Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data

In a two-dimensional domain Ω ? R ², we consider the wave equation with variable velocity c(x ₁, x ₂) degenerating on the boundary Γ = ?Ω as the square root of the distance to the boundary, and construct an asymptotic solution of the Cauchy problem with localized initial data. This problem is related to the so-called “run-up problem” in tsunami wave theory. One main idea (also used by the authors in earlier papers in the one-dimensional case and the two-dimensional case with c ²(x ₁, x ₂) = x ₁) is that the (singular) curve Γ is a caustic of special type. We use this idea to introduce a generalization of the Maslov canonical operator covering the problem with degeneration and obtain efficient formulas for the asymptotic solutions.     

Some solutions of the 3D Laplace equation in a layer with oscillating boundary describing an array of nanotubes with applications to cold field emission. II. Irregular arrays

As in the first part (J. Brüning, S.Yu. Dobrokhotov, D.S. Minenkov, 2011), we construct a family of special solutions of the Dirichlet problem for the Laplace equation in a domain with fast changing boundary. Using these solutions, we construct an analytic model of cold field electron emission from surfaces simulating arrays of vertically aligned nanotubes. Explicit analytic formulas lead to fast computations and also allow us to study the case of random arrays of tubes with stochastic distribution of parameters. We present these results and compare them with numerical approximations given in [1].     

Quantum dynamics in a thin film. II. Stationary states

The spectrum of quantum waveguides simulating thin toroidal tubes and thin spherical surfaces is investigated. Asymptotic formulas are obtained and a geometric classification using the so-called Reeb graphs is carried out.     

Problem of the reversal of a wave for the model equationr_t + rr_x - \frac{{ih}}{2}r_{xx} = 0

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 143–147, June, 1992.     

Resonance phenomena in the nonlinear equations of a proper semiconductor h2Δu=shu

One considers a Steklov-type boundary-value problem for the nonlinear equation of a semiconductor. Under the assumption of the existence on the surface of the semiconductor of a closed geodesic, stable in a linear approximation, one constructs asymptotic solutions which are concentrated in the neighborhood of this geodesic. The obtained solutions are expressed in terms of the known asymptotic eigenfunctions of the Laplace operator on a Riemann manifold and in terms of the multisoliton solutions of the Sine-Gordon equation. Similar solutions are obtained for the mixed boundary-value problem.