Abstract:
Rational solutions and special polynomials associated with the generalized $K_2$ hierarchy are studied. This hierarchy is related to the Sawada–Kotera and Kaup–Kupershmidt equations and some other integrable partial differential equations including the Fordy–Gibbons equation. Differential–difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations $\Gamma$ and $-2\Gamma$ is established. Properties of the polynomials are studied. Differential–difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.
Keywords:
point vortices, special polynomials, generalized $K_2$ hierarchy, Sawada–Kotera equation, Kaup–Kupershmidt equation, Fordy–Gibbons equation.
Citation:
Maria V. Demina, Nikolai A. Kudryashov, “Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations”, Regul. Chaotic Dyn., 16:6 (2011), 562–576
\Bibitem{DemKud11}
\by Maria V. Demina, Nikolai A. Kudryashov
\paper Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations
\jour Regul. Chaotic Dyn.
\yr 2011
\vol 16
\issue 6
\pages 562--576
\mathnet{http://mi.mathnet.ru/rcd457}
\crossref{https://doi.org/10.1134/S1560354711060025}
Linking options:
https://www.mathnet.ru/eng/rcd457
https://www.mathnet.ru/eng/rcd/v16/i6/p562
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J. Ramírez, J.L. Romero, C. Muriel, “Reductions of PDEs to second order ODEs and symbolic computation”, Applied Mathematics and Computation, 291 (2016), 122
M. V. Demina, N. A. Kudryashov, “Polynomial Method for Constructing Equilibrium Configurations of Point Vortices in the Plane”, Model. anal. inf. sist., 19:5 (2015), 50
Anna M Barry, F Hajir, P G Kevrekidis, “Generating functions, polynomials and vortices with alternating signs in Bose–Einstein condensates”, J. Phys. A: Math. Theor., 48:15 (2015), 155205
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M. V. Demina, N. A. Kudryashov, “Rotation, collapse, and scattering of point vortices”, Theor. Comput. Fluid Dyn., 28:3 (2014), 357
Maria V. Demina, Nikolai A. Kudryashov, “Relative Equilibrium Configurations of Point Vortices on a Sphere”, Regul. Chaotic Dyn., 18:4 (2013), 344–355
M. V. Demina, N. A. Kudryashov, “Polynomial method for constructing equilibrium configurations of point vortices in a plane”, Aut. Control Comp. Sci., 47:7 (2013), 545
Kevin A. O'Neil, “Stationary vortex sheets in a stirring flow”, Theor. Comput. Fluid Dyn., 27:6 (2013), 777
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