Zeros of orthogonal polynomials

高等学校計算数学学報 = Numerical mathematics

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

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Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

We consider a recently discovered representation for the general solution of the Sturm–Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm–Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm–Liouville problems, effectively by calculation of the roots of a polynomial. We discuss the examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm–Liouville problems as well as to problems with spectral parameter-dependent boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.

Spectral parameter power series for Sturm–Liouville problems

Relativity Gravitation and World-Structure; the International Series of Monographs on Physics

We study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite-dimensional Hamiltonian systems. The theory is based on Pontryagin’s invariant subspace theorem and extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self-adjoint operators, (ii) the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self-adjoint operators, and (iii) the quadratic forms defined by the two self-adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem. Applicatio...

https://dmpeli.math.mcmaster.ca/PaperBank/CountEigen.pdf

Count of eigenvalues in the generalized eigenvalue problem

We consider the Hamiltonian first-order coupled-mode system that occurs in nonlinear optics, pho- tonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the lin- earized coupled-mode system, which is equivalent to a four-by-four Dirac system with sign-indefinite metric. In the special class of symmetric nonlinear potentials, we construct a block-diagonal repre- sentation of the linearized equations, when the spectral problem reduces to two coupled two-by-two Dirac systems. The block-diagonalization is used in fast numerical computations of eigenvalues with the Chebyshev interpolation algorithm.

https://dmpeli.math.mcmaster.ca/PaperBank/coupled-mode.pdf

Block-Diagonalization of the Symmetric First-Order Coupled-Mode System ∗

To solve the inverse problem for the Sturm-Liouville operator with eigenvalue parameter dependent boundary conditions we reconstruct the spectral distribution function from two spectra of the boundary-value problems with equal Θ(λ) and different real constants in the boundary conditions. The well-known results of A.V. Strauss [5] concerning the connection between the eigenvalue problems with the spectral parameter in the boundary conditions and the theory of generalized resolvents is used.

Inverse Spectral problem for the Sturm-Liouville Operator with Eigenvalue Parameter Dependent Boundary Conditions

We consider a nonlinear fourth-order degenerate parabolic partial differential equation that arises in modeling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blowup. Finally, we present numerical simulations of solutions.

Nonnegative Solutions for a Long-Wave Unstable Thin Film Equation with Convection

We study the time evolution of a thin liquid film coating the outer surface of a sphere in the presence of gravity, surface tension, and thermal gradients. We derive the fourth-order nonlinear partial differential equation that models the thin film dynamics, including Marangoni terms arising from the dependence of surface tension σ on temperature T. We consider two different imposed temperature distributions with axial or radial thermal gradients. We analyze the stability of a uniform coating under small perturbations and carry out numerical simulations in COMSOL for a range of parameter values. In the case of an axial temperature gradient, we find steady states either with uniform film thickness or with the fluid accumulating at the bottom or near the top of the sphere, depending on the total volume of liquid in the film, dictating whether gravity or Marangoni effects dominate. This suggests a potential method for the indirect measurement of dσ/dT by monitoring the thickness profile of the thin film. In the case of a radial temperature gradient, a stability analysis reveals the most unstable non-axisymmetric modes on an initially uniform coating film.

Marina Chugunova

Papers

Count of eigenvalues in the generalized eigenvalue problem

Block-Diagonalization of the Symmetric First-Order Coupled-Mode System ∗

Inverse Spectral problem for the Sturm-Liouville Operator with Eigenvalue Parameter Dependent Boundary Conditions

Nonnegative Solutions for a Long-Wave Unstable Thin Film Equation with Convection

Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients