In mathematics, the Benjamin–Ono equation is a nonlinear partial
integro-differential equation that
describes one-dimensional
internal wave
Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in ...
s in deep water.
It was introduced by and .
The Benjamin–Ono equation is
:
where ''H'' is the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...
.
See also
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Bretherton equation In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:
:u_+u_+u_+u = u^p,
with p integer and p \ge 2. While u_t, u_x and u_ denote partial derivatives of the scalar field ...
References
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External links
Benjamin-Ono equations: Solitons and Shock Waves
Nonlinear partial differential equations
Integrable systems
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