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. 1 2 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, doi:10.1146/annurev.fl.25.010193.000415
  5. ↑ Kudryashov (1991)
  6. ↑ Nayfeh (2004, §5.8)
  7. 1 2 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
  8. ↑ 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

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 
  • 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, doi:10.1016/0375-9601(91)90481-M 
  • Nayfeh, A.H. (2004), Perturbation methods, Wiley–VCH Verlag, ISBN 0-471-39917-5 
This article is issued from Wikipedia - version of the Tuesday, November 10, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.