Gardner Equation Research Articles

Traveling wave solutions for a Gardner equation with distributed delay under KS perturbation

Traveling wave solutions for a Gardner equation with distributed delay under KS perturbation

Using Symmetries to Investigate the Complete Integrability, Solitary Wave Solutions and Solitons of the Gardner Equation

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota’s method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper’s aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems.

Compactons in a class of doubly sublinear Gardner equations

We introduce and study a class of doubly sublinear Gardner equations ut+F(u;n)x+u3x=0 where F(u;n)=u1+n−κ1+2nu1+2n, which for 0<n induce solitons and in the doubly sublinear cases wherein −1/2<n<0, bi-directional compactons propagating in either direction. Their emergence, evolution, chase and head-on interactions are studied.

P-Wave Velocity Log Simulation for Petroelastic Inversion Using Gardner’s Equation, Neural Networks And Ensemble Methods Based on Trees: A Case Study of the Santos Basin, Brazil

One of the main tools for reservoir characterization is analyzing well-log data. The importance of such methods stems from petrophysical properties estimation, such as porosity, which is very important to the oil and gas industry. In scenarios where data is hard to collect, data loss and technical failures during the acquisition impose an extra challenge. Thus mathematical and petrophysical models are good candidates to fill information gaps in the well-log dataset. In such a way, the rock’s petroelastic and petrophysical properties can be successfully estimated. Several studies correlate the velocity of compressional waves (VP) to other basic well data. In this study, we used the Gardner equation and Machine Learning methods such as Neural Networks, Random Forest and Gradient Boosting regressions to generate VPlogs. We used real-world data acquired from twenty wells of the pre-salt formation from Santos Basin in Brazil to train and test the Machine Learning methods and evaluated the data estimated by those models using statistical metrics. We calculated the acoustic impedance from the estimated logs and used it to create a prior model for a petroelastic inversion, which allowed us to estimate the natural logarithm of the acoustic impedance for a seismic volume. The Machine Learning methods presented lesser errors between estimated and measured velocities when compared to Gardner’s equation.

Study of multi solitons, breather soliton structures in the earth's magnetotail region

The investigation of wave packets with varying magnitudes, which predominantly carry energy across different physical mediums, proves to be highly intriguing and unveils numerous nonlinear characteristics within Earth's magnetotail region and magnetosphere, as confirmed by extensive space observations. Nonlinear features in wave dynamics are elucidated through diverse nonlinear structures, among which breather structures hold prominence and manifest across various wave fields. We have considered an unmagnetized collisionless homogeneous hydrodynamical governing equations set in an electron-ion plasma comprising hot and cold ions. In this study, we have explored 1-soliton, 2-soliton, and breather structures in the framework of the Gardner equation (GE). We have derived the GE from the normalized set of governing equations employing the reductive perturbation technique (RPT). The GE, an extension of the Korteweg-de Vries (KdV) equation, encompasses the combined effects of quadratic and cubic nonlinearities. By employing the Hirota bilinear method (HBM), we have obtained multi-soliton solutions and breather soliton solutions. We have noticed the variations in electron temperature ratios and density concentration ratios substantially impact and reshape breather soliton structures. Such alterations in breather structures, influenced by different electron temperatures and concentration ratios, hold potential significance in understanding energy transport within Earth's magnetotail region. The alteration of shapes and soliton structures may also be investigated in the dynamics of wave fields upon interaction such as hydrodynamics, optics, optic fibers, signal processing, etc.

Comparative analysis of bulk and velocity-derived density data in deep wells in the Adriatic Region (Italy)

This work is based on a systematic comparison between two datasets related to density data of geological formations crossed in 13 deep wells. The wells are located in the Adriatic area and reach depths of over 5000 meters. The main lithologies involved include sandstones, marls, clays, evaporite, and carbonate rocks. The first dataset concerns density values obtained from the analysis of sonic logs recorded along the wells, by applying the Gardner relation. The second dataset, on the other hand, refers to actually acquired density log measurements. Differences among the values in the two datasets have been calculated. The aim behind this work is to assess the reliability of rock densities estimated using the Gardner equation by comparing them to measurements obtained through density logs, despite many factors influencing the log density. This comparison of the densities obtained from various lithologies and geological formations leads us to draw, in this region, some initial considerations regarding the applicability and accuracy of Gardner formula, usually considered as standard reference, since the density log is generally available only within the reservoir. In the area we analyzed it is observed that the density values estimated from sonic velocities are underestimated for the Plio-Quaternary formations characterized by clayey-sandy lithologies by at least 0.1 g/cm3; whereas, densities of the carbonate sequences are overestimated by the same extent. Noteworthy, the density estimates deviate from the real values especially for gypsum units, overestimating by a factor of approximately 0.3 g/cm3. The results we obtained emphasize differences in density values when using the Gardner formula and suggest the need for taking into account the possible errors in the specific geological context or instead lithologies, such as those explored in this study.

N-order solutions to the Gardner equation in terms of Wronskians

$N$-order solutions to the Gardner equation (G) are given in terms of Wronskians of order $N$ depending on $2N$ real parameters. We get solutions expressed with trigonometric or hyperbolic functions. When one of the parameters goes to $0$, we succeed to get for each positive integer $N$, rational solutions as a quotient of polynomials in $x$ and $t$ depending on $2N$ real parameters. We construct explicit expressions of these rational solutions for the first orders.

Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma

Integrability, bilinearization, Bäcklund transformations and solutions for a generalized variable-coefficient Gardner equation with an external-force term in a fluid or plasma

Influences of damping, perturbation and variable coefficient on an extended nonlinear Gardner model

Describing the nonlinear wave propagation in fluids has posed a challenging task. The nonlinear Gardner equation stands as a crucial model in fluid dynamics. An extended Gardner model incorporates variable coefficients, damping, and perturbation terms, offering a more comprehensive approach to understanding complex wave dynamics in fluids. In our study, we employ the symbolic computation tool, specifically the Darboux transformation, to derive new analytic solutions up to the Nth order for this extended Gardner equation. By assigning the spectrum parameters in these solutions into real values, we are able to obtain line-like solitons. Moreover, when N⩾2 and pairs of complex conjugate values are assigned to two of the spectrum parameters, breathers emerge. This enables a comprehensive analysis of the dynamics involving solitons, breathers, and their hybrid forms, shedding light on their interactions. Furthermore, our work introduces several noteworthy insights by investigating the influences of three key factors: the damping, perturbation, and variable coefficient. Through a few of specific examples, we can confirm the following facts: (i) The damping factor plays a crucial role in rendering solitons and breathers unstable, causing them to undergo exponential changes in both amplitudes and directions during propagation. (ii) The perturbation function creates non-zero background waves along the time axis, which interact with solitons and breathers. (iii) The variable coefficient can introduce disturbances to solitons and breathers. Our findings disclose that this equation exhibits a wealth of intricate propagation patterns. Various factors can exert varying degrees of influence on solitons and their dynamics. This observation just underscores the complexity inherent in the real world and should be useful in applications.

A STUDY ON THE INVESTIGATION OF THE TRAVELING WAVE SOLUTIONS OF THE MATHEMATICAL MODELS IN PHYSICS VIA (m + (1/G))-EXPANSION METHOD

The Gardner equation models the propagation of dust ion acoustic waves. As a result, it has received extensive attention in the literature. For the first time, the (m + (1/G))-expansion method is used to establish novel exact wave solutions for the Gardner equation. Solutions can exhibit various types of behavior, which can be visualized using 3D and contour graphs. The results obtained demonstrate that the presented method is powerful, useful, practical, and suitable for examining the model.

Construction of conservation laws for the Gardner equation, Landau–Ginzburg–Higgs equation, and Hirota–Satsuma equation

In this paper, two different methods for calculating the conservation laws are used, these are the direct construction of conservation laws and the conservation theorem proposed by Ibragimov. Using these two methods, we obtain the conservation laws of the Gardner equation, Landau–Ginzburg–Higgs equation and Hirota–Satsuma equation, respectively.

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions.

Study of nonlinear waves structures through damped modified Gardner equations in quantum plasmas

Study of nonlinear waves structures through damped modified Gardner equations in quantum plasmas

Existence of Traveling Wave Solutions for the Perturbed Modefied Gardner Equation

Existence of Traveling Wave Solutions for the Perturbed Modefied Gardner Equation

Construction of shock, periodic and solitary wave solutions for fractional-time Gardner equation by Jacobi elliptic function method

Construction of shock, periodic and solitary wave solutions for fractional-time Gardner equation by Jacobi elliptic function method

Data-driven high-order rogue waves and parameters discovery for Gardner equation using deep learning approach

In this paper, an adaptive mix training physics-informed neural networks (A-MTPINNs) model is proposed to study the high-order rogue waves of the famous Gardner equation (GE). Through a series of numerical experiments, we can conclude that this A-MTPINNs model can not only recover the dynamic behavior of the high-order rogue waves of the GE well, but also can guarantee higher learning ability and prediction accuracy than the original PINNs model, and the prediction accuracy is improved by two orders of magnitude. By testing that the A-MTPINNs model can maintain high robustness under different noises for the data-driven solutions of the GE. Furthermore, this model also has good performance for the inverse problem of the Gardner equation by a data-driven discovery approach and also remains robust in noise experiments.

Bi-directional solitons of dual-mode Gardner equation derived from ideal fluid model

In this article, the main goal is to find the exact solution to the dual-mode Gardner equation derived from the ideal fluid model. To achieve this, two distinct methods will be employed: the tan/cot method and the tanh/coth method. Using these methods, the problem can be seen through different angles, providing a full understanding of how the equation works and the various solutions it provides. Furthermore, the periodic, kink and singular solutions will be illustrated through the use of 3D and 2D graphs. This visualization will clearly show how the phase velocity parameter affects the solutions, improving our understanding of wave dynamics within the dual-mode Gardner equation. The findings from this study help in understanding how both solitons and periodic solutions influence wave behavior in shallow water and how light travels through optical fibers.

(3+1)-Dimensional Gardner Equation Deformed from (1+1)-Dimensional Gardner Equation and its Conservation Laws

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given.