A Direct Algorithm Maple Package of One-Dimensional Optimal System for Group Invariant Solutions

AbstractIn this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

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Optimal System and Group Invariant Solutions of the Whitham-Broer-Kaup System

Advances in Mathematical Physics ◽

10.1155/2019/1892481 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Xiaorui Hu ◽

Yongyang Jin ◽

Kai Zhou

Keyword(s):

Lie Algebra ◽

Nonlinear Waves ◽

Optimal System ◽

Invariant Solutions ◽

One Dimensional ◽

Nonlocal Symmetries ◽

Group Invariant ◽

Group Invariant Solutions

From the nonlocal symmetries of the Whitham-Broer-Kaup system, an eight-dimensional Lie algebra is found and the corresponding one-dimensional optimal system is constructed to provide an inequivalent classification. Six types of inequivalent group invariant solutions are demonstrated, some of which reflect the interactions between soliton and other nonlinear waves.

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Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2018-0311 ◽

2019 ◽

Vol 20 (6) ◽

pp. 713-723 ◽

Cited By ~ 2

Author(s):

Pabitra Kumar Pradhan ◽

Manoj Pandey

Keyword(s):

Adjoint Representation ◽

Group Classification ◽

Optimal System ◽

Invariant Solutions ◽

Horizontal Temperature Gradient ◽

One Dimensional ◽

Group Invariant ◽

Group Invariant Solutions ◽

The One ◽

Complete Symmetry Group

AbstractA complete symmetry group classification for the system of shallow water equations with the horizontal temperature gradient, also known as Ripa system, is presented. A rigorous and systematic procedure based on the general invariants of the adjoint representation is used to construct the one-dimensional optimal system of the Lie algebra. The complete inequivalence class of the group invariant solutions are obtained by using the one-dimensional optimal system. One such solution of the Ripa system is used to study the evolutionary behaviour of the discontinuity wave.

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Optimal systems and their group-invariant solutions to geodesic equations

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501354 ◽

2019 ◽

Vol 16 (09) ◽

pp. 1950135

Author(s):

Bismah Jamil ◽

Tooba Feroze ◽

Muhammad Safdar

Keyword(s):

Nonlinear Systems ◽

Separation Of Variables ◽

Noether Symmetries ◽

Invariant Solutions ◽

One Dimensional ◽

Group Invariant ◽

First Order ◽

Geodesic Equations ◽

Integrating Factor ◽

Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

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Optimal system, group invariant solutions and conservation laws of the CGKP equation

Nonlinear Dynamics ◽

10.1007/s11071-017-3392-6 ◽

2017 ◽

Vol 88 (4) ◽

pp. 2503-2511

Author(s):

Lihua Zhang ◽

Fengsheng Xu ◽

Lixin Ma

Keyword(s):

Conservation Laws ◽

Optimal System ◽

Invariant Solutions ◽

Group Invariant ◽

Group Invariant Solutions

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Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

Discrete Dynamics in Nature and Society ◽

10.1155/2019/9609357 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Ben Gao ◽

Yanxia Wang

Keyword(s):

Chaplygin Gas ◽

Group Method ◽

Invariant Solutions ◽

One Dimensional ◽

Group Invariant ◽

Lie Group Method ◽

Two Component ◽

Present Universe ◽

Group Invariant Solutions ◽

Similarity Reductions

In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.

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