From Wikipedia, the free encyclopedia
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eafcd748297bac892b6112f85a768b9ab985aba2)
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.
[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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc289ed417e9f81f56142e57acabb33d532e3c44)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1707f1658b106c6cc86544fb13d6ce6e7fd9d3fe)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88bab0a813b65da35166864a93a16e85792a58b7)
in terms of
functional derivatives involving the Hamiltonian
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21c62fd1280ad6208ccb4adb008f14709cf36db3)
with
the Hamiltonian density – consequently
The Hamiltonian
is the total energy of the system, and is
conserved over time.
[7]
[8]
Notes
- ^
a
b
Bretherton (1964)
-
^
Nayfeh (2004, §§5.8, 6.2.9 & 6.4.8)
-
^
Drazin & Reid (2004, pp. 393–397)
-
^ 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
-
^
Kudryashov (1991)
-
^
Nayfeh (2004, §5.8)
- ^
a
b 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
-
^ 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,
S2CID
250783550
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,
S2CID
123193107
-
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