Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlin...

Modulation instability conditions for the generation of femtosecond pulses represented by perturbative nonlinear Schrödinger equation with higher order dispersion and nonlinear effects is investigated. It is observed that modulation instability occurs even in the normal dispersion regime due to the presence of fourth order dispersion.

In this paper, we develop, analyze and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation which contains nonlinear high order derivatives and its zero-viscosity and zero-dispersion limit. The energy stability for general solutions are proved and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to il...

An exact pulse for the parametrically forced nonlinear Schrodinger equation (NLS) is isolated. The equation governs wave envelope propagation in dispersion-managed fiber lines with positive residual dispersion. The pulse is obtained as a ground state of an averaged variational principle associated with the equation governing pulse dynamics. The solutions of the averaged and original equations a...

We study noise-induced perturbations of dispersion-managed solitons. We do so by first developing soliton perturbation theory for the dispersion-managed nonlinear Schrödinger DMNLS equation, which governs the long-term behavior of optical fiber transmission systems and certain kinds of femtosecond lasers. We show that the eigenmodes and generalized eigenmodes of the linearized DMNLS equation ar...

Study in surface water quality is important. Rivers are one of the main sources of water supply fordrinking, agriculture and industry. Unfortunately, sometime Rivers where wastewater dischargesare considered. For this reason,the pollutant transmission in river is one of the most importantproblems in Environmental Engineering. the Advection Dispersion Equation (ADE) is governedon the pollutant t...

Let us consider the following problem. Imagine a homogeneous nonlinear medium in which only one type of wave with dispersion law co(k) may propagate. Let the nonlinearity of the medium be quadratic while the equation co(k~ + k~) = co(k0 + co(k2) (1) defines a nonzero manifold F whose codimension in kl, k2 space is unity. Equation (1) means that dispersion law co(k) allows decay processes. The n...

Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg-de Vries equation in the small dispersion limit Abstract In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillat...

The carrier-envelope phase slip of an ultrashort pulse circulating in a mode-locked Ti:sapphire laser is analyzed. The laser cavity is modeled by a dispersion- and nonlinearity-managed nonlinear Schrödinger equation. The combined contributions to the phase slip induced by nonlinear phase and nonlinear dispersion are found to approach zero for strong dispersion maps. The dependence of the slip o...

Contaminant transport by liquid ow in a porous medium is modeled by the addition of a stochastic term to Darcy's ow equation. The resulting stochastic diierential equation is studied using results from the theory of diiusions as embodied in the Dynkin formula. The resulting integral equation for the probability distribution of uid elements is solved for the case of a spatially homogeneous mediu...