On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps

We show that certain naturally arising cones over the main component of a moduli space of J₀-holomorphic maps into Pn have a well-defined Euler class. We also prove that this is the case if the standard complex structure J₀ on Pn is replaced by a nearby almost complex structure J. The genus-zero analogue of the cone considered in this paper is a vector bundle. The genus-zero Gromov-Witten invariant of a projective complete intersection can be viewed as the Euler class of such a vector bundle. As shown in a separate paper, this is also the case for the “genus-one part” of the genus-one GW-invariant. The remaining part is a multiple of the genus-zero GW-invariant.     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Validity of the Weakly Nonlinear Solution of the Cauchy Problem for the Boussinesq‐Type Equation

We consider the initial‐value problem for the regularized Boussinesq‐type equation in the class of periodic functions. Validity of the weakly nonlinear solution, given in terms of two counterpropagating waves satisfying the uncoupled Ostrovsky equations, is examined. We prove analytically and illustrate numerically that the improved accuracy of the solution can be achieved at the timescales of the Ostrovsky equation if solutions of the linearized Ostrovsky equations are incorporated into the asymptotic solution. Compared to the previous literature, we show that the approximation error can be controlled in the energy space of periodic functions and the nonzero mean values of the periodic functions can be naturally incorporated in the justification analysis.     

On two classes of autonomous third-order nonlinear ordinary differential equations

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.     

Asymptotically periodic solutions to Volterra integral equations in epidemic models

Through the use of the limiting equation, conditions are given under which a scalar nonlinear Volterra integral equation, of either convolution or nonconvolution type, has an asymptotically periodic solution. Periodicity can be in either the forcing function or the kernel of the integral equation. The results are applied in several models for the spread of gonorrhea to gain insight into the causes of the observed oscillations in the infective populations.     

Nonlinear electrohydrodynamic Kelvin-Helmholtz and Marangoni instabilities near the marginal state

The effects of surface tension and adsorption on the electrohydrodynamic Kelvin-Helmholtz instability are studied. The system is stressed by a normal electric field such that it allows for the presence of surface charges at the interface. The method used is that of multiple scales. The nonlinear Schrödinger equation describing the behavior of the disturbed system is derived. The stability of the perturbed system is discussed both analytically and numerically and the stability diagrams are obtained. At the critical point, a generalized formulation of the evolution equation is developed, which leads to the nonlinear Klein-Gordon equation. The various stability criteria are derived from this equation.     

The WDVV symmetries in two-primary models

From the bi-Hamiltonian standpoint, we investigate symmetries of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations proposed by Dubrovin. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-Hamiltonian systems of hydrodynamic type. In particular, we show that the moduli space of two-primary models under symmetries of the WDVV equations can be parameterized by the polytropic exponent h. We discuss the transformation properties of the free energy at the genus-one level.     

Nonlinear stability and chaos in electrohydrodynamics

Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.     

The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions with the same initial data of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneous to the leading order asymptotic behavior of the Fourier transform of the initial data near k= 0, different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible in comparison with dispersion, and where nonlinear effect are asymptotically negligible in comparison with linear effect, i.e., dissipation and dispersion. They always hold for solutions with "small" initial data. In most circumstances however a new a priori bound on certain negative homogeneous Sobolev norms of solutions is obtained, which implies that any solution, even one which is initially "large" will eventually satisfy the smallness condition, and hence will have the above described asymptotic behavior     

Multiple-scale Homogenization for Weakly Nonlinear Conservation Laws with Rapid Spatial Fluctuations

We consider hyperbolic conservation laws with rapid periodic spatial fluctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are computed asymptotically using multiple spatial and temporal scales to capture the homogenized solution as well as its long-term behavior. We show that the linear problem may be destabilized through interactions between two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation, introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the homogenized leading-order solution are more general than their counterparts for conservation laws having no rapid spatial variations. In particular, these equations may be diffusive for certain general flux vectors. Selected examples are solved numerically to substantiate the asymptotic results.     

Solutions to the random heat equation by the method of successive approximations

An iterative scheme for solving the random heat equation is proposed. Convergence of the method is established. Properties of the solution as well as error estimates are obtained. Indications as to possible application to nonlinear, inhomogeneous, time-dependent, random diffusion problems are given. A specific example of application to random diffusion in the unit interval is treated both analytically and numerically.     

A Numerical Study of the Light Bullets Interaction in the (2+1) Sine-Gordon Equation

The propagation and interaction in more than one space dimension of localized pulse solutions (so-called light bullets) to the sine-Gordon [SG] equation is studied both asymptotically and numerically. Similar solutions and their resemblance to solitons in integrable systems were observed numerically before in vector Maxwell systems. The simplicity of SG allows us to perform an asymptotic analysis of counterpropagating pulses, as well as a fully resolved computation over rectangular domains. Numerical experiments are carried out on single pulse propagation and on two pulse collision under different orientations. The particle nature, as known for solitons, persists in these two space dimensional solutions as long as the amplitudes of initial data range in a finite interval, similar to the conditions on the vector Maxwell systems.     

Analytical approximate solutions for the generalized nonlinear oscillator

He's energy balance method (HEBM) is employed in this article to obtain the analytical approximate solution of the generalized nonlinear oscillator. Existence of periodic solutions is analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A number of numerical simulations are carried out and accuracy of the HEBM is then examined within an error analysis. The exact values of the natural frequency numerically obtained via the elliptic integrals are taken into account as the references bases and the relative error is then evaluated for a range of oscillation amplitudes. Excellent correlation of the approximate frequencies with the exact ones demonstrates that the approximate solutions are quite consistent even for large amplitudes of oscillation.     

Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

We investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.     

A Two-Grid Finite-Element Method for the Nonlinear Schrödinger Equation

In this paper, some two-grid finite element schemes are constructed for solving the nonlinear Schrödinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy.     

On the structure of solutions of a class of hyperbolic systems with several spatial variables in the far field

An asymptotic expansion of the solution to the Cauchy problem for a class of hyperbolic weakly nonlinear systems with many spatial variables is constructed. A parabolic quasilinear equation describing the behavior of the solution at asymptotically large values of the independent variables is obtained. The pseudo-diffusion processes that depend on the relationship between the number of equations and the number of spatial variables are analyzed. The structure of the subspace in which there are pseudo-diffusion evolution processes of the solution in the far field is described.     

Inaccessible States in Time-Dependent Reaction Diffusion

Using asymptotic methods we show that the long-time dynamic behavior in certain systems of nonlinear parabolic differential equations is described by a time-dependent, spatially inhomogeneous nonlinear evolution equation. For problems with multiple stable states, the solution develops sharp fronts separating slowly varying regions. By studying the basins of attraction of Abel's nonlinear differential equation, we demonstrate that the presence of explicit time dependence in the asymptotic evolution equation creates “forbidden regions” where the existence of interfaces is excluded. Consequently, certain configurations of stable states in the nonlinear system become inaccessible and cannot be achieved from any set of real initial conditions.     

DECAY ESTIMATIONS OF THE SOLUTION OF A NONLINEAR PSEUDOPARABOLIC EQUATION

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受弹性点支的任意形状的膜的振动

本文提供了一个求解受弹性点支的任意形状的膜的振动的新方法.将弹性点支反力看作是作用于膜上的未知外力,求出了包含有未知反力的运动方程的精确解,利用弹性点支处位移和反力的线性关系导出频率方程.最后以受弹性点支的圆膜为例给出了其频率方程的具体计算公式,并数值计算了受两个对称弹性点支的圆膜的固有振动频率.     

Qualitative difference between solutions for a model of the Boltzmann equation in the linear and nonlinear cases

We consider the Boltzmann equation in the framework of a nonlinear model for problems of the gas flow in a half-space (the Kramers problem). We prove the existence of a positive bounded solution and find the limit of this solution at infinity. We show that taking the nonlinear dependence of the collision integral on the distribution function into account leads to an asymptotically new solution of the initial equation. To illustrate the result, we present examples of functions describing the nonlinearity of the collision integral.