Bretherton equation

In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:^[1]

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

with $p$ integer and $p \ge 2.$ 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.^[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]

Variational formulations

The Bretherton equation derives from the Lagrangian density:^[6]

$\mathcal{L} = \tfrac12 \left( u_t \right)^2 + \tfrac12 \left( u_x \right)^2 -\tfrac12 \left( u_{xx} \right)^2 - \tfrac12 u^2 + \tfrac{1}{p+1} u^{p+1}$

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:^[7]

$\begin{align} u_t & - \frac{\delta{H}}{\delta v} = 0, \\ v_t & + \frac{\delta{H}}{\delta u} = 0, \end{align}$

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

$H(u,v) = \int \mathcal{H}(u,v;x,t)\; \mathrm{d}x$   and   $\mathcal{H}(u,v;x,t) = \tfrac12 v^2 - \tfrac12 \left( u_x \right)^2 +\tfrac12 \left( u_{xx} \right)^2 + \tfrac12 u^2 - \tfrac{1}{p+1} u^{p+1}$

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.^[7][8]

Notes

1. Bretherton (1964)
2. Nayfeh (2004, §§5.8, 6.2.9 & 6.4.8)
3. Drazin & Reid (2004), pp. 393–397
4. Hammack, J.L.; Henderson, D.M. (1993), "Resonant interactions among surface water waves", Annual Review of Fluid Mechanics 25: 55–97, Bibcode:1993AnRFM..25...55H, doi:10.1146/annurev.fl.25.010193.000415
5. Kudryashov (1991)
6. Nayfeh (2004, §5.8)
7. Levandosky, S.P. (1998), "Decay estimates for fourth order wave equations", Journal of Differential Equations 143 (2): 360–413, Bibcode:1998JDE...143..360L, doi:10.1006/jdeq.1997.3369
8. Esfahani, A. (2011), "Traveling wave solutions for generalized Bretherton equation", Communications in Theoretical Physics 55 (3): 381–386, Bibcode:2011CoTPh..55..381A, doi:10.1088/0253-6102/55/3/01

References

• Bretherton, F.P. (1964), "Resonant interactions between waves. The case of discrete oscillations", Journal of Fluid Mechanics 20 (3): 457–479, Bibcode:1964JFM....20..457B, doi:10.1017/S0022112064001355
• Drazin, P.G.; Reid, W.H. (2004), Hydrodynamic stability (2nd ed.), Cambridge University Press, ISBN 0-521-52541-1
• Kudryashov, N.A. (1991), "On types of nonlinear nonintegrable equations with exact solutions", Physics Letters A 155 (4–5): 269–275, Bibcode:1991PhLA..155..269K, doi:10.1016/0375-9601(91)90481-M
• Nayfeh, A.H. (2004), Perturbation methods, Wiley–VCH Verlag, ISBN 0-471-39917-5