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A uniform asymptotic high frequency solution is developed for the diffraction of plane waves by the junction of two half-planes. One of the half-planes is assumed to be characterized by Seniors resistive-type partially transmissive boundary conditions and the other is soft at the top and hard at the bottom. The related boundary-value problem is formulated as a matrix Wiener-Hopf equation which is solved explicitly through the Daniele-Khrapkov method. Some graphical results are also presented.  相似文献   

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Starting from the general system of difference-differential equations for the amplitudes of the diffracted beams of light, given by Mertens, and using the method of Kuliasko, Mertens and Leroy for the diffraction of light by one supersonic wave, it is possible to reduce the solution of the system of difference-differential equations, to the solution of a partial differential equation. In this way it is possible to calculate the intensities of the ordern and ?n, as a series expansion in ρ. Here we only considered terms up to ρ2. It was also possible to verify the general symmetry properties for the intensities studied by Leroy and Mertens.  相似文献   

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We derive boundary conditions for the phase space energy density of acoustic waves in a half space, in the high frequency limit. These boundary conditions generalize the usual reflection—transmission relations for plane waves and are well suited for the study of wave propagation in bounded randed random media in the radiative transport approximation[15]. The high frequency analysis is based on direct calculations with Fourier integrals in the case of constant coefficients and Wigner measures in general, and it is presented in detail  相似文献   

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In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:
  1. Forρ = 0, the results of Murty’s elementary theory are reestablished.
  2. Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.
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A study and the solution of an extension of the classical Sommerfeld half-plane problem which leads to a pair of integral equations of the Wiener-Hopf type is given. The method of solution is function theoretic in character and employs a combination of the ideas of Wiener and Hopf and Carleman.  相似文献   

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In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)(e2iωtφ,0,0), (0,e2iωtφ,0)(0,e2iωtφ,0), (0,0,e2iωtφ)(0,0,e2iωtφ), where φ is a ground state of the scalar nonlinear Schrödinger equation.  相似文献   

11.
The Cauchy problem is considered for the equation of internal waves to which reduce many problems of the linear theory of waves in a continuously stratified fluid. The theorem of uniqueness is proved, and the formula for explicit representation of solution in terms of integrals whose kernels contain the obtained in /1/ fundamental solution of the internal wave operator and its time derivative are derived. Asymptotic analysis of solution in the “distant zone” is carried out for large values of dimensionless time.  相似文献   

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In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction-diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction-diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder's fixed point theorem, we then show that there exists a constant c>0 such that for each c>c, the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.  相似文献   

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Some model system of equations is examined that comprises two sixth order equations of Sobolev type with the second order time derivative. This system describes explosive instability in plasma accounting for the strong space-time dispersion and nonlinear dependence of polarizability on the electric field strength. The case of the so-called focusing medium is also considered.  相似文献   

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Summary Considered here are model equations for weakly nonlinear and dispersive long waves, which feature general forms of dispersion and pure power nonlinearity. Two variants of such equations are introduced, one of Korteweg-de Vries type and one of regularized long-wave type. It is proven that solutions of the pure initial-value problem for these two types of model equations are the same, to within the order of accuracy attributable to either, on the long time scale during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.This research was supported in part by the National Science Foundation. A considerable portion of the project was completed while the first author was resident at the Institute for Mathematics and Its Applications, University of Minnesota.  相似文献   

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The Galerkin method, together with a second order time discretization, is applied to the periodic initial value problem for $$\frac{\partial }{{\partial t}}(u - (a(x)u_x )_x ) + (f(x,u))_x = 0$$ . Heref(x, ·) may be highly nonlinear, but a certain cancellation effect is assumed for∫f(x, u) x u. Optimal order error estimates inL 2,H 1, andL are derived for a general class of piecewise polynomial spaces.  相似文献   

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Standing wave solutions of coupled nonlinear Hartree equations with nonlocal interaction are considered. Such systems arises from mathematical models in Bose–Einstein condensates theory and nonlinear optics. The existence and non-existence of positive ground state solutions are proved under optimal conditions on parameters, and various qualitative properties of ground state solutions are shown. The uniqueness of the positive solution or the positive ground state solution are also obtained in some cases.  相似文献   

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We consider a boundary value problem for a special system of integro-differential equations with variational derivatives. We establish the relationship between this problem and a system of integral equations with a power-law nonlinearity whose kernels and right-hand sides are random functions. We study the solvability of the boundary value problem. Special cases and examples are considered.  相似文献   

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By reduction to Goursat problems, we obtain various versions of conditions ensuring that the solution of the system in question can be constructed in closed form.  相似文献   

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