Extended Korteweg-de Vries Research Articles

The exact analytical solutions of the (2+1)-dimensional extended Korteweg–de Vries equation using bilinear neural network method and bilinear residual network method

The [Formula: see text]-dimensional extended Korteweg–de Vries (KdV) equation is predominantly utilized to elucidate the propagation of waves that exhibit both dispersive and nonlinear characteristics within the domain of nonlinear physics. This paper employs the bilinear neural network method (BNNM) to derive the exact analytical solutions of the equation. By constructing various bilinear neural network models, we obtain lump solution, breather solution and periodic interaction solution of the equation. The bilinear residual network method (BRNM) is an extension of BNNM. We apply BRNM under specific constraints to obtain breather solution of the equation, thereby offering a broader conceptual framework. Subsequently, various 3D plots, contour plots, density plots and x-curves are used to illustrate the physical properties and dynamic behaviors of these waves.

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.

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches

In this paper, the nonlinear space-time fractional extended Korteweg-de Vries (F-eKdV) equation involving the conformable fractional derivative (CFD) is considered. Bernoulli and Riccati equations as a simplest equation method together with applying modified traveling wave transformation are used to obtain the exact solution of F-eKdV equation. Also, by exp⁡(−Φ(ζ))-expansion method, different kind of exact traveling wave solutions are gained, including periodic-singular, dark-singular and plane-wave solutions. As a result, several soliton solutions of the proposed equation are revealed. These novel soliton waves are constructed using the software package Maple with some given parametric values and displayed graphically in 2-dimensional (2D) and 3-dimensional (3D) plots by using MATLAB. To illustrate how the fractional operator affects the results, some of the acquired solutions are presented for various values of the fractional order α and also compared to their exact solutions in classical case (i.e. α=1). In the given graphical representations, the physical meaning of the reported solutions is illustrated. From the obtained results, it is observed that the methods are effective and dependable to solve nonlinear fractional partial differential equations (NFPDEs).

Numerical investigation on the interaction between internal solitary wave and self-propelled submersible

Internal solitary waves (ISWs) often seriously threaten the survivability of the underwater submersible. Most of the existing investigations focus on the interaction between ISWs and fixed or suspended submersibles. However, the investigation on the interaction between ISWs and self-propelled submersibles is still scarce, which is a more realistic case in the marine engineering. In this paper, a three-dimensional numerical model for the interaction between ISW and self-propelled submersible is developed. Based on the extended Korteweg–de Vries (eKdV) theory, the ISW is generated in a two-layer fluid numerical wave tank by solving the Navier–Stokes (N–S) equation. By introducing the hydrodynamic loadings provided by the ISW environment into the standard operation equation of the submersible, the six degrees-of-freedom motions of the submersible can be obtained. The dynamic overset mesh technology is adopted to simulate the motions of the self-propelled submersible in the ISW fluid field. The present numerical model is validated by comparing with the experiment on a submerged cylinder in the ISW environment. Using this numerical model, we compare the interaction effects of the fixed, suspended, or the self-propelled submersible in ISW and discuss the influences of propulsive forces of the self-propelled submersible on ISW–structure interaction effects. The numerical results show that the loadings and movements of the submersible change remarkably in the surge, heave, and pitch direction. Especially, the submersible with high propulsive forces can pierce the wave surface and reach a large pitch angle with the amplitude of 36°, which further results in a 35% motion speed decrease in the initial propulsion direction.

Optical solutions to a conformable fractional extended KdV model equation

In this paper, we intend to obtain the extended Korteweg-de Vries(K-dV) equation in the followingDταφ1+λ32[32λ4−(C2(1−f)+D2fσ2)]Dξ2αφ1+λ32(52λ6−C3(1−f)−fσ3D3)(Dξαφ1)3+λ32Dξ3αφ1=0by using the reductive perturbation approach and we explore solutions of this equation using the generalized exponential rational function method. The answers obtained in this study are new and have not been reported so far.

Extended Korteweg-de Vries equation for long gravity waves in incompressible fluid without strong limitation to surface deviation

We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate solutions. The derivation of extended KdV equation is based on the Euler equations for inviscid irrotational and incompressible fluid. It is shown that the extended KdV equation reduces to standard KdV equation for small amplitude of the waves. We have also generalized the extended KdV equation for describing the decaying effect of the waves. Quasi-periodic and solitary wave solutions for extended KdV equation with decaying effect are found as well. We also demonstrate that the fundamental approach based on the inverse scattering method is applicable for solving the extended KdV equation in the case when decaying effect is negligibly small. Such case always occur for restricted propagation distances of the waves.

Soliton, periodic and superposition solutions to nonlocal (2+1)-dimensional, extended KdV equation derived from the ideal fluid model

We present new (2+1)-dimensional extended KdV (KdV2) equation derived within an ideal fluid model. Next, we show several families of analytic solutions to this equation. The solutions are expressed by functions of argument ξ=(kx+ly-ωt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi = (k x +l y-\\omega t)$$\\end{document}. We found the soliton solutions in the form Asech2(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {sech}^{2}(\\xi )$$\\end{document}, periodic solutions in the form Acn2(ξ,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {cn}^{2}(\\xi ,m)$$\\end{document} and superposition solutions in the form A2[dn2(ξ,m)±mcn(ξ,m)dn(ξ,m)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{A}{2}[\ ext {dn}^{2}(\\xi ,m)\\pm \\sqrt{m}\\,\ ext {cn}(\\xi ,m)\ ext {dn} (\\xi ,m)]$$\\end{document} analogous to the solutions of (1+1)-dimensional, extended KdV equation and to the solutions to ordinary Korteweg-de Vries equation. On the other hand, the existence of these families of analytical solutions for the highly nonlinear non-local (2+1)-dimensional, extended KdV equation is astounding. The existence of essentially one-dimensional solutions to the (2+1)-dimensional extended KdV equation explains the enormous success of the one-dimensional nonlinear wave equations for the shallow water problem.

Application of different internal solitary wave theories for SAR remote sensing inversion in the northern South China Sea

This study discusses the theoretical method of the inversion of internal solitary waves (ISWs) characteristic parameters using synthetic aperture radar (SAR) images and ocean reanalysis data. Based on the horizontal distribution characteristics of the pixel intensity of the internal wave fringe in the image, an imaging theoretical model based on the extended Korteweg-de Vries (eKdV) and Benjamin-Ono (BO) theories in SAR was constructed, and the peak-to-peak method and curve-fitting method for calculating the characteristic wavelength of internal solitary waves were determined. The results show that the eKdV theoretical model has the advantage of inverting solitary waves with large amplitudes under the same marine stratified structure, and it provides a more reasonable characteristic parameter estimation than the traditional Korteweg-de Vries (KdV) theory. Although the BO theory is theoretically applicable to deep-water internal waves whose wavelength is far less than the water depth, it can still provide good inversion results in practical applications. Compared to the peak-to-peak method, the curve fitting method fully considers the pixel intensity variation characteristics of the internal wave fringe profile of the image, and the calculated characteristic parameters are more reasonable. Using the maximum buoyancy frequency and internal wave vertical mode to construct the marine stratification model of the deep waters of the northeastern South China Sea may have great uncertainty, and it is better to calculate the interface depth using the relative error of the coefficients of two-layer and continuous internal wave equations. In addition, this study also proposes an internal wave inversion method using continuous SAR images, which provides more reference and support for the selection of internal wave inversion models and reliability test of inversion results in a single SAR image.

Nonlinear Jeans-Buneman instability in gravitating complex plasmas

The nonlinear evolutionary dynamics of the hybrid Jeans-Buneman instability (JBI) in charge-fluctuating viscous dust molecular clouds (DMCs) of infinite extension is semi-analytically studied. The cloud macroscopic state is initially considered to be in a quasi-hydrostatic homogeneous equilibrium configuration, modeled in a classical non-relativistic framework, on the Jeans spatiotemporal scales. The global quasi-neutrality, dust-charge fluctuation, viscous dissipation, and inter-species collision effects are presumed to coexist. Such complex plasma situations are practically realizable in the star-forming DMCs in the diffused interstellar media extensively. A local nonlinear perturbative analysis herein yields a unique conjugational pair of coupled extended Korteweg-de Vries (e-KdV) equations on the gravito-electrostatic potential fluctuations. A numerical platform illustrates that the electrostatic and gravitational fluctuations grow as a unique admixture of compressive and rarefactive solitary wave patterns. It is seen that, even for cm-sized heavier dust grains, the gravitational wave amplitudes supersede the electrostatic counterparts. It is demonstrated geometrically that the electrostatic (gravitational) phase portraits are cyclically regular (irregular), unbounded (quasi-bounded), open (quasi-closed), and astable (quasi-stable) with no limit cycle behavior evolving as non(quasi)-homoclinic trajectories. The fluctuation amplitude-growth features with plasma multi-parametric variation is illustratively explored alongside its non-trivial futuristic applicability in bounded astrostructure formation.

Nonlinear streaming instability in viscoelastic quantum dusty plasmas

A theoretic model formalism is developed to investigate the nonlinear gravito-electrostatically coupled streaming instability modes excitable in viscoelastic quantum dusty plasmas. It consists of weakly correlated light electrons, ions; and strongly correlated heavy dust grains. We treat the inertialess electrons as a degenerate quantum fluid, inertial ions as a classical inviscid fluid, and massive dust particles as a classical viscoelastic fluid. It considers a dimensionality-dependent parameter, γ = [(D − 2)/3D], termed as the Bohmian quantum correction prefactor (D: System dimensionality). A weakly nonlinear perturbative analysis against the defined hydrodynamic homogeneous equilibrium procedurally results in a unique conjugational construct of gravito-electrostatically coupled extended pair Korteweg-de Vries (ep-KdV) equations with atypical multiparametric coefficients. A numerical analysis upshots in two distinct classes of nonlinear wave structures: (a) Electrostatic regular periodic pulse-like patterns, and (b) Self-gravitational extended bell-shaped solitons. The amplitude variations of the streaming-induced fluctuations are analyzed especially with the constitutive electron-ion equilibrium streams and with the Galilean reference frame velocity in an elaborate comparative platform alongside geometric trajectory analysis illustratively. The main implications and applications of this semi-analytic study are finally outlined from an extensive perspective.

Experimental and theoretical study of internal solitary wave loads on a submerged slender body

Based on the extended Korteweg de Vries (eKdV) theory and Morison's equation, a theoretical model of internal solitary wave (ISW) loads on a submerged slender body is investigated, and experimental measurements are further carried out in a large stratified fluid flume. The results show that the theoretical results are in good agreement with the experimental results. The horizontal force of the ISW on the slender body is consists of the horizontal dynamic pressure force and the friction force, in which the horizontal dynamic pressure force plays a dominant role. The vertical force includes the vertical wave force and the reduced gravity during the interaction between a slender body and an ISW. The magnitude of the force increases with increasing ISW amplitude, while its direction remains unchanged, and the force is closely related to the submerged depth of the slender body in the density pycnocline, which affects both the magnitude and the direction of the force.

Assessment of KdV and EKdV theories for simulating internal solitary waves in the continental slope of the South China Sea

Korteweg-de Vries (KdV) and Extended KdV (EKdV) are two important nonlinear theoretical models that are most commonly used for the description of internal solitary waves (ISWs). However, most of the previous studies based on the applications of these two models focus only on laboratory-based simulations. Moreover, the modeling data in those studies were rarely compared with the in-situ measurements. In this study, three moorings were deployed to the west of Dongsha Islands of the northern South China Sea (SCS), based on the KdV and EKdV theoretical approaches, to simulate and statistically analyze 854 mode-1 ISWs and 37 mode-2 ISWs captured during the period of observation. The results from this study indicated that both theories can be used to well describe the mode-1 ISWs with η0/H<0.12 (η0 and H are the amplitude of ISW and depth of seawater, respectively). Generally, when the depth of pycnocline was not close to H/2, the EKdV theory was more accurate than the KdV theory in simulating the waveforms and vertical velocities of mode-1 ISWs. However, for mode-1 ISWs with larger amplitudes, the horizontal velocities of the upper layer calculated by the EKdV theory were extremely large. The EKdV theory was no longer applicable when the depth of pycnocline was close to H/2 and η0 exceeded its limiting value of the EKdV theory. For mode-2 ISWs, the waveforms simulated by the EKdV theory were in close agreement with the in-situ measurements. However, neither of the two theories can accurately simulate the velocities. Further, the statistical results of propagation speeds of 49 mode-1 ISWs highlighted that the EKdV theory performed slightly better than the KdV theory in calculating the propagation speed of ISWs. Finally, the MCC (Miyata-Choi-Camassa) theory was introduced as a supplement to simulate larger-amplitude ISWs. The MCC theory was found to show better performance than the above two theories in simulating the ISWs with η0/H>0.15 but the trend was opposite when η0/H<0.15.

Whitham shocks and resonant dispersive shock waves governed by the higher order Korteweg–de Vries equation

The addition of higher order asymptotic corrections to the Korteweg–de Vries equation results in the extended Korteweg–de Vries (eKdV) equation. These higher order terms destabilize the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the eKdV equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

Solitary Wave Interactions with an External Periodic Force: The Extended Korteweg-de Vries Framework

In this work we asymptotically and numerically studied the interaction of large amplitude solitary waves with an external periodic force using the forced extended Korteweg-de Vries equation (feKdV). Regarding these interactions, we found three types of regimes depending on the amplitude of the solitary wave and how its speed and the speed of the external force are related. A solitary wave can remain steady when its crest and the crest of the external force are in phase, it can bounce back and forth remaining close to its initial position when its speed and the external force speed are near resonant, or it can move away from its initial position without reversing its direction. Additionally, we verified that the numerical results agreed qualitatively well within the asymptotic approximation theory for external broad forces.

Simulation of Internal Undular Bores in Rapidly Varying Topography

This paper intends to look at the influence of rapidly varying regions on the propagation of internal undular bores in a two-layer fluid flow. Internal undular bores have been observed in the ocean around the world. The appropriate mathematical model to describe the evolution of internal undular bores in a stratified fluid is the variable-coefficient extended Korteweg-de Vries equation. The governing equation is solved numerically using the method of lines to simulate the propagation of internal undular bores. Our numerical results show that the effects of rapidly varying topography lead to adiabatic and non-adiabatic deformation of the internal undular bores, including the generation of solitary wavetrain, generation of nonlinear wavetrain and generation of rarefaction wave. Besides, multi-phase behaviour is also observed during the evolution.

Rational Solitons in the Gardner-Like Models

Rational solutions of nonlinear evolution equations are considered in the literature as a mathematical image of rogue waves, which are anomalously large waves that occur for a short time. In this work, bounded rational solutions of Gardner-type equations (the extended Korteweg-de Vries equation), when a nonlinear term can be represented as a sum of several terms with arbitrary powers (not necessarily integer ones), are found. It is shown that such solutions describe first-order algebraic solitons, kinks, and pyramidal and table-top solitons. Analytical solutions are obtained for the Gardner equation with two nonlinear terms, the powers of which differ by a factor of 2. In other cases, the solutions are obtained numerically. Gardner-type equations occur in the description of nonlinear wave dynamics in a fluid layer with continuous or multilayer stratification, as well as in multicomponent plasma, and their solutions are used for the interpretation of rogue waves.

Near-critical turbulent free-surface flow over a wavy bottom

Steady plane turbulent free-surface flow over a slightly wavy bottom is considered for very large Reynolds numbers, very small bottom slopes, and Froude numbers close to the critical value 1. As in previous works, the slope and the deviation from the critical Froude number are assumed to be coupled such that turbulence modeling is not required. The amplitudes of the periodic bottom elevations, however, are assumed to be half an order of magnitude larger than in the previous case of bumps or ramps of finite length. Asymptotic expansions give a steady-state version of an extended Korteweg–deVries (KdV) equation for the surface elevation. The extension consists of a forcing term due to the unevenness of the bottom and a damping term due to friction at the bottom. Other flow quantities, such as pressure, flow velocity components, local Froude number and bottom friction force, can be expressed in terms of the surface elevation. Exact solutions of the extended KdV equation, describing stationary cnoidal waves, are obtained for bottoms of particular periodic shapes. As a limiting case, the solitary waves over a bottom ramp are re-obtained in accord with previous results.

Effects of rotation and topography on internal solitary waves governed by the rotating Gardner equation

Abstract. Nonlinear oceanic internal solitary waves are considered under the influence of the combined effects of saturating nonlinearity, Earth's rotation, and horizontal depth inhomogeneity. Here the basic model is the extended Korteweg–de Vries equation that includes both quadratic and cubic nonlinearity (the Gardner equation) with additional terms incorporating slowly varying depth and weak rotation. The complicated interplay between these different factors is explored using an approximate adiabatic approach and then through numerical solutions of the governing variable depth, i.e., the rotating Gardner model. These results are also compared to analysis in the Korteweg–de Vries limit to highlight the effect of the cubic nonlinearity. The study explores several particular cases considered in the literature that included some of these factors to illustrate limitations. Solutions are made to illustrate the relevance of this extended Gardner model for realistic oceanic conditions.

Influence of ions nonextensivity on the dynamics of dust acoustic double layers in a magnetized self-gravitating dusty plasma

The existence and polarity of dust acoustic double layers (DADLs) in a magnetized self-gravitating opposite polarity dusty plasma (OPDP) have been investigated. The constituents of the OPDP model are two species of positively and negatively charged dust grains, Maxwellian electrons, and nonextensive distributed ions. Employing the reduction perturbation method to solve basic fluid equations for dust grains, the extended Korteweg-de Vries (E-KdV) equation was derived, and its solution was extracted and analyzed. Results show that DADLs behavior is influenced by several new relevant configurational parameters, including the nonextensivity of ions, strength, and direction of the static magnetic field, as well as the self-gravitational field. In the proposed OPDP model positive and negative double-layer polarities may be formed. The polarity of OPDP is severely affected by the nonextensivity of plasma ions and the mass ratio of positive dusts to negatives. Results of higher-order nonlinear equation for solitary structure deny the presence of solitons in the physical condition of modeled plasma. In this work, simultaneous effects of an external magnetic field, self-gravitational force, and plasma ions’ nonextensivity have been studied extensively, which can be applied in various material processing plasmas, fusion plasmas, to space plasmas such as cometary tails, and Earth’s mesosphere.