Kernel Formula Approach to the Universal Whitham Hierarchy

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.

💡 Research Summary

The paper presents a unified derivation of the dispersionless Hirota equations for the universal Whitham hierarchy by employing the kernel‑formula technique originally introduced by Carroll and Kodama. After a concise review of the universal Whitham hierarchy—its Lax and Orlov‑Schulman functions, Poisson brackets, and the infinite set of time variables—the authors adapt the kernel identity
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