Applied Mathematics, 2013, 4, 1599-1602 Published Online December 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.412216 Open Access AM Doubly and Triply Periodic Waves Solutions for the KdV Equation* Ying Huang, Dengguo Xu Department of Mathematics, Chuxiong Normal University, Chuxiong, China Email: huang11261001@163.cn Received September 6, 2013; revised October 6, 2013; accepted October 13, 2013 Copyright © 2013 Ying Huang, Dengguo Xu. 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 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. Keywords: KdV Equation; Doubly Periodic Solution; Triply Periodic Solution; Darboux Tr ansformation 1. Introduction The famo us KdV equati on 6 txxxx uuuu0 (1) is a shallow water wave equation early derived by Korteweg de and Vries, its first application was discov- ered in the study of collision-free hydro-magnetic waves in 1960. Subsequently, it has arisen in a number of phy- sical contexts, such as stratified internal waves, ion- acoustic waves, plasma physics, lattice dynamics and so on. Following the further studies of these physical prob- lems, its exact solutions have attracted much attention and have been extensively studied [1-7]. However, in contrast to solitary wave solutions, the analytic periodic solutions represent only a small subclass of its known solutions, and multi-periodic solutions are scarce. It is always useful to seek more and various multi-periodic solutions for recovering interactions among some simple periodic waves in a nonl inear medium. We know that the Darboux transformation method is the main method to construct exact multi-soliton solu- tions, and this method is scarcely used for solving multi- periodic solutions [8-10]. In the paper, not only explicit doubly periodic solutions are available, but also a group of explicit triply periodic solutions is obtained by means of the Darboux transformation method. 2. Doubly Periodic Solutions According to [11], the linear system 01 , 0 42 , x x tx u uu Au (2) is the Lax pair for Equation (1), with the Darboux matrix 2 1 ,, , i ii i Dxt (3) where 42 x uuu i u , are the spectral parameters. The monograph [11] further points out, if is a known solution to Equati on (1), then ,0,1, ii 2 i 2 122 iii uu (4) becomes new solution generated from , with i u 21 22 11 12 ,,,, , ,, ,, ii ii ii iii ii ii axt axt axt axt (5) where, i and i are arbitrary constants, but ii 22 0 , and 22 ,, ,, i ijk xta xt u is the fundamental solution matrix to the lax pair on . i Only solving the fundamental solution matrix of the lax pair corresponding to constant solution 0, it is possible to construct multi-periodic solutions to the KdV Equation (1). Substituting into the system (2) yields u 0 u *This work was supported by the Chinese Natural Science Foundation Grant (11261001) and Yunnan Provincial Department of Education Research Foundation Grant (2012Y130).
Y. HUANG, D. G. XU 1600 0 00 01 , 0 01 =4 2. 0 x t u uu (6) If setting 0 =42 ut , then we can assert that both the system (6) and the following linear system 0 01 0u have exactly the same solutions. Under the condition for 0 u , by the eigenvalue method, we obtain the com- plex-valued fundamental solution matrix to the above system ee , ee ia ia ia ia ia ia (7) where 0 aa u . Because the real and imaginary parts of a complex-valued solution are also solutions, we thus take 0 cos sin ,, sin cos xt aa (8) as the fundamental solution matrix to the the system (6), where a . For simplicity, we setting 22 ,, j iiiiijij aaaa i ,0,1,2ij , . From (5), we have 000 0 00 0000 sincos , cos sin a in the above formula, choosing 00 1, 0 and 00 0, 1 , respectively, we get 00 tan ta0 (9) and 00 0 cot , ca (10) respectively, with (4), the periodic wave solutions 22 11 000 2secuua and 22 12 000 2cscuua are obtained. Now we construct the doubly periodic solutions gen- erated from , thanks to (4), we see that 1 u 2 21 0001 =2222 ,uu 2 (11) we first give 1 , then substitute 0 and 1 into (11). Again according to [11], we can obtain the fundamental solution matrix to the lax pair associated with the known periodic wave solution in the following manner 1 u 0 10 2 00 0 00 1 ,, ,, cossinsincos , xt xt aa PQ (12) where 2 00 0 cos sinPa , 2 00 0 sin cosQa 11 1,0 . After combining (5) and (12), choosing , we get 2 10 0011 101 1 tan . tan t a a (13) Substituting (9) and (13) into (11), we have new dou- bly periodic solution 22 22 010011 21 02 0011 2sec sec . tan tan aa uu aa (14) Again substituting (10) and (13) into (11), we obtain another new doubly periodic solution 22 22 01 0011 22 02 0011 2cscsec . cot tan aa uu aa (15) Similarly, choosing 11 0, 1 , we have 2 10 0011 101 1 cot , cot c a a (16) which implies the doubly periodic solutions 22 22 01 0011 23 02 0011 2 seccsc tancot aa uu aa (17) and 22 22 01 0001 24 02 0011 2csccsc . cot cot aa uu aa (18) Specially, although 23 u is a doubly periodic solu- tion, its structure is very similar to a given two-soliton solution in [1]. 3. Triply Periodic Solutions As shown in [11], the fundamental so lution matrix to the lax pair associated with the doubly periodic wave solu- tion can be given by 2 u 1 21 2 11 1 1 ,,,, ,xt xt (19) substituting (12) into (19 ), in exactly the same manner as in Section 2, we get Open Access AM
Y. HUANG, D. G. XU 1601 12 022 21 20 01022 tan tan t a a and 12 022 21 20 01022 cot . cot c a a Owing to (4) and (11), we have 22 3000 2112 22uu 2 0 u i . (20) Here, we set tan ii Fa , cot ,0,1,2 ii i Gai . Substituting 01 , tt 2t and into (20), we obtain trip- ly periodic solution 22 31 000 2 22 22 12 01001102 2 02 101002 2 22 22 12022 20 010 2 02 101002 2sec 2secsec 2secsec . uua aaFF FF FF aaFF FF FF Similarly, we have 22 32 000 2 22 22 12 01001102 2 02 101002 2 22 22 12 02220010 2 02 101002 2csc 2cscsec 2sec csc, uua aaGF FGG F aaFG FGG F 22 33 000 2 22 22 12 01001102 2 021010 02 2 22 22 12 02220010 2 021010 02 2sec 2sec csc 2secsec , uua aaFF GF FF aaGF GF FF 22 34 000 2 22 22 12 01001102 2 02 101002 2 22 22 12 02220010 2 02 101002 2csc 2csccsc 2sec csc, uua aaGF GGG F aaGG GGG F 22 35 000 2 22 22 12010 01 102 2 02 101002 2 22 22 12 02220010 2 02 101002 2sec 2secsec 2csc sec, uua aaFG FF FG aaFF FF FG 22 36 000 2 22 22 12 01001102 2 02 101002 2 22 22 12022 20 010 2 02 101002 2csc 2csc sec 2csccsc , uua aaGG FG GG aaFG FG GG 22 37 000 2 22 22 12 01001102 2 021010 02 2 22 22 12 02220010 2 021010 02 2sec 2sec csc 2csc sec, uua aaFG GF FG aaGF GF FG and 22 38 000 2 22 22 12010 01 102 2 02 101002 2 22 22 12 02220010 2 02 101002 2csc 2csccsc 2csccsc . uua aaGG GG GG aaGG GG GG REFERENCES [1] M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolutions and Inverse Scattering,” Cambridge Univer- sity Press, Cambridge, 1991, pp. 23-99. http://dx.doi.org/10.1017/CBO9780511623998 [2] J. Álvarez and A. Durán, “Error Propagation When Ap- proximating Multi-Solitons: The KdV Equation with as a Case Study,” Applied Mathematic s and Computation, Vo l. 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,” Non- linear 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 Open Access AM
Y. HUANG, D. G. XU Open Access AM 1602 [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,” Jour- nal of Computational and Applied Mathematic s, 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. Rama dau, “A Small Time Solu- tions for the KdV Equation Using Bubnov-Galerkin Fi- nite Element Method,” Journal of the Egyptian Mathe- matical 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 Equa- tion,” 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 Transfor- mation and Explicit Solutions for Some (2 + 1)-dimen- sional 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 Transforma- tion,” 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 Transfor- mation in Soliton Theory and Its Applications on Geome- try,” Shanghai Scientific and Technical Publishers, Shanghai, 2005.
|