Physical interpretation of the transformation group for isodispersive phases

Clarify the physical interpretation of the transformation group connecting isodispersive phases, particularly when the set of isodispersive phases is non-compact for reflectionless potentials generated by the Lax operator L in the integrable system defined by the Lax pair (L, M).

Background

For uniform backgrounds, the isodispersive phases are related by standard gauge transformations (e.g., U(1) ⊗ SO(3) in spinor condensates), and the corresponding transformation group has a clear physical meaning.

In the tsunami-soliton regime, more complicated quasiperiodic backgrounds arise and the set of isodispersive phases may be non-compact, for which the physical meaning of the connecting transformations is not yet understood.