We consider nonlinear Schrödinger equations in R. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear ...

To study the nonlinear dynamics, such as Hopf bifurcation, of partial differential equations with delay, one needs to consider the characteristic equation associated to the linearized equation and to determine the distribution of the eigenvalues; that is, to study the spectrum of the linear operator. In this paper we study the projectors on the generalized eigenspaces associated to some eigenva...

I investigated a Bose–Einstein condensate confined harmonically to 1D, within the framework of Gross–Pitaevskii mean–field theory. I consider the 1D Gross–Pitaevskii equation with a combined step in its external potential and nonlinear terms. I generalise solitary wave solutions to the equation in its nonlinear limit, to take into account a constant potential and nonlinearity. I then generalise...

Eigenvalues arising in scattering theory have been envisioned as a potential source of target signatures nondestructive testing materials, whereby perturbations the eigenvalues computed for penetrable medium would be used to infer changes its constitutive parameters relative some reference values. We consider recently introduced modification class Stekloff eigenvalues, which inclusion smoothing...

The creep crack problems in damaged materials under mixed mode loading (Mode I and Mode II loading) in the framework of creep-damage coupled formulation are considered. The class of the self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law cr...

We extend to the case of many competing densities the results of the paper [7]. More precisely, we are concerned with an optimal partition problem in N -dimensional domains related to the method of nonlinear eigenvalues introduced by Z. Nehari, [16]. We prove existence of the minimal partition and some extremality conditions. Moreover, in the case of two–dimensional domains we give an asymptoti...

Many statistical applications require an estimate of a covariance matrix and/or its inverse. Whenthe matrix dimension is large compared to the sample size, which happens frequently, the samplecovariance matrix is known to perform poorly and may suffer from ill-conditioning. There alreadyexists an extensive literature concerning improved estimators in such situations. In the absence offurther kn...

A multi-rate discrete-time model, whose response agrees exactly with that of a continuous-time original at all sampling instants for any sampling periods, is developed for a linear system, which is assumed to have multiple real eigenvalues. The sampling rates can be chosen arbitrarily and individually, so that their ratios can even be irrational. The state space model is obtained as a combinati...

The spatially inhomogeneous large N solutions to Kazakov–Migdal model are analyzed. The set of nonlinear differential equations is derived in the continuum limit. In one dimensional case these equations has a natural interpretation in terms of the dynamics of a Fermi gas. The multidimensional case seems to be inconsistent because of its instability related to the collapse of eigenvalues of the ...

This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.