Bilinear Dispersive Estimates Via Space Time Resonances, Dimensions Two and Three

An extended nonlinear Schrödinger equation, with positive fourth-order dispersion and a quintic nonlinearity, is shown to possess a soliton-like solution which exists in resonance with linear dispersive waves, but emits no radiation at all. When this solution is perturbed the emission of radiation begins. An approximate variational analysis and direct numerical integrations are carried out to show that this soliton-like solution is robust enough to resist small perturbations without being dispersed away. Initial conditions which are close to the exact solution evolve into quasi-stationary solutions which emit radiation, but maintain their shapes over extremely long distances.

A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves.

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

<p>In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $ n $-dimensional fractional Hardy operator from $ L^p $ to $ L^{q, \infty} $. Next, we studied the sharp bounds for the $ m $-linear $ n $-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $ L^p $ spaces.</p>

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.
 The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.
 The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

Wave interaction with multiple articulated floating elastic plates

High-order non-reflecting boundary conditions for dispersive waves

Images of short wind‐driven gravity waves were taken from an offshore platform, using a charge coupled device television camera recording diffuse sky radiance reflected from the ocean surface. A two‐dimensional power spectrum was calculated from nine statistically independent images. The resultant ensemble‐averaged spectrum exhibited good statistical stability and provided information on the angular spread and direction of the wave components present. One‐dimensional sampling of each image in a sequence allowed a space‐time image to be constructed which clearly shows the effects of wave dispersion as well as the modulation of the phase velocities of the short wavelength waves by the long wavelength components. An ensemble‐averaged space‐time spectrum, when combined with the directional parameters, is compared with the predictions of linear gravity wave dispersion theory. Two distinct wave systems were present: the local wind driven system showed a space‐time spectrum in agreement with linear theory out to ∼1 cyc/m, but with excess phase velocity at higher spatial frequencies. The second wave system, which was presumably generated by a distant wind field, showed a deficiency in phase velocity when compared to linear theory.

In linear isotropic elasticity, cases exist where pure shear waves are possible even in bounded media; these have a nondispersive mode with propagation speed equal to that in an infinite medium. For these cases, consideration of a two-layer medium shows the existence of dispersion which vanishes only with equality of the propagation speeds. The present study uses the method of asymptotic expansions; a uniformly valid approximation is obtained to describe the speed and dispersive nature of these waves. The elegance of this approach is brought out by derivation of the basic result of this study viz., Jeffrey equation to describe the farfield structure of these waves.

In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.

For weighted spaces of functions of positive smoothness on irregular domains, embedding theorems into weighted Lebesgue spaces are proved.

Wavenumber-frequency spectra of microwave backscatter from the sea surface occasionally show harmonics of the expected linear gravity wave dispersion in both backscattered power (AM) and Doppler velocity (FM) modulations. Harmonics are confined to the peak wave direction, and the ratio of fundamental to second harmonic energy in the velocity modulation indicates increasing nonlinearity with wavenumber, peaking at a wavenumber consistent with observed wave group modulations. Under higher sea state conditions where whitecapping was evident, spectra show weak reverse travelling waves-opposite to the dominant wave direction. Space-time filtering of the image data suggests breaking events are a source for these waves.

This paper is devoted to the study of Fourier series multipliers . An analog of the Marcinkiewicz theorem on multipliers of Fourier series in weighted Lebesgue spaces is obtained.

Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.