Definitions

# Breather

[bree-ther]

A breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards an even distribution of initially localized energy.

A discrete breather is a breather solution on a nonlinear lattice.

The term breather originates from the characteristic that most breathers are localized in space and oscillate (breath) in time. But also the opposite situation: oscillations in space and localized in time, is denoted as a breather.

## Overview

A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon equation and the focusing nonlinear Schrödinger equation are examples of one-dimensional partial differential equations that possess breather solutions. Discrete nonlinear Hamiltonian lattices in many cases support breather solutions. Breathers are solitonic structures. There are two types of breathers: standing or traveling ones. Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called oscillons). A necessary condition for the existence of breathers in discrete lattices is that the breather main frequency and all its multipliers are located outside of the phonon spectrum of the lattice.

## Example of a breather solution for the sine-Gordon equation

The sine-Gordon equation is the nonlinear dispersive partial differential equation

$frac\left\{partial^2 u\right\}\left\{partial t^2\right\} - frac\left\{partial^2 u\right\}\left\{partial x^2\right\} + sin u = 0,$

with the field u a function of the spatial coordinate x and time t.

An exact solution found by using the inverse scattering transform is:

$u = 4 arctanleft\left(frac\left\{sqrt\left\{1-omega^2\right\};cos\left(omega t\right)\right\}\left\{omega;cosh\left(sqrt\left\{1-omega^2\right\}; x\right)\right\}right\right),$

which, for ω < 1, is periodic in time t and decays exponentially when moving away from x = 0.

## Example of a breather solution for the nonlinear Schrödinger equation

The focusing nonlinear Schrödinger equation is the dispersive partial differential equation:

$i,frac\left\{partial u\right\}\left\{partial t\right\} + frac\left\{partial^2 u\right\}\left\{partial x^2\right\} + |u|^2 u = 0,$

with u a complex field as a function of x and t. Further i denotes the imaginary unit.

One of the breather solutions is


` u =`
` left(`
frac{2, b^2 cosh(theta) + 2, i, b, sqrt{2-b^2}; sinh(theta)} {2, cosh(theta)-sqrt{2},sqrt{2-b^2} cos(a, b, x)}
`   - 1`
` right);`
` a; exp(i, a^2, t)`
quadtext{with}quad theta=a^2,b,sqrt{2-b^2};t, which gives breathers periodic in space x and approaching the uniform value a when moving away from the focus time t = 0. These breathers exist for values of the modulation parameter b less than √ 2.