Theory of non-propagating surface-wave solitons

Abstract

An incompressible inviscid fluid contained in a channel in a gravitational field admits soliton-like disturbances where the velocity potential depends upon all three coordinates as well as time, yet its centre of mass can be at rest. These solitons were recently discovered by Wu, Keolian & Rudnick. The calculations are carried out with the multiple-scales approach. Consequences of mass conservation and radiation are discussed.

New Variable Separation Excitations of a (2+1)-Dimensional Broer-Kaup-Kupershmidt System Obtained by an Extended Mapping Approach

“…which can be derived from the standard Painlevé truncated expansion, where the functions H 0 = H 0 (x,t) and v 0 = 0 are seed solutions of the BKK system (8). Based on (9) and the seed solutions, we can straightly obtain a simple relation for H and v = H y .…”

“…Obviously, one of the simplest travelling wave excitations can be easily obtained when χ(x,t) = kx + ct and ϕ(y) = ly, where k, l and c are arbitrary constants. Still based on the derived solutions, we may also derive rich stationary localized solutions, which are not travelling wave excitations or not propagating waves [8], just as Wu et al reported about the non-propagating solitons in 1984 [9]. For instance, when the arbitrary functions are selected to be χ(x,t) = ς (x) + τ(t) and ϕ(y) = g(y), where ς , τ and g are also arbitrary functions of the indicated arguments, then we can obtain many kinds of non-propagating localized solutions like dromion, ring, peakon, compacton solutions, and so on.…”

Nonlinear interfacial waves in a circular cylindrical container subjected to a vertical excitation

“…One of these is concerned with the instability [5][6][7][8][9][10][11] and chaotic behaviors in the container with a closed basin [12][13][14][15][16][17]. The other relates to solitary standing waves observed in a long narrow channel [18,19]. More recently, Feng et al [20] considered surface wave motions in a container with a nearly square base subjected to a vertical oscillation.…”

Nonpropagating Solitary Waves in (2+1)-Dimensional Generalized Dispersive Long Wave Systems

“…Denardo et al (1990) have also observed a kink in the phase of surface wave oscillations on a shallow liquid in a parametrically driven rectangular channel. The forced standing or nonpropagating solitary wave phenomena, such as breather and kinks as mentioned above, can be explained by the cubic noninear Schrodinger equation (NLS; Drazin and Johnson, 1989) which were formulated by Larraza and Putterman (1984) and Miles (1984).…”