Spherical Kadomtsev Petviashvili Equation Research Articles

A Note on the Transformation of Variables of KP Equation, Cylindrical KP Equation and Spherical KP Equation**Supported by the National Natural Science Foundation of China under Grant No. 11771381

In a recent article (Commun. Theor. Phys. 67 (2017) 207), three (2+1)-dimensional equations — KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by using different transformation of variables, respectively. In this short note, by adding an adjustment item to original transformation, three more general transformation of variables corresponding to above three equations have been given. Substituting the solutions of the KdV equation into our transformation of variables, more new exact solutions of the three (2+1)-dimensional equations can be obtained.

Solitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation**Supported by the National Natural Science Foundation of China under Grant No. 11301153 and the Doctoral Foundation of Henan University of Science and Technology under Grant No. 09001562, and the Science and Technology Innovation Platform of Henan University of Science and Technology under Grant No. 2015XPT001

Three (2+1)-dimensional equations–KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the KdV equation have been known already, substituting the solutions of the KdV equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three (2+1)-dimensional equations can be obtained successfully.

Spherical solitons in Earth’S mesosphere plasma

Soliton formation in Earth’s mesosphere plasma is described. Nonlinear acoustic waves in plasmas with two-temperature ions and a variable dust charge where transverse perturbation is dealt with are studied in bounded spherical geometry. Using the perturbation method, a spherical Kadomtsev–Petviashvili equation that describes dust acoustic waves is derived. It is found that the parameters taken into account have significant effects on the properties of nonlinear waves in spherical geometry.

Ion-acoustic solitons in plasma: an application to Saturn’s magnetosphere

Soliton formation in plasma is addressed. Nonlinear acoustic waves in plasma where the combined effects of bounded spherical geometry and the transverse perturbation are dealt with, in two-temperature electron plasma are studied. Using the perturbation method, a spherical Kadomtsev-Petviashvili equation (SKP) that describes the ion acoustic waves is derived. The plasma is modeled by a kappa distribution function for both electrons components as found in Saturn’s magnetosphere. It is found that parameters taken into account have significant effects on the properties of nonlinear waves. The model is applied to Saturn’s magnetosphere, where two temperature superthermal electrons are present. Hence, ion acoustic waves are deduced for three regions, namely, the inner, the intermediate and the outer Saturn’s magnetosphere. We point out, that this work has been motivated by recent observations of Saturn’s magnetosphere.

Ion acoustic solitary waves in nonplanar plasma with two-temperature kappa distributed electrons

Ion acoustic (IA) solitons are studied in a two-temperature kappa-distributed electron plasma. As we expect for double-Boltzmann electrons, solitons of both polarities can exist in this model. Some density ratios support coexistence of both the polarities. We have also derived an deformed Korteweg–de Vries equation for IA waves by using reductive perturbation technique. The influence of geometry and suprathermality effects on IA waves is investigated. This model can support both compressive and rarefractive solitons for some plasma parameters. The low temperature electron component favors the emergence and development of rarefractive solitons. It is found that the amplitude and the velocity of IA solitons increase with superthermal index of hot electrons (κ 2). Also it is clear that the increase rate of velocity and amplitude in spherical geometry is greater than that for cylindrical geometry. We have considered the transverse perturbation for IA solitons with spherical symmetry. It is shown that the evolution of IA solitons is governed by a variable coefficient spherical Kadomtsev–Petviashvili equation.

Nonplanar Ion Acoustic Waves with Nonthermal Electrons

Using the standard reductive perturbation technique, nonlinear cylindrical and spherical Kadomtsev-Petviashvili (KP) equations are derived for the propagation of ion acoustic solitary waves in an unmagnetized collisionless plasma with nonthermal electrons and warm ions. The influence of nonthermally distributed electrons and the effects caused by the transverse perturbation on cylindrical and spherical ion acoustic waves (IAWs) are investigated. It is observed that the presence of nonthermally distributed electrons has a significant role in the nature of ion acoustic waves. In particular, when the nonthermal distribution parameter β takes certain values the usual cylindrical KP equation (CKPE) and spherical KP equation (SKPE) become invalid. One then has to have recourse to the modified CKPE or SKPE. Analytical solutions of both CKPE and SKPE and their modified versions are discussed in the present paper. The present investigation may have relevance in the study of propagation of IAWs in space and laboratory plasmas.

Spherical Kadomtsev–Petviashvili equation for dust acoustic waves with dust size distribution and two-charges-ions

The nonlinear dust acoustic waves in dusty plasmas with negative as well as positive ions and the combined effects of bounded spherical geometry and the transverse perturbation and the size distribution of dust grains are studied. Using the perturbation method, a spherical Kadomtsev–Petviashvili (SKP) equation that describes the dust acoustic waves is deduced.

Multiple Soliton-Like Solutions and Similarity Reductionsof a Spherical Kadomtsev–Petviashvili Equation fromPlasma Physics

A spherical Kadomtsev–Petviashvili (SKP) equationfor dust acoustic or ion-acoustic waves is studied. Similarityreductions of the SKP equation are obtained with the one-parameter(ε) Lie group of infinitesimal transformations andClarkson–Kruskal direct method. The SKP equation is also solved with ageneralized tanh function method.

Reply to: “Comment on: ‘Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation’ ” [Phys. Lett. A 361 (2007) 520

Abstract Dusty plasmas exist almost everywhere in the Universe and relevant nonlinear studies are encouragingly going to non-planar geometry. On our recent construction of a spherical Kadomtsev–Petviashvili model for the dust-ion-acoustic waves in a cosmic dusty plasma [B. Tian, Y.-T. Gao, Phys. Lett. A 340 (2005) 243], Hong [W.P. Hong, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.11.021 , in press] comments that certain interesting coordinate transformations exist, with presentation of a transformed equation (TE) and nebulon solutions. In this Reply, we point out that the TE is valuable to the studies on both cosmic plasmas and applied mathematics. We obtain an auto-Backlund transformation on the TE and more general nebulons. With series of pictures, we discuss nebulon structures out of the TE, and address that there are cosmic plasma systems for which the TE is valid. We remove a constraint set in that Comment so that the TE can be useful for the Saturn-F-ring-typed dusty plasmas as well.

Comment on: “Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation” [Phys. Lett. A 340 (2005) 243

Abstract Tian and Gao [B. Tian, Y.T. Gao, Phys. Lett. A 340 (2005) 243] have recently constructed a spherical Kadomtsev–Petviashvili equation for the dust-ion-acoustic waves with zenith-angle perturbation in a cosmic dusty plasma, and symbolically obtained and discussed spherical structures of the expanding dark, shrinking dark, expanding bright and shrinking bright nebulons. In this Comment, we point out that certain simple coordinate transformations exist, which make the analytic nebulon solutions more easily accessible. We also show that the transformed solutions can result in a couple of nebulon solutions in addition to the second-order nebulons found by Tian and Gao.

Spherical Kadomtsev–Petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation

Plasmas and dust exist almost everywhere in the Universe. For the dust ion-acoustic waves with zenith-angle perturbation, a spherical Kadomtsev–Petviashvili equation is derived with symbolic computation. Spherical structures of the expanding dark nebulon, shrinking dark nebulon, expanding bright nebulon and shrinking bright nebulon are symbolically obtained and discussed. Possibly-observable nebulonic effects for the future space and laboratory plasma experiments are proposed.

Spherical nebulons and B�cklund transformation for a space or laboratory un-magnetized dusty plasma with symbolic computation

Plasmas and dust are a couple of the ubiquitous elements of the Universe. The nonlinear dust-acoustic-wave propagation in a space or laboratory un-magnetized dusty plasma is hereby considered, which is described by a spherical Kadomtsev-Petviashvili equation. By virtue of computerized symbolic computation, we obtain the analytically-expressed spherical nebulons and an auto-Backlund transformation for the electrostatic potential in such a plasma. With figures, we discuss the features of the nebulon structures, and propose some (3+1)-dimensional, possibly-observable nebulonic effects for the future space/laboratory plasma experiments.

Singularity Analysis of Spherical Kadomtsev–Petviashvili Equation

The (2+1)-dimensional spherical Kadomtsev–Petviashvili (SKP) equation of J.-K. Xue [Phys. Lett. A 314 (2003) 479] fails the Painleve test for integrability at the highest resonance, where a nontrivial compatibility condition for recursion relations appears. This compatibility condition, however, is sufficiently weak and thus allows the SKP equation to possess an integrable (1+1)-dimensional reduction, which is found by the method of truncated singular expansion.

A spherical KP equation for dust acoustic waves

The nonlinear dust acoustic waves in dusty plasmas with the combined effects of bounded spherical geometry and the transverse perturbation are studied. Using the perturbation method, a spherical Kadomtsev–Petviashvili (SKP) equation that describes the dust acoustic waves is deduced for the first time. A particular solution of this SKP equation is also obtained. It is shown that the dust acoustic solitary wave can exist in the SKP equation.