Cylindrical KdV Equation Research Articles

Nonclassical Symmetries, Nonlinear Self-adjointness, Conservation Laws and Some New Exact Solutions of Cylindrical KdV Equation

Nonclassical Symmetries, Nonlinear Self-adjointness, Conservation Laws and Some New Exact Solutions of Cylindrical KdV Equation

A Novel Lie Group Classification Method for Generalized Cylindrical KdV Type of Equation: Exact Solutions and Conservation Laws

This paper is concerned with the generalized cylindrical KdV (cKdV) equation, which includes the cKdV types of equations. By a novel Lie group classification method, all of the point symmetries of the variable-coefficient partial differential equations are obtained under the compatibility condition, which is necessary and the basis for the complete group classification. Then, the symmetries of the corresponding nonlinear constant-coefficient PDE and the classical cKdV equation are presented accordingly. Furthermore, the exact solutions to the generalized cKdV type of equation are provided in general case, and the explicit conservation laws of the equation are given through the nonlinearly self-adjoint procedure.

Analytical solutions of cylindrical and spherical dust ion-acoustic solitary waves

Abstract In the present work, employing the conventional reductive perturbation method to the field equations of an unmagnetized dusty plasma consisting of inertial ions, Boltzmann electrons and stationary dust particles in the nonplanar geometry we derived cylindrical(spherical) KdV and mKdV equations. Being aware of the fact that there exists no exact analytical solution for the progressive waves for such evolution equations we presented the exact analytical solution of a generalized form of such evolution equations in the planar geometry and used this solution to obtain an analytical approximate progressive wave solution to the generalized evolution equation. Then the progressive wave solutions for the cylindrical(spherical) KdV and m-KdV equations is obtained as some special cases. The analytical approximate solutions so obtained are compared with the numerical solutions of these equations. The results reveal that both solutions are in a good agreement. One advantage of present analytical approximate solution is that it allows readers to gain information, share understandings, or carry out a physical parametric study on the evolution solution behavior as well as the solution calculation has no CPU time-consuming or round off error.

Cylindrical and spherical solitary waves in an electron-acoustic plasma with vortex electron distribution

We consider the nonlinear propagation of electron-acoustic 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 (spherical) coordinates by employing the reductive perturbation technique. The modified cylindrical (spherical) KdV equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, this evolution equation cannot be reduced to the conventional KdV equation. A new family of closed form analytical approximate solution to the evolution equation and a comparison with numerical solution are presented and the results are depicted in some 2D and 3D figures. The results reveal that both solutions are in good agreement and the method can be used to obtain a new progressive wave solution for such evolution equations. Moreover, the resulting closed form analytical solution allows us to carry out a parametric study to investigate the effect of the physical parameters on the solution behavior of the modified cylindrical (spherical) KdV equation.

Decay Mode Solutions for the Supersymmetric Cylindrical KdV Equation

Abstract Supersymmetric cylindrical KdV equation is presented. Decay mode solutions for the supersymmetric KdV equation are derived by supersymmetric Hirota operator.

On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids

With the inhomogeneities of media taken into account, under investigation is hereby a generalized variable‐coefficient forced Korteweg‐de Vries (vc‐fKdV) equation, which describes shallow‐water waves, internal gravity waves, etc. Under an integrable constraint condition on the variable coefficients, in this paper, the complete integrability of the generalized vc‐fKdV equation is proposed. By virtue of a generalization of Bells polynomials, we systematically present its bilinear representations, Bäcklund transformations, Lax pairs and Darboux covariant Lax pairs, which can be reduced to the ones of some integrable models, such as vcKdV model, cylindrical KdV equation, and an analytical model of tsunami generation. It is very interesting that its bilinear formulism is free for the integrable constraint condition. Besides, researching the Lax equations yield its infinitely conservation laws, all conserved densities and fluxes of them are obtained by explicit recursion formulas. Furthermore, by considering its bilinear formulism with an extra auxiliary variable, we present the soliton solutions and Riemann theta function periodic wave solutions of the equation. According to the constraint among the nonlinear, dispersive, and line‐damping coefficients, we further discuss the solitonic structures and interaction properties by some graphic analysis. Finally, the relationships between the periodic wave solutions and soliton solutions are presented in detail by a limiting procedure.

Propagation of cylindrical and spherical dust–ion acoustic solitary waves in a relativistic dusty plasma

The properties of cylindrical and spherical dust–ion acoustic solitary waves (DIASW) in an unmagnetized dusty plasma comprising of relativistic ions, Boltzmann electrons, and stationary dusty particles are investigated. Under a suitable coordinate transformation, the cylindrical KdV equation can be solved analytically. The change of the DIASW structure due to the effect of geometry, relativistic streaming factor, ion density and electron temperature is studied by numerical calculation of the cylindrical/spherical Kdv equation. It is noted that with ion pressure the effect of relativistic streaming factor to solitary waves structure is different. Without ion pressure, as the relativistic streaming factor decreases, the amplitude of the solitary wave decreases. However, when the ion pressure is taken into account, the amplitude decreases as the relativistic streaming factor increases and is highly sensitive to relativistic streaming factor. Our results may have relevance in the understanding of astrophysical plasmas.

Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients

This paper obtains an exact solitary wave solution of the Korteweg–de Vries equation with power law nonlinearity with time-dependent coefficients of the nonlinear as well as the dispersion terms. In addition, there are time-dependent damping and dispersion terms. The solitary wave ansatz is used to carry out the analysis. It is only necessary for the time-dependent coefficients to be Riemann integrable. As an example, the solution of the special case of cylindrical KdV equation falls out.

Collisional phase shifts of ring dark solitons in inhomogeneous Bose–Einstein condensates

Abstract The head-on collisions of two ring dark solitons in inhomogeneous Bose–Einstein condensates (BECs) with thin disk-shaped potential are studied by the extended Poincare–Lighthill–Kuo (PLK) perturbation method. The result shows that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by two modified cylindrical KdV equations in the respective reference frames. In particular, the analytical phase shifts induced by the head-on collisions between two ring dark solitary waves are derived, and the result shows that the phase shifts change with the radial coordinate r according to the ( 1 + σ r 2 ) r − 1 / 3 law (where σ ∼ ω r 2 / ω z 2 ), which are quite different with the homogeneous case.

Head-on collision of ring dark solitons in Bose–Einstein condensates

The ring dark solitons and their head-on collisions in a Bose–Einstein condensates with thin disc-shaped potential are studied. It is shown that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by the two cylindrical KdV equations in the respective reference frames. By using the extended Poincare–Lighthill–Kuo perturbation method, the analytical phase shifts following the head-on collisions between two ring dark solitary waves are derived. It is shown that the phase shifts decrease with the radial coordinate r according to the r−1/3 law and depend on the initial soliton amplitude and radius.

The adiabatic approximation solutions of cylindrical and spherical dust ion-acoustic solitary waves

By using the equivalent particle theory, the adiabatic approximation solutions of the Korteweg–de Vries type equation (including KdV equation, cylindrical KdV equation and spherical KdV equation) in dust ion-acoustic solitary waves were obtained. The method can be extended to other nonlinear evolution equations.

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.

An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications

By using the homogeneous balance principle, an auto-Backlund transformation (BT) to the generalized KdV equation with variable coefficients is derived. The auto-BT only involves one quadratic homogeneity equation to be solved. Solving the homogeneity equation by use of the ε-expansion method and using the auto-BT, generally speaking, we can obtain an exact solution containing N-solitary wave of the generalized KdV equation with variable coefficients. As an illustrative example, we obtain an exact solution containing 2 solitary wave of the equation in detail. Since the generalized KdV (GKdV) equation, cylindrical KdV equation and variable coefficient KdV equation are all the special cases of the generalized KdV equation with variable coefficients, the corresponding results of these equations are also given respectively.

Solitary waves in an inhomogeneous rod composed of a general hyperelastic material

In this paper, we study analytically the development of a soliton propagating along a circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material density. The purpose is to provide analytical descriptions for the following two phenomena found, respectively, in numerical and perturbation studies: (1) Fission of a soliton. When a soliton moves from a part of the rod with thick cross-sections to a part with thin cross-sections, it will split into two or more solitons; (2) When a soliton propagates along a rod with slowly decreasing radius, it will develop into a solitary wave with a shelf behind. By using a nondimensionalization process and the reductive perturbation technique, we derive a variable-coefficient Korteweg–de Vries (vcKdV) equation as the model equation. The inverse scattering transforms are used to study the vcKdV equation. By considering the associated isospectral problem, the phenomenon of soliton fission is successfully explained. We are able to provide a condition that exactly how many solitons will emerge when a single soliton moves from a thick section to a thin section. Then, by introducing suitable variable transformations, we successfully manage to transform the vcKdV equation into a cylindrical KdV equation. As a result, several exact bounded solutions in terms of Airy function Ai and Bi are obtained. One of the solutions has the shape of a solitary wave with a shelf behind. Thus, it provides an analytical description for the perturbation and experimental results in literature. Comparisons are also made between the analytical solutions and numerical results, and good agreement is found.

Propagation of axi-symmetric nonlinear shallow water waves over slowly varying depth

A problem in nonlinear water-wave propagation on the surface of an inviscid, stationary fluid is presented. The primary surface wave, suitably initiated at some radius, is taken to be a slowly evolving nonlinear cylindrical wave (governed by an appropriate Korteweg–de Vries equation); the depth is assumed to be varying in a purely radial direction. We consider a sech 2 profile at an initial radius (which is, following our scalings, rather large), and we describe the evolution as it propagates radially outwards. This initial profile was chosen because its evolution over constant depth is understood both analytically and numerically, even though it is not an exact solitary-wave solution of the cylindrical KdV equation. The propagation process will introduce reflected and re-reflected components which will also be described. The precise nature of these reflections is fixed by the requirements of mass conservation. The asymptotic results presented describe the evolution of the primary wave, the development of an outward shelf and also an inward (reflected) shelf. These results make use of specific depth variations (which were chosen to simplify the solution of the relevant equations), and mirror those obtained for the problem of 1D plane-waves over variable depth, although the details here are more complex due to the axi-symmetry.

A note on an asymptotic solution of the cylindrical Korteweg–de Vries equation

The solitary wave solution of the cylindrical KdV equation is not generated by the typical initial profile often used, for example, in modern water-wave studies, namely, the familiar sech 2 profile. One reason is that this solution carries zero mass and therefore cannot, alone, describe the evolution of a wave of elevation. This paper describes an alternative approach; this is an asymptotic solution of the cylindrical KdV equation, given a sech 2 initial profile, based on an appropriate small parameter ( ε=1/initial radius, in non-dimensional variables). In terms of the limiting process ε→0, the various components of the resulting wave are described: the leading wave (a pulse), the trailing shelf and the oscillatory transition back to undisturbed conditions. The solution that is obtained takes a very simple form (and is therefore likely to be useful in more complicated scenarios), it satisfies mass conservation and each of the three elements of the solution satisfy the matching principle. The resulting evolution of the leading wave, and the complete structure of the asymptotic solution, are compared with numerical solutions of the cylindrical KdV equation; the agreement is exceptionally good, for both outward and inward propagation.

Exact Solutions of a Variable-Coefficient KdV Equation Arising in a Shallow Water

This paper studies a variable-coefficient KdV (vcKdV) equation arising in a shallow water. For a practically acceptable water bottom, we find that it can be transformed into the cylindrical KdV equation. As a result, several exact bounded solutions are obtained. One of the solutions has the shape of a solitary wave with a shelf behind, which confirms approximate analytical and numerical results in the literature.

Something hidden in the Fourier series and its partial sum: Solitary wave solutions to polynomial nonlinear differential equations

Transformation proposed in the author's recent work is utilized to obtain the polynomial and rational solitary wave solutions of polynomial nonlinear 1 + 1-dimension evolution equations, and thus some special solutions of polynomial nonlinear ODEs. In particular, several theorems are established to provide the necessary and sufficient conditions for the existence of these solutions. With the new transformation, polynomial solitary waves can be hidden in the form of a partial sum of the Fourier series of an intermediate variable (a curvilinear coordinate system), and therefore the balance between instabilities (nonlinear) and dispersive, dissipative, and other effects is well represented in a given polynomial nonlinear differential equation. A direct generalization of the method yields a solitary wave-like solution to the cylindrical KdV equation. In addition, we prove that the main transformation in this paper is equivalent to a simplified sine-Gordon equation. Finally, some new (complex) traveling wave solutions of the KdV equation, the Burgers equation and the modified KdV equation are obtained via our method. Among those new solutions, one of the KdV equation is shown taking similar form to the solitary wave solution of the nonlinear Schrödinger equation, which once again exposes the relationship between the two well-known nonlinear wave equations.

Discrete soliton equations and convergence acceleration algorithms

Some of the well-known convergence acceleration algorithms, when viewed as two-variable difference equations, are equivalent to discrete soliton equations. It is shown that the η-algorithm is nothing but the discrete KdV equation. In addition, one generalized version of the ϱ-algorithm is considered to be an integrable discretization of the cylindrical KdV equation.

Limit Manipulation between the Cylindrical Toda Equation and the Cylindrical KdV Equation

Limit Manipulation between the Cylindrical Toda Equation and the Cylindrical KdV Equation