Symmetry analysis of a system of modified shallow-water equations

We revise the symmetry analysis of a modified system of one-dimensional shallow-water equations (MSWE) recently considered by Raja Sekhar and Sharma [Commun Nonlinear Sci Numer Simulat 2012;20:630–36]. Only a finite dimensional subalgebra of the maximal Lie invariance algebra of the MSWE, which in fact is infinite dimensional, was found in the aforementioned paper. The MSWE can be linearized using a hodograph transformation. An optimal list of inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra is constructed and used for Lie reductions. Non-Lie solutions are found from solutions of the linearized MSWE.     

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.     

Dynamics of a two-layer fluid sloshing problem

A second-order ordinary differential equation, which is a reducedform of the periodically forced extended Korteweg–de Vries(eKdV) equation, is derived in the physical context of sloshinga two-layer fluid tank. In the limit of small dispersion, numericalevidence is given of multiple periodic solutions displayingfast oscillations superimposed on slow periodic waves and ahigher-order Melnikov method is then used to verify the existenceof such solutions. The dynamical behaviour of a similar equationwith more general coefficients is also examined, demonstratingthe existence of periodic and chaotic behaviour. We highlightnew aspects which arise due to the presence of mixed nonlinearity.     

Strongly nonlinear internal soliton load on a small vertical circular cylinder in two-layer fluids

The solutions of MCC theory are used to investigate larger-amplitude strongly nonlinear internal soliton load on a small surface-piercing circular cylinder in two-layer fluids. By comparing the wave profiles and instantaneous horizontal velocities calculated by MCC theory with those of KdV theory and experimental data, we verify the validity of MCC theory for larger-amplitude strongly nonlinear internal soliton. The accelerations are computed, and then force and torque on a small cylinder are estimated based on Morison’s formula for both MCC and KdV theories. Computed results show that the internal soliton force and torque become more and more large and wide with the increase of amplitude for MCC theory. The location of torque crest calculated by MCC theory departs from origin (moving to the right) as the amplitude grows and whenever the inertial term is included or not, the wave forces computed based on the two theories both have small discrepancies for the same amplitude, but when the inertial term is included, the torque obtained by MCC theory will be much larger and the torque obtained by KdV still have a small discrepancy. The reasons are presented in detail. The internal wave force will be underestimated if the traditional KdV theory is used. Therefore, ocean engineers should consider the large-amplitude strongly nonlinear internal soliton load on marine construct carefully.     

Definable predicates of standardness in internal set theory

It is shown that Nelson’s internal set theory IST has no definable predicate that is a proper extension of the standardness predicate and satisfies the carry-over, idealization, and standardization principles. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 803–809, December, 1999.     

Sensitivity of a shallow-water model to parameters

An adjoint based technique is applied to a shallow water model in order to estimate the influence of the model’s parameters on the solution. Among parameters, the bottom topography, initial conditions, boundary conditions on rigid boundaries, viscosity coefficients, Coriolis parameter and the amplitude of the wind stress tension are considered. Their influence is analyzed from three points of view:

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flexibility of the model with respect to a parameter that is related to the lowest value of the cost function that can be obtained in the data assimilation experiment that controls this parameter;

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possibility to improve the model by the parameter’s control, i.e., whether the solution with the optimal parameter remains close to observations after the end of control;

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sensitivity of the model solution to the parameter in a classical sense. That implies the analysis of the sensitivity estimates and their comparison with each other and with the local Lyapunov exponents that characterize the sensitivity of the model to initial conditions.

Two configurations have been analyzed: an academic case of the model in a square box and a more realistic case simulating Black sea currents. It is shown in both experiments that the boundary conditions near a rigid boundary highly influence the solution. This fact points out the necessity to identify optimal boundary approximation during a model development.     

Effect of vertical vibrations on a two-layer system with a deformable interface

The effect of single- and two-frequency vibrations on the behavior of a system consisting of two homogeneous viscous fluids bounded by rigid walls is analyzed. It is assumed that the system as a whole is under vertical vibrations obeying a certain law. An eigenvalue problem is obtained in order to analyze the stability of the relative equilibrium. The case of finite frequencies and arbitrary modulation amplitudes is treated along with the case of high frequencies and small modulation amplitudes. In the former case, the parametric resonance domains are examined depending on the parameters of the system. In the latter case, the high-frequency vibration is shown to create effective surface tension, thus flattening the interface, and can suppress instability when the heavy fluid is over the light one.     

Robust state estimation for Markov jump genetic regulatory networks based on passivity theory

In this article, the robust state estimation problem for Markov jump genetic regulatory networks (GRNs) based on passivity theory is investigated. Moreover, the effect of time‐varying delays is taken into account. The focus is on designing a linear state estimator to estimate the concentrations of the mRNAs and the proteins of the GRNs, such that the dynamics of the state estimation error can be stochastically stable while achieving the prescribed passivity performance. By applying the Lyapunov–Krasovskii functional method, delay‐dependent criteria are established to ensure the existence of the mode‐dependent estimator in the form of linear matrix inequalities. Based on the obtained results, the parameters of the desired estimator gains can be further calculated. Finally, a numerical example is given to illustrate the effectiveness of our proposed methods. © 2015 Wiley Periodicals, Inc. Complexity 21: 214–223, 2016     

Modeling of tsunami generation and propagation by a spreading curvilinear seismic faulting in linearized shallow-water wave theory

The processes of tsunami evolution during its generation in search for possible amplification mechanisms resulting from unilateral spreading of the sea floor uplift is investigated. We study the nature of the tsunami build up and propagation during and after realistic curvilinear source models represented by a slowly uplift faulting and a spreading slip-fault model. The models are used to study the tsunami amplitude amplification as a function of the spreading velocity and rise time. Tsunami waveforms within the frame of the linearized shallow-water theory for constant water depth are analyzed analytically by transform methods (Laplace in time and Fourier in space) for the movable source models. We analyzed the normalized peak amplitude as a function of the propagated uplift length, width and the average depth of the ocean along the propagation path.     

Sufficient conditions for a strong relative minimum in an optimal control problem

In this paper, on the basis of Young's method (Ref. 1), sufficient conditions for a strong relative minimum in an optimal control problem are given. Young's method generalizes geodesic coverings and the simplest Hilbert integral from the standard variational calculus. This paper carries Young's method over to nonparametric problems.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.     

Separable Banach space theory needs strong set existence axioms

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, comprehension, is needed to prove such basic facts as the existence of the weak- closure of any norm-closed subspace of . This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for sentences. En route to our main results, we prove the Krein-\v{S}mulian theorem in , and we give a new, elementary proof of a result of McGehee on weak- sequential closure ordinals.

     

Sufficient conditions for a control to be a strong minimum

The control literature either presents sufficient conditions for global optimality (for example, the Hamilton-Jacobi-Bellman theorem) or, if concerned with local optimality, restricts attention to comparison controls which are local in theL -sense. In this paper, use is made of an exact expression for the change in cost due to a change in control, a natural extension of a result due to Weierstrass, to obtain sufficient conditions for a control to be a strong minimum (in the sense that comparison controls are merely required to be close in theL ₁-sense).     

Robust H{infty} control for Markovian jump nonlinear systems

This paper deals with the robust H control problem of uncertainnonlinear stochastic systems with Markovian jump parameters.We assume that the uncertainties are structured such that suitablebounding functions can be found. A robust controller is thendesigned that will guarantee disturbance attenuation and asymptoticstability for all admissible uncertainties and L₂-bounded disturbances.The solution to the problem is characterized in terms of a setof smooth-positive semidefinite functions satisfying certainHamilton-Jacobi-Isaac (HJI) inequalities with some appropriatescaling functions. Both the case of matched and unmatched uncertaintiesare considered.     

High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems

This paper is concerned with the development of high order methods for the numerical approximation of one-dimensional nonconservative hyperbolic systems. In particular, we are interested in high order extensions of the generalized Roe methods introduced by I. Toumi in 1992, based on WENO reconstruction of states. We also investigate the well-balanced properties of the resulting schemes. Finally, we will focus on applications to shallow-water systems.

     

Multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach

In this paper we study quasilinear problems involving variable exponent growth conditions and nonlocal terms on the whole space R^N. A multiplicity result is established. All the coefficients involved in the terms of the equation depend both on the variable x and the unknown function u. Our main argument is nonsmooth critical point theory.     

Strong coprimality and strong irreducibility of Alexander polynomials

A polynomial f(t) with rational coefficients is strongly irreducible if f(tk) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(tk) and g(tl) are relatively prime for all positive integers k and l. We provide some sufficient conditions for strong irreducibility and prove that the Alexander polynomials of twist knots are pairwise strongly coprime and that most of them are strongly irreducible. We apply these results to describe the structure of the subgroup of the rational knot concordance group generated by the twist knots and to provide an explicit set of knots which represent linearly independent elements deep in the solvable filtration of the knot concordance group.     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. I. General theory

A general algorithm for building a uniform asymptotic solution of the kinetic equations for spatially inhomogeneous reactive gas mixtures is proposed. It solves the problem of irregular asymptotic solution arising in the ordinary Chapman–Enskog method, providing expressions for chemical reaction rates that agree with the mono-molecular reaction theory. We study a quasi-stationary behavior of the system, characterized by the slowly varying gas-dynamic variables which number is greater than the number of integral invariants of the collision operator. The gas-dynamic equations for reacting and relaxing gas mixtures are derived in general form. It is shown that accurate treatment of non-equilibrium processes gives rise to additional terms caused by the strong influence of small perturbations of quasi-equilibrium distribution functions on the kinetics of high-threshold physical and chemical processes. These terms are describing the influence of inelastic collisions, expansion/compression processes and spatial non-uniformity of gas-dynamic variables.