A remark on a perturbed Benjamin-Bona-Mahony type equation and its complete integrability
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In the Letter, we study a perturbed Benjamin-Bona-Mahony nonlinear equation, which was derived for describing shallow water waves and possessing a rich Lie symmetry structure. Based on the gradient-holonomic integrability checking scheme applied to this equation, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible Poisson structure and stated its complete integrability.
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Peregrine DH. Calculations of the development of an undular bore. J. Fluid Mech. 1966; 25:321-330. doi:10.1017/S0022112066001678.
Benjamin TB, Bona JL, Mahony JJ. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Royal Soc. London Ser. A 1972; 272:47-78.
Duran A, Dutykh D, Mitsotakis D. On the Galilean Invariance of Some Nonlinear DispersiveWave Equations. Stud. Appl. Math. 2013; 131:359–388. doi:10.1111/sapm.12015.
Cheviakov Al, Dutykh D, Assylbekuly A. On Galilean invariant and energy preserving BBM-type equations. 2021. (https://www.mdpi.com/journal/symmetry).
Olver PJ. Applications of Lie groups to differential equations, 2 ed.; Vol. 107, Graduate Texts in Mathematics, Springer-Verlag: New York. 1993; 500. doi:10.1007/978-1-4684-0274-2.
Johnson RS. Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 2002; 455:63-82.
Dullin HR, Gottwald GA, Holm DD. Camassa-Holm. Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 2003; 33:73-95.
Novikov S, Manakov SV, Pitaevskii LP, Zakharov VE. Theory of solitons, Springer. 1984.
Moser J, Zehnder E. Notes on Dynamical Systems, Courant Lecture Notes in Math. 2005;12: NYU.
Takhtajan LA, Faddeev LD. Hamiltonian methods in the theory of solitons. Springer. 2007.
Shubin MA. Pseudodifferential Operators and Spectral Theory, Springer. 2002.
Artemovych OD, Balinsky AA, Blackmore D, Prykarpatski AK. Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators. Symmetry. 2018; 10:601. doi:10.3390/sym10110601.
Dubrovin BA. On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour. Comm. Math. Phys. 2006; 267:117-139. https://doi.org/10.1007/s00220-006-0021-5; https://gdeq.org/files/seminar-moscow-21.pdf
Mitropolsky YuA, Bogolubov NN, Prykarpatsky AK, Samoylenko VHr. Integrable dynamical systems. Spectral and differential geometric aspects. K Naukova Dumka. 1987.
Shabat AB, Mikhailov AV. Symmetries - test of integrability, in: Important developments in soliton theory, in: Springer Ser. Nonlinear Dynam., Springer, Berlin. 1993; 355-374.
Blackmore D, Prykarpatsky AK, Samoylenko VHr. Nonlinear dynamical systems of mathematical physics: spectral and differential-geometrical integrability analysis. World Scientific Publ., NJ, USA. 2011.
Prykarpatsky AK, Mykytyuk IV. Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects. Kluwer Academic Publishers, the Netherlands. 1998.
Newell A. Solitons in mathematics and physics. SIAM. 1985.
Magri F. A Simple Model of the Integrable Hamiltonian Equation, J. Math. Phys. 1978; 19(5):1156-1162.
Abraham R, Marsden J. Foundations of mechanics. Benjamin/Cummings Publisher. 2008.
Arnold VI. Mathematical methods of classical mechanics. Springer. 1989.