H. I. Abdel-Gawad

On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients.

In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.

Exact Solutions of Space Dependent Korteweg–de Vries Equation by The Extended Unified Method

Recently the unified method for finding traveling wave solutions of non-linear evolution equations was proposed by one of the authors a . It was shown that, this method unifies all the methods being used to find these solutions. In this paper, we extend this method to find a class of formal exact solutions to Korteweg-de Vries (KdV) equation with space dependent coefficients. A new class of multiple-soliton or wave trains is obtained.

On the Behavior of Solutions of a Class of Nonlinear Partial Differential Equations

The behavior of the steady-state (or the traveling wave) solutions for a class of nonlinear partial differential equations is studied. The nonlinearity in these equations is expressed by the presence of the convective term. It is shown that the steady-state (or the traveling wave) solution may explode at a finite value of the spatial (or the characteristic) variable. This holds whatever the order of the spatial derivative term in these equations. Furthermore, new special solutions of a set of equations in this class are also found.

A method for finding the invariants and exact solutions of coupled non-linear differential equations with applications to dynamical systems

The polynomial invariants of (a set) non-linear differential equations are found by using a direct approach. The integrability of these invariants deserves the integrability of the given set of coupled differential equations. As applications, the Lorenz and Rikitake sets, among others, are studied. New invariants are obtained.

Approximate solutions of the Kuramoto–Sivashinsky equation for periodic boundary value problems and chaos

Abstract An approach to find approximate solutions of the Kuramoto–Sivashinsky (KS) equation for boundary value problems (BVP) is developed. Attention is focused to periodic boundary value problems. This approach is used to find the approximate solutions for stress-free and rigid boundary conditions. In the first case, it is shown that the spatial pattern of the solutions fluctuates chaotically for small times. But it becomes asymptotically regular. The time-averaged solutions are also regular. In contrast, the solution for rigid boundary conditions exhibit robust chaos.

An approach to solutions of coupled semilinear partial differential equations with applications

In this work, an approach for finding the solution of coupled semi-linear diffusion equations for initial value problems is presented. The formal exact solution is found and the Picard iteration is constructed. It is shown that the constructed sequence of solutions converges uniformly for some classes of initial value problems. The problem of dispersion of an oxygen demanding pollutant released into a uniform flow is studied.

On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

Rogue waves are more precisely defined as waves whose height is more than twice the significant wave height. This remarkable height was measured (by Draupner in 1995). Thus, the need for constructing a mechanism for the rogue waves is of great utility. This motivated us to suggest a mechanism, in this work, that rogue waves may be constructed via nonlinear interactions of solitons and periodic waves. This suggestion is consolidated here, in an example, by studying the behavior of solutions of the complex (KdV). This is done here by the extending the solutions of its real version.

On the Integrability and Exact Solutions of a Generalized Korteweg-de-Vries Equation

In Plasma physics, nonlinear ion acoustic waves are shown to be described by the Korteweg-de-Vries equation. When the drag force acted by the waves on the particles is taken into consideration, the governing equation is shown to be a generalized Korteweg-de-Vries equation. The Integrability properties of this equation are discussed. Some exact solutions for this equation are derived by using the techniques of auto-Backlund transformation and polynomial invariants.

A Chemotherapy-Diffusion Model for the Cancer Treatment and Initial Dose Control

A one site chemotherapy agent-difiusion model is proposed which accounts for difiusion of chemotherapy agent, normal and cancer cells. It is shown that, by controlling the initial conditions, consequently an initial dose of the chemotherapy agent, the system is guaranteed to evolute towards a target equilibrium state. Or, growth of the normal cells occurs against decay of the cancer cells. Efiects of difiusion of chemotherapy-agent and cells are investigated through numerical computations of the concentrations in square and triangular cancer sites.