Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Pleba\'nski

We present first heavenly equation of Pleba\'nski in a two-component evolutionary form and obtain Lagrangian and Hamiltonian representations of this system. We study all point symmetries of the two-component system and, using the inverse Noether theorem in the Hamiltonian form, obtain all the integrals of motion corresponding to each variational (Noether) symmetry. We derive two linearly independent recursion operators for symmetries of this system related by a discrete symmetry of both the two-component system and its symmetry condition. Acting by these operators on the first Hamiltonian operator $J_0$ we obtain second and third Hamiltonian operators. However, we were not able to find Hamiltonian densities corresponding to the latter two operators. Therefore, we construct two recursion operators, which are either even or odd, respectively, under the above-mentioned discrete symmetry. Acting with them on $J_0$, we generate another two Hamiltonian operators $J_+$ and $J_-$ and find the corresponding Hamiltonian densities, thus obtaining second and third Hamiltonian representations for the first heavenly equation in a two-component form. Using P. Olver's theory of the functional multi-vectors, we check that the linear combination of $J_0$, $J_+$ and $J_-$ with arbitrary constant coefficients satisfies Jacobi identities. Since their skew symmetry is obvious, these three operators are compatible Hamiltonian operators and hence we obtain a tri-Hamiltonian representation of the first heavenly equation. Our well-founded conjecture applied here is that P. Olver's method works fine for nonlocal operators and our proof of the Jacobi identities and bi-Hamiltonian structures crucially depends on the validity of this conjecture.Comment: Some text overlap with our paper arXiv:1510.03666 is caused by our use here of basically the same method for discovering the Hamiltonian and bi-Hamiltonian structures of the equation, but the equation considered here and the results are totally different from arXiv:1510.0366

Recursions of Symmetry Orbits and Reduction without Reduction

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Amp\`ere equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat K\"ahler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation

Symmetry analysis and exact solutions of modified Brans-Dicke cosmological equations

We perform a symmetry analysis of modified Brans-Dicke cosmological equations and present exact solutions. We discuss how the solutions may help to build models of cosmology where, for the early universe, the expansion is linear and the equation of state just changes the expansion velocity but not the linearity. For the late universe the expansion is exponential and the effect of the equation of state on the rate of expansion is just to change the constant Hubble parameter.Comment: LaTeX2e source file, 14 pages, 7 reference