We develop a representation of reverse-time migration (RTM) in terms of Fourier integral operators, the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse-time co...

With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions (ABCs) for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been u...

We study the influence of thermal fluctuations on the dewetting dynamics of thin liquid films. Starting from the incompressible Navier-Stokes equations with thermal noise, we derive a fourth-order degenerate parabolic stochastic partial differential equation which includes a conservative, multiplicative noise term—the stochastic thin-film equation. Technically, we rely on a long-wave-approximat...

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K-P equation (Kadomtsev & Petviashvili 1970) to incl...

A novel non-paraxial Time-Domain technique based on Pade approximant for modeling ultra short optical pulses has been formulated and tested. The new method is an extension to the parabolic Time-Domain Beam Propagation Method (TDBPM) that solves the time domain wave equation along one direction that allows the time window to follow the evolution of the pulse. In this work the technique was appli...

A short-wave problem of reflection and radiation by an open end of a two-dimensional planar waveguide is studied. The incident mode is assumed to have frequency close to the cut-off. The problem is studied in the parabolic approximation. A recently developed approach based on the embedding formula and the “spectral” equation for the directivity of an edge Green’s function is applied to the prob...

In this paper we give an overview on some methods that are useful to solve a class of integrodi erential inverse problems. Precisely, we present some methods to solve integrodi erential inverse problems of parabolic type that are based on the theory of analytic semigroups, optimal regularity results and xed point arguments. A large class of physical models can be treated with this procedure, fo...

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schrödinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation...

We study the Cauchy problem for a nonlinear damped wave equation. Under suitable assumptions for the nonlinearity and the initial data, we obtain the global solution which satisfies weighted L and L∞ estimates. Furthermore, we establish the higher order asymptotic expansion of the solution. This means that we construct the nonlinear approximation of the global solution with respect to the weigh...

We systematically provide a similarity transformation reducing the (3 + 1)-dimensional inhomogeneous coupled nonlinear Schrödinger (CNLS) equation with variable coefficients and parabolic potential to the (1 + 1)-dimensional coupled nonlinear Schrödinger equation with constant coefficients. Based on the similarity transformation, we discuss the dynamics of the propagation of the three-dimension...