We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real e...

We present a Coulomb gas method to calculate analytically the probability of rare events where the maximum eigenvalue of a random matrix is much larger than its typical value. The large deviation function that characterizes this probability is computed explicitly for Wishart and Gaussian ensembles. The method is general and applies to other related problems, e.g., the joint large deviation func...

The di erential equation dX dt = [X; k(X)] where k is a Toeplitz annihilator has been suggested as a means to solve the inverse Toeplitz eigenvalue problem. Starting with the diagonal matrix whose entries are the same as the given eigenvalues, the solution ow has been observed numerically to always converge a symmetric Toeplitz matrix as t ! 1. This paper is an attempt to understand the dynamic...

The paper is devoted to the eigenvalue problem for a second order strongly elliptic operator. The problem is considered on curved domains, which require interpolated boundary conditions in approximating finite element formulation. The necessary triangulations for solving the eigenvalue problem consists of isoparametric elements of degree n, where n is any integer greater than two. An approximat...

In general, the optimal computational complexity of Arnoldi iteration is O(kN) for solving a generalized eigenvalue problem, with k being the number of dominant eigenvalues and N the matrix size. In this work, we reduce the computational complexity of the Arnoldi iteration from O(kN) to O(N), thus paving the way for full-wave extraction of very large-scale onchip interconnects, the k of which i...