Classical Matrix sine-Gordon Theory

TL;DR

This paper explores the classical integrable matrix sine-Gordon theory, providing exact solutions, solitons, and conservation laws, and extends the understanding of its mathematical structure and physical properties.

Contribution

It introduces the $A_3$-generalization of the matrix sine-Gordon theory, deriving solutions, conservation laws, and Bäcklund transformations using the zero curvature formalism.

Findings

Exact soliton and breather solutions found

Infinite conservation laws established

Bäcklund transformation and superposition principle derived

Abstract

The matrix sine-Gordon theory, a matrix generalization of the well-known sine-Gordon theory, is studied. In particular, the $A_{3}$-generalization where fields take value in $SU(2)$ describes integrable deformations of conformal field theory corresponding to the coset $SU(2) \times SU(2) /SU(2)$. Various classical aspects of the matrix sine-Gordon theory are addressed. We find exact solutions, solitons and breathers which generalize those of the sine-Gordon theory with internal degrees of freedom, by applying the Zakharov-Shabat dressing method and explain their physical properties. Infinite current conservation laws and the B\"{a}cklund transformation of the theory are obtained from the zero curvature formalism of the equation of motion. From the B\"{a}cklund transformation, we also derive exact solutions as well as a nonlinear superposition principle by making use of the Bianchi's permutability theorem.