Presence Of Cubic Nonlinearity Research Articles

Global existence of strong solutions to a biological network formulation model in 2+1 dimensions

In this paper we study the initial boundary value problem for the system\\ $-\mbox{{div}}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right]=s(x),\ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m}=\beta^2(\mathbf{m}\cdot\nabla p)\nabla p$ in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution $(\mathbf{m},p) $ with both $|\nabla p|$ and $|\nabla\mathbf{m}|$ being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for $v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p$, and then suitably applying the De Giorge iteration method to the equation.

A Note on Free Vibration of a Double-beam System with Nonlinear Elastic Inner Layer

In this note, small amplitude free vibration of a double-beam system in presence of inner layer nonlinearity is investigated. The nonlinearity is due to inner layer material and is not related to large amplitude vibration. At first, frequencies of a double-beam system with linear inner layer are studied and categorized as synchronous and asynchronous frequencies. It is revealed that the inner layer does not affect higher modes significantly and mainly affects the first frequency. Then, equation of motion in the presence of cubic nonlinearity in the inner layer is derived and transformed to the form of Duffing equation. Using an analytical solution, the effect of nonlinearity on the frequency for simply-supported and clamped boundary conditions is analyzed. Results show that the nonlinearity effect is not significant and, in small amplitude free vibration analysis of a double-beam system, the material nonlinearity of the inner layer could be neglected.

Development of an Exactly Solvable Model of Resonance Tunneling of Electromagnetic Waves through Gradient Barriers in a Nonuniform Magnetoactive Plasma

An exactly solvable one-dimensional model describing resonance tunneling (reflectionless transmission) of a transverse electromagnetic wave through wide layers of magnetoactive plasma is developed on the basis of the Helmholtz equation. The plasma layers include a set of spatially localized density structures the amplitudes and thicknesses of which are such that approximate methods are inapplicable for their analysis. The profiles of the plasma density structures strongly depend on the choice of the free parameters of the problem that determine the amplitudes of plasma density modulation, characteristic scale lengths of the density structures, their number, and the total thickness of the nonuniform plasma layer. The plasma layers can also include a set of random inhomogeneities. The propagation of electromagnetic waves through such complicated plasma inhomogeneities is analyzed numerically within the proposed exactly solvable model. According to calculations, there are a wide set of inhomogeneous structures for which an electromagnetic wave incident from vacuum can propagate through the plasma layer without reflection, i.e., the complete tunneling of thick plasma barriers takes place. The model also allows one to exactly solve a one-dimensional problem on the nonlinear transillumination of a nonuniform plasma layer in the presence of cubic nonlinearity. It is important that, due to nonlinearity, the thicknesses of the evanescent plasma regions can decrease substantially and, at a sufficiently strong nonlinearity, such regions will disappear completely. The problem of resonance tunneling of electromagnetic radiation through gradient wave barriers is of interest for various applications, such as efficient heating of dense plasma by electromagnetic radiation and transmission of electromagnetic signals from a source located in the near-Earth plasma or deep in the plasma of an astrophysical object through the surrounding evanescent regions.

Predicting the closed-loop stability and oscillation amplitude of nonlinear parametrically amplified oscillators

This work investigates the closed-loop operation of microelectromechanical oscillators in the presence of both cubic (Duffing) nonlinearities and parametric amplification. We present a theoretical model for this system that enables us to predict oscillation amplitude and instability and experimentally verify it using a silicon disk resonator with a quality factor (Q) of 85 000 and a natural frequency of 251 kHz. We determine that, contrary to previous understanding gained from analyzing the open-loop system, the presence of cubic nonlinearities does not limit the maximum stable oscillation amplitude if the resonator is operated in a closed loop. In addition, the stability and amplitude behavior predicted by our theoretical model are independent of the presence or severity of cubic nonlinearities, or on drive amplitude.

Symmetry breaking in binary chains with nonlinear sites

We consider a system of two or four nonlinear sites coupled with binary chain waveguides. When a monochromatic wave is injected into the first (symmetric) propagation channel, the presence of cubic nonlinearity can lead to symmetry breaking, giving rise to emission of antisymmetric wave into the second (antisymmetric) propagation channel of the waveguides. We found that in the case of nonlinear plaquette, there is a domain in the parameter space where neither symmetry-preserving nor symmetry-breaking stable stationary solutions exit. As a result, injection of a monochromatic symmetric wave gives rise to emission of nonsymmetric satellite waves with energies differing from the energy of the incident wave. Thus, the response exhibits nonmonochromatic behavior.

Spatial Soliton Scattering in a Quasi Phase Matched Quadratic Media in Presence of Cubic Nonlinearity

We consider the propagation, fusion and scattering of solitary pulse in a quasi phase mached quadratic media in presence of cubic nonlinearity. The chief motivation is to analyze the mutual influence of phase matching, quadratic and cubic nonlinearities in the evolution of solitary pulse and its robustness. It is observed that the scenario of scattering changes considerably due to the presence of cubic nonlinearity, which is seen to favour the event of trapping of the pulses and their fusion is more favoured in the purely quadratic situation. The basic properties of the pulses, such as width, chirp etc. are also analyzed during their propagation along the fibre. The behaviour of these parameters is seen to be more controlled only in presence of cubic nonlinearity. Lastly we have also studied with some periodic amplification in the system. In this situation the QPM (quasi phase matched) system in absence of cubic term shows much better behaviour.

Propagation of localized longitudinal strain waves in a plate in the presence of cubic nonlinearity

Possible propagating longitudinal strain solitary waves in a plate are shown to be seriously altered when physical cubic nonlinearity is taken into account in the modeling. This also affects an amplification of the wave due to the transverse instability of plane-localized waves and due to the plane-wave interaction.

Longitudinal strain solitary waves in presence of cubic non-linearity

A new governing equation with combined quadratic and cubic non-linearities is obtained to account for longitudinal strain solitary waves in an elastic rod. It is shown that a strain solitary wave solution of this equation arises as a result of balance between quadratic non-linearity and dispersion and exists even in the absence of cubic non-linearity. However, the amplitude, the width and the velocity of the wave are affected by the cubic non-linearity causing, in particular, a narrowing of the wave. This allows to agree better with experiments on strain solitary wave generation.

Quadratic spatial solitons in periodically poled KTiOPO4.

The evolution of fundamental beams into quadratic spatial solitons was investigated in a periodically poled bulk KTiOPO4 crystal near and at phase matching. Various soliton properties, such as trapping efficiency as a function of phase mismatch, poling-induced walk-off, and the effects of pulse width, beam width, and input beam quality, were measured. Effects attributed to the presence of cubic nonlinearities were also observed.