$\text{T}\bar{\text{T}}$-deformed Nonlinear Schr\"odinger

TL;DR

This paper extends the $ ext{T}ar{ ext{T}}$ deformation concept to non-relativistic models, specifically the nonlinear Schrödinger equation, deriving the deformed Lagrangian and analyzing soliton solutions and their properties.

Contribution

It introduces a novel $ ext{T}ar{ ext{T}}$-deformed Lagrangian for non-relativistic models like NLS and explores the resulting soliton solutions and deformation characteristics.

Findings

Deformed Lagrangian contains a square-root form similar to relativistic cases.

The $ ext{T}ar{ ext{T}}$-deformation does not trivially follow from non-relativistic limits of relativistic theories.

Derived Poisson brackets and equations of motion for the deformed models.

Abstract

The $\text{T}\bar{\text{T}}$-deformed classical Lagrangian of a 2D Lorentz invariant theory can be derived from the original one, perturbed only at first order by the bare $\text{T}\bar{\text{T}}$ composite field, through a field-dependent change of coordinates. Considering, as an example, the nonlinear Schr\"odinger (NLS) model with generic potential, we apply this idea to non-relativistic models. The form of the deformed Lagrangian contains a square-root and is similar but different from that for relativistic bosons. We study the deformed bright, grey and Peregrine's soliton solutions. Contrary to naive expectations, the $\text{T}\bar{\text{T}}$-perturbation of nonlinear Schr\"odinger NLS with quartic potential does not trivially emerge from a standard non-relativistic limit of the deformed sinh-Gordon field theory. The $c \rightarrow \infty$ outcome corresponds to a different type of irrelevant deformation. We derive the corresponding Poisson bracket structure, the equations of motion and discuss various interesting aspects of this alternative type of perturbation, including links with the recent literature.