The Symmetric Chiral Field Equation

TL;DR

This paper investigates the symmetric chiral field equation, deriving conservation laws, explicit soliton solutions, and establishing its equivalence to the Sine-Gordon equation, with implications for gravitational solitons.

Contribution

It introduces a recursive formula for conservation laws, explicit N-soliton solutions, and proves the equivalence to the Sine-Gordon equation, advancing understanding of symmetric chiral fields.

Findings

Infinite local conservation laws for the symmetric chiral field.

Explicit N-soliton solutions on arbitrary backgrounds.

Equivalence to the Sine-Gordon equation.

Abstract

The reduction problem of the chiral field equation on symmetric spaces is studied. It is shown that the symmetric chiral field has infinitely many local conservation laws. A recursive formula for these conservation laws is derived and the first associated integral of motion are given explicitly. Furthermore, the Zakharov-Mikhailov (inverse scattering) transform is used to derive explicit formulas for the N-soliton solution on arbitrary diagonal background. The solitons' properties and interactions are analyzed. Such solitons are naturally related to gravitational solitons of Einstein's field equations, and this result is used to clarify why the latter do not have fixed amplitude and velocities (unlike `classical' solitons). Finally, it is proven that the symmetric chiral field (matrix) equation is equivalent to a single scalar equation, which in turn, is equivalent to the Sine-Gordon equation.