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## The pressure field beneath intense surface water wave groups

A weakly-nonlinear potential theory is developed for the description of deep penetrating pressure

fields caused by single and colliding wave groups of collinear waves due to the second-order nonlinear

interactions. The result is applied to the representative case of groups with the sech-shape of

envelope solitons in deep water. When solitary groups experience a head-on collision, the induced due

to nonlinearity dynamic pressure may have magnitude comparable with the magnitude of the linear

solution. It attenuates with depth with characteristic length of the group, which may greatly exceed the

individual wave length. In general the picture of the dynamic pressure beneath intense wave groups looks

complicated. The qualitative difference in the structure of the induced pressure field for unidirectional and

opposite wave trains is emphasized.

The bottom pressure distribution beneath large amplitude waves is studied within linear theory in time and space domain, weakly dispersive Serre–Green–Naghdi system and fully nonlinear potential equations. These approaches are used to compare pressure fields induced by solitary waves, but also by transient wave groups. It is shown that linear analysis in time domain is in good agreement with Serre– Green–Naghdi predictions for solitary waves with highest amplitude A = 0.7h, h being water depth. In the meantime, when comparing results to fully nonlinear potential equations, neither linear theory in time domain, nor in space domain, provide a good description of the pressure peak. The linear theory in time domain underestimates the peak by an amount similar to the overestimation by linear theory in space domain. For transient wave groups (up to A = 0.52h), linear analysis in time domain provides results very similar to those based on the Serre–Green–Naghdi system. In the meantime, linear theory in space domain provides a surprisingly good comparison with prediction of fully nonlinear theory. In all cases, it has to be emphasized that a discrepancy between linear theory in space domain and in time domain was always found, and presented an averaged value of 20%. Since linear theory is often used by coastal engineers to reconstruct water elevation from bottom mounted sensors, the so-called inverse problem, an important result of this work is that special caution should be given when doing so. The method might surprisingly work with strongly nonlinear waves, but is highly sensitive to the imbalance between nonlinearity and dispersion. In most cases, linear theory, in both time and space domain, will lead to important errors when solving this inverse problem.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.