Cylindrical Kadomtsev-Petviashvili Research Articles

[formula omitted]-solitons solutions, Bäcklund transformation and Lax Pair for a generalized variable-coefficient cylindrical Kadomtsev–Petviashvili equation

In this paper, we investigate a generalized variable-coefficient cylindrical Kadomtsev–Petviashvili equation, which characterizes the water waves propagation in fluid dynamics. Via the generalized Laurent series truncated at the constant-level term, an auto-Bäcklund transformation is derived. We establish the bilinear forms through the utilization of the Bell polynomials. Based on the Hirota method, we construct N-soliton solutions. We derive the bilinear Bäcklund transformation and Lax pair by virtue of the Hirota bilinear operators’ exchange formulae and symbolic computation.

Cylindrical Kadomtsev–Petviashvili equation for ion acoustic waves with double spectral indices distributed electrons

We study theoretically and numerically the nonlinear propagation of ion acoustic waves (IAWs) in plasma. The electrostatic potential in electron-ion (e-i) plasma is mimicked by the cylindrical Kadomtsev–Petviashvili equation using the traditional reductive perturbation approach. By applying the direct integration technique, the soliton solution is obtained. The results of simulations carried out numerically show that the amplitude as well as width of the soliton solution are significantly influenced by the plasma parameters. The exploration of IAWs propagation in laboratory as well as space plasmas may benefit from the current work.

Dust magnetoacoustic waves in an inhomogeneous cylindrical four-component dusty plasma in the presence of polarization force

Abstract Dust magnetoacoustic waves have been examined in an inhomogeneous, bounded, cylindrical dusty plasma containing oppositely polarized dust particles. Considering polarization force, dust dynamics in r − θ plane is studied in the presence of inhomogeneous external magnetic field along z axis. At equilibrium, the dusty plasma components are supposed to follow Gaussian density distribution. Using reductive perturbation method (RPM), a variable coefficient cylindrical Kadomtsev–Petviashvili (VCCKP) equation has been derived. For weak azimuthal perturbation, an analytical solution, obtained by Zhang (“Exact solutions of a kdv equation with variable coefficients via exp-function method,” Nonlinear Dynam., vol. 52, nos. 1–2, pp. 11–17, 2008) using Exp-function method, is chosen. Phase velocity of dust magnetoacoustic wave is found to be modified by the density inhomogeneities, polarization force, dust charge state ratio and ion-to-electron temperature ratio. Spatio-temporal evolution of the dust number densities has been noticed. Existence of the compressive electromagnetic solitary waves is observed numerically for the chosen dusty plasma parameter range. The impacts of the inhomogeneity, polarization force, dust charge state ratio and ion-to-electron temperature ratio on the relative amplitude of the dust magnetoacoustic wave are also discussed.

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.

Solitons and lump waves to the elliptic cylindrical Kadomtsev–Petviashvili equation

We construct various kinds of solutions to the elliptic cylindrical Kadomtsev–Petviashvili equation, which can be derived in the context of surface gravity waves. This investigation yields a diverse set of analytical outcomes, encompassing the dynamics of horseshoe-like solitons, lump chains, and lump waves. Based on the Hirota bilinear method, the lump chain is derived under specified conjugate constraint conditions. Additionally, we uncover three distinct types of novel interactions among lump waves through the utilization of the long wave limit method. A detailed analysis is conducted to elucidate the features and patterns inherent in these lump waves. The analytic findings presented may providing a descriptive framework for generic weakly nonlinear and weakly dispersive waves.

Solitons and lumps in the cylindrical Kadomtsev-Petviashvili equation. I. Axisymmetric solitons and their stability.

We revise soliton and lump solutions described by the cylindrical Kadomtsev-Petviashvili (cKP) equation and construct new exact solutions relevant to physical observation. In the first part of this study, we consider basically axisymmetric waves described by the cylindrical Kortweg-de Vries equation and analyze approximate and exact solutions to this equation. Then, we consider the stability of the axisymmetric solitons with respect to the azimuthal perturbations and suggest a criterion of soliton instability. The results of our numerical modeling confirm the suggested criterion and reveal lump emergence in the course of the development of the modulation instability of ring solitons in the unstable case. In the next part of this study, which will follow shortly, we will present exact solutions to the cKP equation describing weakly nonlinear waves in media with positive dispersion subject to the modulation instability of solitons with respect to small azimuthal perturbations.

Solitons and lumps in the cylindrical Kadomtsev-Petviashvili equation. II. Lumps and their interactions.

We study solitary waves in the cylindrical Kadomtsev-Petviashvili equation designated to media with positive dispersion (the cKP1 equation). By means of the Darboux-Matveev transform, we derive exact solutions that describe two-dimensional solitary waves (lumps), lump chains, and their interactions. One of the obtained solutions describes the modulation instability of outgoing ring solitons and their disintegration onto a number of lumps. We also derive solutions describing decaying lumps and lump chains of a complex spatial structure-ripplons. Then, we study normal and anomalous (resonant) interactions of lump chains with each other and with ring solitons. Results obtained agree with the numerical data presented in Part I of this study [Hu et al., Chaos (2024)].

Variable coefficient extended cKP equation for Rossby waves and its exact solution with dissipation

In this article, the variable coefficient (2 + 1)-dimensional extended cylindrical Kadomtsev–Petviashvili (cKP) equation describing Rossby waves was derived from the quasi-geostrophic potential vorticity equation. It is difficult for the variable coefficient cKP equation with dissipation to calculate the exact solution. For obtaining the exact solution, a new transformation was constructed for the first time to reduce the extended cKP equation to the extended KP equation. We emphasize that the exact solution, and not just approximate solution, in Rossby waves flow field can be obtained when dissipation is included. The exact lump and interaction solutions with dissipative effect are given according to the modified Hirota bilinear method, and physics for the evolution of Rossby waves is analyzed based on the obtained solutions. When the dissipative parameter μ0 increases, the structure of the amplitude A changes in the spatial scale y. And when the dissipative parameter increases to a certain value, the structure of Rossby waves tends to be stable. It is pointed out that the dissipative parameter μ0 determines not only the amplitude A of Rossby waves but also structures of Rossby waves flow field, with μ0 acting on the spatial scale y and the timescale t.

Nonplanar ion-acoustic solitons and shocks with superthermal trapped electrons and transverse perturbations

The formation and propagation of nonlinear ion-acoustic (IA) waves are studied in an unmagnetized cold viscous plasma, comprising the inertial ions and superthermal trapped electrons in cylindrical geometry with transverse perturbations. The well-known reductive perturbation method is employed to derive cylindrical trapped Kadomtsev Petviashvili Burgers (CTKPB) and cylindrical trapped Kadomtsev Petviashvili (CTKP) equations with and without dissipation effects, respectively. The exact solutions of CTKPB and CTKP equations are obtained for the first time by utilizing the dependent variable transformation. The soliton and shock structures are found to be significantly affected by the plasma parameters including the trapping efficiency parameter β, the superthermality parameter κ, viscosity µ 0 and the geometrical effects. Importantly, the geometrical effects and transverse perturbations alter the shape of solitons and shocks, resulting into the parabolic nonlinear structures. These structures become more parabolic at later times due to an interplay of transverse and time coordinates (η, τ). The results of the present study might be helpful to understand the characteristics of IA structures in space plasmas, such as auroral regions, where energetic trapped electrons have been observed.

Dynamics of Dusty Pair-Ion-Electron Plasma Modeled by the Cylindrical Kadomtsev-Petviashvili Equations

The nonlinear propagation and stability of dust ion-waves in plasma is analytically and numerically investigated. By using the the standard reductive perturbation method, the electrostatic potential in dusty pair-ion-electron plasma is modelled by cylindrical Kadomtsev-Petviashvili (CKP) equation. The soliton solutions are obtained using the direct integration for single soliton solution and the Hirota bilinear method to find multisoliton solution of the system. It is noticed that the Hirota method better illustrate the physical reality of dust pair-ion plasma since it generalizes different forms of solutions. From the numerical simulations, it is obseved that, the plasma parameters strongly influence the properties of the soliton solution, namely, the amplitude and the width. The analysis of the stability of the soliton solutions revels that the stable solution co-propagates with seven other solutions, eigenmodes of the Legendre equation. These modes contain basic symmetry and axisymmetric configuration consistent with relevant experimental observations in existing experiments.

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).

Time-Space Fractional Model for Complex Cylindrical Ion-Acoustic Waves in Ultrarelativistic Plasmas

In this paper, the fractional order models are used to study the propagation of ion-acoustic waves in ultrarelativistic plasmas in nonplanar geometry (cylindrical). Firstly, according to the control equations, (2 + 1)-dimensional (2D) cylindrical Kadomtsev–Petviashvili (CKP) equation and 2D cylindrical-modified Kadomtsev–Petviashvili (CMKP) equation are derived by using multiscale analysis and reduced perturbation methods. Secondly, using the semi-inverse method and the fractional variation principle, the abovementioned equations are derived the time-space fractional equations (TSF-CKP and TSF-CMKP). Furthermore, based on the fractional order transformation, the 1-decay mode solution of the TSF-CKP equation is obtained by using the simplified homogeneous balance method, and using the generalized hyperbolic-function method, the exact analytic solution of TSF-CMKP equation is obtained. Finally, the effects of the phase speedλ, electron number density (throughβ3) and the fractional orderα,β,ωon the propagation of ion-acoustic waves in ultrarelativistic plasmas are analyzed.

Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation

The cylindrical Kadomtsev-Petviashvili (cKP) equation related to cylindrical geometry, as one type of the variable-coefficient KP equation, is widely used to describe nonlinear phenomena in fluid, plasma and other fields. With symbolic computation, we have derived the multi-soliton solutions, rational solutions, lump solutions and interaction solutions to the cKP equation based on its bilinear representation. The interaction solutions include two types: The interaction between lump and stripe, and the interaction between lump and soliton. Moreover, we have proposed a new approach to search for the interaction solutions, which can decrease the complexity of the associated nonlinear algebraic equations via reducing the number of the variables. The fast calculation approach provides the condition for the predictability of the interaction solution.

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.

Dynamics of the soliton waves, breather waves, and rogue waves to the cylindrical Kadomtsev-Petviashvili equation in pair-ion–electron plasma

A lot of work has been reported to present some numerical results on pair-ion–electron plasmas. However, very few works have reported the corresponding mathematical analytical results in these aspects. In this work, we study a cylindrical Kadomtsev-Petviashvili (CKP) equation, which can be derived from pair-ion–electron plasmas. We further report some interesting mathematical analytical results, including some dynamics of soliton waves, breather waves, and rogue waves in pair-ion–electron plasma via the CKP equation. Using a novel gauge transformation, the Grammian N-soliton solutions of the CKP equation are found analytically. Based on the bilinear transformation method, the breather wave solutions are obtained explicitly under some parameter constraints. Furthermore, we construct the rogue waves using the long wave limit method. In addition, some remarkable characteristics of these soliton solutions are analyzed graphically. According to analytic solutions, the influences of each parameter on the dynamics of the soliton waves, breather waves, and rogue waves are discussed, and the method of how to control such nonlinear phenomena is suggested.

Interaction and resonance of fast magnetoacoustic solitary waves in cylindrical geometry for dense astrophysical plasmas

The cylindrical Kadomtsev-Petviashvili (CKP) equation, also known as Johnson's equation, is derived for dense electron ion plasmas in the small amplitude limit to study the interaction and resonance of two magnetoacoustic solitons in cylindrical geometry. The exact analytical solutions of the CKP equation are derived using Hirota's direct method and a novel gauge transformation. It is observed that cylindrical geometry transforms the line solitons to the horseshoe-like solitary structures. For multi-solitons, it is shown that the interaction parameter for the CKP solitary wave depends upon the plasma parameters of the system. For the present investigation, plasma parameters are chosen that are customarily found in the white dwarf stars. The interaction of magnetoacoustic solitons are shown to change their trajectory and introduce a phase shift. The resonance condition is developed for the CKP equation which shows that the amplitude of resultant solitons becomes four times the amplitude of initial solitons.

Solutions of nonplanar KP-equations for dusty plasma system with GE-method

The propagation of three-dimensional nonlinear dust acoustic waves in a dusty plasma consisting of positive and negative dust grains as well as Boltzmann distributed electrons and ions is investigated. Using a reductive perturbation method, Cylindrical Kadomtsev-Petviashvili and Spherical Kadomtsev-Petviashvili equations, appropriate for describing the evolution of the system, are derived. The Generalized Expansion method is used to find the various solutions of the obtained nonlinear Kadomtsev-Petviashvili equations. New classes of triangular, hyperbolic solitary, rational, and Jacobi elliptical type solutions are obtained and graphically presented. It is found that the proposed dusty plasma model can support solitary waves with compressive and rarefactive potential pulses. Moreover, the effects of plasma parameters on this solitary wave structure are investigated. The current findings are applied to a cosmic-type plasma in different regions of space, viz. cometary tails, mesosphere, and Jupiter's magnetosphere as well as laboratory-type plasma such as low temperature experiments, where a dusty plasma with opposite polarity is dominant.

Cylindrical fast magnetosonic solitary waves in quantum degenerate electron-positron-ion plasma

The nonlinear properties of fast magnetosonic solitary waves in a quantum degenerate electron-positron (e-p) plasma in the presence of stationary ions for neutralizing the plasma background of bounded cylindrical geometry were studied. By employing the standard reductive perturbation technique and the quantum hydrodynamic model for the e-p fluid, the cylindrical Kadomtsev-Petviashvili (CKP) equation was derived for small, but finite, amplitude waves and was given the solitary wave solution for the parameters relevant to dense astrophysical objects such as white dwarf stars. By a suitable coordinate transformation, the CKP equation can be solved analytically. An analytical solution for magnetosonic solitons and periodic waves is presented. The numerical results reveal that the Bohm potential has a main effect on the periodic and solitary wave structures. By increasing the values of the plasma parameters, the amplitude of the solitary wave will be increased. The present study may be helpful in the understanding of nonlinear electromagnetic soliton waves propagating in both laboratory and astrophysical plasmas, and can help in providing good agreement between theoretical results and laboratory plasma experiments.

Propagation of cylindrical ion acoustic waves in a plasma with q-nonextensive electrons with nonthermal distribution

The propagation of ion acoustic waves (IAWs) in a cylindrical collisionless unmagnetized plasma, containing ions and electrons is investigated. The electrons are considered to be nonextensive and follow nonthermal distribution. The reductive perturbation technique (RPT) is used to obtain a nonlinear cylindrical Kadomtsev-Petviashvili (CKP) evolution equation. This equation is solved analytically. The effects of plasma parameters on the IAWs characteristics are discussed in details. Both compressive and rarefactive solitons are found to be created in the proposed plasma system. The profile of IAWs is found to depend on the nonextensive and nonthermal parameters. The present study is useful for understanding IAWs in the regions where mixed electron distribution in space, or laboratory plasmas, exist.