Marina Chugunova

Count of eigenvalues in the generalized eigenvalue problem

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...

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

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.

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

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.

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

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling 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 blow-up. Finally, we present numerical simulations of solutions.

Spectrum of a non-self-adjoint operator associated with the periodic heat equation

Abstract We study the spectrum of the linear operator L = − ∂ θ − ϵ ∂ θ ( sin θ ∂ θ ) subject to the periodic boundary conditions on θ ∈ [ − π , π ] . We prove that the operator is closed in L per 2 ( [ − π , π ] ) with the domain in H per 1 ( [ − π , π ] ) for | ϵ | 2 , its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in L per 2 ( [ − π , π ] ) .

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

Abstract.We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in

Regularized shock solutions in coating flows with small surface tension

\mathcal{L}^2(-\pi, \pi)

Qualitative Analysis of Coating Flows on a Rotating Horizontal Cylinder

. Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.