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Bretherton equation : ウィキペディア英語版 | Bretherton equation In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: : with integer 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 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 : and with the Hamiltonian density – consequently The Hamiltonian is the total energy of the system, and is conserved over time.〔
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