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

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]

through the Euler–Lagrange equation:

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

in terms of functional derivatives involving the Hamiltonian

  and  

with the Hamiltonian density – consequently The Hamiltonian 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, Bibcode:1993AnRFM..25...55H, 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, 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, S2CID 250783550

References

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