Bretherton equation

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

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

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

The original equation studied by Bretherton has quadratic nonlinearity, ${\displaystyle p=2.}$ Nayfeh treats the case ${\displaystyle p=3}$ 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]

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

through the Euler–Lagrange equation:

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

The equation can also be formulated as a Hamiltonian system:[7]

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

in terms of functional derivatives involving the Hamiltonian ${\displaystyle H:}$

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

with ${\displaystyle {\mathcal {H}}}$ the Hamiltonian density – consequently ${\displaystyle v=u_{t}.}$ The Hamiltonian ${\displaystyle H}$ 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, doi:10.1017/CBO9780511616938, 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