A new nonlinear equation in the shallow water wave problem

Abstract

In the paper a new nonlinear equation describing shallow water waves with the topography of the bottom directly taken into account is derived. This equation is valid in the weakly nonlinear, dispersive and long wavelength limit. Some examples of soliton motion for various bottom shapes obtained in numerical simulations according to the derived equation are presented.

Single soliton solution to the extended KdV equation over uneven depth

“…Amazingly they were simple, though governed by a more exact expansion of the Euler equations with several new terms added when compared to the Korteweg-de Vries (KdV) equation [1,[3][4][5]. Our next step is to consider how these results are modified by a rough river or ocean bottom.…”

“…The last three terms are due to a bottom profile. We emphasize, that (1) was derived in [1,5] under a e-mail: p.rozmej@if.uz.zgora.pl the assumption that α, β, δ are small (positive by definition) and of the same order. As usual, α = A/H, i.e., the ratio of the wave amplitude A to the mean water depth H and β = (H/L) 2 , where L is the mean wavelength.…”

Symmetries and conservation laws for the Karczewska–Rozmej–Rutkowski–Infeld equation

“…Recently the authors of [2] have developed a systematic procedure for deriving an equation for surface elevation of shallow water waves for a prescribed relation between the orders of the two expansion parameters. This procedure was mutatis mutandis applied in [6,8] for deriving a fifth-order equation describing unidirectional shallow-water waves in channels with variable bottom geometry.…”

“…Consider the equation in question [6,8], to which we shall refer to as to the KarczewskaRozmej-Rutkowski-Infeld (KRRI) equation,…”

“…where a, b, and d are constants (denoted in [6,8] by α, β and δ), and h = h(x) is a smooth function of x giving the dimensionless channel depth, and the primes indicate xderivatives of h; the dependent variable u, i.e., the dimensionless wave elevation, denoted in [6,8] by η, is a function of x and t.…”