The scattering problem for the Schrödinger equation with a potential linear in time and in space. II. Correctness,smoothness, behavior of the solution at infinity

We study the scattering problem associated with the behavior of whispering gallery waves near the inflection point of the boundary. In order to solve the scattering problem, we prove the theorems of existence, uniqueness and smoothness of the solution. The formal asymptotic behavior is justified for t– and superexponential smallness of the wave field in the shadow zone is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 13–29, 1985.     

The quasihomogeneous function method and Fock's problem

The diffraction of a high-frequency wave by a smooth convex body near the tangency point of the limiting ray to the surface is restated as the scattering problem for the Schrodinger equation with a linear potential on a half-axis. Various prior estimates for the scattering problem are used in order to prove existence, uniqueness, and smoothness theorems. The corresponding solution satisfies the principle of limiting absorption. The formal solution of the corresponding Schrodinger equation in the form of quaslhomogeneous functions is essentially used in our constructions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 144–151, 1985.     

Exact solution of a model of three bound oscillators with Stark nonlinearity

A model of three bound boson oscillators with Stark nonlinearity is introduced and solved by the quantum inverse scattering method. For the trilinear oscillator, the eigenvalue problem is reduced to the spectral problem for the second-order homogeneous differential equation. Bibliography: 11 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 122–128. Translated by B. M. Bekker.     

Formal asymptotic integration for one class of weakly nonlinear infinite-dimensional Hamiltonian systems

An asymptotic integration technique is described for a class of infinite-dimensional Hamiltonian systems. The method will generate a solution with any desired accuracy. The solutions are constructed using the theory of generalized functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 42–51, 1985.In conclusion, I would like to acknowledge the helpful commments of V. B. Babich, I. A. Molotkov, V. F. Lazutkin, M. M. Popov and other participants of the LOMI wave diffraction seminar.     

一类具非线性边值条件的微分方程的奇摄动问题

研究了一类非线性三阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用不动点原理证明了原问题解的存在性及所得形式渐近解的一致有效性.     

Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain G×(0,T), G∈Rn are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that - together with the uniqueness and existence results - makes the problem under consideration to a well-posed problem in the Hadamard sense.     

Application of the functional integration method to laser theory

An approach is proposed for the investigation on nonequilibrium processes, based on the functional integration method. A system of laser type is examined. Expressions for the density matrix of a two-level system in a classical field and for the effective action functional of the radiation field are obtained. The equation for the stationary point of the effective action characterizes the process of laser generation. In the equilibrium limit this equation describes the Bose condensation in Dicke-type models.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 148–165, 1983.In conclusion, the author expresses his gratitude to V. N. Popov for his interest in this work.     

Concentration of the solutions near a limit ray in the neighborhood of an inflection point of the boundary

One considers the question regarding the energy concentration of a wave field in the neighborhood of a limit ray in the problem of the behavior of whispering gallery waves near an inflection point of the boundary. An estimate related to this question is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 155–158, 1988.     

两参数非线性反应扩散积分微分方程的内层激波渐近解

研究了一类两参数非线性反应扩散积分微分奇摄动问题.利用奇摄动方法,构造了问题的外部解、内部激波层、边界层及初始层校正项,由此得到了问题解的形式渐近展开式.最后利用积分微分方程的比较定理证明了该问题解的渐近展开式的一致有效性.     

Justification of an asymptotic formula for solutions of the perturbed Klein-Fock-Gordon equation

A problem with periodic boundary conditions is considered for the Fock-Klein-Gordon equation perturbed by a small nonlinear operator R[t, u, u_x, u_xx]:. It is shown that under certain conditions the solution exists and is close to the known asymptotic solution on the interval 0 t l/.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 84–92, 1981.In conclusion, the author wishes to thank I. A. Molotkov, V. M. Babich, Ya. V. Kurylev, M. M. Popov, V. B. Filippov, and all the participants in the seminar on wave diffraction (LOMI) for valuable advice, pointers, and discussions.     

Focusing problem and the asymptotics of the spectral function of the Laplace-Beltrami operator. II

In the paper a formal high-frequency solution of the problem of a point source of oscillations near a reflecting boundary is constructed. The boundary is geodesically concave which makes it possible to use methods developed in diffraction theory by V. A. Fock and J. B. Keller. On the basis of the formal solution of the problem of a point source of oscillations, it is possible to construct the asymptotics of the spectral functions of the Laplace-Beltrami operator.Translated fromZapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 14–53, 1979.     

Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Asymptotics of the fundamental solution of the cauchy problem for a hyperbolic equation of second order containing a large parameter

An asymptotic expansion of the fundamental solution of the Cauchy problem for a linear hyperbolic equation containing a large parameter is constructed in this paper. The coefficients of the equation are assumed to be infinitely differentiable. The expansion obtained admits termwise differentiation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 93–101, 1981.The author wishes to thank V. M. Babich for posing the problem and for his constant attention to the work.     

Behavior of the formal solution to a mixed problem for the wave equation

The behavior of the formal solution, obtained by the Fourier method, to a mixed problem for the wave equation with arbitrary two-point boundary conditions and the initial condition φ(х) (for zero initial velocity) with weaker requirements than those for the classical solution is analyzed. An approach based on the Cauchy–Poincare technique, consisting in the contour integration of the resolvent of the operator generated by the corresponding spectral problem, is used. Conditions giving the solution to the mixed problem when the wave equation is satisfied only almost everywhere are found. When φ(x) is an arbitrary function from L₂[0, 1], the formal solution converges almost everywhere and is a generalized solution to the mixed problem.     

Asymptotic behavior of the Fourier coefficients for the problem of scattering on contours tano

In this paper the analytic properties of the wave function in the region bounded by the scattering body are studied. This problem is of major importance for the development of the analytic side of the method of separation of variables in problems with noncoordinate boundaries. New asymptotic estimates for the Fourier coefficients of the scattered field are obtained. They indicate the analyticity of the wave function in the physical plane for all Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 27–38, 1976.In conclusion, we wish to thank V. M. Babich, M. M. Popov, and S. Yu. Slavyanov for making valuable suggestions regarding the topic treated here which contributed to the completion of this paper.     

The semilinear second-order hyperbolic equation with data strongly singular at one point

In this paper we study the initial value problem for the scalar semilinear strictly hyperbolic equation in multidimensional space with data strongly singular at one point. Under the assumption of the initial data being conormal with respect to one point and bounded or regular with a certain low degree, the existence of the solution to this problem is obtained; meanwhile, it is proved that the singularity of the solution will spread on the forward characteristic cone of the hyperbolic operator issuing from this point, and the solution is bounded and conormal with respect to this cone.     

Separation of variables as a means of solving the mixed boundary-value problem of viscoelasticity

The quasistatic mixed boundary-value problem is considered for a viscoelastic body with constant Poisson's ratio. A method of reducing viscoelastic problems to the solution of a series of uniform elastic problems is described. The convergence of the method is proved. The method is illustrated with reference to the solution of a specific viscoelastic problem. Tabulated influence functions of the Abel type are used in the solutions.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 588–596, July–August, 1969.     

奇摄动分数阶微分方程的渐近解

本文研究了一类奇摄动非线性分数阶微分方程初值问题.利用伸长变量构造出解的形式展开式,并利用微分不等式理论,证明了解的一致有效的渐近式.所得的结果具有较好精度的近似解.     

An Inverse Problem for the Heat Equation II

Completeness of the set of products of the derivatives of the solutions to the equation ( av ')' m u v = 0, v (0, u ) = 0 is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $ \dot u = (a(x)u')' $ , u ( x , 0) = 0, u (0, t ) = 0, u (1, t ) = f ( t ) known for all t > 0, from the heat flux a (1) u '(1, t ) = g ( t ). Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point x = 0 in our case) does not have a unique solution, in general.