Long-wavelength Solutions Research Articles

Optimized technique and dynamical behaviors of fractional Lax and Caudrey–Dodd–Gibbon models modelized by the Caputo fractional derivative

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Korteweg–de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey–Dodd–Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error’s analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approach is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.

The Basics of Primordial Black Hole Formation and Abundance Estimation

This paper is a biased review of the primordial black hole (PBH) formation and abundance estimation. We first review the three-zone model for PBH formation to help an intuitive understanding of the PBH formation process. Then, for more accurate analyses, we introduce necessary tools such as cosmological long-wavelength solutions, the definition of the mass and compaction function in a spherically symmetric spacetime and peak theory. Combining all these tools, we calculate the PBH mass spectrum for the case of the monochromatic curvature power spectrum as a demonstration.

Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

The aim of this paper is to investigate the approximate solutions of nonlinear temporal fractional models of Gardner and Cahn-Hilliard equations. The fractional models of the Gardner and Cahn-Hilliard equations play an important role in pulse propagation in dispersive media. The time-fractional derivative is observed in the conformable framework. In this orientation, a reliable computationally algorithm is designed and developed by following a residual error and multivariable power series expansion. Basically, the approximate solutions of pulse wave function of the fractional higher-order Gardner and Cahn-Hilliard equations are obtained in the form of a conformable convergent fractional series. Relevant consequences are theoretically and numerically investigated under the conformable sense. Besides, the analysis of the error and convergence of the developed technique are discussed. Some of the unidirectional homogeneous physical applications of the posed models in a finite compact regime are tested to confirm the theoretical aspects, demonstrate different evolutionary dynamics, and highlight the superiority of the novel developed algorithm compared to other existing analytical methods. For this purpose, associated graphs are displayed in two and three dimensions. Growing and decaying modes of the fractional parameters are analyzed for several α values. From a numerical viewpoint, the simulations and results declare that the proposed iterative algorithm is indeed straightforward and appropriate with efficiency for long-wavelength solutions of nonlinear partial differential equations.

A novel analytical algorithm for generalized fifth-order time-fractional nonlinear evolution equations with conformable time derivative arising in shallow water waves

The purpose of this research is to study, investigate, and analyze a class of temporal time-FNEE models with time-FCDs that are indispensable in numerous nonlinear wave propagation phenomena. For this purpose, an efficient semi-analytical algorithm is developed and designed in view of the residual error terms for solving a class of fifth-order time-FCKdVEs. The analytical solutions of a dynamic wavefunction of the fractional Ito, Sawada-Kotera, Lax’s Korteweg-de Vries, Caudrey-Dodd-Gibbon, and Kaup-Kupershmidt equations are provided in the form of a convergent conformable time-fractional series. The related consequences are discussed both theoretically as well as numerically considering the conformable sense. In this direction, convergence analysis and error estimates of the developed algorithm are studied and analyzed as well. Concerning the considered models, specific unidirectional physical experiments are given in a finite compact regime to confirm the theoretical aspects and to demonstrate the superiority of the novel algorithm compared to the other existing numerical methods. Moreover, some representative results are presented in two- and three-dimensional graphs, whilst dynamic behaviors of fractional parameters are reported for several α values. From the practical viewpoint, the archived simulations and consequences justify that the iterative algorithm is a straightforward and appropriate tool with computational efficiency for long-wavelength solutions of nonlinear time-FPDEs in physical phenomena.

Statics correction methods for 3D converted-wave (PS) seismic reflection

One of the most difficult steps in the PS processing sequence is estimation of the S-wave receiver statics. This process is particularly important at the coal scale, due to the need for higher frequency content (better resolution).We present an analysis of three approaches for estimating 3D PS statics. These include a surface-consistent inversion algorithm (analogous to the residual-statics method used in conventional P-wave processing), PPS refraction statics, and a so-called robust statistical method. This analysis is achieved through the use of synthetic models, and a coal-scale 3D-3C survey acquired in the Bowen Basin.The presented datasets demonstrate that the surface-consistent inversion method can become unstable in certain environments. This is likely due to parameter leakage between receiver and structural terms, caused by the highly asymmetric nature of the shallow PS reflection paths. The robust statistical method appears reliable for determination of short-wavelength receiver statics, and hence is useful for continuity enhancement. The PPS refraction approach can provide both short-wavelength and long-wavelength solutions, provided the PPS arrivals can be picked reliably. As with P-wave analysis, a combination of algorithms may provide the most effective production tool for determination of PS receiver statics.

Improved depth conversion with FWI – a case study

In the study region, notoriously heterogeneous shallow carbonates, here between depths of approximately 1.5- 2.5 km, give rise to short-wavelength velocity variations in the overburden, which can cause severe depth undulations at the reservoir level (~3 km depth subsea). As the reservoirs are relatively thin (~30 m), stacked, fluvial-deltaic channel sands with sharp meanders and lateral truncations, even a localised 1% velocity error can produce a 30 m depth error at 3 km, which is not acceptable for development well planning. The resolution of traveltime-tomography is limited and may not resolve overburden variability in a manner suitable for depth conversion in a development setting. Geostatistical scaling using the available well control is one option to further improve depth conversion reliability. This workflow utilizes time-depth data at well locations along with geophysically constrained statistics to derive a stable, geologically plausible background trend that accounts for the majority of depth error. The result is a regionally consistent model that ties the available well control and provides more reliable depth conversion away from wells. The background trend provided by the well-based velocity calibration is however a long-wavelength solution and is unlikely to be able to correct for the short-wavelength depth errors which can arise at depth. FWI on the other hand provides a resolution of less than half a seismic wavelength to capture thin and localized subsurface features. With short-wavelength features accurately resolved, well control can then be used to correct for any residual long-wavelength errors and ensure more reliable depth conversion away from the wells. In this study, reflection tomography and 3D TTI (tilted transverse isotropy) FWI velocity models were geostatistically calibrated to wells and compared for depth conversion accuracy. Using the calibrated reflection tomography model, early appraisal wells were still more than 50 m off prognosis at the top reservoir level. These appraisal wells were just a few hundred metres away from the exploration wells used for control. 3D TTI FWI was then run using the reflection tomography model as input. The same geostatistical calibration to wells was applied to the FWI output and ultimately provided a model with depth conversion accuracy to within 15 m at the reservoir level.

Cosmological long-wavelength solutions and primordial black hole formation

We construct cosmological long-wavelength solutions without symmetry in general gauge conditions compatible with the long-wavelength scheme. We then specify the relationship among the solutions in different time slicings. Applying this general framework to spherical symmetry, we derive the correspondence relation between long-wavelength solutions in the constant mean curvature slicing with conformally flat spatial coordinates and asymptotic quasihomogeneous solutions in the comoving gauge and compare the numerical results of PBH formation in these two different approaches. To discuss the PBH formation, it is convenient and conventional to use $\tilde{\delta}_{c}$, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowest-order long-wavelength expansion. We numerically find that within compensated models, the sharper the transition from the overdense region to the FRW universe is, the larger the $\tilde{\delta}_{c}$ becomes. We suggest that, for the equation of state $p=(\Gamma-1)\rho$, we can apply the analytic formulas for the minimum $\tilde{\delta}_{c, {\rm min}}\simeq [3\Gamma/(3\Gamma+2)]\sin^{2}\left[\pi\sqrt{\Gamma-1}/(3\Gamma-2)\right]$ and the maximum $\tilde{\delta}_{c, {\rm max}}\simeq 3\Gamma/(3\Gamma+2)$. As for the threshold peak value of the curvature variable $\psi_{0,c}$, we find that the sharper the transition is, the smaller the $\psi_{0,c}$ becomes. We analytically explain this feature. Using simplified models, we also analytically deduce an environmental effect that $\psi_{0,c}$ can be significantly larger (smaller) if the underlying density perturbation of much longer wavelength is positive (negative).

Acoustic scattering by a spherical obstacle: Modification to the analytical long-wavelength solution for the zero-order coefficient

Classical long wavelength approximate solutions to the scattering of acoustic waves by a spherical liquid particle suspended in a liquid (an emulsion) show small but significant differences from full solutions at very low k(c)a (typically k(c)a < 0.01) and above at k(c)a > 0.1, where k(c) is the compressional wavenumber and a the particle radius. These differences may be significant in the context of dispersed particle size estimates based on compression wave attenuation measurements. This paper gives an explanation of how these differences arise from approximations based on the significance of terms in the modulus of the complex zero-order partial wave coefficient, A(0). It is proposed that a more accurate approximation results from considering the terms in the real and imaginary parts of the coefficient, separately.

Multiple scattering and visco-thermal effects

For modeling sound propagation in a rigid-framed fluid-saturated porous material it is customary to use frequency-dependent density and compressibility functions. These functions, which describe ‘‘temporal’’ dispersion effects due to inertial/viscous and thermal effects, can be computed by FEM in simple geometries and give complete information about the long-wavelength properties of the medium. When the wavelength is reduced, new effects due to scattering must be considered. To study this, we consider solving the sound propagation problem in a 2-D ‘‘phononic crystal’’ made of an infinite square lattice of solid cylinders embedded in a fluid. An exact multiple-scattering solution is first developed for an ideal saturating fluid and then generalized to the case of visco-thermal fluid, by using the concept of visco-thermal admittances. The condition to use this concept is that the viscous and thermal penetration depths are small compared to the cylinder radius. We validate our results in the long-wavelength regime by direct comparisons with FEM data [A. Cortis, ‘‘Dynamic parameters of porous media,’’ Ph.D. dissertation (Delft U.P., Delft, (2002)]. When frequency increases, differences appear between the long-wavelength solution and the exact multiple-scattering solution, which could be interpreted in terms of ‘‘spatial’’ dispersion effects.

On the Growth of Perturbations as a Test of Dark Energy and Gravity

The strongest evidence for dark energy at present comes from geometric techniques such as the supernova distance-redshift relation. By combining the measured expansion history with the Friedmann equation, one determines the energy density and its time evolution and hence the equation of state of dark energy. Because these methods rely on the Friedmann equation, which has not been independently tested, it is desirable to find alternative methods that work for both general relativity and other theories of gravity. Assuming that sufficiently large patches of a perturbed Robertson-Walker spacetime evolve like separate Robertson-Walker universes, that shear stress is unimportant on large scales, and that energy and momentum are locally conserved, we derive several relations between long-wavelength metric and matter perturbations. These relations include generalizations of the initial-value constraints of general relativity. For a class of theories including general relativity we reduce the long-wavelength metric, density, and velocity potential perturbations to quadratures including curvature perturbations, entropy perturbations, and the effects of nonzero background curvature. When combined with the expansion history measured geometrically, the long-wavelength solution provides a test that could distinguish modified gravity from other explanations of dark energy.

Amplitude of Biot's slow wave scattered by a spherical inclusion in a fluid-saturated poroelastic medium

SUMMARY Spatial heterogeneity of hydrocarbon reservoirs causes significant attenuation and dispersion of seismic waves due to wave-induced flow of the pore fluid between more compliant and less compliant areas. This paper investigates the interaction between a plane elastic wave in a poroelastic medium with a spherical inhomogeneity of another porous material. The behaviour of both the inclusion and the background medium is described by the low-frequency variant of Biot's equations of poroelasticity with the standard boundary conditions at the inclusion surface, and for the inclusion size much smaller than the wavelength of the fast compressional wave. The scattering problem is formulated as a series expansion of displacements expressed in the spherical harmonics. The resulting scattered wavefield consists of the scattered normal compressional and shear waves and Biot's slow wave, which attenuates rapidly with distance from the inclusion and represents the main difference from the elastic case. This study concentrates on the attenuation effects caused by the mode conversion into Biot's slow wave. The solution obtained for Biot's slow wave is well described by the two terms of order n= 0 and n= 2 of the scattering series. The scattering amplitude for the term of order n= 0 is given by a simple expression. The full expression for the term of order n= 2 is very complicated, but can be simplified assuming that the amplitude of the scattered fast (normal) compressional and shear waves are well approximated by the solution of the equivalent elastic problem. This assumption yields a simple approximation for the amplitude of the scattered slow wave, which is accurate for a wide range of material properties and is sufficient for the analysis of the scattering amplitude as a function of frequency. In the low-frequency limit the scattering amplitude of the slow wave scales with ω3/2, and reduces to the asymptotic long-wavelength solution of Berryman (1985), which is valid for inclusions much smaller than the wavelength of Biot' slow wave. For inclusions larger than the wavelength of Biot's slow wave, the scattering amplitude is proportional to ω1/2, which is consistent with the results of Gurevich et al. (1998), which were derived by the Born approximation and therefore were limited to a weak contrast between the inclusion and the background medium. Our general solution, however, does not require these assumptions on frequency and material properties. The obtained results can be used in the analysis of the effective properties, attenuation and dispersion of elastic waves in randomly inhomogeneous porous materials.

The Rigorous Approximation of Long-Wavelength Capillary-Gravity Waves

In a previous paper we proved that long-wavelength solutions of the water-wave problem in the case of zero surface tension split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries (KdV) equation. In this paper we examine the effect of surface tension on this scenario. We find that we obtain three different physical regimes depending on the strength of the surface tension. For weak surface tension, the propagation of the wave packets is very similar to that in the zero surface tension case. For strong surface tension, the evolution is again governed by a pair of KdV equations, but the coefficients in these equations have changed in such a way that the KdV soliton now represents a wave of depression on the fluid surface. Finally, at a special, intermediate value of the surface tension (where the Bond number equals ⅓) the KdV description breaks down and it is necessary to introduce a new approximating equation, the Kawahara equation, which is a fifth order, nonlinear partial differential equation. In each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water-wave problem.

Tomostatics applications for basalt-outcrop land and OBC multi-component surveys

Tomostatics has been recently applied to two types of data sets. The first is from the Middle East region where the near surface layer is exhibiting high-velocity basalt outcrops and the second is a multi-component (4C) OBC data set containing low-velocity gas clouds. Near-surface velocities estimated from turning-ray tomography provide useful information for geologic interpretation and structural imaging. Tomostatics has shown advantages over traditional refraction statics in regions where 1) no refractors can be easily identified, and 2) high velocity materials (e.g. basalt) are overlaid on top of the low velocity sediments immediately below the topography. P-wave velocities that estimated from turning-ray tomography could be used to calculate the traveltime contours. Correlating P-wave and converted-wave traveltimes would allow us to estimate Vp/Vs ratios, leading to a practical approach for converted-wave long-wavelength statics solution.

Tomostatics Applications for Basalt-Outcrop Land and OBC Multi-Component Surveys

Tomostatics have been recently applied to two types of data sets. The first is from the Middle East region where the near surface layer is exhibiting high-velocity basalt outcrops and the second is a multi-component (4C) OBC data set containing lowvelocity gas clouds. Near-surface velocities estimated from turning-ray tomography provide useful information for geologic interpretation and structural imaging. Tomostatics have shown advantages over traditional refraction statics in regions where (1) no refractors can be easily identified, and (2) high velocity materials (e.g. basalt) are overlaid on top of the low velocity sediments immediately below the topography.P-wave velocities estimated from turning-ray tomography can be used to calculate the traveltime contours. Correlating P-wave and converted-wave traveltimes would allow us to estimate Vp/Vs ratios, leading to a practical approach for converted-wave longwavelength statics solution.

Gauge symmetry and localized gravity in M-theory

We discuss the possibility of having gravity ``localized'' in dimension d in a system where gauge bosons propagate in dimension d+1. In such a circumstance - depending on the rate of falloff of the field strengths in d dimensions - one might expect the gauge symmetry in d+1 dimensions to behave like a global symmetry in d dimensions, despite the presence of gravity. Naive extrapolation of warped long-wavelength solutions of general relativity coupled to scalars and gauge fields suggests that such an effect might be possible. However, in some basic realizations of such solutions in M theory, we find that this effect does not persist microscopically. It turns over either to screening or the Higgs mechanism at long distances in the d-dimensional description of the system. We briefly discuss the physics of charged objects in this type of system.

Renormalization of the long-wavelength solution of Einstein’s equation

Using the renormalization group method, we improved the first order solution of the long-wavelength expansion of the Einstein equation. By assuming that the renormalization group transformation has the property of Lie group, we can regularize the secular divergence caused by the spatial gradient terms and absorb it to the background seed metric. The solution of the renormalization group equation shows that the renormalized metric describes the behavior of gravitational collapse in the expanding universe qualitatively well.

Interphase stability of stratified viscous liquids in Couette flow

An exact long-wavelength asymptotic solution is constructed for the problem of interphase stability in the Couette flow of a two-layer system of Newtonian liquids with different viscosities and with shallow depths. It is shown that the dispersion relation for the complex velocity of a wave of disturbances can be represented as an expansion in fractional powers of the wave number which are multiples of 1/3. New stability criteria, different from those obtained previously, are found.

Hamilton-Jacobi Equation for Brans-Dicke Theory and Its Long-Wavelength Solution

Hamilton-Jacobi equation for Brans-Dicke theory is solved by using a long-wavelength approximation. We examine the non-linear evolution of the inhomogeneities in the dust fluid case and the cosmological constant case. In the case of dust fluid, it turns out that the inhomogeneities of space-time grow. In the case of cosmological constant, the inhomogeneities decay, which is consistent with the cosmic no hair conjecture. The inhomogeneities of the density perturbation and the gravitational constant behave similarly with that of space-time.

Flux phase models in three dimensions

We consider a long-wavelength theory for fermions propagating on a two-dimensional lattice in the presence of a background flux distribution and discuss the symmetries of the solutions. These notions are then generalized to three spatial dimensions, where the background flux is due to effective magnetic monopole fields. The nature of the long-wavelength solutions is again discussed.

Reexamination of neutrino oscillation solutions to the solar-neutrino problem.

A deficit of solar neutrinos compared to the standard solar model prediction has been measured in both the Homestake and Kamiokande-II experiments, with quite different detectors. We analyze the parameter space for oscillations of two neutrinos to find regions that can explain both experiments. Long-wavelength vacuum oscillation solutions are found with $\ensuremath{\delta}{m}^{2}\ensuremath{\simeq}0.5\ensuremath{-}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ e${\mathrm{V}}^{2}$ and ${sin}^{2}2\ensuremath{\theta}\ensuremath{\gtrsim}0.7$. Matter-enhanced oscillation solutions are found for (i) nonadiabatic transitions having $\ensuremath{\delta}{m}^{2}{\ensuremath{\theta}}^{2}\ensuremath{\approx}{10}^{\ensuremath{-}8}$ e${\mathrm{V}}^{2}$ and $\ensuremath{\delta}{m}^{2}\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\ensuremath{-}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ e${\mathrm{V}}^{2}$, and (ii) adiabatic transitions having $\ensuremath{\delta}{m}^{2}\ensuremath{\approx}{10}^{\ensuremath{-}7}\ensuremath{-}{10}^{\ensuremath{-}4}$ e${\mathrm{V}}^{2}$ with ${sin}^{2}2\ensuremath{\theta}\ensuremath{\gtrsim}0.5$. The two-neutrino long-wavelength and matter-enhanced solutions predict $^{71}\mathrm{Ga}$ measurements in the ranges 50-80 and 5-115 solar-neutrino units (SNU), respectively, compared to 130 SNU without oscillations. We emphasize the distinctive features of each solution, including their predictions for future $\ensuremath{\nu}\ensuremath{-}e$ scattering experiments and detectable seasonal variations in the long-wavelength solutions. We also find the range of three-neutrino vacuum oscillation solutions.