The integrability of an m-component system of hydrodynamic type, Ut = v(u)ux,
by the generalized hodograph method requires the diagonalizability of the m x m
matrix v(u). The diagonalizability is known to be equivalent to the vanishing of
the corresponding Haantjes tensor. This idea is applied to hydrodynamic chains -
infinite-component systems of hydrodynamic type for which the 00 x 00 matrix v(u)
is 'sufficiently sparse'. For such 'sparse' systems the Haantjes tensor is well-defined,
and the calculation of its components involves only a finite number of summations.
The calculation of the Haantjes tensor is done by using Mathematica to perform
symbolic calculations. Certain conservative and Hamiltonian hydrodynamic chains
are classified by setting Haantjes tensor equal to zero and solving the resulting system
of equations. It is shown that the vanishing of the Haantjes tensor is a necessary
condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic
reductions, thus providing an easy-to-verify necessary condition for the
integrability of such sysyems. In the cases of the Hamiltonian hydrodynamic chains
we were able to first construct one extra conservation law and later a generating
function for conservation laws, thus establishing the integrability.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.