Breather
   HOME

TheInfoList



OR:

In physics, a breather is a
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
s, 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 Breathing (or ventilation) is the process of moving air into and from the lungs to facilitate gas exchange with the internal environment, mostly to flush out carbon dioxide and bring in oxygen. All aerobic creatures need oxygen for cell ...
e) in time. But also the opposite situation: oscillations in space and localized in time, is denoted as a breather.


Overview

A breather is a localized periodic solution of either
continuous media Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
equations or discrete lattice equations. The exactly solvable
sine-Gordon equation The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surf ...
and the focusing
nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
Translated from ''Teoreticheskaya i Matematicheskaya Fizika'' 72(2): 183–196, August, 1987. are examples of one- dimensional
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s that possess breather solutions. Discrete nonlinear Hamiltonian lattices in many cases support breather solutions. Breathers are
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...
ic structures. There are two types of breathers:
standing Standing, also referred to as orthostasis, is a position in which the body is held in an ''erect'' ("orthostatic") position and supported only by the feet. Although seemingly static, the body rocks slightly back and forth from the ankle in the ...
or
traveling Travel is the movement of people between distant geographical locations. Travel can be done by foot, bicycle, automobile, train, boat, bus, airplane, ship or other means, with or without luggage, and can be one way or round trip. Travel ca ...
ones. Standing breathers correspond to localized solutions whose amplitude vary in time (they are sometimes called
oscillon In physics, an oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in granular media result from vertically vibrating a plate with a layer of uniform particles placed freely on top. When the sinusoida ...
s). A necessary condition for the existence of breathers in discrete lattices is that the breather main
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
and all its multipliers are located outside of the
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
of the lattice.


Example of a breather solution for the sine-Gordon equation

The
sine-Gordon equation The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surf ...
is the nonlinear
dispersive partial differential equation In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. E ...
:\frac - \frac + \sin u = 0, with the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''u'' a function of the spatial coordinate ''x'' and time ''t''. An exact solution found by using the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
is: :u = 4 \arctan\left(\frac\right), which, for ''ω < 1'', is periodic in time ''t'' and
decays exponentially A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive rat ...
when moving away from ''x = 0''.


Example of a breather solution for the nonlinear Schrödinger equation

The focusing
nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
is the dispersive partial differential equation: :i\,\frac + \frac + , u, ^2 u = 0, with ''u'' a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
field as a function of ''x'' and ''t''. Further ''i'' denotes the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...
. One of the breather solutions is : u = \left( \frac - 1 \right)a\,e^ with : \theta=a^2\,b\,\sqrt\;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 In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the '' carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informat ...
parameter ''b'' less than . Note that a limiting case of the breather solution is the Peregrine soliton.


See also

* Breather surface *
Soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...


References and notes

{{Reflist Waves