Some group-invariant solutions of potential Kadomtsev–Petviashvili equation by using Lie symmetry approach

2018 ◽  
Vol 92 (2) ◽  
pp. 781-792 ◽  
Author(s):  
Mukesh Kumar ◽  
Atul Kumar Tiwari
Keyword(s):  
Lie Symmetry ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Symmetry Approach ◽  
Petviashvili Equation ◽  
Group Invariant Solutions ◽  
Lie Symmetry Approach

scholarly journals Group Invariant Solutions and Conserved Quantities of a (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1012
Author(s):  
Innocent Simbanefayi ◽  
Chaudry Masood Khalique
Keyword(s):  
Real World ◽  
Nonlinear Partial Differential Equations ◽  
Elliptic Functions ◽  
Lie Symmetry ◽  
Conserved Quantities ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Higher Dimensional ◽  
Petviashvili Equation ◽  
Group Invariant Solutions

In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.

Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation

2013 ◽  
Vol 68 (8-9) ◽  
pp. 510-514 ◽  
Author(s):  
Andrew G. Johnpillai ◽  
Abdul H. Kara ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Symmetry Algebra ◽  
Lie Symmetry ◽  
Point Of View ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Symmetry Approach ◽  
De Vries ◽  
Group Invariant Solutions ◽  
Symmetry Generators

We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

Group Invariant Solutions and Conservation Laws of the Fornberg– Whitham Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 489-496 ◽  
Author(s):  
Mir Sajjad Hashemi ◽  
Ali Haji-Badali ◽  
Parisa Vafadar
Keyword(s):  
Exact Solutions ◽  
Reduction Method ◽  
Lie Symmetry ◽  
First Integrals ◽  
Invariant Solutions ◽  
Analysis Method ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Whitham Equation

In this paper, we utilize the Lie symmetry analysis method to calculate new solutions for the Fornberg-Whitham equation (FWE). Applying a reduction method introduced by M. C. Nucci, exact solutions and first integrals of reduced ordinary differential equations (ODEs) are considered. Nonlinear self-adjointness of the FWE is proved and conserved vectors are computed

scholarly journals Some General New Einstein Walker Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Partial Differential Equations ◽  
Symmetry Group ◽  
Lie Symmetry ◽  
Group Method ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Walker Manifold

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

scholarly journals Invariant Solutions and Conservation Laws of the (2 + 1)-Dimensional Boussinesq Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Wenjuan Rui ◽  
Peiyi Zhao ◽  
Yufeng Zhang
Keyword(s):  
Conservation Laws ◽  
Boussinesq Equation ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Symmetry Approach ◽  
Lie Symmetry Approach ◽  
Conservation Theorem ◽  
Lagrange Approach

Invariant solutions and conservation laws of the (2 + 1)-dimensional Boussinesq equation are studied. The Lie symmetry approach is used to obtain the invariant solutions. Conservation laws for the underlying equation are derived by utilizing the new conservation theorem and the partial Lagrange approach.

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