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Quasi-periodic solutions of \((3+1)\) generalized BKP equation by using Riemann theta functions. (English) Zbl 1410.35166

Summary: This paper is focused on quasi-periodic wave solutions of \((3+1)\) generalized BKP equation. Because of some difficulties in calculations of \(N = 3\) periodic solutions, hardly ever has there been a study on these solutions by using Riemann theta function. In this study, we obtain one and two periodic wave solutions as well as three periodic wave solutions for \((3+1)\) generalized BKP equation. Moreover we analyze the asymptotic behavior of the periodic wave solutions tend to the known soliton solutions under a small amplitude limit.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
14K25 Theta functions and abelian varieties
35B15 Almost and pseudo-almost periodic solutions to PDEs
35G20 Nonlinear higher-order PDEs
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[1] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of Solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423
[2] Bluman, G. W.; Kumei, S., Symmetries and differential equations (1989), Springer Verlag: Springer Verlag New York · Zbl 0698.35001
[3] Miura, M. R., Bäcklund Transformation (1978), Springer Verlag: Springer Verlag Berlin
[4] Belokolos, E. D.; Bobenko, A. I.; Enol’skii, V. Z.; Its, A. R.; Matveev, V. B., Algebro-geometric Approach to Non-linear Integrable Equations (1994), Springer · Zbl 0809.35001
[5] Novikov, S. P., A periodic problem for the Korteweg-de Vries equation, Funct. Anal. Appl., 8, 236-246 (1974) · Zbl 0299.35017
[6] Dubrovin, B. A., Periodic problems for the Korteweg — de Vries equation in the class of finite band potentials, Funct. Anal. Appl., 9, 215-223 (1975) · Zbl 0358.35022
[7] Its, A. R.; Matveev, V. B., Hill’s operators with a finite number of lacunae, Funk. Anal., i Prilozen, 69-70 (1975)
[8] Lax, P. D., Periodic solutions of the KdV equation, Commun. Pure Appl. Math., 28, 141-188 (1975) · Zbl 0295.35004
[9] Nakamura, A., A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution, J. Phys. Soc. Jpn., 47, 1701-1705 (1979) · Zbl 1334.35006
[10] Nakamura, A., A direct method of calculating periodic wave solutions to nonlinear evolution equations. II. Exact one- and two-periodic wave solution of the coupled bilinear equations, J. Phys. Soc. Jpn., 48, 1365-1370 (1980) · Zbl 1334.35250
[11] Hon, Y. C.; Fan, E. G., A kind of explicit quasi-periodic solution and its limit for the TODA lattice equation, Mod. Phys. Lett. B., 22, 547-553 (2008) · Zbl 1151.82320
[12] Cheng, Z.; Hao, X., The periodic wave solutions for a (2 + 1)-dimensional AKNS equation, Appl. Math. Comput., 234, 118-126 (2014) · Zbl 1310.35065
[13] Tian, S. F.; Zhang, H. Q., Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV-Burgers equation, Theor. Math. Phys., 170, 3, 287-314 (2012) · Zbl 1274.35294
[14] Tian, S.; Zhang, H., Riemann theta functions periodic wave solutions and rational characteristics for the (1+1)-dimensional and (2+1)-dimensional Ito equation, Chaos, Solitons Fractals, 47, 27-41 (2013) · Zbl 1258.35011
[15] Lu, B.; Zhang, H., Quasi-periodic wave solutions of (3+1)-dimensional Jimbo-Miwa Equation, Int. J. Nonlinear Sci., 10, 452-461 (2010) · Zbl 1394.35101
[16] Ma, W.; Zhu, Z., Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218, 11871-11879 (2012) · Zbl 1280.35122
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