Classification of quasiperiodic isodispersive phases

Characterize all quasiperiodic isodispersive phases for reflectionless potentials of the Lax operator L in the classical integrable system defined by the Lax pair (L, M) and evolution equations (iu_t = 2(q^* ξ_x − q ξ^*_x), iv_t = 2(ξ ξ_x^* − ξ^* ξ_x)_x, i ξ_t = −2 v ξ + q_x, i q_t = −2 v q − 2 ξ u_x − 4 ξ_x u + 4 ξ (|ξ|^2)_x − ξ_{xxx}) by determining the conserved densities ρ_n^{R,I} required to classify such phases beyond the uniform-phase invariants ρ_1^I = u − 2|ξ|^2 and (ρ_3^I)_{uniform} = |q|^2 + (u^2 + v^2)/2 + 2 u |ξ|^2 − 2 |ξ|^4.

Background

The paper introduces isodispersive phases to describe backgrounds at x → ±∞ that share the same dispersion relation λ(k) for the Lax operator L. While uniform isodispersive phases can be classified by low-order conserved densities, the tsunami-soliton dynamics produce oscillatory and quasiperiodic backgrounds whose classification is nontrivial.

The authors suggest that a complete classification of these quasiperiodic isodispersive phases may require higher-order conserved densities ρ_n^{R,I}, but this requirement and the full classification framework remain unresolved.