scholarly journals Nonlocal Symmetries of Systems of Evolution Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Renat Zhdanov
Keyword(s):  
Evolution Equations ◽  
Lie Symmetry ◽  
Nonlocal Symmetries ◽  
The Core ◽  
Potential Symmetry ◽  
Transformation Of Variables ◽  
Nonlocal Transformation

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. This fact is in the core of our approach to computation of potential and more general nonlocal symmetries of systems of evolution equations having nontrivial Lie symmetry. Several examples are considered.

scholarly journals Nonlocal symmetries of evolution equations

2009 ◽  
Vol 60 (3) ◽  
pp. 403-411 ◽  
Author(s):  
Renat Zhdanov
Keyword(s):  
Evolution Equations ◽  
Nonlocal Symmetries

scholarly journals Lie Symmetry Analysis for Soliton Solutions of Generalised Kadomtsev-Petviashvili-Boussinesq Equation in (3+1)-dimensions

2021 ◽  
Author(s):  
VISHAKHA JADAUN ◽  
Navnit Jha ◽  
Sachin Ramola
Keyword(s):  
Vector Fields ◽  
Evolution Equations ◽  
Invariant Solution ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Closed Form Solutions ◽  
Exact Invariant ◽  
Infinitesimal Transformations

The Lie group of infinitesimal transformations technique and similarity reduction is performed for obtaining an exact invariant solution to generalized Kadomstev-Petviashvili-Boussinesq (gKPB) equation in (3+1)-dimensions. We obtain generators of infinitesimal transformations, which provide us a set of Lie algebras. In addition, we get geometric vector fields, a commutator table of Lie algebra, and a group of symmetries. It is observed that the analytic solution (closed-form solutions) to the nonlinear gKPB evolution equations can easily be treated employing the Lie symmetry technique. A detailed geometrical framework related to the nature of the solutions possessing traveling wave, bright and dark soliton, standing wave with multiple breathers, and one-dimensional kink, for the appropriate values of the parameters involved.

Lie symmetry analysis and dynamical structures of soliton solutions for the (2 + 1)-dimensional modified CBS equation

2020 ◽  
Vol 34 (25) ◽  
pp. 2050221
Author(s):  
S. Kumar ◽  
D. Kumar
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Surface Condition ◽  
Dynamical Behavior ◽  
Lie Symmetry ◽  
Nonlinear Evolution Equations ◽  
Optimal System ◽  
Explicit Solutions ◽  
Nonlinear Odes ◽  
Symmetry Reductions

In this present article, the new [Formula: see text]-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, commutation table, invariant surface condition, Lie symmetry reductions, infinitesimal generators and explicit solutions are constructed. As we all know, an optimal system contains constructively important information about the various types of exact solutions and it also offers clear understandings into the exact solutions and its features. The symmetry reductions of [Formula: see text]-dimensional mCBS equation is derived from an optimal system of one-dimensional subalgebra of the Lie invariance algebra. Then, the mCBS equation can further be reduced into a number of nonlinear ODEs. The generated explicit solutions have different wave structures of solitons and they are analyzed graphically and physically in order to exhibit their dynamical behavior through 3D, 2D-shapes and respective contour plots. All the produced solutions are definitely new and totally different from the earlier study of the Manukure and Zhou (Int. J. Mod. Phys. B 33, (2019)). Some of these solutions are demonstrated by the means of solitary wave profiles like traveling wave, multi-solitons, doubly solitons, parabolic waves and singular soliton. The calculations show that this Lie symmetry method is highly powerful, productive and useful to study analytically other nonlinear evolution equations in acoustics physics, plasma physics, fluid dynamics, mathematical biology, mathematical physics and many other related fields of physical sciences.

Lie symmetry reductions, abound exact solutions and localized wave structures of solitons for a (2 + 1)-dimensional Bogoyavlenskii equation

2021 ◽  
pp. 2150252
Author(s):  
Sachin Kumar ◽  
Monika Niwas
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Evolution Equations ◽  
Dynamical Behavior ◽  
Lie Symmetry ◽  
Nonlinear Evolution Equations ◽  
Mathematical Methods ◽  
Closed Form Solutions ◽  
Symmetry Reductions ◽  
Engineering Sciences

By applying the two efficient mathematical methods particularly with regard to the classical Lie symmetry approach and generalized exponential rational function method, numerous exact solutions are constructed for a (2 + 1)-dimensional Bogoyavlenskii equation, which describes the interaction of Riemann wave propagation along the spatial axes. Moreover, we obtain the infinitesimals, all the possible vector fields, optimal system, and Lie symmetry reductions. The governing Bogoyavlenskii equation is converted into various nonlinear ordinary differential equations through two stages of Lie symmetry reductions. Accordingly, abundant exact closed-form solutions are obtained explicitly in terms of independent arbitrary functions, rational functions, trigonometric functions, and hyperbolic functions with arbitrary free parameters. The dynamical behavior of the resulting soliton solutions is presented through 3D-plots via numerical simulation. Eventually, single solitons, multi-solitons with oscillations, kink wave with breather-type solitons, and single lump-type solitons are obtained. The proposed mathematical techniques are effective, trustworthy, and reliable mathematical tools to work out new exact closed-form solutions of various types of nonlinear evolution equations in mathematical physics and engineering sciences.

Lie and Noether symmetry analysis in Brans–Dicke cosmology

2018 ◽  
Vol 33 (34) ◽  
pp. 1850198 ◽  
Author(s):  
Sourav Dutta ◽  
Santu Mondal
Keyword(s):  
Evolution Equations ◽  
Analytic Solution ◽  
State Parameter ◽  
Lie Symmetry ◽  
Noether Symmetry ◽  
Coordinate Systems ◽  
Group Invariant ◽  
Augmented System ◽  
Equation Of State Parameter ◽  
Group Invariant Solutions

This paper is aimed to study the group invariant solutions of the evolution equations in Brans–Dicke cosmology. In this context, we have considered the flat homogeneous Brans–Dicke (BD) scalar field in the background of flat homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model and have used Lie and Noether symmetry on the augmented system. From Lie symmetry we have determined the unknown potential for two different values of the equation of state parameter w. Then assuming that the Lagrangian admits a Noether symmetry, an analytic solution of the system is obtained in both old and new coordinate systems.

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