A model equation for gravity-capillary waves in deep water is proposed. This model is a quadratic approximation of the deep water potential flow equations and has wavepacket-type solitary wave solutions. The model equation supports line solitary waves which are spatially localized in the direction of propagation and constant in the transverse direction, and lump solitary waves which are spatial...

Several explanations have been proposed to account for the mechanisms of neuroglial interactions involved in neural plasticity. We review experimental results addressing plastic nonlinear interactions between glial membranes and synaptic terminals. These results indicate the necessity of elaborating on a model based on the dynamics of hydroionic waves within the neuropil. These waves have been ...

The well-known Stokes waves refer to periodic traveling under the gravity at free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude with infinite depth are nonlinearly unstable long-wave perturbations. Our approach is based on modulational approximation system and instability mechanism focusing cubic nonlinear Schrödinger equation.

The free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg–de Vries equation...

A Lagrangian for strongly nonlinear unsteady water waves ~including overturning waves! is obtained. It is shown that the system of quadratic equations for the Stokes coefficients, which determine the shape of a steady wave ~discovered by Longuet-Higgins 100 years after Stokes derived his system of cubic equations! directly follows from the canonical system of Lagrange equations. Applications to...

In this paper, we shall study traveling wave solutions for a set of onedimensional nonlinear, nonlocal, evolutionary partial differential equations. This class of equations originally arose at quadratic order in the asymptotic expansion for shallow water waves [4,10]. The famous Korteweg–de Vries equation – which is nonlinear, but local – arises uniquely at linear order in this shallow water wa...

A model for the formation and evolution of longshore sand ridges in waters deeper than the shoaling region suggests that a possible but by no means exclusive agent for the formation of these bars are weakly nonlinear dispersive long waves. The model describes how these water waves, which are represented by wave packets, and a sandy bottom topography, conspire to produce sand ridges. This study ...

Observed crest speeds of maximally steep, breaking water waves are much slower than expected. Our fully nonlinear computations of unsteadily propagating deep water wave groups show that each wave crest approaching its maximum height slows down significantly and either breaks at this reduced speed, or accelerates forward unbroken. This previously noted crest slowdown behavior was validated as ge...

Three–wave nonlinear resonant interaction (TWRI) is typically encountered in the description of any conservative nonlinear medium where the nonlinear dynamics can be considered as a perturbation of the linear wave solution, the lowest–order nonlinearity is quadratic in the field amplitudes, the three–wave resonance can be satisfied [1]. Indeed TWRI is the lowest–order nonlinear effect for a sys...