Two-time Method Research Articles

Two-Timing Methods Valid on Expanding Intervals

The problem \[y'' + \varepsilon f(y,y') + y = 0,\quad y(0) = a,\quad y'(0) = b,\]with f a polynomial is considered. A two-timing method is described which yields series expansions in s for the solution. These expansions are then shown to be generalized asymptotic expansions, uniformly valid on t intervals of the form $[0, k/ \varepsilon ]$. The method of proof also yields the existence of solutions to the above problem on such intervals for k appropriately chosen.

A convergent two-time method for periodic differential equations

Two timing, an ad hoc method for studying periodic evolution equations, can be given a rigorous justification when the problem is in standard form, u = ϵf( t, u). First solve dw dσ = ϵ(I − M) f(σ, w) for w(σ, v) , where M is the mean value operator and v is any initial value. Then w( σ, v) is periodic in σ but does not satisfy the original equation. Now, force a solution u( t), using nonlinear variation of constants, in the form w( σ, v( τ)), where σ = t is the fast time and τ = ϵt is the slow time. With the resulting differential equation for v, one reads off from its nonconstant solutions thè approximate transient behavior of u( t) for times of order ϵ −1. On the other hand, the equilibrium points (constant solutions) v 0 correspond to steady state (periodic solutions) of the original system. Interesting applications, such as to one-dimensional wave equations with cubic damping, can be given.

Elementary excitation spectrum of the Heisenberg ferromagnet

The method of thermodynamic two-time retarded Green function is applied to calculate the elementary excitation spectrum of a simple cubic Heisenberg ferromagnet. An expression for the spin-wave self-energy operator is obtained in terms of the three-particle Green function. The equation of motion for the three-particle Green function is solved in the approximation in which all Green functions involving more than three particles are neglected. In the limit of low temperatures the spectrum reduces to the result of Silberglitt and Harris. This spectrum is valid for all wavelengths and includes the effect of two-spin-wave bound states and scattering resonances. In the limit of long wavelengths the well-known spectrum of Dyson is recovered. The extension of the Green-function method to higher-order calculations and to the study of the effect of multi-spin-wave complexes is straightforward and is briefly discussed.

An Iterative Approach to Nonlinear Dynamic Stability Problems

A two-time iteration scheme for solving nonlinear dynamic stability problems is presented. It not only has distinct computational advantages over the two-time perturbation method of Matkowsky, Kogelman and J. B. Kelley but also possesses a greater range of applicability. A nonlinear diffusion equation is used to illustrate the two-time iteration method and in order to substantiate our results, the same problem is solved numerically by using a finite-difference method.

Stochastic Differential Equations with Applications to Random Harmonic Oscillators and Wave Propagation in Random Media

The two-time method is used to obtain an expansion, valid for $\varepsilon $ small and t large, of the vector solution $u( {t,\varepsilon } )$ of an abstract ordinary differential equation involving $\varepsilon $. The same method is used to get expansions of functions of u. The results are shown to apply to the solutions of stochastic equations. They are used to find the first two moments and the transition probability of the displacement of a harmonic oscillator with spring constant a random function of t. The result contains the condition for mean square stability due to Stratonovich. The results are also applied to one-dimensional wave propagation through a layer with refractive index a random function of position. They are used to find the mean square amplitude reflection and transmission coefficients, which are just the mean power reflection and transmission coefficients. A graph of the mean square transmission coefficient as a function of layer thickness is presented. The results are also compared with those obtainable by other methods.

Comparison of the Modified Method of Averaging and the Two Variable Expansion Procedure

Comparison of the Modified Method of Averaging and the Two Variable Expansion Procedure