Breather

Overview

A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon 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.

Example of breather solutions

Sine-Gordon equation

The sine-Gordon equation is the nonlinear dispersive partial differential equation

:\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 \arctan\left(\frac{\sqrt{1-\omega^2};\cos(\omega t)}{\omega;\cosh(\sqrt{1-\omega^2}; x)}\right),

which, for *ω

Nonlinear Schrödinger equation

The focusing nonlinear Schrödinger equation is the dispersive partial differential equation:

: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 (Kuznetsov-Ma breather) 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,e^{i a^2 t}

with

: \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 . Note that a limiting case of the breather solution is the Peregrine soliton.

References and notes

References

1. M. J. Ablowitz. (1973). "Method for solving the sine-Gordon equation". Physical Review Letters.
2. N. N. Akhmediev. (1987). "First-order exact solutions of the nonlinear Schrödinger equation". Theoretical and Mathematical Physics.
3. N. N. Akhmediev. (1997). "Solitons, non-linear pulses and beams". Springer.
4. Miroshnichenko A, Vasiliev A, Dmitriev S. ''[http://homepages.tversu.ru/~s000154/collision/main.html Solitons and Soliton Collisions]''.
5. 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.
6. (2010). "The Peregrine soliton in nonlinear fibre optics". Nature Physics.