Exponential Nonlinearity Research Articles

Existence and non-existence of ground states of bi-harmonic equations involving constant and degenerate Rabinowitz potentials

Recently, the authors of the current paper established in Chen et al. (Calc Var Partial Differ Equ 59:1–38, 2020) the existence of a ground-state solution to the following bi-harmonic equation with the constant potential or Rabinowitz potential: $$\begin{aligned} (-\Delta )^{2}u+V(x)u=f(u)\ \text {in}\ \mathbb {R}^{4}, \end{aligned}$$ (0.1) when the nonlinearity has the special form \(f(t)=t(\exp (t^2)-1)\) and \(V(x)\ge c>0\) is a constant or the Rabinowitz potential. One of the crucial elements used in Chen et al. (Calc Var Partial Differ Equ 59:1–38, 2020) is the Fourier rearrangement argument. However, this argument is not applicable if f(t) is not an odd function. Thus, it still remains open whether the Eq. (0.1) with the general critical exponential nonlinearity f(u) admits a ground-state solution even when V(x) is a positive constant. The first purpose of this paper is to develop a Fourier rearrangement-free approach to solve the above problem. More precisely, we will prove that there is a threshold \(\gamma ^{*}\) such that for any \(\gamma \in (0,\gamma ^*)\), the Eq. (0.1) with the constant potential \(V(x)=\gamma >0\) admits a ground-state solution, while does not admit any ground-state solution for any \(\gamma \in (\gamma ^{*},+\infty )\). The second purpose of this paper is to establish the existence of a ground-state solution to the Eq. (0.1) with any degenerate Rabinowitz potential V vanishing on some bounded open set. Among other techniques, the proof also relies on a critical Adams inequality involving the degenerate potential which is of its own interest.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

Some existence results for elliptic systems with exponential nonlinearities and convection terms in dimension two

In this paper, we establish the existence of solutions to a class of elliptic systems. The nonlinearities include exponential growth terms and convection terms. The exponential growth term means it could be critical growth at $\infty$. The Trudinger-Moser inequality is used to deal with it. The convection term means it involves the gradient of unknown function. The strong convergence of sequences is employed to overcome the difficulties caused by convection terms. The variational methods are invalid and the Galerkin method and an approximation scheme are applied to obtain four different solutions. Our results supplements those from \cite{Araujo2018}.

Local and global existence of solutions to a time-fractional wave equation with an exponential growth

A time-fractional wave equation with an exponential growth source function is considered. This model can be regarded as a modified version of the one studied by Nakamura and Ozawa (1999). The main feature of this model is that the exponential growth nonlinearity is essentially different from the polynomial growth one, and harder in controlling. We first prove the local existence and uniqueness of mild solutions in an Orlicz space by deriving a nonlinear estimate and Lp−Lq estimates the source term and solution operators, respectively. Furthermore, by additionally using specific techniques for estimating integrals, we show that small data solutions exist globally over time in such Orlicz space. The last theoretical result is about the second global-in-time existence of solutions in Besov spaces. Such a result is based on a different nonlinear estimate which is derived from a logarithmic interpolation inequality. The main ingredients of proof are the efficient application of function spaces such as Lebesgue spaces, Orlicz space,Besov spaces and computational techniques involving generalized integrals. In addition, some numerical examples are provided to illustrate theoretical results.

Electronic equivalent of a pump-modulated erbium-doped fiber laser

We built the electronic equivalent circuit based on the model of a pump-modulated erbium-doped fiber laser. The model equations have two state variables, optical field intensity x and population inversion y, and a harmonic modulation applied to the pump parameter. In addition, the model contains two nonlinear terms, xy and ey. The exponential nonlinearity is implemented electronically using a power series expansion. The circuit dynamics is investigated using bifurcation, Lyapunov spectrum, and phase space analyses. Similar to the fiber laser, the equivalent electronic circuit exhibits very rich dynamics, exhibiting chaos and multiple coexisting periodic orbits. The existence of a saddle–node fixed point is demonstrated.

Blow-up in Coupled Solutions for a $4$-dimensional Semilinear Elliptic Kuramoto–Sivashinsky System

The existence of singular limit solutions is established for a nonlinear elliptic Kuramoto–Sivashinsky system with exponential nonlinearity and Navier boundary conditions, by means of the nonlinear domain decomposition method.

Embryonic aortic arch material properties obtained by optical coherence tomography-guided micropipette aspiration

It is challenging to determine the in vivo material properties of a very soft, mesoscale arterial vesselsof size ∼ 80 to 120 μm diameter. This information is essential to understand the early embryonic cardiovascular development featuring rapidly evolving dynamic microstructure. Previous research efforts to describe the properties of the embryonic great vessels are very limited. Our objective is to measure the local material properties of pharyngeal aortic arch tissue of the chick-embryo during the early Hamburger-Hamilton (HH) stages, HH18 and HH24. Integrating the micropipette aspiration technique with optical coherence tomography (OCT) imaging, a clear vision of the aspirated arch geometry is achieved for an inner pipette radius of Rp = 25 μm. The aspiration of this region is performed through a calibrated negatively pressurized micro-pipette. A computational finite element model is developed to model the nonlinear behaviour of the arch structure by considering the geometry-dependent constraints. Numerical estimations of the nonlinear material parameters for aortic arch samples are presented. The exponential material nonlinearity parameter (a) of aortic arch tissue increases statistically significantly from a = 0.068 ± 0.013 at HH18 to a = 0.260 ± 0.014 at HH24 (p = 0.0286). As such, the aspirated tissue length decreases from 53 μm at HH18 to 34 μm at HH24. The calculated NeoHookean shear modulus increases from 51 Pa at HH18 to 93 Pa at HH24 which indicates a statistically significant stiffness increase. These changes are due to the dynamic changes of collagen and elastin content in the media layer of the vessel during development.

Stability of equilibria of exponential type system of three differential equations under stochastic perturbations

A system of three differential equations with exponential nonlinearity is considered. It is shown that the considered system has both the zero and nonzero (positive or negative) equilibria. It is supposed that the system is exposed to stochastic perturbations that are directly proportional to the deviation of a system state from an equilibrium. Via the general method of Lyapunov functionals construction and the method of linear matrix inequalities sufficient conditions of stability in probability for an each equilibrium are obtained. Numerical simulations and figures are presented to demonstrate the obtained results. The proposed investigation method can be applied for many other types of nonlinear systems.

Steady-states solutions of the Vlasov–Maxwell–Fokker–Planck system of proton channeling in crystals

The paper studies special classes of the stationary solutions of the generalized Vlasov–Maxwell–Fokker–Planck (VMFP) system. We reduce the VMFP equations to a nonlinear elliptic system with exponential nonlinearities. For the Vlasov–Poisson–Fokker–Planck system a new form of stationary states is obtained which generalizes the known ones from the works of K. Dressler and R. Glassey. We consider the one-dimensional case of the elliptic equations, corresponding to the axial symmetry of a crystal. For the associated boundary value problem, the existence of at least one solution is proved by the lower–upper solution method. Besides, we propose an iterative algorithm and perform illustrative numerical calculations. The numerical results are compared with our upper–lower solutions.

Multiple solutions for elliptic equations with exponential nonlinearity term combined with convection term in dimension two

In this paper, we consider the following problem(0.1){−Δu=λ(|u|r−1u+a|∇u|s)+f(x,u)inΩ,u=0on∂Ω, where λ>0, r∈(0,1), s∈(0,2), a∈R and f∈C(Ω×R). The term f can be exponential growth at ∞. Convection term, namely gradient term, makes the problem (0.1) invariational. Under suitable conditions imposed on f, through the approximation scheme we prove that problem (0.1) admits a positive solution if a⩾0 and a negative solution if a⩽0 for λ∈(0,λ⁎) with λ⁎>0. Particularly, problem (0.1) admits a positive solution and a negative solution in the case a=0 for λ∈(0,λ⁎).

Polyharmonic Kirchhoff-type problem without Ambrosetti–Rabinowitz condition

In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti–Rabinowitz condition: { − M ( ∫ Ω | ∇ m u | n m ) Δ n m m u = f ( x , u ) | x | σ in Ω , u = ∇ u = ⋯ = ∇ m − 1 u = 0 on ∂ Ω , where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < σ < n , n ≥ 2 m ≥ 2 , M is a Kirchhoff function and f ( x , u ) has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.

Local solution to an energy critical 2-D stochastic wave equation with exponential nonlinearity in a bounded domain

We prove the existence and the uniqueness of a local maximal solution to an H1-critical stochastic wave equation with multiplicative noise on a smooth bounded domain D⊂R2 with exponential nonlinearity. First, we derive the appropriate deterministic and stochastic Strichartz inequalities in suitable spaces and, then use them in arguments based on fixed point method to show the local well-posedness result. We also present an explosion result for the constructed unique local maximal solution.

The backward nonlinear local Lyapunov exponent and its application to quantifying the local predictability of extreme high-temperature events

The backward nonlinear local Lyapunov exponent and its application to quantifying the local predictability of extreme high-temperature events

An efficient finite element iterative method for solving a nonuniform size modified Poisson-Boltzmann ion channel model

In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution of Model 1 collecting all the singular points and Models 2 and 3 much easier to solve numerically than the original nuSMPBIC model. A damped two-block iterative method is then presented to solve Model 3, along with a novel modified Newton iterative scheme for solving each related nonlinear algebraic system. To this end, an effective nuSMPBIC finite element solver is derived and then implemented as a program package that works for an ion channel protein with a three-dimensional molecular structure and a mixture of multiple ionic species. Numerical results for a voltage-dependent anion channel (VDAC) in a mixture of four ionic species demonstrate a fast convergence rate of the damped two-block iterative method, the high performance of the software package, and the importance of considering nonuniform ion sizes. Moreover, the nuSMPBIC model is validated by the anion selectivity property of VDAC.

Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation

We consider an equation with exponential nonlinearity under the Dirichlet boundary condition. For a one- or two-dimensional domain, a global solution has been obtained. In this paper, to extend the result to a higher dimensional case, we concentrate on the radial solutions in an annulus. First, we construct a time-local solution with an abstract theory of differential equations. Next, we show that decreasing energy exists in this problem. Finally, we establish a global solution for the sufficiently small initial value and parameter by Sobolev embedding and Poincaré inequalities together with some technical estimates. Moreover, when we take the smaller parameter, we prove that the global solution tends to zero as time goes to infinity.

Blow-Up Time of Solutions for a Parabolic Equation with Exponential Nonlinearity

This paper studies a parabolic equation with exponential nonlinearity, which has several applications, for example the self-trapped beams in plasma. Based on a modified concavity method we prove the blow-up of the solution for initial data with high initial energy. We also proposed the solution’s lower and upper bound of the blow-up time for the equation. Our results complement the existing results.

The coupling effects of pore structure and rock mineralogy on the pre-Darcy behaviors in tight sandstone and shale

The pore structure and mineral composition play a crucial role in controlling the water flow behaviors of low-permeability media, which is characterized by threshold pressure gradient (TPG) and nonlinear flow curve. To figure out the coupling effects of these two factors on pre-Darcy flow regimes, water flow experiments were conducted on four tight sandstones, three shales and one fractured shale. The results indicate that both the TPG and pseudo threshold pressure gradient (PTPG) decrease with an increase in total porosity and macropores porosity. The variation of the nonlinear exponent (n) with total porosity and macropore porosity can be described by an exponential function with three parameters. The clay minerals and brittle minerals exert completely opposite influence on the variation of pre-Darcy flow parameters (PTPG, TPG and n). Both TPG/PTPG and n increase with increasing clay content while reduce with increase in the brittle minerals content. Theoretical analyses based on the disjoining pressure demonstrate that immovable water and electro-viscous effect are mainly responsible for water flow deviating from Darcy's law in tight sandstone and shale. Calculation results show that the proportion of immovable water of intact samples ranges from 23.46% to 89.73%, the movable water content ranges from 10.27%–76.54%. The pre-Darcy flow parameters exponentially increase with increase in immovable water content while exponentially decrease with increasing proportion of movable water. For the shale sample used in this work, the immovable water and electro-viscous effect contributing to the pre-Darcy flow account for 81.11%–90.35% and 9.58%–18.51%, respectively. The different flow regimes are controlled by the pore diameter and the thickness of water with shear fluidity. If the diameter of interconnected pores is larger than 10 μm, the Darcy flow regime dominates. When the diameter of minor pore throat of the flow channels are less than 33.82 nm for sandstones and 37.68 nm for shales, the movement of water in this flow channel would generate the TPG.

Chaotic RF Generator for Sub-1-GHz Chaos-Based Communication Systems: Mathematical Modeling and Experimental Validation

In this paper, we have studied, designed and realized a single-transistor chaotic generator with a smooth power spectrum of about −35 dBm in frequency bandwidth up to 1 GHz. The chaos generator model is described by a continuous-time six-dimensional autonomous system assuming an exponential nonlinearity. Chaotic behavior is characterized by bifurcation diagrams, Lyapunov exponents, phase portraits of the attractors and spectra of the oscillations, using both numerical and circuit simulations. Advanced Design System (ADS)-based simulations were carried out to support the theoretical analysis, and to validate the mathematical model. The simulations are carried out using real transistor parameters and taking into account the properties of the substrate, the influence of the board topology and the characteristics of the layout material. This simulation method known as EM/Circuit Co-Simulation allows the simulation results to approach as closely as possible to those of the experiments. In the light of the positive simulation results, the proposed structure is realized to demonstrate its feasibility, and to confirm the numerical results. The prototype is manufactured and mounted on a breadboard using the surface-mount devices and BFP193 bipolar junction transistor. After realizing the generator, we pushed it further towards perfection, through the proposal and realization of a broadband amplifier, in order to gain more bandwidth to improve the spectral characteristic of the generator, which makes it promising for many communication applications such as spread spectrum communication, direct chaotic communication, etc.

Lie group classification of the nonlinear transmission line model and exact traveling wave solutions

A nonlinear transmission line (NLTL) model is very essential tools in understanding of propagation of electrical solitons which can propagate in the form of voltage waves in nonlinear dispersive media. These models are often formulated using nonlinear partial differential equations. One of the basic tools available to study these equations are numerical methods such as finite difference method, finite element method, etc, have been developed for nonlinear partial differential equations. These methods require a great amount of time and memory due to the discretization and usually the effect of round-off error causes loss of accuracy in the results. So in this paper, we use one of the most famous analytical methods the Lie group analysis due to Sophus Lie. One of the advantages of this approach is that requires only algebraic calculations. The main aim of this study is to explore the nonlinear transmission line model with arbitrary capacitor's voltage dependence, through the use of Lie group classification, we show that the specifying form of arbitrary capacitor's voltage are power law nonlinearity, exponential law nonlinearity and constant capacitance. The exact solutions and similarity reductions generated from the symmetries are also provided. Furthermore, translational symmetries were utilized to find a family of traveling wave solutions via the \(\tanh\)-method of the governing nonlinear problem.

Study of the structural properties and temperature-dependent hopping conductivity mechanism to the analysis of nonlinearity exponent in Nd0.7−xCexSr0.3MnO3 manganites

Study of the structural properties and temperature-dependent hopping conductivity mechanism to the analysis of nonlinearity exponent in Nd0.7−xCexSr0.3MnO3 manganites