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Formulas for the asymptotics of some class of integrals of rapidly oscillating functions that generalize the well-known stationary phase method, which were obtained in the previous paper of the author, are applied to integrals arising in the well-known tsunami hydrodynamic piston model in the case of a constant pool bottom. As a result, asymptotic formulas are obtained for the head part of the wave for large values of the time elapsed since the occurrence of the tsunami. These formulas contain some special reference integrals and have different forms depending on combinations of wave and time parameters.

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase.

A linear problem for propagation of gravity waves in the basin having the bottom of a form of a smooth background with added rapid oscillations is considered. The formulas derived below are asymptotic ones; they are quite formal, and we do not discuss the problem concerning their uniformness with respect to these parameters.

The system of equations of gravity surface waves is considered in the case where the basin’s bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom’s oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors’ works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.

In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrodinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum.

The present paper is a survey of some results concerning higher-order elliptic differential operators which degenerate on the boundary of a domain. The principal aspect in the study of such operators is that of investigating the corresponding ordinary equations with parameters which degenerate at a single point. The parametrix for such operators is a degenerating pseudo-differential operator. We investigate the properties of the kernels of these pseudo-differential operators.