WordNet

draw air into, and expel out of, the lungs; "I can breathe better when the air is clean"; "The patient is respiring" (同)take a breath, respire, suspire
reach full flavor by absorbing air and being let to stand after having been uncorked; "This rare Bordeaux must be allowed to breathe for at least 2 hours"
utter or tell; "not breathe a word"
allow the passage of air through; "Our new synthetic fabric breathes and is perfect for summer wear"
be alive; "Every creature that breathes"
impart as if by breathing; "He breathed new life into the old house"
manifest or evince; "She breathes the Christian spirit"
a short respite (同)breather, breathing place, breathing space, breathing spell, breathing time
a slight movement of the air; "there wasnt a breath of air in the room"
the air that is inhaled and exhaled in respiration; "his sour breath offended her"
the process of taking in and expelling air during breathing; "he took a deep breath and dived into the pool"; "he was fighting to his last breath"

PrepTutorEJDIC

《修飾語を伴って》(…の)呼吸をする人(動物) / 《話》一息,一休み
『呼吸する』,息をする / 一息いれる,休息する / 生きている,生存する / 〈風などが〉そよぐ / 〈空気〉‘を'『呼吸する』 / (…に)…‘を'吹き込む《+『名』+『into』+『名』》 / …‘を'口に出す,〈秘密など〉‘を'もらす,ささやく〈ため息など〉‘を'もらす / …‘に'一息入れさせる
〈U〉『息』,『呼吸』;UC〉1回の吸気(め気) / 〈C〉(風の)そよぎ;(…の)かすかなかおり《+『of』+『名』》 / 〈C〉(…の)ささやき(whisper)《+『of』+『名』》

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/06/08 21:38:04」(JST)

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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

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]

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

^ ^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.
^ ^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.
^ N. N. Akhmediev; A. Ankiewicz (1997). Solitons, non-linear pulses and beams. Springer. 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.
^ 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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