Approximate Equations for Long Nonlinear Waves on a Viscous Fluid

Stabilization of the Kawahara equation with localized damping

“…For further considerations see also, for example, Topper and Kawahara [24]. Nonlinear dispersive problems have been object of intensive research (see, for instance, the classical paper of Benjamin et al [1], Biagioni and Linares [3], Bona and Chen [4], Menzala et al [15], Rosier [19], and references therein).…”

Solitary-Wave and Periodic Solutions of the Kuramoto-Velarde Dispersive Equation

“…3, α 0 ≥ 0, α 1 = 0, α 2 = 0, sign α 2 = sign α 3 , is a generalization of the Kuramoto-Velarde (KV) equation corresponding to the case α 1 = 0 and of the dispersive Kuramoto-Sivashinsky (KS) equation which corresponds to the case α 3 = 0. The dispersive (KS) equation is a model equation for long waves on a viscous fluid flowing down an inclined plane [1], as well as for drift waves in plasma [2]. The KV equation is an equation describing slow space-time variations of disturbances at interfaces, diffusion-reaction fronts and plasma instability fronts [3] and [4].…”

Kawahara equation in a quarter-plane and in a finite domain

“…Our study is motivated by physics and numerics: the nonlinear relation, called Kawahara equation [19], is the fifth-order dispersive-type partial differential equation describing one-dimensional propagation of small amplitude long waves in various problems of fluid dynamics and plasma physics [2,30]. This equation is also known as perturbed KdV or the special version of the Benney-Lin equation [3,4].…”

“…It should be noted also that imposed boundary conditions are reasonable both from physical and mathematical point of view, see [7,30] and comments in [15].…”