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Ward hypothesis for integrable systems

Prove the Ward hypothesis by establishing that every integrable system arises as a dimensional reduction of self-dual Yang–Mills theory, extending the currently verified cases (KdV, sine-Gordon, and nonlinear Schrödinger equations) to all integrable models.

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Background

The paper situates the AKNS system within a broader context connecting integrable systems to self-dual Yang–Mills (SDYM) theory and twistor geometry. In this framework, the Ward hypothesis asserts a unifying origin for integrable equations via dimensional reduction of SDYM. While this conjecture has been verified for several canonical models (KdV, sine-Gordon, NLS), its general validity across all integrable systems remains unproven. The authors reference this conjecture to motivate the significance of AKNS as a master system within the unified picture.

References

Particularly compelling is the Ward hypothesis , which posits that all integrable systems may emerge as dimensional reductions of self-dual Yang-Mills theory - a conjecture verified for key models including the KdV, sine-Gordon, and nonlinear Schr\"odinger equations.

— Analyzing the relationship between infinite symmetries and $N$-soliton solutions in the AKNS system (2510.19568 - Hao et al., 22 Oct 2025) in Introduction (Section 1), first paragraph (Page 1)