Ceni Babaoglu

Two-dimensional wave packets in an elastic solid with couple stresses

Abstract The problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the “generalized Davey–Stewartson equations”. Under some restrictions on parameter values, the generalized Davey–Stewartson equations reduce to the Davey–Stewartson and to the non-linear Schrodinger equations. Finally, some special solutions involving sech–tanh–tanh and tanh–tanh–tanh type solitary wave solutions are presented.

Global existence and nonexistence results for a generalized Davey-Stewartson system

We consider a system of three equations, which will be called generalized Davey–Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic–elliptic–elliptic case.

Two-dimensional wave propagation in a generalized elastic solid

The object of the present study is to investigate the propagation of two-dimensional waves in a weakly nonlinear and weakly dispersive elastic solid. The reductive perturbation method is directly applied to a Lagrangian whose Euler‐Lagrange equations give the field equations for a quadratically nonlinear elastic medium with higher order gradients. In the long-wave approximation, it is shown that the long-time behavior of the two transverse waves is governed by the two coupled modified Kadomtsev‐Petviashvili (CMKP) equations. Depending on the choice of the direction of perpendicular dynamics, various forms of the CMKP equations are obtained. Some special solutions are also presented for a simplified form of the CMKP equations. ” 2000 Elsevier Science Ltd. All rights reserved.

Homoclinic Structure for a Generalized Davey-Stewartson System

In this study, we analyze the homoclinic structure for the generalized Davey-Stewartson system with periodic boundary conditions. This system involves three coupled nonlinear equations and describes (2 + 1) dimensional wave propagation in a bulk medium composed of an elastic material with coupled stresses. We first provide linearized stability analysis of the plane wave solutions of the generalized Davey-Stewartson system. Then, give an analytic description of the characteristics of homoclinic orbits near the fixed point by finding soliton type solutions. These solutions are derived via Hirota’s bilinear method. We also show that two of these solutions form a pair of symmetric homoclinic orbits and all these symmetric homoclinic orbit pairs construct the homoclinic tubes.

Symmetry in a parabolic multi-phase overdetermined problem

In this article, we prove symmetry for a parabolic overdetermined problem, related to heat balls. We consider a multi-phase version of classical symmetry problems with nonlinear overdetermined boundary conditions.