Related papers: On the Orr-Sommerfeld Equation with Linear Profile

This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many…

This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schr\"odinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and…

In the paper we develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of first-order ordinary differential equations in spaces of smooth functions on a finite interval. This problems are set…

An infinite sequence of potential well functions is considered. A numerical method is used for the Schr$\ddot{\text{o}}$dinger equation to obtain the energy eigenvalue spectra for a number of these potential well functions. The results for…

Let $\om $ be a bounded domain in an $n$-dimensional Euclidean space $\Bbb R^n$. We study eigenvalues of an eigenvalue problem of a system of elliptic equations: $$ \{\aligned &\Delta {\mathbf u}+ \alpha{\rm grad}(\text{div}{\mathbf…

We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad (-1,1),$$ where $f : {\mathbb R} \to {\mathbb R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary…

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex $k$-plane. This Riemann-Hilbert…

We investigate the spectrum of Schr\"odinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to $-\infty$, a narrow cluster of…

For low enough flow rates, turbulent channel flow displays spatial modulations of large wavelengths. This phenomenon has recently been interpreted as a linear instability of the turbulent flow. We question here the ability of linear…

Analytic solutions for the energy eigenvalues are obtained from a confined potentials of the form $br$ in 3 dimensions. The confinement is effected by linear term which is a very important part in Cornell potential. The analytic eigenvalues…

Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…

The spectrum of the singular indefinite Sturm-Liouville operator $$A=\text{\rm sgn}(\cdot)\bigl(-\tfrac{d^2}{dx^2}+q\bigr)$$ with a real potential $q\in L^1(\mathbb R)$ covers the whole real line and, in addition, non-real eigenvalues may…

We consider the following problem: on any given complete Riemannian manifold $(M,g)$, among all curves which have fixed length as well as fixed end-points and tangents at the end-points, minimise the $L^\infty$ norm of the curvature. We…

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schr\"{o}dinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by…

This paper concerns spectral instability of shear flows in the incompressible Navier-Stokes equations with sufficiently large Reynolds number: $R\to \infty$. It is well-documented in the physical literature, going back to Heisenberg, C.C.…

We consider perturbations of the semiclassical Schr{\"o}dinger equation on a compact Riemannian surface with constant negative curvature and without boundary. We show that, for scales of times which are logarithmic in the size of the…

We study existence and universality of scaling limits for the eigenvalues of a random normal matrix, in particular at points on the boundary of the spectrum. Our approach uses Ward's equation, which is an identity satisfied by the 1-point…

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of $L^2 \mapsto L^2$ type for the operator, as well as for the corresponding maximal function. If…

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this…

For equations with the neutral term strictly delayed we construct the fundamental solution, derive a variation-of-constants formula for inhomogeneous equations, and prove growth estimates. Only unavoidable measure and integration theory, up…