Abstract
In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail.
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References
Bell, E.T. Exponential polynomials. Ann. Math., 35: 258–277 (1934)
Dubrovin, B. Integrable systems in topological field theory. Nucl. Phys. B, 379(3): 627–689 (1992)
Fan, E.G. Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii’s breaking soliton equation in (2+1) dimensions. Phys. Rev. E, 78(3): 036607 (2008)
Fan, E.G. The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A, 375(3): 493–497 (2011)
Furukawa, M., Tokuda, S. Mechanism of stabilization of ballooning modes by toroidal rotation shear in tokamaks. Phys. Rev. Lett., 94(17): 175001 (2005)
Hasegawa, A., Tappert, F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett., 23(3): 142–144 (1973)
Hirota, R., Satsuma, J. A simple structure of superposition formula of the Bäcklund transformation. J. Phys. Soc. Japan, 45(5): 1741–1750 (1978)
Hirota, R. Direct Methods in Soliton Theory. Springer-verlag, Berlin, 2004
Iwao, M., Hirota, R. Soliton solutions of a coupled modified KdV equations. J. Phys. Soc. Japan, 66(3): 577–588 (1997)
Lambert, F., Loris, I., Springael, J., Willox, R. On a direct bilinearization method: Kaup’s higher-order water wave equation as a modified nonlocal Boussinesq equation. J. Phys. A: Math. Gen., 27(15): 5325–5334 (1994)
Lambert, F., Springael, J. Construction of Bäcklund transformations with binary Bell polynomials. J. Phys. Soc. Japan, 66(8): 2211–2213 (1997)
Lambert, F., Leble, S., Springael, J. Binary Bell polynomials and Darboux covariant Lax pairs. Glasgow Math. J., 43A: 53–63 (2001)
Lambert, F., Springael, J. On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations. Chaos, Solitons and Fractals, 12(14–15): 2821–2832 (2001)
Lax, P.D. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure. Appl. Math., 21: 467–490 (1968)
Luo, L. New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation. Phys. Lett. A, 375(7): 1059–1063 (2011)
Luo, L. Bäcklund transformation of variable-coefficient Boiti-Leon-Manna-Pempinelli equation. Appl. Math. Lett., 94: 94–98 (2019)
Ma, W.X., Zhou, R., Gao, L. Exact one-periodic and two-periodic wave solutions to hirota bilinear equations in (2+1) dimensions. Modern Phys. Lett. A, 24(21): 1677–1688 (2009)
Mckean, H.P., Moerbeke, P. van The spectrum of Hill’s equation. Invent. Math., 30(3): 217–274 (1975)
Miao, Q., Wang, Y.H., Chen, Y., Yang, Y.Q. PDEBellII: a Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations. Comput. Phys. Commun., 185(1): 357–367 (2014)
Nakamura, A. A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution. J. Phys. Soc. Japan, 47(5): 1701–1705 (1979)
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. Japan, 48(4): 1365–1370 (1980)
Novikov, S.P. The periodic problem for the Korteweg-de vries equation. Funct. Anal. Appl., 8(3): 236–246 (1975)
Osman, M.S., Ghanbari, B., Machado, J.A.T. New complex waves in nonlinear optics based on the complex Ginzburg-Landau equation with Kerr law nonlinearity. Eur. Phys. J. Plus, 134(1): 20 (2019)
Raissi, M., Perdikaris, P., Karniadakis, G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 378: 686–707 (2019)
Satsuma, J. Higher conservation laws for the Korteweg-de Vries equation through Bäcklund transformation. Progr. Theoret. Phys., 52(4): 1396–1397 (1974).
Satsuma, J., Ablowitz, M.J. Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys., 20(7): 1496–1503 (1979)
Tian, S.F., Zhang, H.Q. On the integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation. J. Phys. A: Math. Theor., 45(5): 055203 (2012)
Wang, Y.H., Chen, Y. Integrability of the modified generalised Vakhnenko equation. J. Math. Phys., 53(12): 123504 (2012)
Wang, Y.H., Chen, Y. Binary Bell polynomial manipulations on the integrability of a generalized (2+1)-dimensional Korteweg-de Vries equation. J. Math. Anal. Appl., 400(2): 624–634 (2013)
Wang, Y.H., Wang, H., Temuer, C. Lax pair, conservation laws, and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation. Nonlinear Dyn., 78(2): 1101–1107 (2014)
Wang, Y.P., Tian, B., Wang, M., Wang, Y.F., Sun, Y., Xie, X.Y. Bäcklund transformations and soliton solutions for a (2+1)-dimensional Korteweg-de Vries-type equation in water waves. Nonlinear Dyn., 81(4): 1815–1821 (2015)
Wang, Y.H., Chen, Y. Bell polynomials approach for two higher-order KdV-type equations in fluids. Nonlinear Anal. RWA, 31: 533–551 (2016)
Wazwaz, A.M. Two new Painlevé-integrable (2+1) and (3+1)-dimensional KdV equations with constant and time-dependent coefficients. Nucl. Phys. B, 954: 115009 (2020)
Yu, S.J., Toda, K., Fukuyama, T. N-soliton solutions to a (2+1)-dimensional integrable equation. J. Phys. A: Math. Gen., 31(50): 10181–10186 (1998)
Zhang, Y., Ye, L.Y., Lv, Y.N., Zhao, H.Q. Periodic wave solutions of the Boussinesq equation. J. Phys. A: Math. Theor., 40(21): 5539–5549 (2007)
Zhang, Y., Wei, W.W., Cheng, T.F., Song, Y. Binary Bell polynomial application in generalized (2+1)-dimensional KdV equation with variable coefficients. Chin. Phys. B, 20(11): 110204 (2011)
Funding
The project is supported by the National Natural Science Foundation of China (No. 12175069 and No. 12235007) and Science and Technology Commission of Shanghai Municipality (No. 21JC1402500 and No. 22DZ2229014).
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Pu, Jc., Chen, Y. Integrability and Exact Solutions of the (2+1)-dimensional KdV Equation with Bell Polynomials Approach. Acta Math. Appl. Sin. Engl. Ser. 38, 861–881 (2022). https://doi.org/10.1007/s10255-022-1020-9
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DOI: https://doi.org/10.1007/s10255-022-1020-9
Keywords
- the bilinear formalism
- bilinear Bäcklund transformations
- Lax pair
- lump solution
- periodic wave solution
- the asymptotic properties