Related papers: The Fourier transform and the wave equation
Due to the space and time dependence of the wave function in the time dependent Schroedinger equation, different boundary conditions are possible. The equation is usually solved as an ``initial value problem'', by fixing the value of the…
We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the…
In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. As is pointed out by [8], in this combination, the frictional damping term is dominant for the viscoelastic one for the…
The linearized Davey-Stewartson equation with varing coefficients is solved by Fourier method. The approach uses the inverse scattering transform for the Davey-Stewartson equation.
Solutions for a class of wave equations with effective potentials are obtained by a method of a Laplace-transform. Quasinormal modes appear naturally in the solutions only in a spatially truncated form; their coefficients are uniquely…
We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according with the behavior…
Nowadays the Euler-Poisson-Darboux equation is extensively studied in several settings. The main questions on avery spaces are explicit solutions for the classical Cauchy problems with the second data null. In this note we will generalize…
The Cauchy problem for the nonlinear wave equation $$\Box u=(\partial u)^2, \qquad u(0)=u_0, u_t(0)=u_1$$ in three space dimensions is considered. The data $(u_0,u_1)$ are assumed to belong to $\widehat{H}^r_s(\R^3) \times…
Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.
The interaction of a wave packet (and in particular the wave front) with a mass barrier is investigated in one dimension. We discuss the main features of the wave packet that are inherent to two-dimensional wave packets, such as compression…
We consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We…
In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we…
We consider the Cauchy problem with smooth and compactly supported initial data for the wave equation in a general class of spherically symmetric geometries which are globally smooth and asymptotically flat. Under certain mild conditions on…
We consider radial solutions to the Cauchy problem for the linear wave equation with a small short-range electromagnetic potential (the "square version" of the massless Dirac equation with a potential) and zero initial data. We prove two a…
The wave equation in quantum mechanics and its general solution in the phase space are obtained.
I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other…
We consider the Cauchy problem in R^n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^{1,1}(R^n) initial data by employing a simple method introduced by the first author. The obtained…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
In this paper we consider the multi-dimensional Quantum Hydrodynamics (QHD) system, by adopting an intrinsically hydrodynamic approach. The present work continues the analysis initiated in [6] where the one dimensional case was studied.…
The analytical solving dynamic problems of elasticity theory for piecewise homogeneous half-space is found. The explicit construction of direct and inverse Fourier's vector transform with discontinuous coefficients is presented. The…