Continuation-minimization methods for stability problems

AbstractWe study the solution branches of stable and unstable bifurcations in certain semilinear elliptic eigenvalue problems with Dirichlet boundary conditions. A secant predictor-line search backtrack corrector continuation method is described to trace the solution curves numerically. Sample numerical results with computer graphic output are reported

A system of ODEs for a Perturbation of a Minimal Mass Soliton

We study soliton solutions to a nonlinear Schrodinger equation with a saturated nonlinearity. Such nonlinearities are known to possess minimal mass soliton solutions. We consider a small perturbation of a minimal mass soliton, and identify a system of ODEs similar to those from Comech and Pelinovsky (2003), which model the behavior of the perturbation for short times. We then provide numerical evidence that under this system of ODEs there are two possible dynamical outcomes, which is in accord with the conclusions of Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a soliton structure, a generic initial perturbation oscillates around the stable family of solitons. For initial data which is expected to disperse, the finite dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit

Construction of C 2 Pythagorean-hodograph interpolating splines by the homotopy method

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic “spline” that interpolates a given sequence of points p 0 , p 1 ,..., p N and end-derivatives d 0 and d N to be reduced to solving a “tridiagonal” system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 N +1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable “looping” behavior (which may be quantified in terms of the elastic bending energy , i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding “ordinary” C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41719/1/10444_2005_Article_BF02124754.pd

Numerical exploitation of symmetric structures in BEM

Boundary integral equations often display invariance with respect to orthogonal transformations. If the domain of integration is symmetric with respect to some orthogonal transformations, then appropriate discretizations lead to system matrices which are equivariant with respect to a group of permutations on the nodes. This property can be exploited to design efficient methods for solving the discretized problem. A definition of a Fourier transform generated by an arbitrary finite group of permutations is used for this purpose. Some nodes of the discretization may be left invariant under some actions. This leads to complications in the numerical treatment which have recently been overcome. We discuss the efficient solution of linear systems of equations and of eigenvalue problems with the Fourier transform method. Previous work has exclusively dealt with system matrices that are square. The case of least squares collocation is taken as a motivation to extend the discussion to the recta..

Numerical Exploitation of Equivariance

5 1 Introduction Many problems in science and mathematics exhibit symmetry phenomena which may be exploited to analyze them, and also to effect a significant cost reduction in their numerical treatment. Usually the symmetry stems from the domain or body on which the problem is considered. The numerical treatment of problems such as partial differential equations and integral equations generally involves discretizations which ought (as far as possible) to incorporate or respect such symmetries. The present paper summarizes some of the recent work of the authors concerning systematic techniques for exploiting symmetry in the numerical treatment of systems of linear equations that arise from discretizing operator equations displaying symmetries. The unifying concept is a generalization of the Fourier transform for arbitrary finite groups. We study here the general case which incorporates nonabelian groups (i.e., having irreducible representations of dimension ?<F