Exponential Nonlinearity Research Articles

Existence of solutions to a Neumann boundary value problem with exponential nonlinearity

This paper is concerned with the solutions to the following sinh-Poisson equation with Hénon term{−Δu+u=ε2|x−q1|2α1⋯|x−qn|2αn(eu−e−u),u>0,inΩ,∂u∂ν=0,on∂Ω, where Ω⊂R2 is a bounded, smooth domain, ε>0, α1,...,αn∈(0,∞)∖N, and q1,...,qn∈Ω are fixed. Given any two non-negative integers k,l with k+l⩾1, it is shown that, for sufficiently small ε>0, there exists a solution uε for which ε2|x−q1|2α1⋯|x−qn|2αn(eu−e−u) asymptotically (i.e. the limit as ε→0) develops k+n interior Dirac measures and l boundary Dirac measures. The location of blow-up points is characterized explicitly in terms of Green's function of Neumann problem and the function k(x)=|x−q1|2α1⋯|x−qn|2αn.

Least energy sign-changing solution for N-Laplacian problem with logarithmic and exponential nonlinearities

In this paper, we establish sufficient conditions for the existence result of least energy sign-changing weak solution to the following quasilinear nonhomogeneous N-Laplacian equation with logarithmic and exponential nonlinearities:{−ΔNu=|u|p−2uln⁡|u|2+μf(u),inΩ,u=0,on∂Ω, where f(t) behaves like eα|t|N/(N−1). By using a main tool of constrained minimization in Nehari manifold, we consider both subcritical and critical exponential growths.

Arbitrary initial energy blow up for fourth-order viscous damped wave equation with exponential-type growth nonlinearity

This letter is concerned with the following damped nonlinear wave equation with exponential nonlinearity utt+Δ2u+u+Δ2ut+ut=f(u)inΩ×(0,∞)where the nonlinearity f is a regular function satisfying f(0)=0 with an exponential growth to fix later. The above wave problem describes a class of essential nonlinear evolution equations appearing in the elastic–plastic-microstructure models. In this letter blow up result is established for arbitrary initial energy with the viscous damping terms.

Existence of Positive Solutions to Weighted Linear Elliptic Equations Under Double Exponential Nonlinearity Growth

Existence of Positive Solutions to Weighted Linear Elliptic Equations Under Double Exponential Nonlinearity Growth

On the even solutions of the Toda system: A degree argument approach

<p style='text-indent:20px;'>In this paper, we show the existence of even solutions with prescribed asymptotic behavior at infinity. Instead of using the integrability of the Toda system, the novel idea here is a degree argument approach. Perturbation theory has also been used in our study. Our method can be generalized to explore non-integrable systems with exponential type nonlinearities.</p>

Robust cryptosystem using a new hyperchaotic oscillator with stricking dynamic properties

This paper presents a digital image cryptosystem utilizing a novel dynamic system with very interesting features. The oscillator is designed by introducing a feedback control law to the third line of the Lorenz oscillator with exponential nonlinearity. This exponential nonlinearity is replaced with hyperbolic sine nonlinearity to induce more complexity in the oscillator. Using some well-known computation analysis tools like Lyapunov spectrum, bifurcation analysis, and phase portraits representations, the dynamic analysis indicates that the oscillator can show chaos or hyperchaos for the same parameter space. In addition, the oscillator is equilibrium free, consequently its attractors are classified as hidden. Finally, the sequences of the oscillator are utilized to design a robust encryption scheme. Our method relies on a discrete orthogonal moment, confusion and diffusion stages. The input image is represented in the transform domain using Hahn orthogonal moments. Chaotic sequences are used to confuse and diffuse the obtaind image. Various security techniques have been used with success to show that our encryption process is powerful to resist malicious attacks.

Global well-posedness for the Vlasov-Poisson system with massless electrons in the 3-dimensional torus

The Vlasov-Poisson system with massless electrons (VPME) is widely used in plasma physics to model the evolution of ions in a plasma. It differs from the Vlasov-Poisson system (VP) for electrons in that the Poisson coupling has an exponential nonlinearity that creates several mathematical difficulties. In particular, while global well-posedness in 3 D is well understood in the electron case, this problem remained completely open for the ion model with massless electrons. The aim of this paper is to fill this gap by proving uniqueness for VPME in the class of solutions with bounded density, and global existence of solutions with bounded density for a general class of initial data, generalising all the previous results known for VP.

Two stage sintering behaviour of Er2O3 doped high performance ZnO varistors

High performance Er2O3 doped ZnO-V2O5 based varistors were prepared by two stage sintering (TSS) method at T1 = 1050 °C for 1/6 h and T2 = 750 and 800 °C for 2.5, 10, 20 and 40 h duration. The microstructural features and electrical properties were studied by XRD, SEM, EDS and a high voltage source unit. The combined effect of TSS and Er2O3 doping is very beneficial to restrain the grain coarsening of varistors. The effect of grain size reduction on nonlinear electrical properties was investigated. The results showed that superior electrical properties were obtained in doped samples via TSS than single stage one. A typical fine-grained (1.7 μm, 97.5 %TD) varistor with sufficiently high nonlinear exponent (α = 154), breakdown field (EB = 15.2 kV cm−1) and reasonably low leakage current density (JL = 0.13 mA cm-2) could be prepared with 0.5 mol.% Er2O3 doping at an optimum sintering condition of T1/T2 = 1050/750 °C for 40 h.

Stumped nature hyperjerk system with fractional order and exponential nonlinearity: Analog simulation, bifurcation analysis and cryptographic applications

This article presents the construction and consumption of hyperjerk system having chaotic nature with the commensurate ordered fractional derivative on rapidly digitalized emerging technologies. The system analyzed analytically using Lipschitz condition and numerically with the help of predictor corrector approaches. Originality of constructed system is confirmed using log error plots which gives satisfactory small values on finite time scale. The dynamical performances of the complex system are termed in multiple phase planes by the mean of their significant nature. Stability and bifurcation analysis around the fractional derivative is also studies for the various parameter of system to check the visibility of chaotic solution. The system is also designed in analog circuit simulator with the aid of operational amplifier, anti-parallel semiconductor diodes for validation of the system. With the help of random number generators (RNGs), binary array generated from the hyperjerk system and plugged in to the NIST 800–22 test suite to measure the randomness. The array having high randomness is used as strong cypher key for the cryptographic execution straightforwardly in both direction encryption and decryption.

Potential predictability of the MJO during easterly and westerly phases of the QBO

The potential predictability of the Madden–Julian Oscillation (MJO) in boreal winter (November–February) is investigated using observational data for the period of 1979–2016. For various MJO indices, nonlinear local Lyapunov exponents are computed to quantify the MJO predictability under the easterly and westerly phases of the Quasi-Biennial Oscillation (easterly: EQBO and westerly: WQBO). All MJO indices exhibit higher predictability during EQBO winters than during WQBO winters. Excluding strong ENSO years from EQBO and WQBO winters has a limited impact on MJO predictability. The highest potential predictability of 43 days during EQBO winters and 37 days during WQBO winters is found for the MJO index obtained from bandpass-filtered (30–80 days) outgoing longwave radiation and wind data. In contrast, the potential predictability of the MJO from the real-time multivariate MJO index is 21 days during EQBO winters and 13 days during WQBO winters. The longer persistence and less disorganization of the MJO during the EQBO winters lead to the higher predictability for EQBO winters, as compared with that for WQBO winters.

Traveling fronts in dissipative granular chains and nonlinear lattices

We consider an infinite chain of particles with nonlinear elastic and dissipative nearest neighbors interactions. Assuming the existence of a traveling front between uniformly compressed (or stretched) states, we obtain jump conditions relating the wave speed and limiting particle velocities to the relative displacements at infinity. Using this result, we characterise compression fronts in chains of touching beads, for viscoelastic contact laws that include a nonlinear elastic force (generalised Hertz contact) and viscous dissipation. We compute compression fronts numerically for the generalised Kuwabara–Kono model in which the viscous contact force is proportional to the derivative of the elastic force, without precompression of the chain. Steady fronts are obtained both as the end result of the compression of one end of the chain and using a shooting method which provides numerically exact traveling waves. Depending on the magnitudes of contact damping and strain applied downstream, we obtain either overdamped (monotonic) or underdamped (oscillatory) compression fronts. To explain this transition and approximate the front profiles, we consider a continuum limit valid when the exponent of the contact nonlinearity is close to (and above) unity. Using multiscale expansions, we formally derive two different amplitude equations for long waves, a Burgers equation with logarithmic nonlinearity, and a logarithmic Korteweg–de Vries (KdV)–Burgers equation for small contact damping. Both models possess traveling front solutions that are in good agreement with the front profiles computed numerically in the granular chain. The analysis of the logarithmic KdV–Burgers equation allows one to approximate the critical damping corresponding to the transition from underdamped to overdamped fronts.

A multi-model study of atmosphere predictability in coupled ocean\u2013atmosphere systems

Of great importance for guiding numerical weather and climate predictions, understanding predictability of the atmosphere in the ocean − atmosphere coupled system is the first and critical step to understand predictability of the Earth system. However, previous predictability studies based on prefect model assumption usually depend on a certain model. Here we apply the predictability study with the Nonlinear Local Lyapunov Exponent and Attractor Radius to the products of multiple re-analyses and forecast models in several operational centers to realize general predictability of the atmosphere in the Earth system. We first investigated the predictability characteristics of the atmosphere in NCEP, ECMWF and UKMO coupled systems and some of their uncoupled counterparts and other uncoupled systems. Although the ECMWF Integrated Forecast System shows higher skills in geopotential height over the tropics, there is no certain model providing the most precise forecast for all variables on all levels and the multi-model ensemble not always outperforms a single model. Improved low-frequency signals from the air − sea and stratosphere − troposphere interactions that extend predictability of the atmosphere in coupled system suggests the significance of air − sea coupling and stratosphere simulation in practical forecast development, although uncertainties exist in the model representation for physical processes in air − sea interactions and upper troposphere. These inspire further exploration on predictability of ocean and stratosphere as well as sea − ice and land processes to advance our understanding of interactions of Earth system components, thus enhancing weather − climate prediction skills.

Statistics of the stress, strain-rate and spin fields in viscoplastic polycrystals

This paper provides estimates for the stress, strain-rate and spin field statistics in viscoplastic polycrystals. They are obtained by combining the fully optimized second-order (FOSO) homogenization method with the self-consistent estimate for linear polycrystalline aggregates. The FOSO linearization method allows the consistent estimation of the field statistics in nonlinear composites and polycrystals directly from those in suitably optimized linear comparison composites (LCC). For the LCC, a novel generalized approach is used to extract the field statistics of the stress, the strain-rate and also of the continuum-spin fields (including the second moments). Then, the stress-field statistics are used to extract the plastic-spin statistics, which, together with the continuum-spin statistics, provide consistent estimates for the microstructural-spin statistics. Applications to power-law viscoplastic FCC polycrystals and ice-like HCP polycrystals are considered, and the overall and intragranular field fluctuations are computed and found to increase with nonlinearity exponent and grain anisotropy. In particular, the covariance tensors of the continuum-spin fluctuations within the grains of viscoplastic polycrystals, which are characterized here for the first time by means of nonlinear homogenization methods, are found to be quite significant, even when the corresponding averages of the continuum-spin in the grains are negligible, and thus have a strong influence on the corresponding microstructural-spin fluctuations.

Decreasing predictability as a precursor indicator for abrupt climate change

Abrupt climate change has an important impact on sustainable economic and social development, as well as ecosystem. However, it is very difficult to predict abrupt climate changes because the climate system is a complex and nonlinear system. In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change. The performance of NLLE as an early warning signal is first verified by those simulated abrupt changes based on four folding models. That is, NLLE in all experiments showed an almost monotonous increasing trend as a dynamic system approached its tipping point. For a well-studied abrupt climate change in North Pacific in 1976/1977, it is also found that NLLE shows an almost monotonous increasing trend since 1970 which give up to 6 years warning before the abrupt climate change. The limit of the predictability for a nonlinear dynamic system can be quantitatively estimated by NLLE, and lager NLLE of the system means less predictability. Therefore, the decreasing predictability may be an effective precursor indicator for abrupt climate change.

Pressure Losses in Power-Law Fluid Flow through a Tube of Variable Cross-Section

Steady laminar flow of an incompressible power-law fluid is a tube with an obstacle of given shape is numerically simulated. The mathematical description of the process is based on the vortex transport equation and the Poisson equation for the stream function, while the rheological properties of the medium are described by the Ostwald—de Waele power law. The steady solution of the problem is obtained using a time-dependent method based on the finite-difference approximation of the governing equations. The pressure distribution is determined by numerically solving the Poisson equation. A parametric investigation of the kinematic and dynamic flow parameters, as functions of the control parameters of the problem, is performed for non-Newtonian media. The effect of the Reynolds number, the nonlinearity exponent, and the obstacle geometry on the coefficient of the local fluid resistance is demonstrated.

When Bingham meets Bratu: mathematical and computational investigations

In this article, we discuss the numerical solution of the Bingham-Bratu-Gelfand (BBG) problem, a non-smooth nonlinear eigenvalue problem associated with the total variation integral and an exponential nonlinearity. Using the fact that one can view the nonlinear eigenvalue as a possible Lagrange multiplier associated with a constrained minimization problem from Calculus of Variations, we associate with the BBG problem an initial value problem (dynamical flow), well suited to time-discretization by operator-splitting. Various mathematical results are proved, including the convergence of a finite element approximation of the BBG problem. The operator-splitting/finite element methodology discussed in this article is robust and easy to implement. We validate the implementation by first solving the classical Bratu-Gelfand problem, obtaining and reporting results consistent with those found in the literature. We then explore the full capability of the implementation by solving the viscoplastic BBG problem, obtaining and reporting results for several values of the plasticity yield. We conclude by exhibiting and discussing the bifurcation diagrams corresponding to these same values of the plasticity yield, and by reporting and examining some finer details of the solver discovered during the course of our investigation.

CREEP OF A THICKWALLED QUASILINEAR VISCOELASTIC TUBE UNDER A CONSTANT EXTERNAL AND INTERNAL PRESSURE

We consider the creep problem for a quasilinear viscoelastic model of a thickwalled tube, loaded with constant internal and external pressure; the material is supposed to be incompressible. An exact solution to this problem was received by one of the authors in previous papers, assuming the state of a tube to be plain deformation; hereby we study properties of this solution for arbitrary material functions of quasilinear viscoelasticity constitutive relation. A criterion of stress stationarity is derived; the stress field of a thickwalled tube under a constant pressure evolves in time in the case of unbounded creep function and arbitrary nonlinearity function, except some particular types. The monotonicity of stress field components is studied: the radial stress monotonicity depends only on internal and external pressure values (for internal pressure, greater than an external one, it is negative and increases in radii). For other stress components, there are derived sufficient conditions of monotonicity. For an exponential nonlinearity function and unbounded creep function, a creep curve is determined to be concave up at the initial moment, and concave down during prolonged observation; the creep curve of a bipower nonlinearity function model may change its convexity. The stressstrain state of a model with a bounded creep function is proved to be bounded.

NONLINEAR THERMAL ANALYSIS FOR THE INTER-SHAFT BEARING UNDER THE DYNAMIC LOAD

As an important supporting part between the higher pressure rotor and the lower pressure rotor in a dual-rotor system, inner and outer races are rotating with the lower and higher pressure rotors, the heat transfer of the inter-shaft bearing is more prominent. This paper studies the nonlinear thermal behaviors of the inter-shaft bearing under the nonlinear dynamic load. The dynamic load of the inter-shaft bearing is defined based on the dynamic responses of the dual-rotor system, where the nonlinear factors of the inter-shaft bearing such as the radial clearance, the fractional exponential nonlinearity and the parameter excitation are taken into consideration. Thus the nonlinear behaviors such as the jump and the bi-stable phenomena happen to the dynamic load. According to Palmgren's empirical formula, a heat transfer model of the inter-shaft bearing under the dynamic load is established with considering the viscosity-temperature relationship of the lubricant. Steady-state temperatures of the inter-shaft bearing are obtained by the numerical integration. It shows that the nonlinear thermal behaviors of the inter-shaft bearing such as the jump and the bi-stable phenomena are caused by the nonlinear behaviors of the dynamic load. Furthermore, effects of the rotation speed ratio, the unbalances, the inter-shaft bearing's radial clearance, Hertz contact stiffness and roller number on the temperatures and the nonlinear thermal behaviors of the inter-shaft bearing are analyzed. The results show that the unbalances, the radial clearance and the stiffness only affect the nonlinear thermal behaviors of the inter-shaft bearing; the rotation speed ratio and the roller number have a magnificent influence on both. This discovery implies that the dynamic load is more appropriate to describe the actual load of the inter-shaft bearing than the static load. The thermal behaviors of the inter-shaft bearing become nonlinear, which is much more complicated than before, due to nonlinear dynamic characteristics of the dual-rotor system.

Cauchy problem for hyperbolic equations of third order with variable exponent of nonlinearity

We investigate weak solutions of the Cauchy problem for the third order hyperbolic equations with variable exponent of the nonlinearity. The problem is considered in some classes of functions namely in Lebesgue spaces with variable exponents. The sufficient conditions of the existence and uniqueness of the weak solutions to given problem are found.

A generalised power-law formulation for the modelling of damping and stiffness nonlinearities

In this paper, a single-degree-of-freedom dynamic model is described, with displacement- and velocity-dependent nonlinearities represented by power laws. The model is intended to support the dynamic identification of structural components subjected to harmonic excitation. In comparison to other analytical expressions, the data-driven estimation of the nonlinear exponents provides a large versatility, making the generalised model adaptable for a wide number of different nonlinearities in both stiffness and damping. For instance, the proposed damping formulation can naturally accommodate air drag (quadratic) damping as well as dry friction. Differently to purely data-driven methods (e.g. black boxes), the obtained model is fully inspectable. The proposed formulation is here applied to the large oscillations of a prototype highly flexible wing and fitted on its steady state response in the frequency domain. These large-amplitude flap-wise bending oscillations are known to be affected by nonlinearities in both the stiffness (nonlinear hardening) and the velocity-dependent damping terms. The model is validated against experiments for different structural configurations and input amplitudes, as both these nonlinearities are energy-dependent.