Phase-plane solutions of Langmuir's equation

Long Time Stability for Solutions of a Β-plane Equation

We prove stability for arbitrarily long times of the zero solution for the so-called β-plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the 2d incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of no...

Solutions of diffusion equation for point defects

An analytical solution of the equation describing diffusion of intrinsic point defects in semiconductor crystals has been obtained for a one-dimensional finite-length domain with the Robin-type boundary conditions. The distributions of point defects for different migration lengths of defects have been calculated. The exact analytical solution was used to verify the approximate numerical solutio...

WAVELET SOLUTIONS OF THE KLEIN-GORDON EQUATION

Stability of Plane Wave Solutions in Complex Ginzburg-Landau Equation with Delayed Feedback

We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg–Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delay...

Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus

It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltoni...