# Breather

For the component in an internal combustion engine, see Crankcase ventilation system.

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.[1] But also the opposite situation: oscillations in space and localized in time[clarification needed], is denoted as a breather.

## Overview

Sine-Gordon standing breather is a swinging in time coupled kink-antikink 2-soliton solution.
Large amplitude moving sine-Gordon breather.

A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon equation[1] and the focusing nonlinear Schrödinger equation[2] are examples of one-dimensional partial differential equations that possess breather solutions.[3] Discrete nonlinear Hamiltonian lattices in many cases support breather solutions.

Breathers are solitonic structures. There are two types of breathers: standing or traveling ones.[4] 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:[1]

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

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 [5] 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 [2]

$u = \left( \frac{2\, b^2 \cosh(\theta) + 2\, i\, b\, \sqrt{2-b^2}\; \sinh(\theta)} {2\, \cosh(\theta)-\sqrt{2}\,\sqrt{2-b^2} \cos(a\, b\, x)} - 1 \right)\; a\; \exp(i\, a^2\, t) \quad\text{with}\quad \theta=a^2\,b\,\sqrt{2-b^2}\;t,$

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. Note that a limiting case of the breather solution is the Peregrine soliton.[6]