Cylindrical Korteweg-de Vries Research Articles

Localized wave structures: Solitons and beyond

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves.

Highly localized horseshoe ripplons and solitons in positive dispersion media

In this study, we systematically review various ripplon solutions to the Kadomtsev–Petviashvili equation with positive dispersion (KP1 equation). We show that there are mappings that allow one to transform the horseshoe solitons and curved lump chains of the KP1 equation into circular solitons of the cylindrical Korteweg–de Vries (cKdV) equation and two-dimensional solitons of the cylindrical Kadomtsev–Petviashvili (cKP) equation. Then, we present analytical solutions that describe new nonlinear highly localized ripplons of a horseshoe shape. Ripplons are two-dimensional waves with an oscillatory structure in space and a decaying character in time; they are similar to lumps but non-stationary. In the limiting case, the horseshoe ripplons reduce to solitons decaying with time and having bent fronts. Such entities can play an important role in the description of strong turbulence in plasma and other media.

Data-driven Ai- and Bi-soliton of the cylindrical Korteweg-de Vries (cKdV) equation via prior information PINNs (PIPINNs)

Abstract In this paper, by modifying loss function MS E and training area of the physics-informed neural networks (PINNs), we proposed a neural networks models: prior information PINNs (PIPINNs). We demonstrated the advantages of PIPINNs by simulating Ai- and Bi-soliton solutions of the cylindrical Korteweg-de Vries (cKdV) equation. Numerical experiments showed that our proposed model is not only simulate these solitons of cKdV equation, but also significantly improve the simulation capability. Compared with original PINNs, the prediction accuracy of our proposed models are improved by one to three orders of magnitude. Moreover, the accuracy of PIPINNs is improved again by adding the restriction of conservation of energy.

Bäcklund transformation for analyzing a cylindrical Korteweg–de Vries equation and investigating multiple soliton solutions in a plasma

The presence of the geometrical curvature makes the planar Korteweg–de Vries (KdV) equation inadequate to describe the propagation of nonlinear waves. In many scientific disciplines including plasma physics, nonlinear optics, oceanography, and communications, the cylindrical KdV (CKdV) equation becomes the appropriate choice for modeling these waves. Motivated by these applications, the Bäcklund transformation is used to analyze and find an analytical solution to the CKdV equation in the present investigation. For the first time, the multi-soliton solutions, including single-, two-, and three-soliton solutions, are investigated, and a general scheme is given to find N-soliton solutions of the CKdV equation in the context of plasma physics. Numerous researchers may find the given solutions helpful in understanding the mechanism of the generation of multi-solitons in their laboratory experiments and may also engender interest in the space physics community to look for these structures in the data coming from a variety of satellites roaming in space.

Solitary wave solution of (3+1)-dimensional cylindrical Kadomstev–Petviashvili equation in warm opposite polarity dusty plasma

A theoretical investigation is done to study the propagation of dust-acoustic (DA) solitary waves in a dusty plasma system containing dynamic positively as well as negatively charged warm dust species and nonthermally distributed ion and electron species. The standard reductive perturbation method is employed to derive the (3+1)-dimensional cylindrical Kadomstev–Petviashvili (KP) equation (also known as cylindrical Korteweg–de Vries equation), which admits solitary wave solution for small but finite amplitudes. The parametric regimes for the existence of DA solitary waves are obtained. Two modes, namely, slow and fast are observed corresponding to different phase velocities. The slow mode has negative solitary pulses, whereas the fast mode contains both positive and negative DA solitary waves. It is shown that the ratio of positively charged dust to negatively charged dust number density significantly modifies the basic features of the DA solitary waves. The noteworthy effects of other parameters (viz. the nonthermal parameter and ion-to-electron temperature ratio) on the basic properties (namely, phase speed, polarity, amplitude, and width) of the DA solitary waves are also mentioned. The fundamental features and the underlying physics of the electrostatic solitary structures that are relevant to cometary tails, upper mesosphere, Jupiter’s magnetosphere, etc., are briefly discussed.

Three-Dimensional Cylindrical Dust-Acoustic Solitary Pulses in Warm Nonthermal Plasma

The nonlinear propagation of 3-D cylindrical dust-acoustic (DA) solitary waves in an unmagnetized dusty plasma system consisting of mobile adiabatic positively charged dust (PCD) grains, nonthermal ions, and electrons is investigated. The basic properties of nonlinear DA solitary structures, which exist in such a warm dusty plasma medium, are studied. The ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3+1$</tex-math> </inline-formula> )-dimensional cylindrical Kadomstev–Petviashvili equation (cKPE) (also known as cylindrical Korteweg-de Vries (KdV) equation) is derived by using the reductive perturbation method (RPM). The possible region of the formation of solitons is shown. The effects of different plasma parameters (viz., the nonthermal parameter, the dust-to-ion temperature ratio, the ion-to-dust number density ratio, etc.) on the characteristics of the DA solitary waves are studied, and also the graphical representation of the cylindrical DA solitary structures is presented. The polarity of DA solitary waves changes with different plasma parameters (viz., the ion-to-electron temperature ratio, the nonthermality of ions and electrons, the ion-to-dust number density, etc.). These results may improve our understanding of DA solitary structures in space and laboratory plasma devices where nonthermal dusty plasma occurs.

Solitary Waves and Their Interactions in the Cylindrical Korteweg–De Vries Equation

We consider approximate, exact, and numerical solutions to the cylindrical Korteweg–de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the interaction of solitary waves of different types.

Airy Kernel Determinant Solutions to the KdV Equation and Integro-Differential Painlevé Equations

We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlev\'e equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlev\'e V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation.

Canonical structure of the cylindrical Korteweg-de Vries equation for dust ion acoustic waves in unmagnetised plasma

Canonical structure of the cylindrical Korteweg-de Vries equation for dust ion acoustic waves in unmagnetised plasma

Evolution of truncated and bent gravity wave solitons: the Mach expansion problem

Abstract

3-D Cylindrical Waves in a Self-Gravitating Degenerate Quantum Plasma

A theoretical investigation is carried out for understanding the properties of three-dimensional degenerate pressure-driven (DPD) gravitoelectrostatic solitary waves (SWs) in a general (but realistic) self-gravitating degenerate quantum plasma (SGDQP) medium whose constituents are noninertial degenerate electron and inertial degenerate heavy and light ions/nuclei/elements. The formation of solitary electrostatic and self-gravitational potential structures in such an SGDQP system is examined. The standard reductive perturbation method is employed to derive the (3+1)-D cylindrical Korteweg-de Vries equation (also known as cylindrical Kadomtsev-Petviashvili equation). The parametric regimes for the existence of solitary electrostatic and self-gravitational potential structures are found. It is shown that the present plasma model supports the DPD SWs with positive and negative electrostatic potentials as well as self-gravitational potentials. The effects of the degenerate plasma particles are found to significantly modify the basic properties (viz., the amplitude, width, and speed) of the electrostatic and self-gravitational SWs. The present investigation can be of relevance to the electrostatic and self-gravitational solitary pulses observed in various plasma space environments (viz., white dwarfs, neutron stars, and so on).

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

Trace formulas for the one-dimensional Stark operator and integrals of motion for the cylindrical Korteweg–de Vries equation

Inverse problem of the cylindrical Korteweg-de Vries equation for dust ion acoustic waves in unmagnetised plasma

Inverse problem of the cylindrical Korteweg-de Vries equation for dust ion acoustic waves in unmagnetised plasma

Large time asymptotics for the cylindrical Korteweg–de Vries equation. I.

This paper is the first part of our study the asymptotic behavior of the solutions of the cylindrical Korteweg–de Vries equation (cKdV). In this first part, we consider the solutions from Schwarz’s class and calculate the large time asymptotics using the method based on the asymptotic solution of the associated direct scattering problem. This approach was first suggested, in the cases of usual KdV and NLS equations, in the late 70s by Zakharov and Manakov. In a sequel to this paper, we plan to calculate the large time asymptotics of some other classes of solutions of the cKdV equation which exhibit the oscillatory type behavior, and we will also evaluate the short time asymptotics of the solutions of the cKdV equation. In the second part we will use the Defit–Zhou nonlinear steepest descent method for oscillatory Riemann–Hilbert problems.

Three-Dimensional Self-Gravito-Acoustic Solitary Waves in a Degenerate Quantum Plasma System

Three-dimensional self-gravito-acoustic solitary waves (SGASWs) in a general (but realistic) self-gravitating degenerate quantum plasma media consisting of heavy nuclei/elements and degenerate electrons are studied. The reductive perturbation method, which is valid for small but finite amplitude SGASWs, is employed to derive the (3+1)-dimensional cylindrical Korteweg–de Vries (cKdV) equation (also known as cylindrical Kadomtsev–Petviashvili equation). To show the parametric regime corresponding to the existence of the localized SGASWs the energy integral equation, which is derived from cKdV equation, is used. Moreover, the effects of the physical parameters on the SGASWs are discussed. The implication of our results for space and laboratory plasmas is briefly discussed.

(3 + 1)-Dimensional cylindrical Korteweg-de Vries equation in a self-gravitating degenerate quantum plasma system

Three-dimensional degenerate pressure driven self-gravito-acoustic solitary waves (SGASWs) in a general (but realistic) self-gravitating degenerate quantum plasma (SGDQP) medium consisting of noninertial degenerate electron and inertial degenerate heavy and light ions/nuclei/elements are studied. The reductive perturbation method, which is valid for small but finite amplitude SGASWs, is employed to derive the (3 + 1)-dimensional cylindrical Korteweg-de-Vries equation (also known as the cylindrical Kadomtsev-Petviashvili equation) which admits a solitary wave solution. Two modes, namely, slow and fast are observed corresponding to different phase velocities. Both compressive (positive) and rarefactive (negative) SGASWs are found to exist in the SGDQP system for fast and slow modes. It has been shown that the presence of both degenerate light and heavy nuclei modifies the nature of SGASWs and allows both compressive and rarefactive SGASWs to exist in such a SGDQP medium. Moreover, the effects of the physical parameters on the SGASWs are discussed. The implication of our results for space and laboratory plasmas is briefly discussed.

Local discontinuous Galerkin method for modeling the nonplanar structures (solitons and shocks) in an electronegative plasma

Using the hydrodynamic equations of cold inertial positive ions with the Maxwellian distribution for light negative ion and electron densities and the Poisson equation, the family of nonplanar (cylindrical and spherical) Korteweg-de Vries (KdV) equations, i.e., the KdV, modified KdV, and extended KdV (EKdV), are obtained for small but finite amplitude ion-acoustic waves. The nonplanar EKdV equation is used to analyze the time-dependent planar and nonplanar soliton and shock structures. It is well-known that the exact solutions of the family of nonplanar KdV equations are not possible. Therefore, a local discontinuous Galerkin (LDG) method is developed for solving the nonplanar EKdV equation numerically. According to this method, the initial and boundary conditions for the solitary and shock waves are accurately identified. The L2 stability of the LDG method is proved for the general nonlinear case. The existence regions for both solitary and shock excitations have been defined precisely depending on the laboratory plasma parameters. Moreover, the effects of the negative-ion temperature ratio and the negative ion concentration on the profiles of the nonlinear structures (solitons and shocks) are examined. The effect of the geometrical divergence on the pulse profile is also reported which indicates that the localized pulses deform as time goes on. Furthermore, it is found that the amplitude of cylindrical structures (solitons and shocks) is larger than that of planar ones but smaller than that of the spherical ones. Moreover, in cylindrical geometry, the nonlinear structures travel slower than in the spherical ones. The implications of our results agree with the experimental observations.

Localized Airy Wave Packets in a Self-Defocusing Kerr Medium

Families of vortex-like and Gaussian-like localized Airy wave packets, propagating in a self-defocusing Kerr medium, are discovered. Their propagation is described by the cylindrical Korteweg–de Vries (CKdV) equation, reduced from the nonlinear Schrodinger equation by utilizing the reductive perturbation technique. The CKdV equation is solved using the Hirota bilinear method to obtain analytical Airy wave packet families of different order. In particular, we focus on the distribution of optical intensity via numerical simulations of the Airy wave packet solutions, obtained for different initial phases and amplitudes, at given propagation distances. In distinction to the usual ring solitons, the Airy wave packets display a novel internal structure, which reveals novel nonlinear phenomena during the propagation of packets.

Painlevé Test, Bäcklund Transformation and Consistent Riccati Expansion Solvability for two Generalised Cylindrical Korteweg-de Vries Equations with Variable Coefficients

Abstract In this paper, the integrability of the (2+1)-dimensional cylindrical modified Korteweg-de Vries equation and the (3+1)-dimensional cylindrical Korteweg-de Vries equation with variable coefficients arising in dusty plasmas in its generalised form was studied by two different techniques: the Painlevé test and the consistent Riccati expansion solvability. The integrability conditions and Bäcklund transformations are constructed. By using Bäcklund transformations and the solutions of the Riccati equation many new exact solutions are found for the two equations in this study. Finally, the application of the obtained solutions in dusty plasmas is investigated.

Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg–de Vries models arising in dusty plasmas by using direct reduction method

In this paper, generalized models for both (2+1)-dimensional cylindrical modified Korteweg–de Vries (cmKdV) equation with variable coefficients and (3+1)-dimensional variable coefficients cylindrical Korteweg–de Vries (cKdV) equation are studied by direct reduction method. A direct reduction to nonlinear ordinary differential equations in the form of Riccati equations obtained for the considered equations under some integrability conditions. The search for solutions for the reduced Riccati equations has yielded many Jacobi elliptic wave solutions, solitary and periodic wave solutions for both (2+1)-dimensional cmKdV and (3+1)-dimensional cKdV equations. Physical application for the obtained solutions as dust ion acoustic waves in plasma physics is given