References
-
Abraham R., Marsden J.
Foundations of Mechanics.
Benjamin Cummings, New York, 1994.
-
Alekseev A., Malkin A.Z.
Symplectic structure of the modili space of flat connection on a Riemann surface.
Comm. Math. Phys. 1995, 169, 99-119.
-
Alonso L.M., Shabat A.B.
Hydrodynamic reductions and solutions of a universal hierarchy.
Theoret. and Math. Phys. 2004, 104 (1), 1073-1085.
doi: 10.1023/B:TAMP.0000036538.41884.57
-
Arnold V.I.
Mathematical Methods of Classical Mechanics.
Springer, New York, 1978.
-
Arnold V.I.
Sur la geometrie differerentielle des groupes de Lie de dimension infinie et ses applications
a l'hydrodynamique des fluides parfaits.
Ann. Inst. Fourier (Grenoble) 1966, 16, 319-361.
-
Arnold V.I., Khesin B.A.
Topological methods in hydrodynamics.
Springer, New York, 1998.
-
Artemovych O.D., Balinsky A.A., Blackmore D., Prykarpatski A.K.
Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky-Novikov Type Symmetry Algebras and Related Hamiltonian Operators.
Symmetry 2018, 10 (11), 601.
doi: 10.3390/sym10110601
-
Artemovych O.D., Blackmore D., Prykarpatski A.K.
Examples of Lie and Balinsky-Novikov algebras related to Hamiltonian operators.
Topol. Algebra Appl. 2018, 6, 43-52.
doi: 10.1515/taa-2018-0005
-
Artemovych O.D., Blackmore D., Prykarpatski A.K.
Poisson brackets, Novikov-Leibniz structures and integrable Riemann hydrodynamic systems.
J. Nonlinear Math. Phys. 2017, 24 (1), 41-72.
doi: 10.1080/14029251.2016.1274114
-
Audin M.
Lectures on gauge theory and integrable systems.
In: Hurtubise J., Lalonde F. (Eds.) Gauge Theory and Symplectic Geometry,
Kluwer, 1-48, 1997.
-
Blackmore D., Prykarpatsky A.K., Samoylenko V.H.
Nonlinear Dynamical Systems of Mathematical Physics.
World Scientific Publisher Co. Pte. Ltd., Hackensack, USA, 2011.
doi: 10.1142/7960
-
Blackmore B., Prykarpatsky Y., Golenia J., Prykapatski A.
Hidden Symmetries of Lax Integrable Nonlinear Systems.
Applied Mathematics 2013, 4 (10), 95-116.
doi: 10.4236/am.2013.410A3013
-
Błaszak M.
Classical R-matrices on Poisson algebras and related dispersionless systems.
Phys. Lett. A 2002, 297 (3-4), 191-195.
doi: 10.1016/S0375-9601(02)00421-8
-
Błaszak M.
Multi-Hamiltonian theory of dynamical systems.
Springer, Berlin, 1998.
-
Błaszak M., Szablikowski B.M.
Classical R-matrix theory of dispersionless systems: II. (2 + 1) dimension theory.
J. Phys. A: Math. Gen. 2002, 35 (48), 10325.
doi: 10.1088/0305-4470/35/48/309
-
Bogdanov L.V., Dryuma V.S., Manakov S.V.
Dunajski generalization of the second heavenly equation: dressing method and the hierarchy.
J. Phys. A: Math. Theor. 2007, 40 (48), 14383-14393.
-
Bogdanov L.V., Pavlov M.V.
Linearly degenerate hierarchies of quasiclassical SDYM type.
J. Math. Phys. 2017, 58, 093505.
doi: 10.1063/1.5004258
-
Doubrov B., Ferapontov E.V.
On the integrability of symplectic Monge-Ampère equations.
J. Geom. Phys. 2010, 60 (10), 1604-1616.
-
Doubrov B., Ferapontov E.V., Kruglikov B., Novikov V.S.
On integrability in Grassmann geometries: integrable systems associated with fourfolds in $Gr(3,5)$.
Proc. Lond. Math. Soc. (3) 2018, 116 (5), 1269-1300.
doi: 10.1112/plms.12114
-
Dunajski M.
Anti-self-dual four-manifolds with a parallel real spinor.
Proc. Roy. Soc. Edinburgh Sect. A 2002, 458, 1205-1222.
doi: 10.1098/rspa.2001.0918
-
Dunajski M., Mason L.J., Tod P.
Einstein-Weyl geometry, the dKP equation and twistor theory.
J. Geom. Phys. 2001, 37 (1-2), 63-93.
-
Ferapontov E.V., Kruglikov B.
Dispersionless integrable systems in 3D and Einstei-Weyl geometry.
J. Differential Geom. 2012, 97 (2), 215-254.
doi: 10.4310/jdg/1405447805
-
Ferapontov E.V., Moss J.
Linearly degenerate PDEs and quadratic line complexes.
Comm. Anal. Geom. 2015, 23 (1), 91-127.
doi: 10.4310/CAG.2015.v23.n1.a3
-
Godbillon C.
Geometrie Differentielle et Mecanique Analytique.
Hermann Publ., Paris, 1969.
-
Hentosh O.Ye., Prykarpatsky Ya.A., Blackmore D., Prykarpatski A.K.
Lie-algebraic structure of Lax-Sato integrable heavenly equations and the Lagrange-d'Alembert principle.
J. Geom. Phys. 2017, 120, 208-227.
-
Holm D., Kupershmidt B.
Poisson structures of superfluids.
Phys. Lett. 1982, 91 (A), 425-430.
doi: 10.1016/j.geomphys.2017.06.003
-
Holm D., Marsden J., Ratiu T., Weinstein A.
Nonlinear stability of fluid and plasma equilibria.
Phys. Rep. 1985, 123 (1-2), 1-116.
-
Kambe T.
Geometric theory of fluid flows and dynamical systems.
Fluid Dyn. Res. 2002, 30, 331-378.
-
Kulish P.P.
An analogue of the Korteweg-de Vries quation for the superconformal algebra.
J. Math. Sci. 1988, 41 (2), 970-975.
doi: 10.1007/BF01247091
-
Kupershmidt B.A., Ratiu T.
Canonical Maps Between Semidirect Products with Applications to Elasticity and Superfluids.
Commun. Math. Phys. 1983, 90, 235-250.
doi: 10.1007/BF01205505
-
Kuznetsov E.A., Mikhailov A.V.
On the topological meaning of canonical Clebsch variables.
Phys. Lett. A 1980, 77 (1), 37-38.
doi: 10.1016/0375-9601(80)90627-1
-
Manakov S.V., Santini P.M.
On the solutions of the second heavenly and Pavlov equations.
J. Phys. A 2009, 42 (40), 404013.
-
Marsden J., Ratiu T., Schmid R., Spencer R., Weinstein A.
Hamiltonian systems with symmetry, coadjoint orbits, and plasma physics.
Atti Acad. Sci. Torino Cl. Sci. Fis. Math. Natur. 117, 1983, 289-340.
-
Marsden J., Ratiu T., Weinstein A.
Reduction and Hamiltoninan structures on duals of semidirect product Lie algebras.
Contemp. Math. 1984, 28, 55-100.
-
Marsden J., Weinstein A.
Reduction of symplectic manifolds with symmetry.
Rep. Math. Phys. 1974, 5, 121-130.
-
Mikhalev V.G.
On the Hamiltonian formalism for Korteweg-de Vries type hierarchies.
Funct. Anal. Appl. 1992, 26 (2), 140-142.
doi: 10.1007/BF01075282
-
Misiolek G.
A shallow water equation as a geodesic flow on the Bott-Virasoro group.
J. Geom. Phys. 1998, 24 (3), 203-208.
-
Ovsienko V.
Bi-Hamilton nature of the equation $u_{tx}=u_{xy}u_{y}-u_{yy}u_{x}$.
arXiv:0802.1818v1
-
Ovsienko V., Roger C.
Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1.
Commun. Math. Phys. 2007, 273, 357-378.
-
Pavlov M.V.
Integrable hydrodynamic chains.
J. Math. Phys. 2003, 44 (9), 4134-4156.
-
Plebański J.F.
Some solutions of complex Einstein equations.
J. Math. Phys. 1975, 16 (12), 2395-2402.
-
Pressley A., Segal G.
Loop groups.
Clarendon Press, London, 1986.
-
Prykarpatski A.K., Hentosh O.Ye., Prykarpatsky Ya.A.
Geometric Structure of the Classical Lagrange-d'Alambert Principle and its Application to Integrable NonlinearDynamical Systems.
Mathematics 2017, 5 (75), 1-20.
doi: 10.3390/math5040075
-
Prykarpatsky A.K., Mykytyuk I.V.
Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects.
Kluwer Academic Publishers, Netherlands, 1998.
-
Reyman A.G., Semenov-Tian-Shansky M.A.
Integrable Systems.
The Computer Research Institute Publ., Moscow-Izhvek, 2003. (in Russian)
-
Schief W.K.
Self-dual Einstein spaces and a discrete Tzitzeica equation. A permutability theorem link.
In: Clarkson P.A., Nijhoff F.W. (Eds.) Symmetries and Integrability of Difference Equations,
London Mathematical Society, Lecture Note Series 255, Cambridge University
Press, 1999, 137-148.
-
Schief W.K.
Self-dual Einstein spaces via a permutability theorem for the Tzitzeica equation.
Phys. Lett. A 1996, 223 (25), 55-62.
-
Semenov-Tian-Shansky M.A.
What is a classical R-matrix?
Funct. Anal. Appl. 1983, 17, 259-272.
doi: 10.1007/BF01076717
-
Sergyeyev A., Szablikowski B.M.
Central extensions of cotangent universal hierarrchy: (2+1)-dimensional bi-Hamiltonian systems
Phys. Lett. A 2008, 372 (47), 7016-7023.
doi: 10.1016/j.physleta.2008.10.020
-
Strachan I.A.B., Szablikowski B.M.
Novikov algebras and a classification of multicomponent Camassa-Holm equations.
Stud. Appl. Math. 2014, 133, 84-117.
doi: 10.1111/sapm.12040
-
Szablikowski B.
Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities.
SIGMA Symmetry Integrability Geom. Methods Appl. 2016, 12, 022, 14 pages.
doi: 10.3842/SIGMA.2016.022
-
Takasaki K., Takebe T.
Integrable Hierarchies and Dispersionless Limit.
Rev. Math. Phys. 1995, 7 (05), 743-808.
-
Takasaki K., Takebe T.
SDiff(2) Toda equation — Hierarchy, Tau function, and symmetries.
Lett. Math. Phys. 1991, 23 (3), 205-214.
doi: 10.1007/BF01885498
-
Takhtadjian L.A., Faddeev L.D.
Hamiltonian Approach in Soliton Theory.
Springer, Berlin-Heidelberg, 1987.
-
Warner F.W.
Foundations of Diffderentiable Manifolds and Lie Groups.
Springer, Ney York, 1983.
-
Weinstein A.
Sophus Lie and symplectic geometry.
Expos. Math. 1983, 1, 95-96.
-
Weinstein A.
The local structure of Poisson manifolds.
J. Differential Geom. 1983, 18, 523-557.