A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

Details A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems A Coordinate-Free Construction for a Class of Integrable Hydrodynamic-Type Systems

2008-03-03

arxiv.org/abs/0803.0308v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

11 pages, LaTeX 2e, major revision

Scientific paper

Using a (1,1)-tensor L with zero Nijenhuis torsion and maximal possible number (equal to the number of dependent variables) of distinct, functionally independent eigenvalues we define, in a coordinate-free fashion, the seed systems which are weakly nonlinear semi-Hamiltonian systems of a special form, and an infinite set of conservation laws for the seed systems. The reciprocal transformations constructed from these conservation laws yield a considerably larger class of hydrodynamic-type systems from the seed systems, and we show that these new systems are again defined in a coordinate-free manner, using the tensor L alone, and, moreover, are weakly nonlinear and semi-Hamiltonian, so their general solution can be obtained by means of the generalized hodograph method of Tsarev.

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