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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.

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.

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

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:

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

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

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

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{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.

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Last updated on Wednesday October 01, 2008 at 20:03:11 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 01, 2008 at 20:03:11 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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