Abstract
By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave (OWW) equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions {X iβ } i=1,2,··· ,n β=1,2,··· ,N depending on a finite number of partial derivatives of the nonlocal variables v β and a restriction i.e., \(\sum\limits_{i,\alpha ,\beta } {\left( {\tfrac{{\partial \xi ^i }} {{\partial v^\beta }}} \right) + \left( {\tfrac{{\partial \eta ^\alpha }} {{\partial v^\beta }}} \right)} \ne 0\) ≠ 0, i.e., \(\sum\limits_{i,\alpha ,\beta } {\left( {\tfrac{{\partial G^\alpha }} {{\partial v^\beta }}} \right)} \ne 0\). Furthermore, the authors investigate some structures associated with the Olver water wave (AOWW) equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, Itˆo and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson, I. M., Kamran, N. and Olver, P. J., Internal, external and generalized symmetries, Adv. Math., 100, 1993, 53–100.
Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer-Verlag, New York, 1989.
Bluman, G. W., Temuerchaolu and Anco, S., New conservation laws obtained directly from symmetry action on a known conservation law, J. Math. Anal. Appl., 322, 2006, 233–250.
Bluman, G. W., Tian, S. F. and Yang, Z. Z., Nonclassical analysis of the nonlinear Kompaneets equation, J. Eng. Math., 84, 2014, 87–97.
Chern, S. S. and Tenenblat, K., Pseudo-spherical surfaces and evolution equations, Stud. Appl. Math., 74(1), 1986, 55–83.
Chen, Y. and Hu, X. R., Lie symmetry group of the nonisospectral Kadomtsev-Petviashvili equation, Z. Naturforsch A., 62(a), 2009, 8–14.
Dodd, R. K. and Gibbon, J. D., The prolongation structure of a higher order Korteweg-de Vries equation, Proc. R. Soc. Lond. A, 358, 1978, 287–296.
Fan, E. G., Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method, J. Phys. A: Math. Gen., 35, 2002, 6853–6872.
Galas, F., New nonlocal symmetries with pseudopotentials, J. Phys. A: Math. Gen., 25, 1992, L981–L986.
Guthrie, G. A. and Hickman, M. S., Nonlocal symmetries of the KdV equation, J. Math. Phys., 26, 1993, 193–205.
Huang, Q. and Qu, C. Z., Symmetries and invariant solutions for the geometric heat flows, J. Phys. A: Math. Theor., 40, 2007, 9343–9360.
Hernandez-Heredero, R. and Reyes, E. G., Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation, J. Phys. A: Math. Theor., 42, 2009, 182002.
Ito, M., An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders, J. Phys. Soc., 49, 1980, 771–778.
Ibragimov, N. H., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1, CRC Press, Boca Raton, Florida, 1994.
Kaup, D. J., On the inverse scattering problem for cubic eigenvalue problems of the class Ψxxx +6QΨx + 6RΨ = *λΨ, Stud. Appl. Math., 62, 1980, 189–216.
Kupershmidt, B. A., A super Korteweg-de Vries equation: An integrable system, Phys. Lett. A, 102, 1984, 213–215.
Kudryashov, N. A. and Sukharev, M. B., Exact solutions of a non-linear fifth-order equation for describing waves on water, J. Appl. Math. Mech., 65, 2001, 855–865.
Lax, P. D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21, 1968, 467–490.
Levi, D. and Winternitz, P., Lie point symmetries and commuting flows for equations on lattices, J. Phys. A: Math. Gen., 35, 2002, 2249–2262.
Lou, S. Y., A (2+1)-dimensional extension for the sine-Gordon equation, J. Phys. A: Math. Gen., 26, 1993, L789–L791.
Lou, S. Y. and Hu, X. B., Infinitely many Lax pairs and symmetry constraints of the KP equation, J. Math. Phys., 38, 1997, 6401–6427.
Noether, E., Invariante variations probleme, Nachr. Konig. Gesell. Wissen. Gottingen, Math. Phys. Kl., 1918, 235–257 (see Transport Theory and Stat. Phys. 1, 1971, 186–207 for an English translation).
Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd edition, Springer-Verlag, New York, 1993.
Olver, P. J., Hamiltonian and non-Hamiltonian models for water waves, Lecture Notes in Physics, vol. 195, Springer-Verlag, New York, 1984.
Qu, C. Z., Potential symmetries to systems of nonlinear diffusion equations, J. Phys. A: Math. Theor., 40, 2007, 1757–1773.
Qu, C. Z. and Zhang, C. R., Classification of coupled systems with two-component nonlinear diffusion equations by the invariant subspace method, J. Phys. A: Math. Theor., 42, 2009, 475201.
Reyes, E. G., On nonlocal symmetries of some shallow water equations, J. Phys. A: Math. Theor., 40, 2007, 4467–4476.
Reyes, E. G., Nonlocal symmetries and the Kaup-Kupershmidt equation, J. Math. Phys., 46, 2005, 073507.
Sawada, K. and Kotera, K., A method for finding N-soliton solutions of the KdV and KdV-like equation, Prog. Theor. Phys., 51, 1974–1355.
Tian, S. F. and Zhang, H. Q., Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations, J. Math. Anal. Appl. 371, 2010, 585–608.
Tian, S. F. and Zhang, H. Q., Lax pair, binary Darboux transformation and new grammian solutions of nonisospectral Kadomtsev-Petviashvili equation with the two-singular-manifold method, J. Nonlinear Math. Phys., 17(4), 2010, 491–502.
Tian, S. F. and Zhang, H. Q., A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations, Commun. Nonlinear Sci. Numer. Simulat., 16, 2011, 173–186.
Tian, S. F. and Zhang, H. Q., Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV-Burgers equation, Theor. Math. Phys., 170(3), 2012, 287–314.
Tian, S. F. and Zhang, H. Q., On the integrability of a generalized variable-coefficient Kadomtsev- Petviashvili equation, J. Phys. A: Math. Theor., 45, 2012, 055203.
Tian, S. F. and Zhang, H. Q., On the integrability of a generalized variable-coefficient forced Korteweg-de Vries equation in fluids, Stud. Appl. Math., 132, 2014, 212–246.
Vinogradov, A. M. and Krasil’shchik, I. S., A method of calculation of higher symmetries of nonlinear evolution equations and nonlocal symmetries, Dokl. Akad. Nauk SSSR, 253, 1980, 1289–1293.
Yan, Z. Y., The new tri-function method to multiple exact solutions of nonlinear wave equations, Phys. Scr., 78, 2008, 035001.
Zhang, S. L., Lou, S. Y. and Qu, C. Z., New variable separation approach: Application to nonlinear diffusion equations, J. Phys. A: Math. Gen., 36, 2003, 12223–12242.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 11301527, 11371361), the Fundamental Research Funds for the Central Universities (No. 2013QNA41) and the Construction Project of the Key Discipline of Universities in Jiangsu Province During the 12th Five-Year Plans (No. SX2013008).
Rights and permissions
About this article
Cite this article
Tian, S., Zhang, Y., Feng, B. et al. On the Lie algebras, generalized symmetries and darboux transformations of the fifth-order evolution equations in shallow water. Chin. Ann. Math. Ser. B 36, 543–560 (2015). https://doi.org/10.1007/s11401-015-0908-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-015-0908-6