We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a-posteriori error analysis, and (3) it does not require numerically integrating th...

We construct a family of numerical methods for the Pauli equation charged particles in time-dependent, homogeneous magnetic field. These are described general setting comprising systems multiple and extend usual splitting Fourier grid approach. The issue is that field causes to rotate. corresponding rotations wave function highly incompatible with approach used standard Schrödinger equation. Mo...

A Green's function solution is derived for calculating the acoustic field generated by phased array transducers of arbitrary shape when driven by a single frequency continuous wave excitation with spatially varying amplitude and phase. The solution is based on the Green's function for the homogeneous wave equation expressed in the spatial frequency domain or k-space. The temporal convolution in...

The cubic nonlinear Schroedinger equation (NLS) describes the space-time evolution of narrow-banded wave trains in one space and one time (1 + 1) dimensions. The richness of nonlinear wave motions described by NLS is exemplified by the fully nonlinear envelope soliton and “breather” solutions, which are fully understood only in terms of the general solution of the equation as described by the i...

Thus white noise is uniform in Fourier space, at least for frequencies below the inverse correlation time. Another way of looking at this is that the distribution of white noise is smooth in real space. Therefore it is also uniform with the same distribution under any linear transformation whose Jacobian is 1. The Jacobian of a symmetric Fourier transform (using the prefactor 1/ √ 2π) must have...

We exhibit here, for a scalar theory, an apparently non-trivial Wilsonian fixed point, which surprisingly describes a free theory. This modest note is an observation which can be of interest in the framework of functional methods in Quantum Field Theory. The search for Wilsonian fixed points is important in Quantum Field Theory, as these represent configurations independent of the energy scale ...

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers’ equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two...

The effect of uniform vertical magnetic field on the onset of steady Rayleigh-Bènard-Marangoni convection in a relatively hotter or cooler layer of electrically conducting liquid is studied theoretically by means of modified linear stability analysis. The top surface of the layer is non-deformable free where surface tension gradients arise on account of variation of temperature, and the bottom ...