Lie symmetry analysis, optimal system and conservation laws of a new (2+1)-dimensional KdV system
TL;DR: In this paper, Lie point symmetries of a new (2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple, and the one-dimensional optimal system associated with corresponding Lie algebra is obtained.
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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TL;DR: In this paper , a method called $$\Big (\frac{G'}{G^2}\Big )$$ -extension method is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative.
Abstract: The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called $$\Big (\frac{G'}{G^2}\Big )$$ -extension method which works well and reveals exact solutions is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative. This method is a very useful approach and extremely utility compared to other analytical methods. With the proposed method, there are three unique types of solutions such as hyperbolic, trigonometric and rational solutions. This approach can similarly be applied to other nonlinear fractional models.
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TL;DR: In this article , the extended G′/G method is used to investigate the space-time fractional Burger-like equation and the space time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics.
Abstract: In this study, the extended G′/G method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. A symbolic computation program called Maple was used to implement the method in a dependable and effective way. There are also a few graphs provided for the solutions. Using the suggested method to solve these equations, we have provided many new exact solutions that are distinct from those previously found. By offering insightful explanations of many nonlinear systems, the study’s findings add to the body of literature. The results revealed that the suggested method is a valuable mathematical tool and that using a symbolic computation program makes these tasks simpler, more dependable, and quicker. It is worth noting that it may be used for a wide range of nonlinear evolution problems in mathematical physics. The study’s findings may have an influence on how different physical problems are interpreted.
1 citations
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TL;DR: In this article , generalized fractional porous medium equations (GFPMEs) with integer order derivative were investigated and the complete group classification of the equations in consideration were performed with respect to their arbitrary parameters.
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01 Jan 1986
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Abstract: 1 Introduction to Lie Groups- 11 Manifolds- Change of Coordinates- Maps Between Manifolds- The Maximal Rank Condition- Submanifolds- Regular Submanifolds- Implicit Submanifolds- Curves and Connectedness- 12 Lie Groups- Lie Subgroups- Local Lie Groups- Local Transformation Groups- Orbits- 13 Vector Fields- Flows- Action on Functions- Differentials- Lie Brackets- Tangent Spaces and Vectors Fields on Submanifolds- Frobenius' Theorem- 14 Lie Algebras- One-Parameter Subgroups- Subalgebras- The Exponential Map- Lie Algebras of Local Lie Groups- Structure Constants- Commutator Tables- Infinitesimal Group Actions- 15 Differential Forms- Pull-Back and Change of Coordinates- Interior Products- The Differential- The de Rham Complex- Lie Derivatives- Homotopy Operators- Integration and Stokes' Theorem- Notes- Exercises- 2 Symmetry Groups of Differential Equations- 21 Symmetries of Algebraic Equations- Invariant Subsets- Invariant Functions- Infinitesimal Invariance- Local Invariance- Invariants and Functional Dependence- Methods for Constructing Invariants- 22 Groups and Differential Equations- 23 Prolongation- Systems of Differential Equations- Prolongation of Group Actions- Invariance of Differential Equations- Prolongation of Vector Fields- Infinitesimal Invariance- The Prolongation Formula- Total Derivatives- The General Prolongation Formula- Properties of Prolonged Vector Fields- Characteristics of Symmetries- 24 Calculation of Symmetry Groups- 25 Integration of Ordinary Differential Equations- First Order Equations- Higher Order Equations- Differential Invariants- Multi-parameter Symmetry Groups- Solvable Groups- Systems of Ordinary Differential Equations- 26 Nondegeneracy Conditions for Differential Equations- Local Solvability- In variance Criteria- The Cauchy-Kovalevskaya Theorem- Characteristics- Normal Systems- Prolongation of Differential Equations- Notes- Exercises- 3 Group-Invariant Solutions- 31 Construction of Group-Invariant Solutions- 32 Examples of Group-Invariant Solutions- 33 Classification of Group-Invariant Solutions- The Adjoint Representation- Classification of Subgroups and Subalgebras- Classification of Group-Invariant Solutions- 34 Quotient Manifolds- Dimensional Analysis- 35 Group-Invariant Prolongations and Reduction- Extended Jet Bundles- Differential Equations- Group Actions- The Invariant Jet Space- Connection with the Quotient Manifold- The Reduced Equation- Local Coordinates- Notes- Exercises- 4 Symmetry Groups and Conservation Laws- 41 The Calculus of Variations- The Variational Derivative- Null Lagrangians and Divergences- Invariance of the Euler Operator- 42 Variational Symmetries- Infinitesimal Criterion of Invariance- Symmetries of the Euler-Lagrange Equations- Reduction of Order- 43 Conservation Laws- Trivial Conservation Laws- Characteristics of Conservation Laws- 44 Noether's Theorem- Divergence Symmetries- Notes- Exercises- 5 Generalized Symmetries- 51 Generalized Symmetries of Differential Equations- Differential Functions- Generalized Vector Fields- Evolutionary Vector Fields- Equivalence and Trivial Symmetries- Computation of Generalized Symmetries- Group Transformations- Symmetries and Prolongations- The Lie Bracket- Evolution Equations- 52 Recursion Operators, Master Symmetries and Formal Symmetries- Frechet Derivatives- Lie Derivatives of Differential Operators- Criteria for Recursion Operators- The Korteweg-de Vries Equation- Master Symmetries- Pseudo-differential Operators- Formal Symmetries- 53 Generalized Symmetries and Conservation Laws- Adjoints of Differential Operators- Characteristics of Conservation Laws- Variational Symmetries- Group Transformations- Noether's Theorem- Self-adjoint Linear Systems- Action of Symmetries on Conservation Laws- Abnormal Systems and Noether's Second Theorem- Formal Symmetries and Conservation Laws- 54 The Variational Complex- The D-Complex- Vertical Forms- Total Derivatives of Vertical Forms- Functionals and Functional Forms- The Variational Differential- Higher Euler Operators- The Total Homotopy Operator- Notes- Exercises- 6 Finite-Dimensional Hamiltonian Systems- 61 Poisson Brackets- Hamiltonian Vector Fields- The Structure Functions- The Lie-Poisson Structure- 62 Symplectic Structures and Foliations- The Correspondence Between One-Forms and Vector Fields- Rank of a Poisson Structure- Symplectic Manifolds- Maps Between Poisson Manifolds- Poisson Submanifolds- Darboux' Theorem- The Co-adjoint Representation- 63 Symmetries, First Integrals and Reduction of Order- First Integrals- Hamiltonian Symmetry Groups- Reduction of Order in Hamiltonian Systems- Reduction Using Multi-parameter Groups- Hamiltonian Transformation Groups- The Momentum Map- Notes- Exercises- 7 Hamiltonian Methods for Evolution Equations- 71 Poisson Brackets- The Jacobi Identity- Functional Multi-vectors- 72 Symmetries and Conservation Laws- Distinguished Functionals- Lie Brackets- Conservation Laws- 73 Bi-Hamiltonian Systems- Recursion Operators- Notes- Exercises- References- Symbol Index- Author Index
8,118 citations
"Lie symmetry analysis, optimal syst..." refers methods in this paper
...A 287 257–67 [5] Ovsiannikov L V 1982 Group Analysis of Differentional Equations (New York: Academic) [6] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer) [7] Ibragimov N H 2007 J. Math....
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...To classify group invariant solutions, one should classify the subalgebra of the Lie algebra generated by Lie point symmetries, so the definition of one-dimensional optimal system was came up with by Ovsiannikov [5, 9, 10] using a global matrix for the adjoint transformation, and then Olver [6, 16] introduced one method, which was used invariants and adjoint operators to simplify a general element in Lie algebra as much as possible....
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TL;DR: In this paper, a general theorem on conservation laws for arbitrary differential equations is proved, where the adjoint equation inherits all symmetries of the original equation and the conservation law does not require the existence of a Lagrangian.
762 citations
TL;DR: A direct and systematic algorithm is proposed to find one- dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra.
Abstract: A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on different values of all the invariants, the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system, with no further proof. To leave the algorithm clear, we illustrate each stage with a couple of well-known examples: the Korteweg-de Vries equation and the heat equation. Finally, we apply our method to the Novikov equation and use the found optimal system to investigate the corresponding invariant solutions.
98 citations
TL;DR: In this article, a generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painleve analysis and it has been found that this equation passes painleve test for $$\alpha =\beta $$ which implies affirmation toward the complete integrability.
Abstract: In present work, new form of generalized fifth-order nonlinear integrable equation has been investigated by locating movable critical points with aid of Painleve analysis and it has been found that this equation passes Painleve test for $$\alpha =\beta $$
which implies affirmation toward the complete integrability. Lie symmetry analysis is implemented to obtain the infinitesimals of the group of transformations of underlying equation, which has been further pre-owned to furnish reduced ordinary differential equations. These are then used to establish new abundant exact group-invariant solutions involving various arbitrary constants in a uniform manner.
77 citations
TL;DR: In this paper, the similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations, and some exact analytic solutions are considered by the power series method.
Abstract: In this paper, variable coefficients Kawahara equation (VCKE) and variable coefficients modified Kawahara equation (VCMKE), which arise in modeling of various physical phenomena, are studied by Lie group analysis. The similarity reductions and exact solutions are derived by determining the complete sets of point symmetries of these equations. Moreover, some exact analytic solutions are considered by the power series method. Further, a generalized -expansion method is applied to VCKE and VCMKE for constructing some new exact solutions. As a result, hyperbolic function solutions, trigonometric function solutions and some rational function solutions with parameters are furnished. Copyright © 2012 John Wiley & Sons, Ltd.
68 citations