A. E. Merzon

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well-known Sommerfeld integral, belongs to the Sobolev space H1(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two-dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright

High frequency asymptotics of edge waves on a beach of nonconstant slope

The paper is devoted to the construction of asymptotic expansions for frequencies of water waves trapped by a beach of nonconstant slope as the longshore wave number tends to infinity. The existence of such trapping modes in this asymptotic regime was proved recently in [A.-S. Bonnet-Ben Dhia and P. Joly, SIAM J. Appl. Math., 53 (1993), pp. 1507--1550]. For a beach of constant slope the formulas obtained reduce to the classical result of Ursell. Some generalizations to other geometries (waves trapped by afloating semisubmerged cylinder and by shores of a channel of finite width) are indicated. The main analytical tool used for the construction of the asymptotics is an explicit solution of a mixed boundary value problem for the Helmholtz equation in an angle with nonhomogeneous boundary conditions.

General boundary value problems in region with corners

We consider the following boundary value problems:

On asymptotic completeness for scattering in the nonlinear lamb system

Au\left( x \right) = \sum\limits_{\left| a \right| \leqslant 2}^n {{\alpha _a}} \left( x \right)\partial _x^\alpha u\left( x \right) = f\left( x \right),x \in M,\quad Bu\left( x \right) = \sum\limits_{\left| a \right| \leqslant m}^n {{b_a}} \left( x \right)\partial _x^\alpha u\left( x \right) = g\left( x \right),x \in \partial M,

Rayleigh–Bloch waves trapped by a periodic perturbation: exact solutions

(1.1) x∈∂M.

Asymptotic completeness of scattering in the nonlinear Lamb system for nonzero mass

We make more precise the Limiting Amplitude Principle in the two-dimensional scattering of an incident plane harmonic wave by a wedge. We find the long-time asymptotic regime of convergence of the amplitude of the cylindrical wave diffracted by the vertex of a wedge to the limiting amplitude of the solution to the corresponding stationary problem. The asymptotics turns out to be uniform on compacta and depends on the magnitude of the wedge and the profile of the incident wave. The cases of Dirichlet-Dirichlet and Dirichlet-Neumann boundary conditions are considered.

On the behavior of the edge diffracted nonstationary wave in scattering by wedges near the front

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points.We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points. 1 Supported partly by Alexander von Humboldt Research Award, Austrian Science Fund (FWF): P22198-N13, and the grants of DFG and RFBR, Supported by CONACYT and CIC of UMSNH and FWF-project P19138-N13.

The completely indeterminate Caratheodory matrix problem in the Rq[a, b] class

We continue to investigate a nonstationary scattering by wedges [1]–[4]. In this paper we consider a nonstationary scattering of plane waves by a “hard-soft” wedge. We give a method for the proof of the existence and uniqueness of solution to the corresponding DN-Cauchy problem in appropriate functional spaces. We show also that the Limiting Amplitude Principle holds.