Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

Abstract

An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second order perturbative approach in the weakly nonlinear, dispersive and long wavelength limit. Only treating all these terms in the second order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.

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

“…On the other hand, the authors of [6] have conjectured that in general the KRRI equation (1) is not integrable by the inverse scattering transform. We provide strong evidence to support this conjecture by rigorously proving that (1) is not symmetry integrable (see e.g.…”

“…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.…”

“…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.…”

A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions

“…It is interesting to note that the extended KdV equation, as first called by Marchant and Smyth [31], of second order in small parameters, was subjected to useful studies in [32][33][34] and some of the references therein. In Ref.…”