Boris Vainberg

Linear Water Waves: A Mathematical Approach

Preface Part I. Time-Harmonic Waves: 1. Greens functions 2. Submerged obstacles 3. Semisubmerged bodies, I 4. Semisubmerged bodies, II 5. Horizontally-periodic trapped waves Part II. Ship Waves on Calm Water: 6. Greens functions 7. The Neumann-Kelvin problem 8. Two-dimensional problem Part III. Unsteady Waves: 9. Submerged obstacles: existence 10. Waves due to rapidly stabilizing and high-frequency disturbances Bibliography Name index Subject index.

On Scattering of Solitons for the Klein–Gordon Equation Coupled to a Particle

We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein–Gordon equation coupled to a charged relativistic particle. The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution of the free Klein–Gordon equation. It is assumed that the charge density satisfies the Wiener condition which is a version of the “Fermi Golden Rule”. The proof is based on an extension of the general strategy introduced by Soffer and Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.

Scattering Solutions in Networks of Thin Fibers: Small Diameter Asymptotics

Small diameter asymptotics is obtained for scattering solutions in a network of thin fibers. The asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph Γ . We calculate the Lagrangian gluing conditions at vertices

Radiation conditions for the difference schrödinger operators

{v\in \Gamma }

Remarks on interior transmission eigenvalues, Weyl formula and branching billiards

for the problems on the limiting graph. If the frequency of the incident wave is above the bottom of the absolutely continuous spectrum, the gluing conditions are formulated in terms of the scattering data for each individual junction of the network.

Bounds on positive interior transmission eigenvalues

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.

On the Structure of Eigenfunctions Corresponding to Embedded Eigenvalues of Locally Perturbed Periodic Graph Operators

The problem of determining a unique solution of the Schrödinger equation on the lattice is considered, where Δ is the difference Laplacian and both f and q have finite supports. It is shown that there is an exceptional set So of points on for which the limiting absorption priciple fails, even for unperturbed operator (q(x)=0). This exceptional set consists of the points when d is even and when d is odd. For all Values of , the radiation conditions are found which single out the same solutions of the problem as the ones determined by the limiting absorption principle. These solutions are conbinations of several waves propagating with different frequencies, and the number of waves depends on the value of λ.

On spectral asymptotics for domains with fractal boundaries

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) − 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).