Cylindrical Korteweg-de Vries Equation 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.

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.

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

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.

(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.

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.

Planar and nonplanar electron acoustic solitons in dissipative quantum plasma

The nonlinear propagation of electron acoustic waves (EAWs) in an unmagnetized dissipative quantum plasma is studied both in an unbounded planar geometry and a bounded nonplanar geometry. With the help of the reductive perturbation technique, damped Korteweg-de Vries (KdV), cylindrical and spherical KdV equations are derived by using the quantum hydrodynamic set of equations for EAWs. The linear dispersion relation is found to be modified by the collision between electrons and ions. The nondimensional parameter δ, the equilibrium density ratio of the cold to hot electron component, plays a crucial role in the formation of both compressive and rarefactive solitons. Approximate analytical time evolution solutions of planar and nonplanar damped KdV equations are presented. Numerically, the effects of several physical parameters on the properties of EAWs are discussed in the planar and nonplanar geometry. It is also found that electron acoustic solitons in the nonplanar geometry significantly differ from the planar geometry.

Dark ring soliton in two-dimensional nonlinear self-defocusing medium

We investigate the control of dark ring solitons with multi-layer ring-shapes, which are the one- and two-soliton solutions of the Korteweg-de Vries (KdV) equation. By using the reductive perturbation method, a cylindrical KdV equation for the two-dimensional self-defocusing nonlinear Schrödinger (NLS) equation is obtained, which possesses dark soliton solutions. By picking different orders of solutions of the KdV equation, the profiles of the intensity of the dark nonlinear waves display multi-layer ring-shapes, and can be controlled by the initial phase and the amplitude. Our results are expected to provide a theoretical basis for the control of dark ring solitons in nonlinear medium.

Classification of Optimal Group-Invariant Solutions: Cylindrical Korteweg–de Vries Equation

Classification of optimal group-invariant solutions has been carried out for cylindrical Korteweg–de Vries equation. It is a nonlinear evolution equation often found in studies of fluid mechanics and plasmas. An optimal system of subalgebras is obtained for the group of Lie generators associated with the equation, and then, the group-invariant solutions are obtained for each member in the optimal class. The procedure for calculating other group-invariant solutions from the optimal class is also illustrated with an example.

Dark spatiotemporal optical solitary waves in self-defocusing nonlinear media

Dark three-dimensional spatiotemporal solitons or the “dark light bullets” in the self-defocusing nonlinear media with equal diffraction and dispersion lengths are demonstrated analytically. Our results show that the main characteristic of the dark light bullets can be described by the cylindrical Korteweg–de Vries (CKdV) equation. The dark wave packets are composed of the single-layer and multilayer toroidal rings. For the multilayer rings, there exist small inner rings enclosed in a large ring-shaped toroidal structure, when one chooses different orders of the soliton solutions of the CKdV equation. The radius of dark ring solitons increases with the propagation distance. Present results provide a feasible method for controlling the fundamental structure of these beams in the self-defocusing nonlinear media.

Exact solution of the cylindrical Korteweg–de Vries equation for dust ion acoustic wave in unmagnetised plasma

The cylindrical Korteweg–de Vries equation is derived for dust ion acoustic waves in unmagnetized plasma by employing reductive perturbation technique. Lie group analysis is employed to obtain exact solution of the cylindrical Korteweg–de Vries equation. The original non-planar system is mapped by invertible point transformations to the equivalent autonomous planar form to obtain exact solutions. It is found that the non-planar cylindrical geometry differs from that of the planar one-dimensional geometry.

Group formalism of Lie transformations to time-fractional partial differential equations

A systematic method is given to derive Lie point symmetries of time-fractional partial differential equation with Riemann–Liouville fractional derivative and its applicability illustrated through (i) time-fractional diffusive equation and (ii) time-fractional cylindrical Korteweg–de Vries equation. Using the Lie point symmetries obtained, we show that each of them has been transformed into ordinary differential equation of fractional order with a new independent variable. We also explain how exact or invariant solutions can be derived from the obtained point symmetries.

A note on the cylindrical solitary waves in an electron-acoustic plasma with vortex electron distribution

In the present work, we consider the propagation of nonlinear electron-acoustic non-planar waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical coordinates through the use reductive perturbation method in the long-wave approximation. The modified cylindrical Korteweg–de Vries equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, which is fractional, this evolution equation cannot be reduced to the conventional Korteweg–de Vries equation. An analytical solution to the evolution equation, by use of the method developed by Demiray [Appl. Math. Comput. 132, 643 (2002); Comput. Math. Appl. 60, 1747 (2010)] and a numerical solution by employing a spectral scheme are presented and the results are depicted in a figure. The numerical results reveal that both solutions are in good agreement.