Pair Of Nonlinear Equations Research Articles

Comparison Results for Nonlinear Equations Involving A-Harmonic Operator

In this paper, we derive comparison theorems for a pair of nonlinear equations involving A-harmonic operator. In the special cases, our results give the well-known Leighton-type comparison results concerning the Finsler p-Laplacian, the p-Laplacian, and the pseudo p-Laplacian equations.

Theory for large-amplitude electrostatic ion shocks in quantum plasmas

We present a generalized nonlinear theory for large-amplitude electrostatic (ES) ion shocks in collisional quantum plasmas composed of mildly coupled degenerate electron fluid of arbitrary degeneracy and nondegenerate strongly correlated ion fluid with arbitrary atomic number. For our purposes, we use the inertialess electron momentum equation including the electrostatic force, pressure gradient, and relevant quantum forces, as well as a generalized viscoelastic ion momentum (GVIM) equation for strongly correlated nondegenerate ions. The ion continuity equation, in the quasineutral approximation, then closes our nonlinear system of equations. When the electric field force is eliminated from the GVIM equation by using the inertialess electron momentum equation, we then obtain a GVIM and ion continuity equations, which exhibit nonlinear couplings between the ion number density and the ion fluid velocity. The pair of nonlinear equations is numerically solved to study the dynamics of arbitrarily-large-amplitude planar and nonplanar ES shocks arising from a balance between harmonic generation nonlinearities and the ion fluid viscosity for a wide range of plasma mass densities and ion atomic numbers that are relevant for the cores of giant planets (viz., Jupiter) and compact stars (viz., white dwarfs). Our numerical results reveal that the ES shock density profiles strongly depend on the plasma number density and composition (the atomic-number) parameters. Furthermore, ion density perturbations propagate with Mach numbers which significantly depend on the studied plasma fractional parameters. It is concluded that the dynamics of the ES shocks in the superdense degenerate plasma is quite different in the core of a white dwarf star from that in the lower density crust region.

Upper-hybrid wave-driven Alfvénic turbulence in magnetized dusty plasmas

The nonlinear dynamics of coupled electrostatic upper-hybrid (UH) and Alfvén waves (AWs) is revisited in a magnetized electron-ion plasma with charged dust impurities. A pair of nonlinear equations that describe the interaction of UH wave envelopes (including the relativistic electron mass increase) and the density as well as the compressional magnetic field perturbations associated with the AWs are solved numerically to show that many coherent solitary patterns can be excited and saturated due to modulational instability of unstable UH waves. The evolution of these solitary patterns is also shown to appear in the states of spatiotemporal coherence, temporal as well as spatiotemporal chaos, due to collision and fusion among the patterns in stochastic motion. Furthermore, these spatiotemporal features are demonstrated by the analysis of wavelet power spectra. It is found that a redistribution of wave energy takes place to higher harmonic modes with small wavelengths, which, in turn, results in the onset of Alfvénic turbulence in dusty magnetoplasmas. Such a scenario can occur in the vicinity of Saturn's magnetosphere as many electrostatic solitary structures have been observed there by the Cassini spacecraft.

Vertically fractured transversely isotropic media: dimensionality and deconstruction

ABSTRACTA vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x1, x2‐plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi‐symmetry about the x3‐axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters − its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non‐orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non‐linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined.

Effects of overburden on joint spacing in layered rocks

When a layered system of rock beds is subjected to a sufficiently large extensional strain, joints form in the competent layers. Previous analyses have shown that the ratio between joint spacing and competent layer thickness decreases as the applied strain increases. Further, if the entire interface between the competent layer and the matrix fails in shear (slip), no new joints can form and a lower bound on the joint spacing is reached. In this paper, a joint spacing analysis is developed to explicitly account for the effects of overburden depth. The resulting model is a pair of non-linear equations that can be solved for the characteristic joint spacing as a function of layer thickness, applied strain, and overburden depth. The model results show that, for a given applied strain, the joint spacing first decreases and then increases with depth. This behavior is controlled by the opposing effects of depth-increasing shear strength along the competent layer–matrix interface and depth-increasing compressive prestress. The analysis also reveals that the lower bound (saturation) joint spacing is strongly dependent on depth. Within the choice of realistic physical parameters, predicted values of the saturation-spacing-to-layer-thickness ratio span the range of values observed in the field.

Dust Convective Cells in a Magnetized Dust-Dust Plasma

We present linear and nonlinear properties of dust convective cells in magnetized dusty plasmas whose constituents are positively and negatively charged dust grains. A pair of nonlinear equations is derived by employing the guiding center drifts for dust grains, the dust continuity and Poisson's equations, together with the parallel component of equations of motion for charged dust grains. The governing equations are then used to study the dust convective cell frequency spectrum and the dynamics of nonlinearly interacting dust convective cells.

Global attractivity in an RBC survival model of Wazewska and Lasota

We obtain a condition for the positive equilibrium to be a global attractor of the survival model of red blood cells proposed by Wazewska and Lasota. Our technique is novel in the sense that a pair of nonlinear equations is utilized, and our result improves earlier results in [3] and [4].

Retrieval of Tropical Cirrus Thermal Optical Depth, Crystal Size, and Shape Using a Dual-View Instrument at 3.7 and 10.8μm

Abstract In this paper the authors derive thermal optical depth at 3.7 and 10.8 μm for tropical cirrus utilizing Along Track Scanning Radiometer data under nighttime conditions. By analytically solving the equation of radiative transfer, inclusive of scattering, a pair of nonlinear equations can be solved for the optical depth. Stable and unique solutions for the optical depth are found by combining nadir and forward (55°) views. The accuracy of the analytic solution is compared with solutions from a numerical radiative transfer model assuming an isothermal cirrus cloud. Numerical solutions for two nonisothermal cirrus clouds are also compared with the analytic solution for an equivalent mean cloud temperature. The numerical model uses appropriate hexagonal column scattering phase functions at 3.7 and 10.8 μm. The largest analytic model error is shown to occur at nadir, and it is shown that this error is not overly sensitive to crystal size and is independent of cloud-top temperature. A correction to the ...

Alfvén vortices in nonuniform dusty magnetoplasmas.

The Alfv\'en waves are shown to be linearly coupled with finite-frequency convective cell modes in a nonuniform cold dusty magnetoplasma. The dynamics of weakly interacting finite-amplitude electromagnetic oscillations is governed by a pair of nonlinear equations. The latter admit dipolar vortices as possible stationary solutions. The present results can have relevance to the nonlinear electromagnetic wave propagation in a space environment such as cometary tails or interstellar clouds.

Dynamic roughening of directed lines.

We study the fluctuations of a stretched string, e.g., a vortex line, moving in a random medium. A pair of nonlinear equations are proposed to describe the evolution of longitudinal and transverse coordinates. The dynamic scaling of the fluctuations is studied analytically (by renormalization group) and numerically. In most cases the fluctuations are superdiffusive, governed by a dynamic exponent z=3/2.

Lagrangian Formulation for Penny-Shaped and Perkins-Kern Geometry Models

Summary Basic theories for vertical penny-shaped and Perkins-Kern (PK) geometry models have been developed with a Lagrangian formulation combined with a virtual-work analysis. The Lagrangian formulation yields a pair of nonlinear equations in Rf or Lf and bf, the fracture radius or length and half-width. By introduction of a virtual-work analysis, a simple equation is obtained that can be solved numerically. This equation is written in a form that can be used to determine fracture geometry when the fluid-loss coefficient of the fracturing fluid is known. Also, this equation, coupled with a material-balance equation after shut-in, can be used to analyze pressure-decline data after shut-in to determine the effective fluid-loss coefficient and fracture geometry. Geometry prediction of the new theory agrees well with other methods for the same models. A unique advantage of this approach, however, is that the geometry estimated from prefracturing design is consistent with the geometry determined from the pressure-decline analysis. Thus, it allows the engineers in the field to predict precisely the amount of fracturing fluid required for a designed fracture radius or length in a full-scale treatment from information gained in a minifracture treatment.

Electromagnetic drift modes in an inhomogeneous electron gas.

A pair of nonlinear equations is derived which describes the dynamics of the electromagnetic drift oscillations in a nonuniform magnetized electron gas. It is shown that the nonlinear electromagnetic drift modes can propagate in the form of dipole vortices.

Lie symmetries, nonlinear equations of motion and new Ermakov systems

It is shown that a Lagrangian of the form L=1/1( rho 2- omega 2(t) rho 2)+G(t)F(k(t) rho ), said to be in factored form, yields an equation of motion that is equivalent to the most general equation derivable via Noether's theorem from the unfactored Lagrangian L=1/2 rho 2-P( rho ,t). In view of this equivalence, the theory of extended Lie groups is applied to the factored nonlinear equation of motion p+ omega 2(t) rho =G(t)F(k(t) rho ) to obtain its Lie symmetries. The latter are obtained when G(t) and k(t), initially arbitrary, are determined in terms of a function x(t) which satisfies the auxiliary equation x+ omega 2(t)x=K/x3. It is then possible with the auxiliary equation and the equation of motion to form a coupled pair of nonlinear equations, an Ermakov system, whose first integral is not invariant under the action of the symmetry group, in contrast to previous Ermakov systems.

Numerical integrators for stiff and highly oscillatory differential equations

Some L-stable fourth-order explicit one-step numerical integration formulas which require no matrix inversion are proposed to cope effectively with systems of ordinary differential equations with large Lipschitz constants (including those having highly oscillatory solutions). The implicit integration procedure proposed in Fatunla [11] is further developed to handle a larger class of stiff systems as well as those with highly oscillatory solutions. The same pair of nonlinear equations as in [11] is solved for the stiffness/oscillatory parameters. However, the nonlinear systems are transformed into linear forms and an efficient computational procedure is developed to obtain these parameters. The new schemes compare favorably with the backward differentiation formula (DIFSUB) of Gear [13], [14] and the blended linear multistep methods of Skeel and Kong [24], and the symmetric multistep methods of Lambert and Watson [17].