Cylindrical Kadomtsev-Petviashvili Equation Research Articles

Auto-Bäcklund transformation and new exact solutions of the generalized variable-coefficients two-dimensional Korteweg–de Vries model

Describing plasma physics, cluster physics, and geophysical fluid dynamics, a generalized variable-coefficient two-dimensional Korteweg–de Vries (GVCKdV) model is hereby under investigation. In this paper, an extended-variable-coefficient homogeneous balance method with symbolic computation is used to obtain an auto-Bäcklund transformation, some rational solutions, and new soliton-type solutions for the GVCKdV model. The solutions of the well-known cylindrical Kadomtsev-Petviashvili equation can be recovered as special cases of the results obtained here.

Auto-Bäcklund Transformation andSoliton-Type Solutions of the Generalized Variable-CoefficientKadomtsev–Petviashvili Equation

Using the truncated Painlevé expansion, an auto-Bäcklundtransformation and soliton-type solutions of the generalizedvariable-coefficient Kadomtsev–Petviashvili (GKP) equation are obtainedby symbolic computation. Since the cylindrical Korteweg-de Vries (cKdV)equation, the cylindrical KP (cKP) equation and the generalized cKP(GcKP) equation are all special cases of the GKP equation, we can alsoobtain the corresponding results of these equations.

Cylindrical Kadomtsev–Petviashvili equation for relativistically magnetosonic solitary wave in the collisionless plasma

In the presence of an applied static and uniform magnetic field, a cylindrical Kadomtsev–Petviashvili (CKP) equation is derived for a relativistically magnetosonic solitary wave propagating in a collisionless plasma in the case of weak relativistic limit. The equation is solved in a stationary frame to obtain a particular solution. It is shown that the externally imposed magnetic field greatly influences the phase velocity, amplitude and width of the CKP soliton.

Variable-coefficient KP equation and solitonic solution for two-temperature ions in dusty plasma

In this Letter, the cylindrical KP equation, which is a variable-coefficient equation, is obtained by using the reductive perturbation method for unmagnified, collisionless and two-temperature ions in dusty plasma. With the help of generalized projected Riccati equation expansion method, the cylindrical KP equation is solved and many solitonic solutions are obtained. Some important parameters for the existence of the compressive and rarefactive solitons are discussed and the effects caused by transverse perturbations are also discussed.

Abundant families of Jacobi elliptic function-like solutions for a generalized variable coefficients 2D KdV equation via the extended mapping method

Based on the symbolic computation system Mathematica and an extended mapping method, we study a generalized variable coefficients 2D KdV equation and obtain new families of exact solutions, which include Jacobi elliptic function-like, combined Jacobi elliptic function-like, soliton-like, kink-bell soliton-like and triangular function-like solutions. The solutions of the well-known cylindrical Kadomtsev–Petviashvili equation can be recovered as special case of the 2D KdV equation.

Cylindrical nebulons, symbolic computation and Bäcklund transformation for the cosmic dust acoustic waves

In a cosmic dusty plasma, the dust-acoustic-wave propagation may be described by a cylindrical Kadomtsev-Petviashvili equation. In this Letter, for such modeling of environments like supernova shells, Saturn’s F-ring, etc., cylindrical nebulons and an auto-Bäcklund transformation are presented via symbolic computation. Nebulon structures are discussed, and possibly observable effects are proposed for cosmic plasmas.

The nonlinear evolution of ring dark solitons in Bose–Einstein condensates

The dynamics of the ring dark soliton in a Bose–Einstein condensate (BEC) with thin disc-shaped potential is investigated analytically and numerically. Analytical investigation shows that the ring dark soliton in the radial non-symmetric cylindrical BEC is governed by a cylindrical Kadomtsev–Petviashvili equation, while the ring dark soliton in the radial symmetric cylindrical BEC is governed by a cylindrical Korteweg–de Vries equation. The reduction to the cylindrical KP or KdV equation may be useful to understand the dynamics of a ring dark soliton. The numerical results show that the evolution properties and the snaking of a ring dark soliton are modified significantly by the trapping.

Auto-Bäcklund Transformation and Exact Solutions to the Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients**The project supported by National Natural Science Foundation of China under Grant No. 10072013 and the State Key Basic Research Development Program under Grant No. G1998030600

By using the extended homogeneous balance method, a new auto-Bäcklund transformation(BT) to the generalized Kadomtsev–Petviashvili equation with variable coefficients (VCGKP) are obtained. And making use of the auto-BT and choosing a special seed solution, we get many families of new exact solutions of the VCGKP equations, which include single soliton-like solutions, multi-soliton-like solutions, and special-soliton-like solutions. Since the KP equation and cylindrical KP equation are all special cases of the VCGKP equation, and the corresponding results of these equations are also given respectively.

Cylindrical dust acoustic waves with transverse perturbation

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

SIMILARITY REDUCTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT KADOMTSEV–PETVIASHVILI EQUATION WITH SYMBOLIC COMPUTATION

We use computerized symbolic computation to study the similarity reductions of a generalized Kadomtsev–Petviashvili (KP) equation, with its space- and time-dependent coefficients arising in plasma physics and fluid mechanics. We obtain a couple of reductions involving those coefficient functions, one of which is to the fourth Painlevé equation, while the other is to one of the second Painlevé equation, the first Painlevé equation or a Weierstrass elliptic function equation. Our results agree with the Painlevé Conjecture. The cylindrical KP equation is presented as an example, which describes nonlinear cylindrical water waves in shallow water with a weak azimuthal dependence.

Cylindrical solitary waves and their interaction in Bénard-Marangoni layers

We consider a motionless Newtonian and incompressible shallow liquid layer bounded below by a bottom plate where temperature is held fixed and above with a free deformable surface whose tension is linearly temperature dependent and on which the heat flux is fixed. When the dimensionless surface tension gradient, measured by a Marangoni number, slightly exceeds its critical value, radially symmetric, long-wavelength excitations obey a dissipative cylindrical Korteweg--de Vries (DCKdV) equation. A dissipative cylindrical Kadomtsev-Petviashvili (DCKP) equation is also derived for nearly radially symmetric disturbances. Exact solitary wave solutions of the DCKP equation are found and the solitary wave solutions of the DCKdV equation are also discussed. Finally the head-on collision between two concentric cylindrical solitary waves is considered and its solitonic character is displayed.

On an analytical method and soliton-typed solutions for certain variable-coefficient partial differential equations in nonlinear mechanics

Among the currently interesting nonlinear topics are the variable-coefficient generalizations of the Kadomtsev-Petviashvili equation (GvcKPs), which are of practical value in mechanical and physical sciences. In this note, we propose a generalized variable-coefficient tanh method with computerized symbolic manipulation for a GvcKP to construct certain soliton-typed analytical solutions. In illustration, the nearly concentric Korteweg-de Vries (KdV) equation is presented as an example, and the cylindrical KP equation as a counter example. Similar success not shown here is on the variable-coefficient KdV equations.

Nonlinear dynamics of vortices in ultraclean type-II superconductors: Integrable wave equations in cylindrical geometry.

Due to their short coherence lengths and relatively large energy gaps, the high-transition temperature superconductors are very likely candidates as ultraclean materials at low temperature. This class of materials features significantly modified vortex dynamics, with very little dissipation at low temperature. The motion is then dominated by wave propagation, being in general nonlinear. Here two-dimensional vortex motion is investigated in the ultraclean regime for a superconductor described in cylindrical geometry. The small-amplitude limit is assumed, and the focus is on the long-wavelength limit. Results for both zero and nonzero Hall force are presented, with the effects of nonlocal vortex interaction and vortex inertia being included within London theory. Linear and nonlinear problems are studied, with a predisposition toward the more analytically tractable situations. For a nonlinear problem in 2+1 dimensions, the cylindrical Kadomtsev-Petviashvili equation is derived. Hall angle measurements on high-${\mathit{T}}_{\mathit{c}}$ superconductors indicate the need to investigate the properties of such a completely integrable wave equation. \textcopyright{} 1996 The American Physical Society.

Truncated Painlevé expansion and a wide-ranging type of generalized variable-coefficient Kadomtsev-Petviashvili equations

Among the topics attracting much attention in mathematical physics are the variable-coefficient generalizations of the well-known Kadomtsev-Petviashvili equation (GVCKP). We show that the application of the truncated Painlevé expansion and symbolic computation leads to a new class of analytical solutions to a wide-ranging type of GVCKPs, and an auto-Bäcklund transformation. The examples presented are solutions to the cylindrical Kadomtsev-Petviashvili equation and the nearly concentric Korteweg-de Vries equation.

Soliton-Typed Solutions to the Generalized Cylindrical Kadomtsev-Petviashvili Equation with Variable Coefficients

Soliton-Typed Solutions to the Generalized Cylindrical Kadomtsev-Petviashvili Equation with Variable Coefficients

Deformations of algebraic curves and integrable nonlinear evolution equations

A brief exposition of applications of the methods of algebraic geometry to systems integrable by the IST method with variable spectral parameters is presented. Usually, theta-functional solutions for these systems are generated by some deformations of algebraic curves. The deformations of algebraic curves are also related with theta-functional solutions of Yang-Mills self-duality equations which contain special systems with a variable spectral parameter as a special reduction. Another important situation in which the deformations of algebraic curves naturally occur is the KdV equation with string-like boundary conditions. Most important concrete examples of systems integrable by the IST method with variable spectral parameter having different properties within a framework of the behavior of moduli of underlying curves, analytic properties of the Baker-Akhiezer functions, and the qualitative behavior of the solutions, are vacuum axially symmetric Einstein equations, the Heisenberg cylindrical magnet equation, the deformed Maxwell-Bloch system, and the cylindrical KP equation.

Generalization to cylindrical coordinate system for the higher order KdV and KP equation

Two new model equations which we call a higher order cylindrical KdV equation and a higher order cylindrical KP equation are proposed when we consider the 2-dimensional modulation of water waves with surface tension in cylindrical coordinate system. Some exact (2+1)-dimensional solitary wave solutions have been obtained by relating them to the higher order KdV equation through an appropriate variable transformation.

Model Equations and Their Solitary Wave Solutions in Cylindrical Coordinate System11Project Supported by National Natural Science Foundation of China and the Foundation of Higher Education of China.

The method of multiple-scales is used to investigate the evolution of a weak nonlinear internal waves between two-layer fluids in cylindrical coordinate system. Two reduced model wave equations, which we call a modified cylindrical KdV equation for axially symmetric case and a modified cylindrical KP equation for non-axially symmetric case, are derived and their solitary wave solutions are also obtained by relating them to the modified KdV equation by means of an appropriate variable transformation.

The cylindrical Kadomtsev-Petviashvili equation; its Kac-Moody-Virasoro algebra and relation to KP equation

We show that a class of integrable nonlinear differential equations in 2+1 dimensions, including the physically important cylindrical Kadomtsev-Petviashvili equation, has a symmetry algebra with a specific Kac-Moody-Virasoro structure. The isomorphism between this algebra and the symmetry algebra of the KP equation, is used to transform the entire class into the KP equation, by a Lie point transformation. This transformation makes it possible to solve completely the Cauchy problem for the cylindrical KP equation.