Complete Whitham theory and finite-genus analysis for the Manakov system

Establish a satisfactory characterization of finite-genus solutions, solve the initial-value problem for the Manakov system with periodic boundary conditions, and develop a comprehensive Whitham modulation theory for the slow modulation of its periodic and quasi-periodic wave solutions.

Background

The paper places its genus-zero analysis in the context of several broader unresolved problems for the integrable two-component Manakov system. In particular, the authors identify gaps in the characterization of finite-genus solutions, the periodic-boundary initial-value problem, and the modulation theory of periodic and quasi-periodic waves.

The results of the paper address the dispersionless, genus-zero case by deriving the four-component Manakov–Whitham system, identifying its local Riemann invariants, and classifying plane-wave modulational stability. They therefore do not resolve the broader finite-genus and fully periodic problems listed here.

References

On the other hand, several fundamental questions concerning the Manakov system remain open. Prominent examples of this state of affairs are the lack of a satisfactory characterization of its finite-genus solutions, the lack of a satisfactory solution of the initial value problem with periodic boundary conditions, and the lack of a comprehensive Whitham modulation theory, (i.e., a systematic theory of the slow modulation of its periodic and quasi-periodic wave solutions).

— On the dispersionless limit of the Manakov system, its Riemann invariants, and the modulational stability of its counterpropagating plane waves  (2609.26669 - Adriazola et al., 22 Sep 2026) in Introduction