New hierarchies of isospectral and non-isospectral integrable NLEEs derived from the Harry–Dym spectral problem

By making use of our previous framework, we display the hierarchies of generalized nonlinear evolution equations (GNLEEs) associated with the Harry–Dym spectral problem, and construct the corresponding generalized Lax representations (GLR) in this paper. It will be clearly seen that the generalized hierarchies can include not only the well-known Harry–Dym hierarchy of isospectral NLEEs but also other new integrable hierarchies of isospectral and non-isospectral NLEEs. Through choosing the so-called ‘seed’ function these new hierarchies of NLEEs give some new integrable evolution equations, which are very likely to have potential applications in theoretical and experimental physics. All of these hierarchies of NLEEs possess the GLR.