# Time dynamics of Bethe ansatz solvable models

## TL;DR

This paper introduces a method to compute the time evolution of Bethe ansatz solvable models by deriving dynamical Bethe equations, enabling exact solutions for models like the Bose-Hubbard dimer and Tavis-Cummings.

## Contribution

It develops a novel approach to determine the dynamics of exactly solvable models using time-dependent Bethe parameters and differential equations.

## Key findings

- Derivation of dynamical Bethe equations for time evolution.
- Exact solutions for Bose-Hubbard dimer and Tavis-Cummings models.
- Extension beyond the Gaudin class of models.

## Abstract

We develop a method for finding the time evolution of exactly solvable models by Bethe ansatz. The dynamical Bethe wavefunction takes the same form as the stationary Bethe wavefunction except for time varying Bethe parameters and a complex phase prefactor. From this, we derive a set of first order nonlinear coupled differential equations for the Bethe parameters, called the dynamical Bethe equations. We find that this gives the exact solution to particular types of exactly solvable models, including the Bose-Hubbard dimer and Tavis-Cummings model. These models go beyond the Gaudin class, and offers an interesting possibility for performing time evolution in exactly solvable models.