Kernel Formula Approach to the Universal Whitham Hierarchy

TL;DR

This paper derives the dispersionless Hirota equations for the universal Whitham hierarchy using the kernel formula approach, verifies associativity equations, and constructs an algebra with structure constants expressed via residue formulas.

Contribution

It introduces a kernel formula method to derive dispersionless Hirota equations and verifies associativity, providing a new algebraic realization for the universal Whitham hierarchy.

Findings

Derived dispersionless Hirota equations from kernel formulas

Verified associativity equations within the hierarchy

Constructed an algebra with residue-based structure constants

Abstract

We derive the dispersionless Hirota equations of the universal Whitham hierarchy from the kernel formula approach proposed by Carroll and Kodama. Besides, we also verify the associativity equations in this hierarchy from the dispersionless Hirota equations and give a realization of the associative algebra with structure constants expressed in terms of the residue formulas.