On a certain nonlinear equation with fast oscillating coefficients (Q1902786)

The problem of the so-called ``echo-effect'' occupies a noticeable place in the studies of both transition processes in spin-systems and transition signals in poly-crystalline ferroelectrics. The mathematical model of this phenomenon for ferroelectrics, which takes into account a nonlinear dependence between deformation and dipole moment, has the following form \[ \ddot x(t) + \omega^2 x(t) = A[\chi_{[0,\Delta]} (t) \sin \omega t + \chi_{[\tau, \tau + \Delta]} (t) \sin \omega (t - \tau)] + Bx^2(t) \sin \omega t, \quad t\in [0,T], \tag{1} \] \[ x(0) = \dot x(0) = 0. \tag{2} \] Here \(\chi_I\) is the characteristic function of a segment \(I\); \(A, B \) are constants, and the constants \(\omega\), \(\Delta\), \(\tau\) and \(T\) satisfy the conditions \({1\over \omega} \ll \Delta\), \(\Delta \ll \tau\), \(T > 2\tau\). Equation (1) describes resonance oscillations of individual crystals. In the case where the number of resonance crystals is distributed near the resonance frequency \(\omega_0\) of the spectrometer by the normal law with dispersion \(\sigma^2\), the observed signal \(J(t,x)\) is determined by the following equality \[ J(t,x) = {1\over \sqrt {2\pi} \sigma} \int^{\omega_0 + 3\sigma}_{\omega_0 - 3\sigma} \text{exp} \{-(\omega -\omega_0)^2/2\sigma^2 \}x (t,\omega) d\omega, \] where \(x(t,\omega)\) is a solution to problem (1), (2). The study of the echo-effect can be reduced to the investigation of the behavior of the function \(J(t,x)\) in a neighborhood of \(t = 2\tau\). We offer an iteration process which enables us to construct a quantum libet precise approximation of \(J(t, x)\) and to demonstrate that the echo-effect appears even at the second step of iterations.