Propagation of long-crested water waves. Ⅱ. Bore propagation

Abstract

This essay is concerned with long-crested waves such as those arising in bore propagation. Such motions obtain on rivers when a surge of water invades an otherwise constantly flowing stretch and in the run-up of waves in the near-shore zone of large bodies of water. The dominating feature of the motion is that, in a standard xyz−coordinate system in which z increases in the direction opposite to which gravity acts and x increases in the principal direction of propagation, the depth of the fluid approaches a constant value h 0 > 0 as x → +∞ and another value h 1 > h 0 as x → −∞. In an earlier work, the authors developed theory for an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in that article applies to the sort of initial data arising in modeling bore propagation. However, well-posedness on the longer, Boussinesq time scale was not dealt with in the case of bore propagation, though such results were established for motions where h 1 = h 0 .We argue that without a well-posedness theory at least on the Boussinesq time scale, such models for bore-propagation may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.The theory makes its way via the derivation of suitable approximations with which to compare the full solution. An interesting feature of the theory is the determination of dynamical boundary behavior that is not prescribed, but which the solution necessarily satisfies.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach

“…The water cycle has been believed to result in the distribution of water on land surface, to purify water, to support plant growth, to facilitate agriculture and to sustain aquatic ecosystem [2]. Water waves have been seen as one of the most common phenomena in nature, the studies of which help the energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc [3][4][5][6][7][8][9][10]. On the water waves, each year, hundreds of papers have been published, a few of which, e.g.…”

“…On the water waves, each year, hundreds of papers have been published, a few of which, e.g. are [3][4][5][6][7][8][9][10] recently. More recent nonlinear-physics contributions have been seen as well [11][12][13].…”