Variable-coefficient Korteweg-de Vries Research Articles

Experimental and theoretical studies of sea ice effects on internal solitary waves

Internal solitary waves in polar regions have attracted much interest recently. It is important to understand how sea ice affects them as this may have a profound influence on human activities and the environment. In this study, experiments on internal solitary waves with and without two types of sea ice (ice sheet and ice keel) are presented, as well as corresponding simulations using the Korteweg-de Vries (KdV) equation, the Benjamin-Ono (BO) equation, and the variable-coefficient Korteweg-de Vries (vKdV) equation, which is a derivation of the KdV equation. Comparison between experiments without sea ice and simulations using the KdV and BO equations proves the suitability of the former over the latter for this study. The experiments with sea ice and theoretical simulations using the vKdV equation provide evidence for wave deformation, oscillation occurring in the rear of the wave, and a decrease in amplitude. The latter suggests possibilities of energy dissipation or the emission of small amplitude linear waves. The sharp vertices of the ice result in occasional inconsistency with the vKdV predictions. Nonetheless, the vKdV equation is still suitable for modeling internal solitary waves under sea ice, giving generally accurate results that can assist further studies. This is the first time the vKdV equation has been applied to investigate the impacts of sea ice on internal solitary waves.

Exact solutions of a variable coefficient KdV equation: Power law in time-coefficients

The Lie symmetry analysis of a power law in-time coefficients Korteweg–de Vries (KdV) equation is performed with the aim of specifying the model parameters (powers of t). That is, the symmetries of the resulting subclasses of the underlying equation are obtained. Further, symmetry reductions and some exact solutions are obtained.

Integrability, lump solutions, breather solutions and hybrid solutions for the (2+1)-dimensional variable coefficient Korteweg-de Vries equation

Integrability, lump solutions, breather solutions and hybrid solutions for the (2+1)-dimensional variable coefficient Korteweg-de Vries equation

VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations – variable coefficient physics-informed neural network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviate the “vanishing gradient” problem and unify linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine–Gordon (vSG), the generalized variable coefficient Kadomtsev–Petviashvili equation (gvKP), the variable coefficient Korteweg–de Vries equation (vKdV), the variable coefficient Sawada–Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects: the necessity of ResNet; the relationship between the convexity of variable coefficients and learning; anti-noise analysis; the unity of forward and inverse problems/relationship with standard PINN.

The Reduction Transformation of the Generalized Variable-coefficient KdV Equation

In this manuscript, we have studied the reduction transformation of the generalized variable-coefficient Korteweg-de Vries(KdV) equation by the modifed Clarkson-Kruskal(CK) direct method, and have established the connection between the generalized variable-coefficient KdV equation and the constant- coefficient KdV equation. After complicated calculations, a new transformation is obtained, which transforms the generalized one-dimensional KdV equation with variable-coefficients into the corresponding KdV equation with constant-coefficients. As we know, the new transformation has not been studied in current literature. Based on the transformation, the solution of variable-coefficient KdV equation can be obtained directly through the constant-coefficient KdV equation, and it is helpful to explore the similarity reduction and exact solution of variable-coefficient KdV equation. Furthermore, a special example is given to verify the correctness of the transformation we have proposed.

Some novel physical structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries system

In this paper, we study the novel nonlinear wave structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system by its analytic solutions. Its N-soliton solutions are obtained via Hirota’s bilinear method, and in particular, the hybrid solutions of lump, breather and line solitons are derived by the long wave limit method. In addition to soliton solutions, similarity reduction, including similarity solutions (also known as group-invariant solutions) and novel non-autonomous rational third-order Painlevé equations, is achieved through symmetry analysis. The analytic results, together with illustrative wave interactions, show interesting physical features, that may shed some light on the study of other variable-coefficient nonlinear systems.

Multiple soliton solutions and other solutions of a variable-coefficient Korteweg–de Vries equation

In this paper, we construct multiple soliton solutions of a time-dependent variable-coefficient Korteweg–de Vries equation in the form of bright, dark and singular soliton solutions. Furthermore, the periodic solutions are derived. These will be achieved via the aid of solitary wave ansatz methods. In addition, other exact solutions of a time-dependent variable-coefficient Korteweg–de Vries equation are attained using symmetry reductions. In one special case, the power series method is employed to derive a solution of the aforesaid equation. We also examine the conservation laws of the aforementioned equation using the variational approach. A brief physical representation of the derived conserved vectors will be conversed.

RETRACTED ARTICLE: In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation

RETRACTED ARTICLE: In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation

In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation

In nonlinear optics, fluid mechanics, plasma physics or atmospheric science: symbolic computation on a generalized variable-coefficient Korteweg-de Vries equation

Numerical Solutions of Variable-Coefficient Fractional-in-Space KdV Equation with the Caputo Fractional Derivative

In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically simulated. Especially, the Fourier spectral method is used to solve the fractional-in-space vcKdV equation with breather soliton. From numerical simulations and compared with other methods, it can be easily seen that our method has low computational complexity and higher precision.

Shock waves in a strongly coupled inhomogeneous dusty plasma

Nonlinear theory for certain dust acoustic waves has been developed for a strongly coupled inhomogeneous collisionless dusty plasma containing dust particles of different sizes. The generalized hydrodynamic set of partial differential equations is considered to describe the fluid model of the present nonlinear system. A reductive perturbation technique is employed on this nonlinear set of equations to derive a variable coefficient Korteweg-deVries (KdV)–Burgers equation with a forcing term. The generalized expansion method is performed to obtain an analytic expression for the solution. This analytic solution describes the monotonic kink-type shock waves, due to the dominant nature of the dissipation term over the dispersion term. In addition to the effects of inhomogeneity and grain size distribution, the effects of nonthermal ions, dust–dust correlation and ion temperature on the wave propagation are investigated. Also, a steady-state monotonic shock solution of the KdV-Burgers equation is obtained numerically. The obtained results are discussed in the context of some dusty plasmas, previously reported experimentally or theoretically.

Lie Group Classification of Generalized Variable Coefficient Korteweg-de Vries Equation with Dual Power-Law Nonlinearities with Linear Damping and Dispersion in Quantum Field Theory

Many physical phenomena in fields of studies such as optical fibre, solid-state physics, quantum field theory and so on are represented using nonlinear evolution equations with variable coefficients due to the fact that the majority of nonlinear conditions involve variable coefficients. In consequence, this article presents a complete Lie group analysis of a generalized variable coefficient damped wave equation in quantum field theory with time-dependent coefficients having dual power-law nonlinearities. Lie group classification of two distinct cases of the equation was performed to obtain its kernel algebra. Thereafter, symmetry reductions and invariant solutions of the equation were obtained. We also investigate various soliton solutions and their dynamical wave behaviours. Further, each class of general solutions found is invoked to construct conserved quantities for the equation with damping term via direct technique and homotopy formula. In addition, Noether’s theorem is engaged to furnish more conserved currents of the equation under some classifications.

Consistent Riccati expansion solvability, symmetries, and analytic solutions of a forced variable-coefficient extended Korteveg–de Vries equation in fluid dynamics of internal solitary waves* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11775047, 11775146, and 11865013).

We study a forced variable-coefficient extended Korteweg–de Vries (KdV) equation in fluid dynamics with respect to internal solitary wave. Bäcklund transformations of the forced variable-coefficient extended KdV equation are demonstrated with the help of truncated Painlevé expansion. When the variable coefficients are time-periodic, the wave function evolves periodically over time. Symmetry calculation shows that the forced variable-coefficient extended KdV equation is invariant under the Galilean transformations and the scaling transformations. One-parameter group transformations and one-parameter subgroup invariant solutions are presented. Cnoidal wave solutions and solitary wave solutions of the forced variable-coefficient extended KdV equation are obtained by means of function expansion method. The consistent Riccati expansion (CRE) solvability of the forced variable-coefficient extended KdV equation is proved by means of CRE. Interaction phenomenon between cnoidal waves and solitary waves can be observed. Besides, the interaction waveform changes with the parameters. When the variable parameters are functions of time, the interaction waveform will be not regular and smooth.

Ablowitz–Kaup–Newell–Segur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg–de Vries equation in plasma physics, fluid dynamics or atmospheric science

For a variable-coefficient Korteweg–de Vries equation in a lake/sea, two-layer liquid, atmospheric flow, cylindrical plasma or interactionless plasma, in this paper, we derive the bilinear Bäcklund transformation, non-isospectral Ablowitz–Kaup–Newell–Segur system and infinite conservation laws for the wave amplitude under certain constraints among the external force, dissipation, nonlinearity, dispersion and perturbation.

The barotropic Rossby waves with topography on the earth’sδ-surface

AbstractThe Rossby solitary waves in the barotropic vorticity model which contains the topography on the earth’sδ-surface is investigated. First, applying scale analysis method, obtained the generalized quasi-geostrophic potential vorticity equation (QGPVE). Using The Wentzel–Kramers–Brillouin (WKB) theory, the evolution equation of Rossby waves is the variable-coefficient Korteweg–de Vries (KdV) equation for the barotropic atmospheric model. In order to study the Rossby waves structural change to exist in some basic flow and topography on theδ-surface approximation, the variable coefficient of KdV equation must be explicitly, Chebyshev polynomials is used to solve a Sturm-Liouville-type eigenvalue problem and the eigenvalue Rossby waves, these solutions show that the parameterδusually plays the stable part in Rossby waves and slow down the growing or decaying of Rossby waves with the parameterβ.

Reduction and analytic solutions of a variable-coefficient Korteweg–de Vries equation in a fluid, crystal or plasma

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.

Periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries equation

In this paper, we can obtain periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries (gvc-KdV) equation for the first time by using the multivariate transformation technique. Periodic solutions are doubly and triply periodic solutions, and the decay mode solutions include the 1-decay solution and the 2-decay mode solution. Furthermore, we will also study the effect of variation of parameters for solutions systematically, relating to different forms of expression, through graphic display.

Propagation of shock structures in inhomogeneous dusty plasmas with dust size distribution and nonadiabatic dust charge fluctuation

Propagation of nonlinear shock structures in inhomogeneous dusty plasmas with power law dust size distribution and nonadiabatic dust charge fluctuation has been investigated. Using reductive perturbation expansion, the hydrodynamic equations are reduced to a modified variable coefficient Korteweg–de Vries (MKdV)–Burgers equation governing the characteristic of shock structures. Two different types of shock structures exist in this system and transform each other. The numerical results show that the inhomogeneity, the power law dust size distribution and the nonadiabatic dust charge fluctuation have strong influence on the nonlinear properties of shock structures and the transition condition between monotone shock structures and oscillatory shock structures.

Multi-waves, breather wave and lump–stripe interaction solutions in a (2 $$+$$ 1)-dimensional variable-coefficient Korteweg–de Vries equation

Under investigation is a ( $$2 + 1$$ )-dimensional variable-coefficient Korteweg–de Vries equation, which is used to describe the electrostatic wave potential in plasmas or shallow-water waves in fluids. Our work is mainly divided into three aspects: (1) lump–stripe interaction solutions are obtained by the Hirota’s bilinear form with variable coefficients; (2) multi-waves solutions are presented based on three waves method with variable coefficients; (3) breather wave solutions are given by the homoclinic breather approach. Furthermore, interaction phenomenon of these obtained solutions are demonstrated by some 3D- and contour plots.

Solitary wave and modulational instability in a strongly coupled semiclassical relativistic dusty pair plasma with density gradient

Using a set of fluid equations to describe the inertial dust grain component in a dense collision-less unmagnetised plasma under the influence of weakly relativistic semiclassical electrons and positrons, the propagation of dust-acoustic wave is studied in the strong coupling regime when the dust density is non-uniform. Our main aim is to analyse the role of semiclassical and relativistic environment (frequently encountered in astrophysical context) on various features of strongly coupled dusty plasma. The semiclassical environment of the electrons and positrons is assumed to be described by the Chandrasekhar equation of state. Our second aim is to see the effect of spatial variation of the dust equilibrium density, which is known to occur due to the deformation of the Debye sheath which in turn leads to polarisation force. A new type of nonlinear Schrodinger equation with spatially varying coefficient is deduced and its modulational stability is studied in detail. In the last section, we have taken recourse to Madelung picture to deduce a variable coefficient Korteweg–de Vries equation from this new nonlinear Schrodinger equation.