Initial supercritical behavior of buckled transversely isotropic elastic medium

The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr ₁ x ₁sinr ₂ x ₂. The uncertainty of the mode of buckling consists in the fact that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r ₁ = r ₂) and the corrugation-type mode (when one of the wave numbers r ₁ or r ₂ vanishes). The initial supercritical equilibrium is shown as being stable.     

Solitary waves of the EW and RLW equations

Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long–wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank–Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the L₂-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.     

Effects of imperfection of ionic channels and exposure to electromagnetic fields on the generation and propagation of front waves in nervous fibre

This work deals with the effects of perturbations such as imperfection of ionic channels and exposure to electromagnetic field on the generation and propagation of wave fronts in a nervous fibre. The initial excitation inserted in the fibre is a pulse of amplitude a and width 1/k. The domain of initial values of a and k leading to front waves generation are delineated for each type of perturbation. Links of the results to biological facts are given. It is found that imperfections of ionic channels strongly modify the velocity of propagation and can even lead to propagation failure.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases

The object of the present investigation is to study the anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases. The velocity of propagation of the wave frcnt is determined. A set of differential equations governing the growth and decay of weak discontinuities are obtained and solved. It is found that if the sonic wave is a compressive wave of order 1, then it terminates into a shock wave after a critical timet _c which has been determined. It is also observed that the effects of heat conduction and dissociation are to decrease the duration of time by which a weak discontinuity will generate into a shock wave.     

Shock Waves in Dispersive Eulerian Fluids

The long-time behavior of an initial step resulting in a dispersive shock wave (DSW) for the one-dimensional isentropic Euler equations regularized by generic, third-order dispersion is considered by use of Whitham averaging. Under modest assumptions, the jump conditions (DSW locus and speeds) for admissible, weak DSWs are characterized and found to depend only upon the sign of dispersion (convexity or concavity) and a general pressure law. Two mechanisms leading to the breakdown of this simple wave DSW theory for sufficiently large jumps are identified: a change in the sign of dispersion, leading to gradient catastrophe in the modulation equations, and the loss of genuine nonlinearity in the modulation equations. Large amplitude DSWs are constructed for several particular dispersive fluids with differing pressure laws modeled by the generalized nonlinear Schrödinger equation. These include superfluids (Bose–Einstein condensates and ultracold fermions) and “optical fluids.” Estimates of breaking times for smooth initial data and the long-time behavior of the shock tube problem are presented. Detailed numerical simulations compare favorably with the asymptotic results in the weak to moderate amplitude regimes. Deviations in the large amplitude regime are identified with breakdown of the simple wave DSW theory.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.     

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L₂ and L^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

Propagation of harmonic waves in prestressed fiber viscoelastic thick tubes filled with a viscous dusty fluid

In this work, propagation of harmonic waves in initially stressed cylindrical viscoelastic thick tubes filled with a Newtonian fluid is studied. The tube, subjected to a static inner pressure P_i and a positive axial stretch λ, will be considered as an incompressible viscoelastic and fibrous material. The fluid is assumed as an incompressible, viscous and dusty fluid. The field equations for the fluid are obtained in the cylindrical coordinates. The governing differential equations of the tube’s viscoelastic material are obtained also in the cylindrical coordinates utilizing the theory of small deformations superimposed on large initial static deformations. For the axially symmetric motion the field equations are solved by assuming harmonic wave solutions. A closed form solution can be obtained for equations governing the fluid body, but due to the variability of the coefficients of resulting differential equations of the solid body, such a closed form solution is not possible to obtain. For that reason, equations for the solid body and the boundary conditions are treated numerically by the finite-difference method to obtain the effects of the thickness of the tube on the wave characteristics. Dispersion relation is obtained using the long wave approximation and, the wave velocities and the transmission coefficients are computed.     

Asymptotic behavior at infinity for solutions of Emden-Fowler type equations

The semilinear equation Δu = |u|^σ?1 u is considered in the exterior of a ball in ? ⁿ , _n ≥ 3. It is shown that if the exponent σ is greater than a “critical” value (= n/n?2), then for x → ∞ the leading term of the asymptotics of any solution is a linear combination of derivatives of the fundamental solution. It is shown that there exist solutions with the indicated leading term of an asymptotics of such a type.     

Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model

For a reaction-diffusion system that serves as a 2-species Lotka-Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c^?, called the minimum wave speed, such that, for each c larger than or equal to c^?, the reaction-diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c?c^?. It has been shown that the minimum wave speed for this system is identical to another important quantity - the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c^? not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.     

Electric Field-induced dynamical evolution of spiral wave in the regular networks of Hodgkin-Huxley neurons

An additional gradient force is often used to simulate the polarization effect induced by the external field in the reaction-diffusion systems. The polarization effect of weak electric field on the regular networks of Hodgkin-Huxley neurons is measured by imposing an additive term V_E on physiological membrane potential at the cellular level, and the dynamical evolution of spiral wave subjected to the external electric field is investigated. A statistical variable is defined to study the dynamical evolution of spiral wave due to polarization effect. In the numerical simulation, 40000 neurons placed in the 200 × 200 square array with nearest neighbor connection type. It is found that spiral wave encounters death and the networks become homogeneous when the intensity of electric field exceeds the critical value, otherwise, spiral wave keeps alive completely. On the other hand, breakup of spiral wave occurs as the intensity of electric field exceeds the critical value in the presence of weak channel noise, otherwise, spiral wave keeps robustness to the external field completely. The critical value can be detected from the abrupt changes in the curve for factors of synchronization vs. control parameter, a smaller factor of synchronization is detected when the spiral wave keeps alive.     

Synchrony of two uncoupled neurons under half wave sine current stimulation

Two uncoupled Hindmarsh–Rose neurons under different initial discharge patterns are stimulated by the half wave sine current; and the synchronization mechanism of the two neurons is discussed by analyzing their membrane potentials and their interspike interval (ISI) distribution. Under the half wave sine current stimulation, the two uncoupled neurons under different initial conditions, whose parameter r (the parameter r is related to the membrane penetration of calcium ion, and reflects the changing speed of the slow adaptation current) is different or the same, can realize discharge synchronization (phase synchronization) or the full synchronization (state synchronization). The synchronization characteristics are mainly related to the frequency and the amplitude of the half wave sine current, and are little related to the parameter r and the initial state of the two neurons. This investigation shows the mechanism of the current’s amplitude and its frequency affecting the synchronization process of neurons, and the neurons’ discharge patterns and synchronization process can be adjusted and controlled by the current’s amplitude and its frequency. This result is of far reaching importance to study synchronization and encode of many neurons or neural network, and provides the theoretic basis for studying the mechanism of some nervous diseases such as epilepsy and Alzheimer’s disease by the slow wave of EEG.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Composite dynamical behaviors in a simple series–parallel LC circuit

In this paper, we report a variety of dynamical behaviors exhibited in a compact series–parallel LC circuit system comprising of two active elements, one linear negative conductance and one ordinary junction diode with piecewise linear v ? i characteristics. For convenience, we consider the amplitude (E_f) and frequency (f) of the driving force as control parameters amongst various other parameters. We observe the phenomenon of antimonotonicity, torus breakdown to chaos, bubbles to chaos, period doubling to chaos and emergence of multiple attractors which follow a progressive sequence, etc. As an overview to understand many more variety of bifurcations and attractors, the construction of two parameter phase diagram is also shown pictorially. The chaotic dynamics of this circuit is realized by laboratory experiment, numerical and analytical investigations and found that the results are in good agreement with each other.     

The destruction of tori in volume-preserving maps

Invariant tori are prominent features of symplectic and volume-preserving maps. From the point of view of chaotic transport the most relevant tori are those that are barriers, and thus have codimension one. For an n-dimensional volume-preserving map, such tori are prevalent when the map is nearly “integrable,” in the sense of having one action and n − 1 angle variables. As the map is perturbed, numerical studies show that the originally connected image of the frequency map acquires gaps due to resonances and domains of nonconvergence due to chaos. We present examples of a three-dimensional, generalized standard map for which there is a critical perturbation size, ε_c, above which there are no tori. Numerical investigations to find the “last invariant torus” reveal some similarities to the behavior found by Greene near a critical invariant circle for area preserving maps: the crossing time through the newly destroyed torus appears to have a power law singularity at ε_c, and the local phase space near the critical torus contains many high-order resonances.     

Mathematical model for coupled roll and yaw motions of a floating body in regular waves under resonant and non-resonant conditions

The prediction of resonance is very important with respect to the vessels stability in the early stages of design. In this paper, an efficient modeling approach is presented to determine coupled roll and yaw motions of a symmetric and slender floating body when the influences of small amplitude regular waves are dominant. The angular motions described in time domain by considering all internal and external forces are transformed into frequency domain to obtain motion characteristics. We adopt a semi-analytical treatment to obtain roll and yaw motions and derive system instability due to roll resonance. To compute hydrodynamic forces, we employ strip theory method where frequency dependent sectional added-mass, damping and restoring coefficients are derived from the Frank’s close-fit curve. Numerical experiments carried out for a vessel of mass 19,190 ton under the action of wave of frequencies 0.56 and 0.76 rad/s with zero and non-zero initial conditions are reported and the effect of various parameters on system stability is investigated. Model results indicate that damping factor (ς) plays a pivotal role when wave encountering frequency (ω) and undamped natural frequency (β) are nearly equal. The essence of this study lies in the efficient modeling technique to evaluate damping factor and critical encountering frequency regime for a given ship particulars when experimentally derived resonance zone is absent.     

Asymptotic Analysis of Reacting Materials with Saturated Explosion. I. Low-Frequency Waves

We derive and study asymptotic equations describing the behavior of low-frequency disturbances in a chemically reacting gas, whose amplitude is inversely proportional to the large activation energy of the reaction. We make an assumption of small heat release, so that fuel depletion is included in the model at leading order. For homogeneous combustion, introducing this assumption removes the thermal-runaway singularity which always forms in models previously used. We demonstrate that for spatially varying solutions, wave-propagation effects permit a singularity still to form, if the initial data has a step discontinuity with amplitude larger than a critical value. We suggest that appearance of this singularity in the model always signifies the birth of a detonation wave, and we describe the evolution of the wave beyond the singularity time.     

Bifurcations of limit circles and center conditions for a class of non-analytic cubic Z 2 polynomial differential systems

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z ₂-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z ₂-equivariant non-analytic cubic differential systems.