The long-wave limit for the water wave problem I. The case of zero surface tension

Abstract

The Korteweg‐de Vries equation, Boussinesq equation, and many other equations can be formally derived as approximate equations for the two‐dimensional water wave problem in the limit of long waves. Here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension. We prove that the solutions of the water wave problem in the long‐wave limit split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg‐de Vries equation. Our result allows us to describe the nonlinear interaction of solitary waves. © 2000 John Wiley & Sons, Inc.

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

“…Requiring the initial data (and thus the solutions of the KdV equations) to lie in weighted Sobolev spaces guarantees that the nonlinear interaction between the four traveling waves can be neglected. As a matter of fact, this condition on the sufficient decay in space of the initial data appears also naturally for the KdV approximation of the one-layer problem, as we see in [6,38].…”

“…The latter model states that any solution of the one-layer water wave problem in the long-wave limit splits up into two counter-propagating waves, each of them evolving independently as a solution of a KdV equation. A justification of such models has been investigated among others by Craig [13], Schneider and Wayne [38] and Bona et al [6]. The study of internal waves followed quickly.…”

Linear stability and instability of ion-acoustic plasma solitary waves

“…A natural question then is how far the dynamics of the Korteweg-de Vries equation can actually be found in this particular model for an ion-acoustic plasma. Up to now, only formal derivations of the Korteweg-de Vries limit seem to be known [12,32]; we refer to [16,9,30,4] for validation of the long-wavelength limit in slightly different contexts. One of the most striking phenomena in the Korteweg-de Vries equation-which largely motivated its discoveryare solitary waves.…”

Waves in Ferromagnetic Media

“…If (0, 0) is a singular point of X, a long-wave asymptotic analysis yields Korteweg-de Vries equations, where the dispersive phenomena are described by third-order differential expressions [11]. A similar situation arises in the water-wave problem, where the long-wave limit yields two counterpropagating waves, each described by a Korteweg-de Vries equation [14,15].…”

“…If, instead of the scalar product, we take the vector product, we obtain an expression for M 2 in terms of M 2 · M 0 and H 2 , 14) where the vector q is given in terms of M 1 and H 1 ,…”