Breather

Example of breather solutions

Sine-Gordon equation

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

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

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:

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

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

Nonlinear Schrödinger equation

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

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

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

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

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.