Harry-Dym Hierarchy: Hereditary Operator and Ablowitz-Ramani-Segur Conjecture

Using Lax representation, hereditary operator related to the Harry-Dym ( HD ) hierarchy is constructed. Also Lie analysis and Lax pair are used to drive the conserved quantities for the HD hierarchy. Lie symmetry method is also applied to study Painlev´e test for the family of HD hierarchy. It is found that for this family the Painlev´e property can be at most sufficient for integrability, but not necessary. This fact has been proved earlier for just first member of this family.

Keywords:

Lie analysis , conservation laws , hereditary operator , Painlev´e property , ARS conjecture