Bifurcating vortex solutions of the complex Ginzburg-Landau equation

It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial $2\pi$-periodic vortex solutions that have $2n$ simple zeros (``vortices'') per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutions of the CGL equation.Comment: To appear in Discrete and Continuous Dynamical Systems (1999); 10 pages, 2 figure

Oscillation of two-dimensional linear second-order differential systems

AbstractThis article is concerned with the oscillatory behavior at infinity of the solution y:[a, ∞) → R2 of a system of two second-order differential equations, y″(t) + Q(t) y(t) = 0, tϵ[a, ∞); Q is a continuous matrix-valued function on [a, ∞) whose values are real symmetric matrices of order 2. It is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix ∝at Q(s) ds tends to infinity as t → ∞. This proves a conjecture of D. Hinton and R. T. Lewis (Rocky Mountain J. Math. 10 (1980), 751–766) for the two-dimensional case. Furthermore, it is shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity

Improved error estimates for the perturbed Galerkin method applied to a class of generalized eigenvalue problems

AbstractWe consider the generalized eigenvalue problem x-Kx = μBx in a complex Banach space E. Here, K and B are bounded linear operators, B is compact, and 1 is not in the spectrum of K. If {En: n = 1, 2,…} is a sequence of closed subspaces of E and Pn: E → En is a linear projection which maps E onto En, then we consider the sequence of approximate eigenvalue problems {xn-PnKxn = μPnBxn in En: n = 1, 2,…}. Assuming that ‖K-PnK‖ → 0 and ‖B-PnB‖ → 0 as n → ∞, we prove the convergence of sequences of eigenvalues and eigenelements of the approximate eigenvalue problem to eigenvalues and eigenelements of the original eigenvalue problem, and establish upper bounds for the errors. These error bounds are sharper than those given by Vainikko in Ref. 2 for the more general problem x = μTx in E, T linear and compact, and the sequence of approximate problems {xn = μTnxn in En: n=l, 2,…}, and do not involve the operator Sn=Tn-PnT/En