Reflective prolate-spheroidal operators and the KP/KdV equations

Area of Science:

• Integrable Systems
• Mathematical Physics
• Signal Processing

Background:

• Commuting integral and differential operators link signal processing, random matrix theory, and integrable systems.
• Previous constructions focused on specific cases, omitting families like Korteweg-de Vries (KdV) equation rational solutions.

Purpose of the Study:

• To prove a general theorem that integral operators associated with wave functions in the infinite-dimensional adelic Grassmannian reflect differential operators.
• To establish new methods for constructing commuting integral and differential operators for Kadomtsev-Petviashvili (KP) hierarchies.

Main Methods:

• Utilizing symmetries of Grassmannians of Kadomtsev-Petviashvili (KP) wave functions.
• Applying a [Formula: see text] rotation argument for generalized Fourier transforms.
• Analyzing truncated generalized Laplace and Fourier transforms of bispectral wave functions.

Main Results:

• A general theorem states integral operators associated with Wilson's adelic Grassmannian wave functions reflect differential operators.
• Integral operators in singular value computations for generalized Laplace transforms of rank 1 bispectral wave functions reflect differential operators.
• Integral operators in singular value computations for generalized Fourier transforms of KP wave functions commute with differential operators.

Conclusions:

• The study provides a unified framework for constructing commuting integral and differential operators.
• New collections of integral operators with prolate-spheroidal properties are generated, including those for KdV and KP hierarchies.
• The findings advance the understanding of integrable systems and their applications in mathematical physics and signal processing.