14,712 research outputs found
Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator
Partial Fourier transforms are used to find explicit formulas for two
remarkable fundamental solutions for a generalized Tricomi operator. These
fundamental solutions reflect clearly the mixed type of the operator. In order
to prove these results, we establish explicit formulas for Fourier transforms
of some type of Bessel functions
Vortex line representation for flows of ideal and viscous fluids
It is shown that the Euler hydrodynamics for vortical flows of an ideal fluid
coincides with the equations of motion of a charged {\it compressible} fluid
moving due to a self-consistent electromagnetic field. Transition to the
Lagrangian description in a new hydrodynamics is equivalent for the original
Euler equations to the mixed Lagrangian-Eulerian description - the vortex line
representation (VLR). Due to compressibility of a "new" fluid the collapse of
vortex lines can happen as the result of breaking (or overturning) of vortex
lines. It is found that the Navier-Stokes equation in the vortex line
representation can be reduced to the equation of the diffusive type for the
Cauchy invariant with the diffusion tensor given by the metric of the VLR
Ab-initio multimode linewidth theory for arbitrary inhomogeneous laser cavities
We present a multimode laser-linewidth theory for arbitrary cavity structures
and geometries that contains nearly all previously known effects and also finds
new nonlinear and multimode corrections, e.g. a bad-cavity correction to the
Henry factor and a multimode Schawlow--Townes relation (each linewidth
is proportional to a sum of inverse powers of all lasing modes). Our theory
produces a quantitatively accurate formula for the linewidth, with no free
parameters, including the full spatial degrees of freedom of the system.
Starting with the Maxwell--Bloch equations, we handle quantum and thermal noise
by introducing random currents whose correlations are given by the
fluctuation--dissipation theorem. We derive coupled-mode equations for the
lasing-mode amplitudes and obtain a formula for the linewidths in terms of
simple integrals over the steady-state lasing modes.Comment: 24 pages, 7 figure
Exact Analytical Solution of the N-dimensional Radial Schrodinger Equation with Pseudoharmonic Potential via Laplace Transform Approach
The second order -dimensional Schr\"odinger equation with pseudoharmonic
potential is reduced to a first order differential equation by using the
Laplace transform approach and exact bound state solutions are obtained using
convolution theorem. Some special cases are verified and variation of energy
eigenvalues as a function of dimension are furnished. To give an
extra depth of this letter, present approach is also briefly investigated for
generalized Morse potential as an example.Comment: 16 pages.Published version has some figure
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