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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)
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)
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)
  • 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)
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)
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)
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)