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- Gulcin M. Muslu, H. A. Erbay
- Mathematics and Computers in Simulation
- 2005

The generalized nonlinear Schrödinger (GNLS) equation is solved numerically by a split-step Fourier method. The first, second and fourth-order versions of the method are presented. A classical problem concerning the motion of a single solitary wave is used to compare the first, second and fourth-order schemes in terms of the accuracy and the computational… (More)

- H. Borluk, Gulcin M. Muslu, H. A. Erbay
- Mathematics and Computers in Simulation
- 2007

Two numerical methods are presented for the periodic initial-value problem of the long wave–short wave interaction equations describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium. The first one is the relaxation method, which is implicit with second-order accuracy in both… (More)

- Nilay Duruk, Albert Erkip, H. A. Erbay
- 2006

where β is a positive constant, u(x, t) is a real-valued function of two real variables and g(u) is a given function u. At the microscopic level the higher order Boussinesq equation was derived in [1] for the longitudinal vibrations of a dense lattice, in which a unit length of the lattice contains a large number of lattice points. Thus, in the above higher… (More)

Many equations can be expressed as a cubic nonlinear Schrödinger (NLS) equation with additional terms, such as the Davey-Stewartson (DS) system [1]. As it is the case for the NLS equation, the solutions of the DS system are invariant under the pseudo-conformal transformation. For the elliptic NLS, this invariance plays a key role in understanding the… (More)

- Hüsnü Ata Erbaya, Saadet Erbaya, Albert Erkipb, H. A. Erbay
- 2015

We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of asymptotic wave models describing the unidirectional propagation of small-but-finite amplitude long waves. When the kernel function is the… (More)

- H. A. Erbay, S. Erbay
- 2014

In this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations: utt − Luxx = B(±|u|u)xx, p > 1. The main characteristic of this class of equations is the existence of two sources of dispersion, characterized by two coercive pseudo-differential operatorsL and B. Members of… (More)

- H. A. Erbay, S. Erbay, A. Erkip
- 2013

In this article we study global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type nonlinearities, utt − Luxx = B(−|u|u)xx, (p > 1), where the nonlocality enters through two pseudo-differential operators L and B. We establish thresholds for global existence versus blow-up using the potential well… (More)

- Hüsnü Ata Erbay, Ceni Babaoglu, H. A. Erbay, Albert Erkip, Ayse Hümeyra Bilge
- 2011

In this study the following Cauchy problem is considered: utt − uxx − Luxx = (g(u))xx, x ∈ R, t > 0, u(x, 0) = φ(x), ut(x, 0) = ψ(x), where g is a sufficiently smooth nonlinear function and L is the linear operator defined by F (Lv) (ξ) = l (ξ)Fv (ξ) . Here F denotes the Fourier transform with respect to variable x and l(ξ) is the Fourier transform of the… (More)

In this paper we investigate traveling wave solutions of a non-linear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a… (More)

- Thiab R. Taha, H. A. Erbay
- 2006

Nonlinear evolution equations are of tremendous interest in both theory and applications. In this talk we introduce parallel algorithms for numerical simulations of CMKdV, NLS and and CNLS equations in 1+1 and 1+2 dimensions. The parallel methods are implemented on multiprocessor system. Numerical experiments have shown that these methods give accurate… (More)