Bretherton equation

In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:
with formula_2 integer and formula_3 While formula_4 and formula_5 denote partial derivatives of the scalar field formula_6
The original equation studied by Bretherton has quadratic nonlinearity, formula_7 Nayfeh treats the case formula_8 with two different methods: Whitham's averaged Lagrangian 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. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.
Variational formulations.
The Bretherton equation derives from the Lagrangian density:
through the Euler–Lagrange equation:
The equation can also be formulated as a Hamiltonian system: 
in terms of functional derivatives involving the Hamiltonian formula_12
with formula_15 the Hamiltonian density – consequently formula_16 The Hamiltonian formula_17 is the total energy of the system, and is conserved over time.