Bretherton equation
# Bretherton equation

In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:

    $u_{tt}+u_{xx}+u_{xxxx}+u=u^{p},$

with $p$ integer and $p\geq 2.$ While $u_{t},u_{x}$ and $u_{xx}$ denote partial derivatives of the scalar field $u(x,t).$

The original equation studied by Bretherton has quadratic nonlinearity, $p=2.$ Nayfeh treats the case $p=3$ 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:

    ${\mathcal {L}}={\tfrac {1}{2}}\left(u_{t}\right)^{2}+{\tfrac {1}{2}}\left(u_{x}\right)^{2}-{\tfrac {1}{2}}\left(u_{xx}\right)^{2}-{\tfrac {1}{2}}u^{2}+{\tfrac {1}{p+1}}u^{p+1}$

through the Euler–Lagrange equation:

    ${\frac {\partial }{\partial t}}\left({\frac {\partial {\mathcal {L}}}{\partial u_{t}}}\right)+{\frac {\partial }{\partial x}}\left({\frac {\partial {\mathcal {L}}}{\partial u_{x}}}\right)-{\frac {\partial ^{2}}{\partial x^{2}}}\left({\frac {\partial {\mathcal {L}}}{\partial u_{xx}}}\right)-{\frac {\partial {\mathcal {L}}}{\partial u}}=0.$

The equation can also be formulated as a Hamiltonian system:

    ${\begin{aligned}u_{t}&-{\frac {\delta {H}}{\delta v}}=0,\\v_{t}&+{\frac {\delta {H}}{\delta u}}=0,\end{aligned}}$

in terms of functional derivatives involving the Hamiltonian $H:$

    $H(u,v)=\int {\mathcal {H}}(u,v;x,t)\;\mathrm {d} x$ and ${\mathcal {H}}(u,v;x,t)={\tfrac {1}{2}}v^{2}-{\tfrac {1}{2}}\left(u_{x}\right)^{2}+{\tfrac {1}{2}}\left(u_{xx}\right)^{2}+{\tfrac {1}{2}}u^{2}-{\tfrac {1}{p+1}}u^{p+1}$

with ${\mathcal {H}}$ the Hamiltonian density – consequently $v=u_{t}.$ The Hamiltonian $H$ is the total energy of the system, and is conserved over time.

