Breather

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.

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

${\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}-{\frac {\partial ^{2}u}{\partial x^{2}}}+\sin u=0,}$

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:[1]

${\displaystyle u=4\arctan \left({\frac {{\sqrt {1-\omega ^{2}}}\;\cos(\omega t)}{\omega \;\cosh({\sqrt {1-\omega ^{2}}}\;x)}}\right),}$

which, for ω < 1, is periodic in time t and decays exponentially when moving away from x = 0.

The focusing nonlinear Schrödinger equation[5] is the dispersive partial differential equation:

${\displaystyle i\,{\frac {\partial u}{\partial t}}+{\frac {\partial ^{2}u}{\partial x^{2}}}+|u|^{2}u=0,}$

with u a complex field as a function of x and t. Further i denotes the imaginary unit.

One of the breather solutions is [2]

${\displaystyle u=\left({\frac {2b^{2}\cosh(\theta )+2ib{\sqrt {2-b^{2}}}\sinh(\theta )}{2\cosh(\theta )-{\sqrt {2}}{\sqrt {2-b^{2}}}\cos(abx)}}-1\right)a\,e^{ia^{2}t}}$

with

${\displaystyle \theta =a^{2}\,b\,{\sqrt {2-b^{2}}}\;t,}$

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]

• Breather surface
• Soliton