Soliton interaction for the extended Korteweg-de Vries equation

Abstract

Soliton interactions for the extended Korteweg-de Vries (KdV) equation are examined. It is shown that the extended KdV equation can be transformed (to its order of approximation) to a higher-order member of the KdV hierarchy of integrable equations. This transformation is used to derive the higher-order, two-soliton solution for the extended KdV equation. Hence it follows that the higher-order solitary-wave collisions are elastic, to the order of approximation of the extended KdV equation. In addition, the higher-order corrections to the phase shifts are found. To examine the exact nature of higher-order, solitary-wave collisions, numerical results for various special cases (including surface waves on shallow water) of the extended KdV equation are presented. The numerical results show evidence of inelastic behaviour well beyond the order of approximation of the extended KdV equation; after collision, a dispersive wavetrain of extremely small amplitude is found behind the smaller, higher-order solitary wave.

Limit cycles for a class of fourth-order differential equations

“…Extended fifth-order Korteweg-de Vries equations have been considered in [6,9,10,16,17,23]. This equation has been used to describe chains of coupled nonlinear oscillators [25] and most notably gravity-capillary shallow water waves [4,14,30].…”

“…These periodic solutions are circles on the plane X = Y = 0 of the unperturbed system (23). All theses periodic orbits have period 2π in the variable t. These periodic orbits filled a plane minus the origin.…”

“…We shall apply Theorem 7 to the differential system (23). We note that this system (23) can be written as system (7) taking…”

Propagation regimes of interfacial solitary waves in a three-layer fluid

“…In this case, stationary (in a properly chosen moving coordinate system) solitary waves normally do not interact elastically. They may be generated from a suitably chosen wave system and they often radiate their energy during their motion and interact with other components of the wave field in a complicated manner (Marchant and Smyth, 1996;Osborne et al, 1998;Osborne, 2010).…”

Application of Hyperbola Function Method to the Family of Third Order Korteweg-de Vries Equations

“…The KdV equation is used to model the disturbance of the surface of shallow water in the presence of solitary waves. It is a generic model for the study of weakly nonlinear long waves, incorporating leading order nonlinearity and dispersion [13]. The mKdV equation appears in electric circuits and multi-component plasmas [14].…”