Korteweg-de Vries Research Articles

Nonlinear Jeans instability analysis of gravitating astrofluids in Eddington-inspired Born-Infeld gravity framework

The nonlinear Jeans instability dynamics in viscoelastic astrofluids within the Eddington-inspired Born-Infeld (EiBI) gravitational framework excitable in star-forming interstellar cloud fluids is semi-analytically investigated. Application of multiple scaling techniques here results in a unique form of the Korteweg-de Vries (KdV) equation moderated with a peculiar set of distinctive fluid polyparametric coefficients. A numerical illustrative platform portrays the excitation of diversified compressive solitary-chain wave patterns. The evolutionary existence of such structural patterns is further confirmed geometrically with the help of illustrative phase-plane analyses. The various parametric stabilizers and destabilizers, alongside accelerating and decelerating factors on the clouds, are illustratively analysed. It is seen that the composite cloud stability is significantly influenced by the strength polarity of the gravitational EiBI-parameter (χ). The dependency of the obtained solitary wave patterns on the EiBI-polarity is further confirmed in a numerical simulation platform illustratively. Our analysis also highlights that the influence of material density on the system stability is dependent on the EiBI-polarity. The nonlinear spatiotemporal evolution of the instability reveals a unique type of soliton fission phenomenon, indicative of inherently produced additional disturbances within the astrofluid. Our obtained results could be widely helpful in application to the EiBI-centric diverse astrocosmic phenomena, such as bounded structure formation and evolution in diverse complex astroenvirons.

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

Fully Discrete Finite Difference Schemes for the Fractional Korteweg-de Vries Equation

Classifications of bosonic supersymmetric third and fifth order systems

This manuscript explores the extensions and classifications of the bosonic supersymmetric systems. For the third order bosonic superfield equations, four types of integrable supersymmetric extensions are identified, including the B-type (trivial) supersymmetric modified Korteweg–de Vries equation, the supersymmetric Sharma–Tasso–Olver equation, and an A-type (non-trivial) supersymmetric potential Korteweg–de Vries equation. In the case of the fifth order bosonic supersymmetric systems, nine kinds of extensions are discovered, with six being B-type and three being A-type. Notably, several equations such as the supersymmetric Sawada–Kotera equation, the supersymmetric Kaup–Kupershmidt equation and the supersymmetric Fordy–Gibbons equation are classified as B-type extensions. Despite this classification, these supersymmetric systems are shown to be connected to linear integrable couplings. The findings have implications for various fields including string theory and dark matter and highlight the importance of understanding bosonic supersymmetric systems. The obtained supersymmetric systems are solved via bosonization method. Applying the bosonization procedure to every one of supersymmetric systems, one can find various dark equation systems. These dark equation systems can be solved by means of the solutions of the classical equations and some graded linear couplings including homogeneous and nonhomogeneous symmetry equations.

Multiple solitons and breathers on periodic backgrounds in the complex modified Korteweg–de Vries equation

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg–de Vries (KdV) of the mean-field analysis. Published by the American Physical Society 2024

Kinetic Alfvén solitary waves in astrophysical plasmas

The magnetospheric plasma (hot and thin) and the solar wind plasma (cold and dense) are separated by the Earth’s magnetopause, in which plasmas of both origins coexist. Different types of plasma diffusions are found due to this plasma mixing, and kinetic Alfvén solitary waves (KASWs) are one of them. In this work, a theoretical approach is taken to study the fundamental properties of heavy ion acoustic KASWs (HIAKASWs) in a magnetized plasma system whose constituents are nonextensive q-distributed two temperature electrons with dynamical heavy ions. The perturbations of the magnetized collisionless plasma system are investigated using the reductive perturbation technique to deduce the Korteweg–de Vries (K–DV) and modified K–DV (MK–DV) equations to determine the fundamental characteristics of small, but finite amplitude HIAKASWs. The presence of nonextensive electrons, magnetic field, obliquity angle (the angle between the external magnetic field and wave propagation), plasma particle number densities, and the temperature of various plasma species are observed to significantly alter the fundamental properties of HIAKASWs. The findings of our present study may be useful for comprehending the nonlinear wave properties in diverse interstellar plasma environments.

Soliton interaction in a two-temperature electron plasma with trapping and superthermality effects

Head-on collision of the two small-amplitude electron-acoustic (EA) solitons is studied in an unmagnetized collisionless plasma in the presence of superthermal (hot) trapped electrons. For this purpose, using a well-known extended Poincare–Lighthill–Kuo (PLK) method, a pair of the trapped Korteweg–de Vries (tKdV) equations is derived to investigate the soliton trajectories and phase shifts. The latter are found dependent on amplitudes of the interacting solitons, effectively altering with hot-electron superthermality and plasma parameters. Typical parameters for the electron diffusion region (EDR) and day-side auroral zone have been selected to examine the impact of hot-electron superthermality, trapping parameter, hot-to-cold electron number density ratio, and cold-to-hot electron temperature ratio on the profiles of potential excitations and phase shifts of interacting solitons. It is found that phase speed of the EA waves becomes altered by varying the κ–parameter, strongly modifying the nonlinearity and dispersive coefficients in a superthermal trapped plasma. However, particle trapping phenomenon does not affect the linear phase speed but introduces a fractional nonlinearity in the tKdV equations of two interacting solitons. The impact of the adiabatic and isothermal pressures is also highlighted to show new modifications in the propagation characteristics of two interacting solitons.

Fuzzy uncertainty modeling of generalized Hirota–Satsuma coupled Korteweg–de Vries equation

This article explores the solitary wave solutions of a generalized Hirota–Satsuma Coupled Korteweg–de Vries (HSCKdV) equation. The HSCKdV equation is a mathematical model that describes certain types of long waves, particularly those found in shallow water. The generalized HSCKdV equation is solved exactly using the Homotopy Perturbation Transform Method (HPTM). By applying this technique, the authors obtain solutions in the form of a convergent power series. These solutions offer an understanding of the characteristics of solitary waves within the domain of shallow water waves. The HSCKdV equation has been solved using the adomian decomposition method, and the results have been compared with those obtained from the HPTM. This comparison demonstrates the effectiveness of the HPTM in solving such nonlinear equations. Further, the HSCKdV equation is extended to a fuzzy version considering the initial condition as a fuzzy parameter. Uncertainty in the initial condition is addressed by representing it using triangular and trapezoidal fuzzy numbers. The generalized fuzzy HSCKdV equation is subsequently tackled using the fuzzy HPTM (FHPTM) providing fuzzy bound solutions. Using the FHPTM, we explain the fuzzy results, highlighting how the solitary wave splits into two solitary waves and noting that the lower and upper bound solutions are interchanged due to negative fuzzy results.

A new study on fractional Schamel Korteweg–De Vries equation and modified Liouville equation

Traveling wave solutions of fractional partial differential equations have great importance in the literature. The diversity of solutions plays an important role in understanding the physical structure of the model it represents. For this reason, two important differential equations with the fractional order, which have a significant role in applied sciences and can model real-life problems most accurately, have been solved by the generalized (G′G)-expansion method in this study. This method is a generalization of the classical (G′G)-expansion method. With this developed method, the non-linear fractional Schmael Korteweg–De Vries equation and fractional modified Liouville differential equations are discussed to find their exact solutions. In this way, new exact solutions of these equations that were not previously included in the literature have been found. The presented method has been applied to these two equations for the first time, and various new traveling wave solutions have been obtained. Thus, the study goes beyond other studies. To understand the physical behavior of these new exact solutions, three-dimensional graphs have been drawn according to different parameter values.

New type of solutions for the modified Korteweg–de Vries equation

In this letter we report a new type of multi-soliton solutions for the modified Korteweg–de Vries (mKdV) equation. These solutions contain τ functions of the trigonometric solitons and classical solitons simultaneously. A new bilinear form of the mKdV equation is introduced to derive these solutions. The obtained solutions display as solitons living on a periodic background, which are analyzed and illustrated.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Painlevé integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

Assessment of unidirectional internal solitary wave models in a two-layer fluid system through an improved experimental method

The suitability of unidirectional internal solitary wave (ISW) models in a two-layer fluid system is evaluated using an improved experimental method. A new method for generating internal solitary waves in the laboratory tank is proposed, utilizing a piston-type wave maker in a density stratified two-layer fluid. The prescribed velocities of the pistons are determined by the mean velocities derived from the adjusted High-order Unidirectional (aHOU) model for the upper and lower layers. Experimental conditions encompass a range of wave Reynolds numbers between 1.8×105 and 2.3×105. The repeatability and amplitude control of the proposed wave-making method are verified. The unidirectional models considered in this study encompass the Korteweg–de Vries (KdV), Gardner and aHOU model. Laboratory experiments and theoretical computations have been systematically compared in this study to examine the wave characteristics inherent in typical unidirectional internal solitary wave models. It is found that Reynolds number (Re) and nonlinear parameter (α=|a|/h) serve as suitable criteria for delineating the range of applicability of unidirectional models. According to these criteria, the KdV theory is suitable for ISWs with amplitudes α<0.02 across all Reynolds number cases, while the Gardner model is valid for cases where α<0.02 and Reω>2.2×105, with α<LimG, where LimG represents the non-dimensional limiting amplitude. The aHOU model exhibits the capability to capture fully nonlinear ISWs across all Reynolds numbers considered in this study.

The effect of higher-order dissipation on solutions of the generalized korteweg-de vries equation

The effect of higher-order dissipation on solutions of the generalized korteweg-de vries equation

Cauchy matrix approach for generalized semi-discrete lattice potential Korteweg–de Vries equations

Starting from a new scattering relation, we generate a generalized semi-discrete lattice potential Korteweg–de Vries(gsd-lpKdV) equation from a determining equation set(DES) and its Lax presentation. Further, the gsd-lpKdV equation with self-consistent sources(gsd-lpKdVSCS) is worked out by introducing two discrete variables to the plane wave factors. Specially, explicit formulae for kink solitons, Jordan-block solutions and diagonal-Jordan-block solutions of gsd-lpKdVSCS are obtained based on the Cauchy matrix approach. Moreover, We illustrate the dynamics of one-kink soliton, two-kink soliton, high order Jordan-block solutions and the diagonal-Jordan-block solution.

Boundary stabilisation of stochastic delay impulsive Korteweg–de Vries–Burgers equations

Boundary stabilisation is addressed for stochastic delay impulsive Korteweg–de Vries–Burgers (SDIKdVB) equations. Initially, a novel boundary controller is devised. Utilizing the Lyapunov–Krasovskii functional approach and the impulsive theory, a sufficient condition is derived for achieving mean-square exponential stabilisation (MSES) of SDIKdVB equations. Secondly, the scenario is investigated in which uncertainties exist in the system parameters, and a sufficient condition is presented to attain the robust MSES. Additionally, the investigation is conducted on the impact of the diffusion term, the dispersion term, impulsive strengths and impulsive intervals on MSES. Lastly, the effectiveness of the theoretic findings is demonstrated via practical applications in both biomechanics and physics.

Landau quantization effects on damping Kawahara solitons in electron–positron–ion plasma in rotating ionized medium

For the dynamics of three-dimensional electron–positron–ion plasmas, a fluid quantum hydrodynamic model is proposed by considering Landau quantization effects in dense plasma. Ion–neutral collisions in the presence of the Coriolis force are also considered. The application of the reductive perturbation technique produces a wave evolution equation represented by a damped Korteweg–de Vries equation. This equation, however, is insufficient for describing waves in our system at very low dispersion coefficients. As a result, we considered the highest-order perturbation, which resulted in the damped Kawahara equation. The effects of the magnetic field, Landau quantization, the ratio of positron density to electron density, the ratio of positron density to ion density, and the direction cosine on linear dispersion laws as well as soliton and conoidal solutions of the damped Kawahara equation are explored. The understanding from this research can contribute to the broader field of astrophysics and aid in the interpretation of observational data from white dwarfs.

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Abstract In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

Bi-Hamiltonian structure of a super KdV equation of Kupershmidt

In this paper, we study a super Korteweg–de Vries (sKdV) equation proposed by Kupershmidt which possesses a Lax operator with three fully nonlocal terms. The Lax operator is reformulated so that it is of the super constrained modified Kadomtsev–Petviashvili (scmKP) type. By calculating the bi-Hamiltonian structure of the scmKP hierarchy and employing Dirac reduction, we obtain the bi-Hamiltonian structure of the sKdV equation. We also present a spectral problem of its modified system.