623 research outputs found
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
Universal description of three two-component fermions
A quantum mechanical three-body problem for two identical fermions of mass
and a distinct particle of mass in the universal limit of zero-range
two-body interaction is studied. For the unambiguous formulation of the problem
in the interval ( and ) an additional parameter determining the wave function near
the triple-collision point is introduced; thus, a one-parameter family of
self-adjoint Hamiltonians is defined. The dependence of the bound-state
energies on and in the sector of angular momentum and parity is calculated and analysed with the aid of a simple model
On a class of second-order PDEs admitting partner symmetries
Recently we have demonstrated how to use partner symmetries for obtaining
noninvariant solutions of heavenly equations of Plebanski that govern heavenly
gravitational metrics. In this paper, we present a class of scalar second-order
PDEs with four variables, that possess partner symmetries and contain only
second derivatives of the unknown. We present a general form of such a PDE
together with recursion relations between partner symmetries. This general PDE
is transformed to several simplest canonical forms containing the two heavenly
equations of Plebanski among them and two other nonlinear equations which we
call mixed heavenly equation and asymmetric heavenly equation. On an example of
the mixed heavenly equation, we show how to use partner symmetries for
obtaining noninvariant solutions of PDEs by a lift from invariant solutions.
Finally, we present Ricci-flat self-dual metrics governed by solutions of the
mixed heavenly equation and its Legendre transform.Comment: LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions
of the Legendre transformed mixed heavenly equation and Ricci-flat metrics
governed by solutions of this equation are added. Eq. (6.10) on p. 14 is
correcte
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