# Breather

In physics, 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 (breathe) in time. But also the opposite situation: oscillations in space and localized in time[clarification needed], is denoted as a breather. This breather pseudospherical surface corresponds to a solution of a non-linear wave-equation.

## Overview Sine-Gordon standing breather is a swinging in time coupled kink-antikink 2-soliton solution.

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

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

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

$i\,{\frac {\partial u}{\partial t}}+{\frac {\partial ^{2}u}{\partial x^{2}}}+|u|^{2}u=0,$ with u a complex field as a function of x and t. Further i denotes the imaginary unit.

One of the breather solutions is 

$u=\left({\frac {2b^{2}\cosh(\theta )+2ib{\sqrt {2-b^{2}}}\sinh(\theta )}{2\cosh(\theta )-{\sqrt {2}}{\sqrt {2-b^{2}}}\cos(abx)}}-1\right)a\,e^{ia^{2}t}$ with

$\theta =a^{2}\,b\,{\sqrt {2-b^{2}}}\;t,$ 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. Note that a limiting case of the breather solution is the Peregrine soliton.