Bretherton Equation

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

:$u_+u_+u_+u = u^p,$

with $p$ integer and $p \ge 2.$ While $u_t, u_x$ and $u_$ denote partial derivatives of the scalar field $u(x,t).$

The original equation studied by Bretherton has quadratic function, quadratic nonlinearity, $p=2.$ Ali H. Nayfeh, Nayfeh treats the case $p=3$ with two different methods: Gerald B. Whitham, Whitham's averaged Lagrangian method and the method of multiple scales.

The Bretherton equation is a model equation for studying weakly-nonlinear dispersion (water waves), 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 (field theory), Lagrangian density:

:$\mathcal = \tfrac12 \left( u_t \right)^2 + \tfrac12 \left( u_x \right)^2 -\tfrac12 \left( u_ \right)^2 - \tfrac12 u^2 + \tfrac u^$

through the Euler–Lagrange equation:

:$\frac \left( \frac \right) + \frac \left( \frac \right) - \frac \left( \frac \right) - \frac = 0.$

The equation can also be formulated as a Hamiltonian system:

:$\begin u_t & - \frac = 0, \\ v_t & + \frac = 0, \end$

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

:$H(u,v) = \int \mathcal(u,v;x,t)\; \mathrmx$ and $\mathcal(u,v;x,t) = \tfrac12 v^2 - \tfrac12 \left( u_x \right)^2 +\tfrac12 \left( u_ \right)^2 + \tfrac12 u^2 - \tfrac u^$

with $\mathcal$ the Hamiltonian density – consequently $v=u_t.$ The Hamiltonian $H$ is the total energy of the system, and is conservation law, conserved over time.

Notes

References

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Nonlinear partial differential equations
Exactly solvable models