In
mathematics, the Bretherton equation 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 othe ...
partial differential equation introduced by
Francis Bretherton
Francis Patton Bretherton (6 July 1935 – 27 June 2021) was an applied mathematician and a professor emeritus of the Department of Atmospheric and Oceanic Sciences at the University of Wisconsin, Madison.
Early career
After graduating from ...
in 1964:
:
with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
and
While
and
denote
partial derivatives of the
scalar field
The original equation studied by Bretherton has
quadratic nonlinearity,
Nayfeh treats the case
with two different methods:
Whitham's averaged Lagrangian
In continuum mechanics, Whitham's averaged Lagrangian method – or in short Whitham's method – is used to study the Lagrangian dynamics of Slowly varying envelope approximation, slowly-varying wave trains in an inhomogeneous (moving) transmissi ...
method and the
method of multiple scales.
The Bretherton equation is a model equation for studying weakly-nonlinear
wave dispersion. It has been used to study the interaction of
harmonics by
nonlinear resonance In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is ...
. Bretherton obtained analytic solutions in terms of
Jacobi elliptic function In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tri ...
s.
[
]
Variational formulations
The Bretherton equation derives from the Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
density:
:
through the Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
:
:
The equation can also be formulated as a Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
:
:
in terms of functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
s involving the Hamiltonian
: and
with the Hamiltonian density – consequently The Hamiltonian is the total energy of the system, and is conserved over time.
Notes
References
*
*
*
*
{{ref end
Nonlinear partial differential equations
Exactly solvable models