The dynamical sine-Gordon model

TL;DR

This paper introduces and analyzes the well-posedness of the two-dimensional dynamical sine-Gordon equation with parameter beta, employing different mathematical techniques depending on the regime of beta, and connects it to physical interface models.

Contribution

It establishes the well-posedness of the dynamical sine-Gordon model in 2D for a range of beta using Wick renormalization, Da Prato-Debussche, and regularity structures, and links it to interface fluctuation models.

Findings

Well-posedness for beta^2 in (0, 16pi/3)

Application of Da Prato-Debussche method for beta^2 in (0, 4pi)

Use of regularity structures for beta^2 in [4pi, 16pi/3)

Abstract

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter $\beta$, which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when $\beta^2 \in (0,\frac{16\pi}{3})$ the Wick renormalised equation is well-posed. In the regime $\beta^2 \in (0,4\pi)$, the Da Prato-Debussche method applies, while for $\beta^2 \in [4\pi,\frac{16\pi}{3})$, the solution theory is provided via the theory of regularity structures (Hairer 2013). We also show that this model arises naturally from a class of $2+1$-dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed by Lacoin, Rhodes and Vargas (2013).