title: Shallow-water soliton dynamics beyond the Korteweg-de Vries equation.
abstract: An alternative way for the derivation of the new Korteweg-de Vries (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.
authors.affiliation: Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland.
authors.forename: Anna
authors.initials: A
authors.lastname: Karczewska
authors.affiliation: Institute of Physics, Faculty of Physics and Astronomy University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland.
authors.forename: Piotr
authors.initials: P
authors.lastname: Rozmej
authors.affiliation: National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa, Poland.
authors.forename: Eryk
authors.initials: E
authors.lastname: Infeld
mesh_terms: D009038:Motion; D017711:Nonlinear Dynamics; D014867:Water
chemical_list: D014867:Water