# Fractional Volterra Hierarchy

## TL;DR

This paper introduces the fractional Volterra hierarchy, a new integrable system generalizing the Volterra lattice, with formal definitions, tau functions, and multi-soliton solutions, motivated by cubic Hodge integrals and Calabi-Yau conditions.

## Contribution

It defines the fractional Volterra hierarchy using Lax pairs and Hamiltonian formalism, and constructs its tau functions and multi-soliton solutions, expanding integrable systems theory.

## Key findings

- Defined the fractional Volterra hierarchy mathematically.
- Constructed explicit tau functions for the hierarchy.
- Presented multi-soliton solutions demonstrating integrability.

## Abstract

The generating function of cubic Hodge integrals satisfying the local Calabi-Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional generalization of the Volterra lattice hierarchy, so we name it the fractional Volterra hierarchy. In this paper, we give the definition of this integrable hierarchy in terms of Lax pair and Hamiltonian formalisms, construct its tau functions, and present its multi-soliton solutions.