Direct linearization approach to discrete integrable systems associated with ℤ_�� graded Lax pairs

Area of Science:

• Mathematical Physics
• Integrable Systems
• Difference Equations

Background:

• Fordy and Xenitidis introduced novel discrete integrable systems using graded Lax pairs.
• These systems, while significant, lacked explicit solutions or detailed structural analysis.

Purpose of the Study:

• To establish a connection between Fordy-Xenitidis (FX) discrete systems and linear integral equations.
• To reveal the underlying solution structure of these FX discrete systems.
• To present the bilinear form and general tau function for the FX integrable difference equations.

Main Methods:

• Establishing a link between FX discrete systems and specific linear integral equations.
• Deriving the bilinear form of the FX integrable difference equations.
• Analyzing the general tau function associated with these systems.

Main Results:

• A clear link is established between the coprime case of FX discrete systems and linear integral equations.
• The solution structure of the FX discrete systems is revealed through this connection.
• The bilinear form and general tau function for the FX integrable difference equations are presented.
• Connections between FX models and the discrete Gel'fand-Dikii hierarchy are elucidated.

Conclusions:

• The study successfully reveals the solution structure of Fordy-Xenitidis discrete systems.
• The findings provide a deeper understanding of the relationships between these novel integrable systems and established mathematical hierarchies.