and Applied Mathematics, Vol. 233, No. 6, 2010, pp. 1612-1618.

http://dx.doi.org/10.1016/j.cam.2009.02.076

[7] N. K. Ameine and M. A. Ramadau, “A Small Time Solutions for the KdV Equation Using Bubnov-Galerkin Finite Element Method,” Journal of the Egyptian Mathematical Society, Vol. 19, No. 3, 2011, pp. 118-125.

http://dx.doi.org/10.1016/j.joems.2011.10.005

[8] X. M. Li and A. H. Chen, “Darboux Transformation and Multi-Soliton Solutions of Boussinesq-Burgers Equation,” Physics Letters A, Vol. 342, No. 5-6, 2005, pp. 413-420.

http://dx.doi.org/10.1016/j.physleta.2005.05.083

[9] Y. Wang, L. J. Shen and D. L. Du, “Darboux Transformation and Explicit Solutions for Some (2 + 1)-dimensional Equation,” Physics Letters A, Vol. 366, No. 3, 2007, pp. 230-240.

http://dx.doi.org/10.1016/j.physleta.2007.02.043

[10] H. X. Wu, Y. B. Zeng and T. Y. Fan, “Complexitons of the Modified KdV Equation by Darboux Transformation,” Applied Mathematics and Computation, Vol. 196, No. 2, 2008, pp. 501-510.

http://dx.doi.org/10.1016/j.amc.2007.06.011

[11] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformation in Soliton Theory and Its Applications on Geometry,” Shanghai Scientific and Technical Publishers, Shanghai, 2005.