In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:[1] with
denote partial derivatives of the scalar field
The original equation studied by Bretherton has quadratic nonlinearity,
Nayfeh treats the case
[2] 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.
[3][4] Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.
[1][5] The Bretherton equation derives from the Lagrangian density:[6] through the Euler–Lagrange equation: The equation can also be formulated as a Hamiltonian system:[7] in terms of functional derivatives involving the Hamiltonian
is the total energy of the system, and is conserved over time.