Recursions of Symmetry Orbits and Reduction without Reduction

Abstract

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Ampère 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 Kähler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation.

References in corpus (5)

• On the integrability of symplectic Monge-Ampére equations
• On a class of second-order PDEs admitting partner symmetries
• General heavenly equation governs anti-self-dual gravity
• Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics
• Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Cited by in corpus (5)

• Recursion Operators and Nonlocal Symmetries for Integrable rmdKP and rdDym Equations
• General heavenly equation governs anti-self-dual gravity
• Recursion Operators and Tri-Hamiltonian Structure of the First Heavenly Equation of Plebański
• Recursion operators and bi-Hamiltonian structure of the general heavenly equation
• Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors