Bretherton equation
In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964:[1]
- [math]\displaystyle{ u_{tt}+u_{xx}+u_{xxxx}+u = u^p, }[/math]
with [math]\displaystyle{ p }[/math] integer and [math]\displaystyle{ p \ge 2. }[/math] While [math]\displaystyle{ u_t, u_x }[/math] and [math]\displaystyle{ u_{xx} }[/math] denote partial derivatives of the scalar field [math]\displaystyle{ u(x,t). }[/math]
The original equation studied by Bretherton has quadratic nonlinearity, [math]\displaystyle{ p=2. }[/math] Nayfeh treats the case [math]\displaystyle{ p=3 }[/math] 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]
- [math]\displaystyle{ \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} }[/math]
through the Euler–Lagrange equation:
- [math]\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. }[/math]
The equation can also be formulated as a Hamiltonian system:[7]
- [math]\displaystyle{ \begin{align} u_t & - \frac{\delta{H}}{\delta v} = 0, \\ v_t & + \frac{\delta{H}}{\delta u} = 0, \end{align} }[/math]
in terms of functional derivatives involving the Hamiltonian [math]\displaystyle{ H: }[/math]
- [math]\displaystyle{ H(u,v) = \int \mathcal{H}(u,v;x,t)\; \mathrm{d}x }[/math] and [math]\displaystyle{ \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} }[/math]
with [math]\displaystyle{ \mathcal{H} }[/math] the Hamiltonian density – consequently [math]\displaystyle{ v=u_t. }[/math] The Hamiltonian [math]\displaystyle{ H }[/math] is the total energy of the system, and is conserved over time.[7][8]
Notes
- ↑ 1.0 1.1 (Bretherton 1964)
- ↑ (Nayfeh 2004)
- ↑ (Drazin Reid)
- ↑ Hammack, J.L.; Henderson, D.M. (1993), "Resonant interactions among surface water waves", Annual Review of Fluid Mechanics 25: 55–97, doi:10.1146/annurev.fl.25.010193.000415, Bibcode: 1993AnRFM..25...55H
- ↑ (Kudryashov 1991)
- ↑ (Nayfeh 2004)
- ↑ 7.0 7.1 Levandosky, S.P. (1998), "Decay estimates for fourth order wave equations", Journal of Differential Equations 143 (2): 360–413, doi:10.1006/jdeq.1997.3369, Bibcode: 1998JDE...143..360L
- ↑ Esfahani, A. (2011), "Traveling wave solutions for generalized Bretherton equation", Communications in Theoretical Physics 55 (3): 381–386, doi:10.1088/0253-6102/55/3/01, Bibcode: 2011CoTPh..55..381A
References
- Bretherton, F.P. (1964), "Resonant interactions between waves. The case of discrete oscillations", Journal of Fluid Mechanics 20 (3): 457–479, doi:10.1017/S0022112064001355, Bibcode: 1964JFM....20..457B
- 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, doi:10.1016/0375-9601(91)90481-M, Bibcode: 1991PhLA..155..269K
- Nayfeh, A.H. (2004), Perturbation methods, Wiley–VCH Verlag, ISBN 0-471-39917-5
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