Using computational methods, it is found that the two-dimensional nonlinear Schrödinger (NLS) equation with a quasicrystal lattice potential admits multiple dipole and vortex solitons. The linear and the nonlinear stability of these solitons is investigated using direct simulations of the NLS equation and its linearized equation. It is shown that certain multiple vortex structures on quasicrystal lattices can be linearly unstable but nonlinearly stable. These results have application to investigations of localized structures in nonlinear optics and Bose-Einstein condensates.

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Nonlinear Schrödinger (NLS) equation with external potentials (lattices) possessing crystal and quasicrystal structures are studied. The fundamental solitons and band gaps are computed using a spectral fixed-point numerical scheme. Nonlinear and linear stability properties of the fundamental solitons are investigated by direct simulations and the linear stability properties of the fundamental solitons are confirmed by analysis the linearized eigenvalue problem.

In the present work, we examine the propagation of weakly nonlinear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid. Considering that the arteries are initially subjected to a large static transmural pressure P 0 and an axial stretch λz and, in the course of blood flow, a finite time dependent displacement is added to this initial field, the nonlinear equation governing the motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is studied. After obtaining the general equation in the long-wave approximation, by a proper scaling, it is shown that this general equation reduces to the well-know nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained.Mathematics Subject Classification (1991). 76B25, 73D35, 73G25.

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Damping of periodic waves in the classically important nonlinear wave systems-nonlinear Schrödinger, Korteweg-deVries (KdV), and modified KdV-is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m = m(t). The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m = 0 to soliton waves when m = 1.