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Bretherton equation

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

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

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

The original equation studied by Bretherton has quadratic nonlinearity, p = 2. {\displaystyle p=2.} Nayfeh treats the case p = 3 {\displaystyle p=3} with two different methods: Whitham's averaged Lagrangian method and the method of multiple scales.

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. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.

Variational formulations

The Bretherton equation derives from the Lagrangian density:

L = 1 2 ( u t ) 2 + 1 2 ( u x ) 2 − 1 2 ( u x x ) 2 − 1 2 u 2 + 1 p + 1 u p + 1 {\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:

∂ ∂ t ( ∂ L ∂ u t ) + ∂ ∂ x ( ∂ L ∂ u x ) − ∂ 2 ∂ x 2 ( ∂ L ∂ u x x ) − ∂ L ∂ u = 0. {\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:

u t − δ H δ v = 0 , v t + δ H δ u = 0 , {\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 H : {\displaystyle H:}

H ( u , v ) = ∫ H ( u , v ; x , t ) d x {\displaystyle H(u,v)=\int {\mathcal {H}}(u,v;x,t)\;\mathrm {d} x}   and   H ( u , v ; x , t ) = 1 2 v 2 − 1 2 ( u x ) 2 + 1 2 ( u x x ) 2 + 1 2 u 2 − 1 p + 1 u p + 1 {\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 H {\displaystyle {\mathcal {H}}} the Hamiltonian density – consequently v = u t . {\displaystyle v=u_{t}.} The Hamiltonian H {\displaystyle H} is the total energy of the system, and is conserved over time.

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, S2CID 250783550

References

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