Statistical Mechanics of the Integrable Models

Abstract

There is an infinity of classically integrable models. The only ones we can consider here, and these only briefly, are: the sine-Gordon (s-G) model

$${\phi _{{\rm{xx}}}}{}^ - {\phi _{{\rm{tt}}}} = {{\rm{m}}^2}\sin \phi ,$$

(1.1) the sinh-Gordon (sinh-G) model

$${\phi _{{\rm{xx}}}}{}^ - {\phi _{{\rm{tt}}}} = {{\rm{m}}^2}\sinh \phi ,$$

(1.2) and the repulsive and attractive non-linear Schrödinger (NLS) models

$${}^ - {\rm{i}}{\phi _{\rm{t}}} = {\phi _{{\rm{xx}}}}{}^ - 2{\rm{c}}\phi {\left| \phi \right|^2}.$$

(1.3) The “attractive” NLS has real coupling constant c < 0; the “repulsive” has c > 0; φ is complex. In (1.1) and (1.2) m is a mass (ħ = c = 1) and φ is real. These 4 integrable models are in one space and one time (1+1) dimensions. There are integrable models in 2+1 dimensions we cannot discuss here [1].