# Provable bounds for the Korteweg-de Vries reduction in multi-component Nonlinear Schrodinger Equation

## TL;DR

This paper rigorously derives bounds and conditions under which the multi-component VNLS can be approximated by a KdV system, providing insights into hydrodynamic effects in Bose gases and related fields.

## Contribution

It offers a rigorous linear analysis and a perturbative reduction of VNLS to KdV, clarifying the connection and conditions for validity in multi-component systems.

## Key findings

- Rigorous theorems on eigenvalues and eigenvectors of the linear problem.
- Validation of KdV approximation through soliton solutions.
- Relevance to Bose gases, nonlinear optics, and related fields.

## Abstract

We study the dynamics of multi-component Bose gas described by the Vector Nonlinear Schr\"{o}dinger Equation (VNLS), aka the Vector Gross--Pitaevskii Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to coupled hydrodynamic equations in terms of multiple density and velocity fields. Using a multi-scaling and a perturbation method along with the Fredholm alternative, we reduce the problem to a Korteweg de-Vries (KdV) system. This is of great importance to study more transparently, the obscure features hidden in VNLS. This ensures that hydrodynamic effects such as dispersion and nonlinearity are captured at an equal footing. Importantly, before studying the KdV connection, we provide a rigorous analysis of the linear problem. We write down a set of theorems along with proofs and associated corollaries that shine light on the conditions of existence and nature of eigenvalues and eigenvectors of the linear problem. This rigorous analysis is paramount for understanding the nonlinear problem and the KdV connection. We provide strong evidence of agreement between VNLS systems and KdV equations by using soliton solutions as a platform for comparison. Our results are expected to be relevant not only for cold atomic gases, but also for nonlinear optics and other branches where VNLS equations play a defining role.