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April 16, 20260 citationsOpen Access

Painleve and Tau-Function Geometry of Derivative-Order Ladders: Hamiltonian Ladders, Hirota Transmutation, and Exact Sigma-State Recovery

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MAMohammad Abu-Ghuwaleh

Key Points

  • The central aim is to explore the tau-function geometry associated with derivative-order ladders and Painleve equations.
  • Developed the concept of Hamiltonian derivative-order ladders.
  • Defined sigma-form closure relations for ladder flows.
  • Derived bilinear transmutation formulas using Hirota's method.
  • Extended the geometry to multitime tau ladders linked with integrable hierarchies.
  • Established methods for exact sigma-state recovery from finite ladder windows.
  • Introduced a new framework for understanding derivative-order geometry through tau-functions.
  • Demonstrated the effectiveness of bilinear transmutation formulas in this context.
  • Validated the approach by showing exact recovery of sigma-states from ladder windows.

Abstract

This preprint develops the tau-function face of derivative-order geometry and explores its connections with Painleve equations. It defines Hamiltonian derivative-order ladders and sigma-form closure relations for ladder flows. The paper derives bilinear (Hirota) transmutation formulas and extends derivative-order geometry to multitime tau ladders associated with integrable hierarchies. It also establishes exact sigma-state recovery from finite ladder windows.

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Cite This Study

Mohammad Abu-Ghuwaleh (2026) studied this question.

synapsesocial.com/papers/69e07dc72f7e8953b7cbeb48https://doi.org/10.5281/zenodo.19569478
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