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A representation formula for second-order nonhomogeneous nonlinear ordinary differential equations (ODEs) has been recently constructed by M. Frasca using its Green’s function, i.e. the solution of the corresponding nonlinear differential equation with a Dirac delta function instead of its nonhomogeneity. It has been shown that the first-order term–the convolution of the nonlinear Green’s function and the right-hand side, analogous to the Green’s representation formula for linear equations — provides a numerically efficient solution of the original equation, while the higher order terms add complementary corrections. The cases of square and sine nonlinearities have been studied by Frasca. Some new cases of explicit determination of nonlinear Green’s function have been studied previously by us. Here, we gather nonlinear equations and their explicitly determined Green’s functions from existing references, as well as investigate new nonlinearities leading to implicit determination of nonlinear Green’s function. Some transformations allowing to reduce second-order nonlinear partial differential equations (PDEs) to nonlinear ODEs are considered, meaning that Frasca’s method can be applied to second-order PDEs as well. We perform a numerical error analysis for a generalized Burgers’ equation and a nonlinear wave equation with a damping term in comparison with the method of lines.

The anisotropic characters of simple cubic lattice are investigated in this paper. Both the linear and nonlinear wave propagating in this lattice have been studied. The dispersion relation has been studied numerically. It is shown that the dispersion relation strongly depends on the directions of wave propagation. Generally, the direction of waves has the inclination angle α with respect to particle displacement. There are compressional waves α = 0 or transverse waves α = π/2 only for some special cases. The nonlinear waves in this lattice have also been studied. The anisotropic characters of this lattice for the nonlinear waves have also been shown. The compressional and transverse nonlinear solitons have also been studied. The characters of both solitons, such as amplitude and width, have been investigated.

We describe the non-symmetrical spatial-distributed electric field near the thin optic layer with nonlinear properties that separated two focusing and defocusing media with Kerr-type nonlinearity differing by values of refractive index. The nonlinear Schrödinger equation (NLSE) with nonlinear potential containing two parameters describes the distribution of electric field strength in adiabatic approximation. The problem is reduced to the solution of NLSEs at the half-spaces with the nonlinear boundary conditions at the interface plane. We obtain five new types of nonlinear stationary states describing the confinement effect of electric field strength across the interface. The field confinement effect is the localization of nonlinear spatially periodic wave during the transition from a one half-space to another one where the field distribution is monotonically damping from the interface. We derive and analyze the frequencies of field confinement effect possibility in dependence of media and interface characteristics.

This paper describes a full-custom mixed-signal chip that embeds digitally programmable analog parallel processing and distributed image memory on a common silicon substrate. The chip was designed and fabricated in a standard 0.5 μm CMOS technology and contains approximately 500 000 transistors. It consists of 1024 processing units arranged into a 32×32 grid. Each processing element contains two coupled CNN cores, thus, constituting two parallel layers of 32×32 nodes. The functional features of the chip are in accordance with the 2nd Order Complex Cell CNN-UM architecture. It is composed of two CNN layers with programmable inter- and intra-layer connections between cells. Other features are: cellular, spatial-invariant array architecture; randomly selectable memory of instructions; random storage and retrieval of intermediate images. The chip is capable of completing algorithmic image processing tasks controlled by the user-selected stored instructions. The internal analog circuitry is designed to operate with 7-bits equivalent accuracy. The physical implementation of a CNN containing second order cells allows real-time experiments of complex dynamics and active wave phenomena. Such well-known phenomena from the reaction–diffusion equations are traveling waves, autowaves, and spiral-waves. All of these active waves are demonstrated on-chip. Moreover this chip was specifically designed to be suitable for the computation of biologically inspired retina models. These computational experiments have been carried out in a developmental environment designed for testing and programming the analogic (analog-and-logic) programmable array processors.

Using the method of dynamical systems for the generalized Schrödinger equation, the bright soliton solution, dark soliton solution, uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained. Exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.

Fatigue cracks generated by repeated loads cause structural failures. Such cracks grow continuously and at an increasing speed owing to the concentration of stresses near the crack tips. Therefore, the early detection of fatigue cracks is imperative in the field of structural-health monitoring for the safety of structures exposed to dynamic loading. In particular, the detection of those cracks subjected to compression is known as a challenging problem in the nondestructive inspection area. The nonlinear ultrasonic modulation technique is effective for the detection of microcracks smaller than the size of a wavelength because this technique uses the deformation of waves passing through the crack surfaces. However, the technique has not been thoroughly verified for detecting cracks subjected to external forces. In this study, nonlinear ultrasonic modulation tests are performed on two types of crack specimens under compressive forces. The results show that in fatigue-cracked specimens, the cracks can be detected using modulated waves even under strong compressions. With artificial cracks, buckling occurs at a relatively low compression, and the amounts of modulated waves rapidly increase due to the bending of the specimen before buckling failure takes place. In this study, the crack detection methodology under compression is proposed and experimentally verified. The proposed method might be beneficial to find cracks under compression in various structural components.

A new set of differential equations with the free surface elevation and the depth-integrated volume flux being the dependent variables is derived for description of monochromatic waves on steady currents over gradually varying bottoms. The derivation is based on the vertical integration of the continuity equation and the equation of motion for general free-surface flows. A similarity law on the dynamic pressure and the horizontal velocity, modified from a relevant relation for small amplitude waves on uniform currents, has been introduced. A finite-difference scheme with second-order accuracy is proposed for solutions of the equations. The computational algorithm is proved to be conditionally stable and easy to implement. Key points on specification of boundary and initial conditions are discussed in details. Computations are carried out to investigate the nonlinear effects on wave shoaling over a uniform slope with following or opposing current in presence. A case test of the proposed model is also performed and satisfactory agreement obtained between the computational results and reported laboratory data.

The characteristics of nonlinear wave decay over a fluidized bed in coastal waters were investigated by applying the HHT method to analyze the measurements from wave flume experiments. The Ensemble EMD (EEMD) was adopted to decompose the wave and pore pressure data into the intrinsic mod functions (IMF). In a resonantly fluidized response, the results of Hilbert amplitude spectrum of waves derived by the HHT illustrate that wave decay obviously occurs mainly on the fundamental component. In following non-resonantly fluidized tests, the wave decay becomes relaxed with thinner fluidized-layer. Furthermore, the wave decay is immediately due to fluidized response, but the decay along propagation over a fluidized bed is relatively insignificant. The wave-decay ratio is found to be directly proportional to the magnitude of Ursell number (Ur).