A comparison of solutions of two model equations for long waves.

*(English)*Zbl 0534.76024
Fluid dynamics in astrophysics and geophysics, Lect. Appl. Math. 20, 235-267 (1983).

[For the entire collection see Zbl 0515.00025.]

(Authors’ summary.) Considered are the partial differential equations (*): \(u_ t+u_ x+uu_ x+Lu=0\) where L denoes \(\partial^ 3_ x\) (equation A) or \(-\partial^ 2_ x\partial_ t\) (equation B). In (*) u is a real-valued function defined for all real x and for \(t\geq 0\), and interest will be focused on solutions of (A) and (B) that correspond to the initial condition that u(x,0) is a given function. Equation (A) is the Korteweg-de Vries equation while (B) is the model studied, for example, by T. B. Benjamin, the first author and J. J. Mahony [Philos. Trans. R. Soc. Lond., Ser. A 272, 47-78 (1972; Zbl 0229.35013)]. It has been argued in the last-quoted reference and elsewhere that either (A) or (B) can be used with equal justification to model various physical phenomena.

To establish this claim an exact relation connecting solutions of (A) and (B) is derived, showing that the two models yield predictions whose difference, over significant time scales, is only of such small order that it is formally neglected by either model.

Complementing the theoretical study are added some numerical experiments based on (B). These experiments suggest that the aforementioned theoretical estimates are sharp, and that they are valid up to the time scale for which either equation formally ceases to be an accurate model of underlying physical phenomena. The experiments also indicate that (B) has the property, which is well known for (A), that certain classes of initial data evolve into a sequence of solitary waves followed by a dispersive wave train.

(Authors’ summary.) Considered are the partial differential equations (*): \(u_ t+u_ x+uu_ x+Lu=0\) where L denoes \(\partial^ 3_ x\) (equation A) or \(-\partial^ 2_ x\partial_ t\) (equation B). In (*) u is a real-valued function defined for all real x and for \(t\geq 0\), and interest will be focused on solutions of (A) and (B) that correspond to the initial condition that u(x,0) is a given function. Equation (A) is the Korteweg-de Vries equation while (B) is the model studied, for example, by T. B. Benjamin, the first author and J. J. Mahony [Philos. Trans. R. Soc. Lond., Ser. A 272, 47-78 (1972; Zbl 0229.35013)]. It has been argued in the last-quoted reference and elsewhere that either (A) or (B) can be used with equal justification to model various physical phenomena.

To establish this claim an exact relation connecting solutions of (A) and (B) is derived, showing that the two models yield predictions whose difference, over significant time scales, is only of such small order that it is formally neglected by either model.

Complementing the theoretical study are added some numerical experiments based on (B). These experiments suggest that the aforementioned theoretical estimates are sharp, and that they are valid up to the time scale for which either equation formally ceases to be an accurate model of underlying physical phenomena. The experiments also indicate that (B) has the property, which is well known for (A), that certain classes of initial data evolve into a sequence of solitary waves followed by a dispersive wave train.

Reviewer: H.S.Takhar

##### MSC:

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76M99 | Basic methods in fluid mechanics |

86A05 | Hydrology, hydrography, oceanography |

35Q99 | Partial differential equations of mathematical physics and other areas of application |