About 40 years ago, Snodgrass and other oceanographers (1966) tracked ocean swell propagating across the entire Pacific Ocean. At about the same time, several investigators (including Benjamin and Feir) showed that a uniform train of plane waves of finite amplitude on deep water is unstable....

Comprehensive numerical simulations of pulse solutions of the cubic–quintic Ginzburg–Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of...

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation...

We develop a geometrical approach for the relative sliding (shear) between filaments in a bundle subjected to bending and twisting deformations, with applications to motility in flagellated cells. Particular examples for helical and toroidal shapes, and combinations of these, are discussed. The...

We construct a discrete model for a particular PDE having nonlinear advection, diffusion, and reaction. The use of the author’s nonstandard finite difference methods form the basis for this construction. We demonstrate that the solutions to the scheme satisfy positivity and boundedness...

We study numerically stabilized solutions of the two-dimensional Schrödinger equation with a cubic nonlinearity. We discuss in detail the numerical scheme used and explain why choosing the right numerical strategy is very important to avoid misleading results. We show that stabilized solutions...

The tanh technique is used to solve exactly a set of nonlinear coupled equations small describing a problem arising in geochemistry. Next a coupled problem originating from the field of (deterministic) random walk theory with reaction kinetics is investigated but now solved approximately. The...

We study a quasi-local approximation for a nonlocal nonlinear Schrödinger equation. The problem is closely related to several applications, in particular to Bose–Einstein condensates with attractive two-body interactions. The nonlocality is approximated by a nonlinear dispersion term, which...

The work reported here is a sequel to our paper (Physica A 304 (2002) 395), wherein we have obtained the invariant solutions of the generalized one-dimensional Fokker–Planck equation via the so-called potential symmetry method. Herein, the application of the same method has been carried over...