Painleve Analysis Research Articles

Painleve analysis of travelling wave solutions and analysis of energy estimates for a nonlinear Sobolev-type equation

Painleve analysis of travelling wave solutions and analysis of energy estimates for a nonlinear Sobolev-type equation

The Painleve Analysis of an Ordinary Differential Equation Which Describes a Traveling-Wave Solution of the Oskolkov–Benjamin–Bona–Mahony–Burgers Equation

The Painleve Analysis of an Ordinary Differential Equation Which Describes a Traveling-Wave Solution of the Oskolkov–Benjamin–Bona–Mahony–Burgers Equation

Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

On the solitary wave solution of the viscosity capillarity van der Waals p-system along with Painleve analysis

This work proposes two well known schemes, painleve analysis test and auxiliary equation mapping (AEM) algorithm to find new solitary wave solutions for van der Waals p-system (VDW) in the viscosity capillarity version. For the clarity of physical features, new solitary wave solutions like periodic, kink and anti kink type, doubly periodic, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions, etc. are articulated coupled with graphically patterns 2D, 3D and density plots to visualize the dynamics of our model. This study reveals that the methods used are simple, reliable, unambiguous and concise as compared to many other techniques for solving nonlinear partial differential equations (NLPDE). A new direct way to solve NLPDEs is provided using this approach. The method allows one to obtain new accurate solutions of the solitary wave that cannot be obtained using other methodologies, and it can be implemented on a computer using computer algebraic systems such as Mathematica or Maple.

New extended Kadomtsev–Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions

In this paper, we develop a new extended Kadomtsev–Petviashvili (eKP) equation. We use the Painleve analysis to confirm the integrability of the eKP equation. We derive the bilinear form, multiple soliton solutions and lump solutions via using the Hirota’s direct method. Moreover, the soliton, breather and lump interaction solutions for this model are also obtained as well. Graphs are drawn to illustrate the abundant dynamical behaviors of the obtained solutions.

Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen–Lee–Liu model

In this paper, we will compute the conservation laws (CLs) of Chen–Lee–Liu equation (CLLE) with the help of scaling invariance technique. We will use Euler and Homotopy operators for the evaluation of conserved densities and fluxes. We will also obtain optical dromions with the help of Unified method (UM). By using this method, we will get domain walls, solitary wave and elliptic wave solutions. Moreover, we will investigate the stability as well as the integrability of the governing model by using linear stability technique and Painleve analysis respectively.

Soliton solutions, Painleve analysis and conservation laws for a nonlinear evolution equation

In this paper, we investigate a reputed nonlinear partial differential equation (NLPDE) known as geophysical Korteweg-de Vries (GPKdV) equation. We implement a renowned Unified method (UM) of nonlinear (NL) sciences for the extraction of polynomial and rational function solutions of GPKdV equation, which degenarate to various wave solutions like solitary, soliton (dromions) and elliptic wave solutions. Further more, for the analysis of the integrability of our governing model, we apply Painlevé (P) algorithm to check the singularities structure of the model. The fulfillment of all the requirements of the P test indicates the solvability of the governing equation with the help of inverse scattering transformation (IST) or some linear techniques. Moreover, we calculate conservation laws (CLs) in polynomial form as conserved fluxes and densities by implementing dilation symmetry. We utilize Euler and Homotopy operators for the evaluation of the intended conserved quantities.

Bubbles interactions in fluidized granular medium for the van der Waals hydrodynamic regime

This paper investigates the fluidized granular materials (FGM) with the van der Waals normal form (VDWF) under the effects of friction and viscosity. The system of macroscopic balance is presented, including the mass, momentum, and energy equations of local densities. For two different types of collisions, elastic and inelastic collisions, analytical solutions of the nonlinear PDEs governing the granular model are investigated using the hydrodynamic equations for granular matter motion. The integrability of the proposed model is analyzed by applying the Painleve analysis. Moreover, the Backlund transformation (BT) is established using the Painleve truncation expansion. New traveling wave solutions of the VDWF within FGM are obtained by using the BT, tanh function, Jacobi elliptic function methods to study the phase separation phenomenon. As two pairs of rarefaction and shock waves emerge and travel away giving the appearance of bubbles, the resulting solutions of the proposed model show a behavior similar to those found in the molecular dynamic simulations. The dispersion relation and their properties to the model equation are investigated. Besides, stability analysis of the VDWF in its ODE form is demonstrated using the phase portrait classifications. Finally, using two- and three-dimensional graphics for seeking model solutions under the influence of friction and viscosity, qualitative agreements with previous related works are shown.

Propagation of dust-acoustic nonlinear waves in a superthermal collisional magnetized dusty plasma

The nonlinear features of dust-acoustic waves (DAWs) propagating in a multicomponent, collisional, magnetized dusty plasma whose constituents are negative dust grains, superthermal ions, and electrons are investigated. The hydrodynamic fluid equations are reduced to a damped Zakharov–Kuznetsov (DZK) equation, using the reductive perturbation technique. The DZK equation is solved using both the generalized $$(G^{{\prime }}/G)$$ –expansion method and the Painleve analysis method. Each method gives a class of solutions. This indicates that these methods are sufficient to give all possible solutions of the DZK equation. These solutions successfully describe different kinds of nonlinear waves such as explosive, soliton, shocklike, periodical, and cnoidal waves. Additionally, the effects of different plasma parameters such as relative densities and temperatures as well as superthermality parameters of ions and electrons on the behavior of the obtained diverse waves are investigated. The relevance of the present investigation can be used to understand the cosmic dust-laden plasmas.

New Meromorphic Solutions of the Cubic Nonlinear Schrödinger Equation by Using the Complex Method

We show abundant simply periodic solutions, trigonometric solutions, hyperbolic function solutions and Weierstrass elliptic solutions of the reduction of the cubic nonlinear Schrodinger equation by the complex method with Painleve analysis, and some solutions appear to be new. At last, we give some computer simulations to illustrate our main results.

Comment on “Painleve analysis and integrability of the trapped ionic system” by M. Benkhali, J. Kharbach, I. El Fakkousy, W. Chatar, A. Rezzouk, and M. Ouazzani-Jamil

Comment on “Painleve analysis and integrability of the trapped ionic system” by M. Benkhali, J. Kharbach, I. El Fakkousy, W. Chatar, A. Rezzouk, and M. Ouazzani-Jamil

Modified Vakhnenko–Parkes equation with power law nonlinearity: Painlevé analysis, analytic solutions and conservation laws

The Vakhnenko equation governs the propagation of high-frequency waves in a relaxation medium. The Painleve analysis of modified Vakhnenko–Parkes equation is performed by using the Kruskal approach. The Lie symmetry formalism is also applied to derive symmetries of generalised modified Vakhnenko–Parkes equation with power law nonlinearity. The exact solutions of this equation are obtained in the form of power series; and other explicit exact solutions in the form of trigonometric, hyperbolic and Jacobi elliptic functions are obtained for modified Vakhnenko–Parkes equation. The bright soliton solutions of generalised modified Vakhnenko–Parkes equation are also obtained. The conservation laws are obtained with the help of multiplier approach. The 3D plots and contour plots are shown by considering the possibilities of arbitrary function occurring in the solutions.

Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model

Microtubules (MTs) are series of protein and protofilaments. These are very important in cell divisioning, designing and nonlinear phenomena in the field of engineering and science. In this article, we study multiple soliton interactions with the help of Hirota bilinear method. We also study dromions for MTs model with the help of the extended $$(G'/G)$$ -expansion method. At the end, we will discuss integrability of model through the Painleve analysis.

Kadomtsev–Petviashvili hierarchy: two integrable equations with time-dependent coefficients

In this paper, we investigate two members of the Kadomtsev–Petviashvili (KP) hierarchy, each with time-dependent coefficients. We use the Painleve analysis and the WTC–Kruskal method to study the compatibility conditions to ensure the integrability of each equation. We use the simplified Hirota’s method to derive multiple soliton solutions for each equation.

Painlevé analysis, group classification and exact solutions to the nonlinear wave equations

This paper is concerned with the general regular long-wave (RLW) types of equations. By the combination of Painleve analysis and Lie group classification method, the conditional Painleve property (PP) and Backlund transformations (BTs) of the nonlinear wave equations are provided under some conditions. Then, all of the point symmetries of the nonlinear RLW types of equations are obtained, the exact solutions to the equations are investigated. Particularly, some explicit solutions are provided by the special function and Φ-expansion method.

Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan — Kundu — Lakshmanan Equation

This paper considers the Radhakrishnan — Kundu — Laksmanan (RKL) equation to analyze dispersive nonlinear waves in polarization-preserving fibers. The Cauchy problem for this equation cannot be solved by the inverse scattering transform (IST) and we look for exact solutions of this equation using the traveling wave reduction. The Painleve analysis for the traveling wave reduction of the RKL equation is discussed. A first integral of traveling wave reduction for the RKL equation is recovered. Using this first integral, we secure a general solution along with additional conditions on the parameters of the mathematical model. The final solution is expressed in terms of the Weierstrass elliptic function. Periodic and solitary wave solutions of the RKL equation in the form of the traveling wave reduction are presented and illustrated.

Inhomogeneous exchange within higher-dimensional ferrites: The singularity structure analysis and pattern formations

Abstract In this work the propagation of nonlinear electromagnetic short waves in a ferromagnetic medium with inhomogeneous exchange is discussed. It is shown that such waves propagate perpendicular to the magnetization density. The evolution of waves under the influence of perturbations in one transverse dimension is considered. As result, we derive a new (2+1)-dimensional evolution system in which painleve analysis reveals its integrability properties. In the wake of such analysis, using the arbitrary functions to enter into the Laurent series of solutions to the above system, we present some typical class of excitations.

Generation of soliton, cnoidal, and periodic waves during pumping GaAs by an electron beam

Abstract Increasing the lifetime of plasma semiconductors that exposed to an electron beam is an important challenge. The dynamical features of growing nonlinear waves, which transfer the energy through the semiconductors to produce defects, are examined. For this purpose, a quantum fluid model including degenerate pressures, exchange-correlation effects, and Bohm potentials are considered for GaAs semiconductor plasma composed of electrons, holes, and electron beam. The basic equations are reduced to Kadomtsev-Petviashvili (KP) equation. The latter is solved analytically using the Painleve analysis method, which yields three different nonlinear solutions describe cnoidal, periodic, and soliton waves. It is found that positive and negative nonlinear wave potentials can exist. The polarity of the wave depends on an electron streaming velocity and moving frame speed. Furthermore, controlling the streaming electron density, electron streaming velocity, and direction of transverse perturbation has great importance to minimize the effects of unfavorable nonlinear waves. The relevance of electron streaming velocity and transverse perturbation direction on nonlinear wave existence is highlighted.

Group classification, conservation laws and Painlevé analysis for Klein–Gordon–Zakharov equations in (3 $$+$$ + 1)-dimension

In this paper, we study Klein–Gordon–Zakharov equations which describe the propagation of strong turbulence of the Langmuir wave in a high-frequency plasma. Using the symbolic manipulation tool Maple, the classifications of symmetry algebra are carried out, and the construction of several local non-trivial conservation laws based on a direct method of Anco and Bluman is illustrated. Starting with determination of symmetry algebra, the one- and two-dimensional optimal systems are constructed, and optimality is also established using various invariant functions of full adjoint action. Apart from classification and construction of several conservation laws, the Painleve analysis is also performed in a symbolic manner which describes the non-integrability of equations.

Symmetry Reduction Related by Nonlocal Symmetry and Explicit Solutions for the Whitham-Broer-Kaup System

The nonlocal symmetry for the Whitham-Broer-Kaup (WBK) equation is obtained by the Painleve analysis. With the introduction of two auxiliary dependent variables, the nonlocal symmetry is localized to the Lie point symmetry. The reciprocal transformation is established by the localization nonlocal symmetry procedure and can be used to construct a link between the WBK equation and the usual Burgers equation. By using the similarity reductions related with the nonlocal symmetry of the Burgers equation, we derive the symmetry invariant solutions for the WBK equation. Furthermore, the power-series solutions are computed by using the power-series theory.