On compressing sinh-Gordon solutions

TL;DR

This paper introduces a method to compress solutions of the sinh-Gordon equation into small regions using approximate non-linear transformations, enabling efficient representation and analysis of these solutions in anti-de Sitter space.

Contribution

It presents a novel approach to approximate sinh-Gordon solutions via piecewise linear strings, demonstrating how these transformations preserve dynamics and enable detailed study of specific solutions.

Findings

Compressed sinh-Gordon solutions exhibit a comb-like structure.

Transformation commutes with time evolution in the large-N limit.

Detailed analysis of static cosh-Gordon solutions and their compressed forms.

Abstract

This paper is concerned with a class of approximate non-linear transformations that compress solutions of the (generalized) sinh-Gordon equation into parametrically small regions in two-dimensional spacetime. Given the sinh-Gordon field near a time-slice, a long Nambu-Goto string can be constructed in three-dimensional anti-de Sitter space. The string is then approximated to arbitrary accuracy by a slightly smoothed piecewise linear string of N segments. The corresponding sinh-Gordon field has a comb-like structure and its size is controlled by the amount of smoothing applied to the segmented string. In a (singular) large-N limit, the transformation commutes with time evolution. As an example, a static cosh-Gordon solution is discussed in detail. The corresponding smooth and segmented string solutions are obtained and the compressed cosh-Gordon potential is investigated.