quadratic and cubic nonlinearities
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Abstract The nonlinear acoustic approach is assessed for applications as a nondestructive tool for reconstructing stress-strain curves and quantifying the ultimate tensile strength for variety of materials. The direct algorithm uses the polynomial stress-strain expansion up to the third power of strain and the literature data on the second-order nonlinearity parameters to calculate relevant segments of the stress-strain curves. Since the third-order nonlinearity parameters are unknown for majority of materials the calculations used an iteration scheme to obtain closer approximations to the experimental data available from static tensile tests. The solution to the inverse problem identifies the range of the nonlinearity parameters for a given tensile strength and enables to categorize the contribution of the quadratic and cubic nonlinearities in mechanical response for different materials.

Abstract Wave propagation in a network of buckled beams, which represents a finite dissipative periodic structure with quadratic and cubic nonlinearities, is studied. The aforementioned structure is harmonically driven externally and parametrically at one end with forcing frequencies lying within its stop band, one above and one below. Numerical calculations show the occurrence of supratransmission, a sudden increase in the energy transmitted across the finite structure, after a certain forcing amplitude of the external excitation. In essence, this nonlinear wave propagation mechanism for the discrete nonlinear periodic structure occurs due to loss of stability of the periodic solutions that are initially localized to the driven end of the structure (nonlinear instability). It is found that small parametric excitation can considerably decrease the required threshold for the onset of energy transmission within the stop band.

Our aim in this research work is to formulate the exact traveling and solitary wave solutions of nonlinear diffusion reaction (DR) equation with quadratic and cubic nonlinearities by implementing the new technique which is a modified mathematical method. We have investigated the density independent nonlinear diffusion equation with convective flux term. As a result, we have found a variety of new exact traveling and solitary wave solutions in the form of dark solitons, bright solitons, combined dark-bright solitons, traveling wave, periodic wave solutions and we also represent the physical structure of the obtained solutions by two- and three-dimensional graphics by using the Mathematica software. This work proves the power, reliability and fruitfulness of this new technique.

The asymptotic perturbation method is used to analyze the nonlinear vibrations and chaotic dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs and the time-varying stiffness. Based on the expressions of the electromagnetic force resultants, the influences of some parameters, such as the cross-sectional area Aα of one electromagnet and the number N of windings in each electromagnet coil, on the electromagnetic force resultants are considered for the rotor-AMB system with 16-pole legs. Based on the Newton law, the governing equation of motion for the rotor-AMB system with 16-pole legs is obtained and expressed as a two-degree-of-freedom system with the parametric excitation and the quadratic and cubic nonlinearities. According to the asymptotic perturbation method, the four-dimensional averaged equation of the rotor-AMB system is derived under the case of 1 : 1 internal resonance and 1 : 2 subharmonic resonances. Then, the frequency-response curves are employed to study the steady-state solutions of the modal amplitudes. From the analysis of the frequency responses, both the hardening-type nonlinearity and the softening-type nonlinearity are observed in the rotor-AMB system. Based on the numerical solutions of the averaged equation, the changed procedure of the nonlinear dynamic behaviors of the rotor-AMB system with the control parameter is described by the bifurcation diagram. From the numerical simulations, the periodic, quasiperiodic, and chaotic motions are observed in the rotor-active magnetic bearing (AMB) system with 16-pole legs, the time-varying stiffness, and the quadratic and cubic nonlinearities.

A simplified operator correspondence scheme is derived to address nonlinear quantum systems within the framework of the [Formula: see text]-representation. The simplified method is applied to a general nonlinear quantum oscillator model that has been used in the literature to describe nonlinear quantum optical and matter wave systems. The [Formula: see text]-representation evolution equation for the model is derived for arbitrary nonlinearity exponents. It is shown that in the high temperature case, the [Formula: see text]-representation is sufficient to describe the model and its associated systems such that there is no need to use alternative, mathematically more involved representations. Systems with quadratic and cubic nonlinearities are considered in more detail. Distributions for the energy levels and photon and particle numbers are obtained within the framework of the [Formula: see text]-representation. Moreover, the electrical field oscillation frequency dependency is studied numerically when interpreting the model as model for quantum optical nonlinear oscillators.