# Breather

Breather

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, 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].

## References and notes

1. ^ a b c M. J. Ablowitz; D. J. Kaup ; A. C. Newell ; H. Segur (1973). "Method for solving the sine-Gordon equation". Physical Review Letters 30 (25): 1262–1264. Bibcode 1973PhRvL..30.1262A. doi:10.1103/PhysRevLett.30.1262.
2. ^ a b N. N. Akhmediev; V. M. Eleonskiǐ; N. E. Kulagin (1987). "First-order exact solutions of the nonlinear Schrödinger equation". Theoretical and Mathematical Physics 72 (2): 809–818. Bibcode 1987TMP....72..809A. doi:10.1007/BF01017105.  Translated from Teoreticheskaya i Matematicheskaya Fizika 72(2): 183–196, August, 1987.
3. ^ N. N. Akhmediev; A. Ankiewicz (1997). Solitons, non-linear pulses and beams. Springer. ISBN 978-0-412-75450-0.
4. ^ Miroshnichenko A, Vasiliev A, Dmitriev S. Solitons and Soliton Collisions.
5. ^ The focusing nonlinear Schrödinger equation has a nonlinearity parameter κ of the same sign (mathematics) as the dispersive term proportional to 2u/∂x2, and has soliton solutions. In the de-focusing nonlinear Schrödinger equation the nonlinearity parameter is of opposite sign.
6. ^ Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J.M. (2010). "The Peregrine soliton in nonlinear fibre optics". Nature Physics 6 (10): 790. Bibcode 2010NatPh...6..790K. doi:10.1038/nphys1740.

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