Hydrodynamic reductions of the heavenly equation

Details Hydrodynamic reductions of the heavenly equation Hydrodynamic reductions of the heavenly equation

2003-01-31

arxiv.org/abs/nlin/0301048v1

Class.Quant.Grav. 20 (2003) 2429-2442

Nonlinear Sciences

Exactly Solvable and Integrable Systems

14 pages

Scientific paper

10.1088/0264-9381/20/11/331

We demonstrate that Pleba\'nski's first heavenly equation decouples in infinitely many ways into a triple of commuting (1+1)-dimensional systems of hydrodynamic type which satisfy the Egorov property. Solving these systems by the generalized hodograph method, one can construct exact solutions of the heavenly equation parametrized by arbitrary functions of a single variable. We discuss explicit examples of hydrodynamic reductions associated with the equations of one-dimensional nonlinear elasticity, linearly degenerate systems and the equations of associativity.

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