Gardner Equation Research Articles

Density - Velocity Relationship and Prediction of Lithology Variation Using Physical Analysis in Kf-4 Well of The Kifl Oil Field, Yamama Formation, South of Iraq

The density-velocity relation is an important tool used to predict one of these two parameters from the other. A new empirical density –velocity equation was derived in Kf-4 well at Kifl Oil Field, south of Iraq. The density was derived from Gardner equation and the results obtained were compared with the density log (ROHB) in Kl-4 well. The petrophysical analysis was used to predict the variations in lithology of Yamama Formation depending on the well logs data, such as density, gamma, and neutron logs. The physical analysis of rocks depended on the density, Vp, and Vs values to estimate the elastic parameters, i.e. acoustic impedance (AI) and Vp/Vs ratio, to predict the lithology and hydrocarbon indicators. According to the results of physical properties, Yamama Formation is divided into five units in Kf-4 well at Kifl Oil Field. The lithology of Yamama Formation was found to consist of limestone, dolomite, shale, and anhydrite rocks.

The Wick-type explicit solutions of the nonlinear stochastic Wick-type fractional Gardner equation

In this article, we consider the nonlinear stochastic Wick-type fractional Gardner equation and the generalized fractional Gardner equation with coefficient variables. We obtain new Wick-type explicit solutions of the nonlinear stochastic Wick-type fractional Gardner equation and new explicit solutions of the generalized fractional Gardner equation with coefficients variables by employing the improved system method. Additionally, dynamical behaviors of the obtained solutions are provided graphically under suitable functions of white noises and physical parameters. We show that the improved system method is powerful and effective to find explicit solutions of the considered equations. It is also recommended to use the improved system method to solve the fractional nonlinear partial differential equations as well as nonlinear partial differential equations.

The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations

The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.

Ion Acoustic Solitary Waves and Double-Layer Propagation in an Unmagnetized Plasma With Degenerate Electrons

The nonlinear propagation of ion-acoustic (IA) solitons and double layers (DLs) in a nonrelativistic degenerate plasma is investigated. This plasma model consists of negative ions, positive ions, and degenerate electrons. The Gardner equation is derived using the reductive perturbation technique. Both the Gardner solitons and DLs exist in our model. The characteristics of these waves are analyzed. It is found that the properties of these structures depend on the densities, the temperature, and the mass ratio of the ions and also on the chemical potential of the degenerate electrons. Additionally, it is found that the system sustains two different IA modes with different phase velocities known as slow and fast IA modes. These investigations can be useful in space plasma situations that contain opposite polarity ions such as compact astrophysical objects like white dwarfs.

Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves

Abstract Propagation of dust ion-acoustic (DIA) Gardner wave in a dusty electron–positron–ion (e–p–i) plasma is investigated. This plasma consists of q-distributed electrons and positrons, warm ions, and dust grains. The effects of the electron nonextensivity, positron nonextensivity, and fractional parameter on the properties of DIA Gardner wave are investigated. Space fractional Gardner (SFG) equation is derived using the semi inverse technique. An efficient modified G′/G-expansion method is presented to solve the SFG equation. It is found that the amplitude of the DIA Gardner wave increases with an increase in space fractional parameter β $\left(\beta \right)$ and spatial parameter ζ $\left(\zeta \right)$ . On other hands, the DIA Gardner wave shape can be modulated using the space fractional parameter β $\left(\beta \right)$ . Our results may help understand the astrophysical environments such as star magnetospheres, solar flares, and galactic nuclei.

Ion-acoustic Gardner solitons in negative ion plasmas

The ion-acoustic Gardner solitons in negative ion plasma have been studied. The Gardner equation is deduced by employing the reductive perturbation technique. We have investigated the ion-acoustic Gardner solitons in negative ion plasma in the parametric regime where the KdV equation is no longer valid to study soliton solution. It is found that for a given set of parameter values with negative ion concentration ( α ) less than the critical value ( α c ), i.e. α < α c ; the system supports compressive Gardner soliton, whereas for α > α c , rarefactive Gardner soliton exists. It is also investigated that Gardner solitons exist only for the fast mode, whereas no such structure is found in the case of the slow mode. The present study may be helpful to understand the nonlinear ion-acoustic solitary waves in space plasma and laboratory experiment plasmas also can be helpful for the study of Ionospheric auroral and magnetospheric plasmas where negative ions are present.

Waves in strongly nonlinear Gardner-like equations on a lattice

We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka–Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.

Envelope solitary wave and periodic wave solutions in a Madelung fluid description of generalized derivative resonant nonlinear Schrödinger equation

We have exploited the theory of Madelung fluid description in order to retrieve the correspondence between the envelope soliton like solutions of a generalized derivative resonant nonlinear Schrödinger equation (GDRNLSE) and the soliton like solutions of a stationary generalized Gardner equation. The motion under consideration was with stationary profile current velocity only and under suitable assumptions on current velocity in context of corresponding boundary conditions of fluid velocity and parametric constraints, the results were obtained. We mainly derived bright and dark (which involves grey and black) envelope soliton like solutions along with periodic wave envelope solutions having a phase depending on both space and time of the generalized derivative resonant nonlinear Schrödinger equation recovered from the corresponding solitary waves and periodic wave solutions of the stationary generalized Gardner equation.

An Improved Prediction Method of Soil-Water Characteristic Curve by Geometrical Derivation and Empirical Equation

Much attention has been paid on the soil-water characteristic curve (SWCC) during decades because it plays great roles in unsaturated soil mechanics. However, it is time-consuming and costly to obtain a series of entire saturation-suction data by experiments. The curves acquired by directly fitting empirical equations to limited experimental data are greatly different from the actual SWCC, and the relevant soil parameters obtained by inaccurate curve are also incorrect. Thus, an improved prediction method for more accurate entire SWCC was established. This novel method was based on the analysis of shape characteristics of SWCC with three critical points S , C 1 , and C 2 under the hypothesis of geometrical symmetric relation. The theoretical computation was specifically deduced under conventional Gardner, VG, and FX models, respectively, and then inferred on different soil types of 45 collected SWCC datasets. This geometrical symmetric relation exhibited well in all these three conventional empirical equations, especially in Gardner equation. Finally, a series of filer paper tests on sand, silt, and clay were also carried out to acquire entire SWCC curve for the verification and evaluation of the proposed geometrical method. Results show that this improved prediction method effectively decreases deviation resulting from directly fitting empirical equations to limited data of wide types of soils. The averaged improvement was larger under VG equation than under Gardner and FX equation. It proved that the accuracy of predicting greatly depends on the shape characteristic point of maximum curve curvature (point C 2 ), other than the number of points. This research provides a novel computation method to improve prediction accuracy even under relative less experimental data.

Global Solutions and Stability Properties of the 5th Order Gardner Equation

In this work, we deal with the initial value problem of the 5th-order Gardner equation in \({\mathbb {R}}\), presenting the local well-posedness result in \(H^2({\mathbb {R}})\). As a consequence of the local result, in addition to \(H^2\)-energy conservation law, we are able to prove the global well-posedness result in \(H^2({\mathbb {R}})\). Finally as a direct application, we prove that some globally defined functions, e.g. breather solutions of 5th order Gardner equation, are \(H^2({\mathbb {R}})\) stable.

Toverline{T} deformations of non-relativistic models

The light-cone gauge approach to Toverline{T} deformed models is used to derive the Toverline{T} deformed matrix nonlinear Schrödinger equation, the Landau-Lifshitz equation, and the Gardner equation. Properties of one-soliton solutions of the Toverline{T} deformed nonlinear Schrödinger and Korteweg-de Vries equations are discussed in detail. The NLS soliton exhibits the recently discussed phenomenon of widening/narrowing width of particles under the Toverline{T} deformation. However, whether the soliton’s size is increasing or decreasing depends not only on the sign of the deformation parameter but also on soliton and potential parameters. The Toverline{T} deformed KdV equation admits a one-parameter family of one-soliton solutions in addition to the usual velocity parameter. The extra parameter modifies the properties of the soliton, in particular, it appears in the dispersion relation.

Investigation of Exact Solutions of some Nonlinear Evolution Equations via an Analytical Approach

This study investigates exact analytical solutions of some nonlinear partial differential equations arising in mathematical physics. To this reason, the Kudryashov-Sinelshchikov equation, the ZK-BBM equation and the Gardner equation have been considered. With the implementation of the trial solution algorithm, solitary wave, bright, dark and periodic exact traveling wave solutions of the considered equations have been attained. The solutions have been checked and graphs have been given via package programs to see the behavior of the waves.

Non‐linear coherent structures in multi‐species dusty plasma

AbstractIn a fluid description, theoretical investigation has been carried out to explore how the presence of two‐temperature electron fluids (hot and cold) influences the characters of dust acoustic solitary waves, double layers, bipolar structures, etc. in a multi‐component dusty plasma, composed of isothermal electrons and ions, along with the negatively charged, cold dusts. A reductive perturbation method is used to derive the well‐known Korteweg‐de Vries (KdV) equation, which describes the basic nature of the dust acoustic solitary waves. Two values of critical ion density exist, which distinguish the regions of compressive and rarefactive solitary waves in parametric space. The critical points directly depend on the densities and temperatures of the electron fluids. With increasing temperature of hot electron fluid, one of the two critical ion densities decreases, and the other one increases. Ultimately, they merge with each other when the temperature of hot electrons reaches a particular value. The amplitude of the dust acoustic soliton gets modified due to the presence of two‐temperature electron fluids. The study is extended further by deriving modified KdV and Gardner equations at the vicinity of critical points. The Gardner equation reveals the existence of double layer and bipolar structures. In these processes, the effects higher‐order nonlinearities are incorporated methodically, by implementing appropriate stretching of the coordinates. Sagdeev's pseudopotential analysis shows that large amplitude rarefactive solitary waves only exist when the corresponding Mach number (M) lies within 1 &lt; M &lt; Mc. The upper bound, Mc increases with the hot electron temperature and cold electron density. The results presented in this investigation are believed to be applicable in the laboratory and astrophysical plasmas.

On the spectral stability of periodic traveling waves for the critical Korteweg-de Vries and Gardner equations

In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For the first equation, we show that both positive and zero mean periodic traveling wave solutions possess a threshold value which may provides us a rupture in the spectral stability. Concerning the second equation, we establish the existence of periodic waves using a Galilean transformation on the periodic cnoidal solution for the modified Korteweg-de Vries equation and for both equations, the threshold values are the same. The main advantage presented in our paper concerns in solving some auxiliary initial value problems to obtain the spectral stability.

Interaction of Gardner solitons in plasmas: applications in the Saturn’s magnetosphere

Multi-soliton solutions of the Gardner equation (GE) have been obtained using the Hirota’s bilinear formalism and, to the best of our knowledge, have been studied in the context of plasmas for the first time. The results have been used to study the electrostatic waves on the ion time scale in a two-electron temperature (TET) kappa distributed plasma in the light of the data obtained from Saturn’s Magnetosphere. The most important result that has come out of the investigation of multi-soliton solution of GE is that both overtaking and head on interactions may occur owing to the simultaneous presence of quadratic and cubic nonlinearities in the GE. The plasma parameters of the system have been found to influence the spatial scale of interaction. The results of kappa distributed plasma have been compared with the Maxwellian case and the key differences with regard to the parametric regimes for the existence of ion acoustic Gardner solitons and the soliton interaction for the two cases have also been explored in detail.

Theoretical and experimental investigation on strongly nonlinear internal solitary waves moving over slope-shelf topography

The variable-coefficient modified Gardner equation is proposed to describe large amplitude internal solitary waves (ISWs) propagating over slope-shelf topography. The model is adapted from weakly nonlinear Gardner equation by adjusting its original coefficients such that the model is capable of representing the crucial characteristics of large amplitude ISWs, including the effective wavelength and wave speed. To numerically solve the nonlinear equation, the fourth-order Runge-Kutta method is applied for temporal discretization, second-order central finite difference numerical scheme is implemented for the space discretization and the iterative scheme is applied to treat the coupling term of time and space. The numerical method is then verified by comparing fairly with laboratory experiments. It's found the variable-coefficient modified Gardner model is a reliable theoretical tool to describe the behavior of large amplitude internal solitary waves passing through bottom topography.

The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, α>0, q∈R. The far field of such solitons may have three options. For q>0 and α>1 the analysis re-confirmed that all traveling solitons have “light” exponentially decaying tails and propagate to the right. If q<0 and α<1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu is negative, the set of solutions may include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic solitons and compactons with any of the three types of tails. The solutions usually have a single-hump structure but if Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.

Ion acoustic wave group dynamics near critical non-thermal parameter in non-thermal plasmas

Weakly nonlinear ion acoustic wave group dynamics at critical non-thermal parameter is investigated in the frame work of Gardner equation in non-thermal plasmas. Also the wave modulation due to self-interactions near the critical parameter is investigated by a nonlinear Schrodinger equation. The non-thermal parameter modifies the wave dispersion and the parameter space of wave modulation. At the boundary of this parameter space, the wave group dynamics is shown to be governed by a modified nonlinear Schrodinger equation. These equations are solved numerically with the help of a pseudo-spectral scheme and the solutions predict a wide class of nonlinear structures related to wave and wave group dynamics of the weakly nonlinear wave. The numerical solutions and the growth rate of the modified nonlinear Schrodinger equation confirm that the ion acoustic wave is modulationally unstable near the marginal state. The results are discussed in the context of astrophysical observations in auroral region.

Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

• Nonlinear Modified Gardner Equation • Lie group analysis • Power series solution • Conservation laws • quantum electron-positron-ion magneto plasmas • Ion-acoustic waves In this paper, the nonlinear modified Gardner (mG) equation is under consideration which represents the super nonlinear proliferation of the ion-acoustic waves and quantum electron-positronion magneto plasmas. The considered model is investigated with the help of Lie group analysis. Lie point symmetries are computed under the invariance criteria of Lie groups and symmetry group for each generator is reported. Furthermore, the one-dimensional optimal system of subalgebras is developed by adjoint technique and then we compute the similarity reductions corresponding to each vector field present in the optimal system, with the help of similarity reduction method we have to convert the PDE into the ODE. Some exact explicit solutions of obtained ordinary differential equations were constructed by the power series technique. With the aid of the Galilean transformation, the model is transformed into a planer dynamical system and the bifurcation behaviour is recorded. All practicable types of phase portraits with regard to the parameters of the problem considered are plotted. Meantime, sensitivity is observed by utilizing sensitivity analysis. In addition, the influence of physical parameters is studied by the application of an extrinsic periodic power. With additional perturbed term, quasi-periodic and quasi-periodic-chaotic behaviours is reported.

Verification of Gardner's equation and derivation of an empirical equation for anhydrite rocks in Sirte basin, Libya: case study

Bulk density is a physical property of rocks measured in the laboratory on rock samples or obtained from oil field logging tools. When bulk density is not measured, a synthetic bulk density log can be calculated, for which Gardner's equation is the most widely used. However, Gardner's equation might not be appropriate for regions in which the density–velocity relationship does not conform to Gardner's curves. Here, we verified the applicability of Gardner's equation to calculation of synthetic bulk density of anhydrite rocks in the Sirte Basin (Libya) and compared the results to those obtained from an equation derived from the available measured bulk density and sonic logs. We used fifteen wells to calibrate Gardner's equation and three wells to derive an equation for the anhydrite rocks. The anhydrite rocks were 10–510 feet thick. The bulk density calculated by Gardner's equation differed only slightly from the measured log values, with the exception of the eastern part of the Sirte Basin. The average of the differences in bulk density between the measured values and Gardner's equation results were 0.022–0.040 g/cm3, and between the measured values and the derived equation results 0.002–0.045 g/cm3, both with a standard error of about 0.01 of the bulk density estimated results. We conclude that while Gardner's equation is more appropriate for estimating the bulk density of anhydrite rocks in the eastern part of the basin, the derived equation could be more appropriate for the western region.