In physics, a breather is a
nonlinearwave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for
infinitesimalamplitudes, 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.
This breather pseudospherical surface corresponds to a solution of a non-linear wave-equation.Pseudospherical breather surface
Overview
Sine-Gordon standing breather is a swinging in time coupled kink-antikink 2-soliton solution.Large amplitude moving
sine-Gordon breather.
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
phononspectrum of the lattice.
Example of a breather solution for the sine-Gordon equation
with u a
complex field as a function of x and t. Further i denotes the
imaginary unit.
One of the breather solutions (Kuznetsov-Ma breather) 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]
^
abN. 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.
S2CID18571794. Translated from Teoreticheskaya i Matematicheskaya Fizika 72(2): 183–196, August, 1987.
^N. N. Akhmediev; A. Ankiewicz (1997). Solitons, non-linear pulses and beams. Springer.
ISBN978-0-412-75450-0.