Auxiliary variables for nonlinear equations with softly broken symmetries

TL;DR

This paper introduces a method for solving nonlinear equations with softly broken symmetries by adding auxiliary variables, improving solution efficiency, and provides a Mathematica package implementing this approach.

Contribution

The paper presents a novel procedure for solving equations with softly broken symmetries by incorporating symmetry generators as auxiliary variables.

Findings

Method improves solution finding for softly broken symmetry problems.

Application to false vacuum decay physics demonstrates effectiveness.

Provides a Mathematica package for implementation.

Abstract

General methods of solving equations deal with solving N equations in N variables and the solutions are usually a set of discrete values. However, for problems with a softly broken symmetry these methods often first find a point which would be a solution if the symmetry were exact, and is thus an approximate solution. After this, the solver needs to move in the direction of the symmetry to find the actual solution, but that can be very difficult if this direction is not a straight line in the space of variables. The solution can often be found much more quickly by adding the generators of the softly broken symmetry as auxiliary variables. This makes the number of variables more than the equations and hence there will be a family of solutions, any one of which would be acceptable. In this paper we present a procedure for finding solutions in this case, and apply it to several simple examples and an important problem in the physics of false vacuum decay. We also provide a Mathematica package that implements Powell's hybrid method with the generalization to allow more variables than equations.