The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u=u∈(X/T ). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/∈ and slow...

Nonlinear stochastic optimal control problems are fundamental in control theory. A general class of such problems can be reduced to computing the principal eigenfunction of a linear operator. Here, we describe a new method for finding this eigenfunction using a moving least-squares function approximation. We use efficient iterative solvers that do not require matrix factorization, thereby allow...

Marseille Cedex 9 France One-dimensional quantum scattering for potentials de ned as measures Charles-Antoine GUERIN1 Abstract We generalize the basic one-dimensional scattering formalism to potentials de ned as measures and retrieve the classical results that hold for smooth potentials. We introduce a set of generalized eigenfunctions for the corresponding Schr odinger operator and study thei...

A wave function of the N -component KP Hierarchy with continuous flows determined by an invertible matrix H is constructed from the choice of an MN -dimensional space of finitely-supported vector distributions. This wave function is shown to be an eigenfunction for a ring of matrix differential operators in x having eigenvalues that are matrix functions of the spectral parameter z. If the space...

Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator H = √ −∆ + V (x) with |V (x)| ≤ C〈x〉−σ, σ > 3/2, are considered. We compute the integral kernels of the boundary values R± 0 (λ) = ( √ −∆− (λ± i0))−1, and prove that the generalized eigenfunctions φ±(x, k) are bounded on R x × {k | a ≤ |k| ≤ b}, where [a, b] ⊂ (0,∞)\σp(H), and σp(H) is the set of eigenvalues of ...

We propose new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The...