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Journal ArticleDOI: 10.1080/17455030.2019.1574410

Einstein's vacuum field equation: Painlevé analysis and Lie symmetries

04 Mar 2021-Waves in Random and Complex Media (Informa UK Limited)-Vol. 31, Iss: 2, pp 199-206
Abstract: The current study deals with investigation of Einstein's vacuum field equation for exploring movable critical points. We employ first the Painleve analysis, and then we use the auto-Backlund transf...

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Topics: Einstein (58%)
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10 results found



Journal ArticleDOI: 10.1007/S11071-019-04955-1
Abdul-Majid Wazwaz1, Lakhveer Kaur2Institutions (2)
01 Jul 2019-Nonlinear Dynamics
Abstract: In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painleve test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.

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Topics: Soliton (59%), Integrable system (50%)

36 Citations


Journal ArticleDOI: 10.1142/S0217979220502215
Sachin Kumar1, D. Kumar1Institutions (1)
Abstract: In this present article, the new (2 + 1)-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, comm...

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Topics: Lie group (62%), Symmetry (physics) (61%), Soliton (60%) ... show more

14 Citations


Open accessJournal ArticleDOI: 10.1016/J.AEJ.2020.12.040
Adil Jhangeer1, Hadi Rezazadeh, Reza Abazari2, Kenan Yildirim3  +2 moreInstitutions (5)
Abstract: The paper investigates Calogero-Degasperis-Fokas (CDF) equation, an exactly solvable third order nonlinear evolution equation (Fokas, 1980). All possible functions for the unknown function F ( ν ) in the considered equation are listed that contains the nontrivial Lie point symmetries. Furthermore, nonlinear self-adjointness is considered and for the physical parameter A ≠ 0 the equation is proved not strictly self-adjoint equation but it is quasi self-adjoint or more generally nonlinear self-adjoint equation. In addition, it is remarked that CDF equation admits a minimal set of Lie algebra under invariance test of Lie groups. Subsequently, Lie symmetry reductions of CDF equation are described with the assistance of an optimal system, which reduces the CDF equation into different ordinary differential equations. Besides, Lie symmetries are used to indicate the associated conservation laws. Also, the well-known ( G ′ / G ) -expansion approach is applied to obtain the exact solutions. These new periodic and solitary wave solutions are feasible to analyse many compound physical phenomena in the field of sciences.

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Topics: Lie group (61%), Ordinary differential equation (61%), Lie algebra (59%) ... show more

7 Citations


Journal ArticleDOI: 10.1088/1572-9494/ABFCB8
Mengmeng Wang1, Shoufeng Shen1, Li-Zhen Wang2Institutions (2)
Abstract: In this paper, Lie point symmetries of a new (2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system, associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear self-adjointness is provided and conservation laws of this KdV system are constructed.

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2 Citations


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21 results found



Journal ArticleDOI: 10.1063/1.525721
Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

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1,823 Citations


Open accessBook
01 Jan 1974-
Abstract: 1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of Transformations Abstract Definition.- 1.3. One-Parameter Group in the Plane.- 1.4. Proof That a One-Parameter Group Essentially Contains Only One Infinitesimal Transformation and Is Determined by It.- 1.5. Transformations Symbol of the Infinitesimal Transformation U.- 1.6. Invariant Functions and Curves.- 1.7. Important Classes of Transformations.- 1.8. Applications to Differential Equations Invariant Families of Curves.- 1.9. First-Order Differential Equations Which Admit a Group Integrating Factor Commutator.- 1.10. Geometric Interpretation of the Integrating Factor.- 1.11. Determination of First-Order Equations Which Admit a Given Group.- 1.12. One-Parameter Group in Three Variables More Variables.- 1.13. Extended Transformation in the Plane.- 1.14. A Second Criterion That a First-Order Differential Equation Admits a Group.- 1.15. Construction of All Differential Equations of First-Order Which Admit a Given Group.- 1.16. Criterion That a Second-Order Differential Equation Admits a Group.- 1.17. Construction of All Differential Equations of Second-Order Which Admit a Given Group.- 1.18. Examples of Application of the Method.- 2. Partial Differential Equations.- 2.0. Partial Differential Equations.- 2.1. Formulation of Invariance for the Special Case of One dependent and Two Independent Variables.- 2.2. Formulation of Invariance in General.- 2.3. Fundamental Solution of the Heat Equation Dimensional Analysis.- 2.4. Fundamental Solutions of Heat Equation Global Affinity.- 2.5. The Relationship Between the Use of Dimensional Analysis and Stretching Groups to Reduce the Number of Variables of a Partial Differential Equation.- 2.6. Use of Group Invariance to Obtain New Solutions from Given Solutions.- 2.7. The General Similarity Solution of the Heat Equation.- 2.8. Applications of the General Similarity Solution of the Heat Equation,.- 2.9. -Axially-Symmetric Wave Equation.- 2.10. Similarity Solutions of the One-Dimensional Fokker-Planck Equation.- 2.11. The Green's Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind - An Example of a P.D.E. in Three Variables Invariant Under a Two-Parameter Group.- 2.12. Infinite Parameter Groups - Derivation of the Poisson Kernel.- 2.13. Far Field of Transonic Flow.- 2.14. Nonlinear and Other Examples.- 2.15. Construction of Partial Differential Equations Invariant Under a Given Multi-parameter Group.- Appendix. Solution of Quasilinear First-Order Partial Differential Equations.- Bibliography. Part 1.- Bibliography. Part 2.

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998 Citations


Open accessPosted Content
P. S. Negi1Institutions (1)
Abstract: We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in ref. [20] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with the pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration . On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the structure of general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion [20], in fact, can not meet the requirement of the `actual mass' set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution. The criterion [20] provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic stellar structures like star clusters.

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791 Citations


Open accessJournal ArticleDOI: 10.1016/J.MCM.2003.12.010
Abdul-Majid Wazwaz1Institutions (1)
Abstract: In this paper, we establish exact solutions for nonlinear wave equations. A sine-cosine method is used for obtaining traveling wave solutions for these models with minimal algebra. The method is applied to selected physical models to illustrate the usage of our main results.

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Topics: Wave equation (53%)

571 Citations


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No. of citations received by the Paper in previous years
YearCitations
20217
20202
20191