hep-th 9607074 Multiple-Scale Analysis of Quantum Systems

Further, we also describe the breathing modes for various order of perturbation. At the end, we compare the solutions obtained via perturbation and numerical methods of parametric-driven sine-Gordon equation with phase shifts. Finally, we concluded that the modes of the breathing decay to a constant in both cases. Beginning with new material on the development of cutting-edge asymptotic methods and multiple scale methods, the book introduces this method in time domain and provides examples of vibrations of systems. Clearly written throughout, it uses innovative graphics to exemplify complex concepts such as nonlinear stationary and nonstationary processes, various resonances and jump pull-in phenomena. It also demonstrates the simplification of problems through using mathematical modelling, by employing the use of limiting phase trajectories to quantify nonlinear phenomena.

It was founded that the algebraic mode decay from discrete to continuous spectrum. In the paper they considered the direct driven vase, while ours is the parameter with extra phase shift. The method of multiple scales is applied to the analysis of a curved box model for the spirally coiled cochlea. The fluid motion is fully three-dimensional and the basilar membrane movement is represented by a single mode of deflection. Coiling parameters for the human cochlea are determined from the experimental data of von Békézy [Experiments in Hearing (McGraw-Hill, New York, 1960)]. The approximate solution is numerically calculated to yield comparisons with the uncoiled case.

Straightforward perturbation-series solution

Here, it is extended to study the Heisenberg operator equations of motion and the Schrödinger equation for the quantum anharmonic oscillator. In the former case, it leads to a system of coupled operator differential equations, which is solved exactly. In the latter case, multiple-scale analysis elucidates the connection between weak-coupling perturbative and semiclassical nonperturbative aspects of the wave function. In this paper, a hierarchical RVE-based continuum-atomistic multi-scale procedure is developed to model the nonlinear behavior of nano-materials. The atomistic RVE is accomplished in consonance with the underlying atomistic structure, and the inter-scale consistency principals, i.e. kinematic and energetic consistency principals, are exploited. To ensure the kinematic compatibility between the fine- and coarse-scales, the implementation of periodic boundary conditions is elucidated for the fully atomistic method.

The first equation is a two-mode Sharma-Tasso-Olver equation, and the second is a two-mode fourth-order Burgers equation. We show that multiple kink solutions for each model exist only for specific values of the nonlinearity and dispersion parameters included in the models. We will use the Cole-Hopf transformation combined with the simplified Hirota’s method to conduct this analysis. In mathematics and physics, multiple-scale analysis comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables.

Propagation of weakly nonlinear magnetoacoustic waves

Multiple-scale analysis is employed for the study of nonlinear wave propagation in periodic layered media. In a first step, wave propagation in each individual layer is modeled by a corresponding equivalent nonlinear transmission line. The multiple-scale analysis is then employed to establish a system of nonlinear equations for the amplitudes of the forward and backward waves in the transmission lines of the model mentioned. This system of nonlinear equations is solved with the aid of a continuation technique for derivation of transmission characteristics of the periodic structure. To study optical bistability, a parameter-switching algorithm is utilized for obtaining the solutions of the system including both stable and unstable ones. For the sake of verification, we have also utilized a nonlinear finite-difference time-domain method to analyze the wave propagation in the aforementioned structure.

The nonlinear waves described by the nonlinear equation are numerically investigated. With the aid of atomistic multiscale modelling and analytical approaches, buckling strength has been determined for carbon nanofibres/epoxy composite systems. Various nanofibres configurations considered are single walled carbon nano tube and single layer graphene sheet and SLGS/SWCNT hybrid systems. Computationally, both eigen-value and non-linear large deformation-based methods have been employed to calculate the buckling strength.

Nanoindentation simulation of coated aluminum thin film using quasicontinuum method

Five basic effects corresponding to different possible dependencies of the modulation amplitude on position, velocity, and slow time are selected . These effects offer a possibility for designing a high-frequency control of the slow motions with specified properties. For example, high-frequency excitation in a system with a nonlinear friction can essentially increase the effective damping. The results are also of significance for system identification and diagnostics.

• Some exact solutions of the nonlinear equation are found for integrable and non-integrable cases.
• Relevant to disciplines across engineering and physics, the asymptotic method combined with the multiple scale method is shown to be an efficient and intuitive way to approach mechanics.
• For the investigation of modulational instability problems three parameters P/Q,Kmax and Γmax play the central role.
• The spatio-temporal nonhomogeneous direct drive type sine-Gordon equation with phase shift and a double-well potential was considered.
• It was shown that there is an instability in which semi fluxons are spontaneously generated.
• This phenomenon of super current is called Josephson effect, and the device is known as Josephson junction .
• Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves.

Along with predicting the existence of the isolas, this technique also reveals the nature of the responses, thus simplifying the process of finding isolas using numerical continuation. Using the reductive perturbation method we show that the modified Burgers equation governs the propagation of a weakly nonlinear slow magnetoacoustic https://wizardsdev.com/en/news/multiscale-analysis/ wave near the equilibrium state with zero transverse magnetic field. The dubbed oscillations built in the Minkowski background for some nonlinear potentials such as axionic sine-Gordon potential are a counterpart of the real scalar field. The bosonic fields with mass such as axions, axion-like candidates, and hidden photons.

High Energy Physics – Theory

The material properties of coarse-scale are modeled with the nonlinear finite element method, in which the stress tensor and tangent modulus are computed using the Hill-Mandel principal through the atomistic RVE. In order to clearly represent the mechanical behavior of the fine-scale, the stress-strain curves of the atomistic RVE undergoing distinct type of deformation modes are delineated. These results are then assessed to obtain the proper fine-scale parameters for the multi-scale analysis. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm.

In this investigation, the approach for analyzing the structures of graphene/carbon fiber/epoxy composite at various length scales was documented. The methodology employed for evaluating the thermomechanical loadings was also specified. This investigation presents the study of ion acoustic shocks in a plasma comprising warm inertial ions, and superthermal hot electrons penetrated by an electron beam. Tanh-method is employed to derive its solution to study the ion acoustic shock structures. Further, by considering the contribution of higher order effects, the inhomogeneous Burgers-type equation is derived and its solution results in the formation of humped-type IA shocks.

The multiscale models are validated by observing reasonably well similarities in the load–depth curves obtained from multiscale and full MD simulations. Refining the element size down to atomic spacing resulted in high computational efforts while the analysis results do not improve significantly. Also, it is shown that by defining the thermostat in the atomistic part, wave reflections are eliminated at the interface of atomic and continuum domains. It is shown that by selecting appropriate dimensions of the atomic domain, there is no need to use nonlinear elasticity in the continuum region. Significant advances in nanoscale research have enabled the continuous miniaturization of devices.

Following the review, we highlight a new multiple-scale method, the bridging scale, and compare it to existing multiple-scale methods. Next, we show example problems in which the bridging scale is applied to fully non-linear problems. Concluding remarks address the research needs for multiple-scale methods in general, the bridging scale method in particular, and potential applications for the bridging scale. We develop two new equations which describe propagation of two different wave modes simultaneously.