Breather
In physics, a breather is a nonlinear[disambiguation needed] 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 (
Overview
A breather is a localized
Breathers are
Example of a breather solution for the sine-Gordon equation
The sine-Gordon equation is the nonlinear dispersive partial differential equation
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]
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:
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]
with
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]
See also
References and notes
- ^ .
- ^ S2CID 18571794. Translated from Teoreticheskaya i Matematicheskaya Fizika 72(2): 183–196, August, 1987.
- ISBN 978-0-412-75450-0.
- ^ Miroshnichenko A, Vasiliev A, Dmitriev S. Solitons and Soliton Collisions.
- ^ 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.
- .