scholarly journals Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Gang-Wei Wang ◽  
Tian-Zhou Xu
Keyword(s):  
Lie Algebra ◽  
Group Analysis ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Optimal System ◽  
Explicit Solutions ◽  
Kawahara Equation ◽  
Lie Symmetry Method ◽  
Symmetry Method ◽  
Similarity Reductions

The simplified modified Kawahara equation with variable coefficients is studied by using Lie symmetry method. Then we obtain the corresponding Lie algebra, optimal system, and the similarity reductions. At last, we also give some new explicit solutions for some special forms of the equations.

Symmetry analysis and invariant solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili equation

2018 ◽  
Vol 15 (08) ◽  
pp. 1850125 ◽  
Author(s):  
Vishakha Jadaun ◽  
Sachin Kumar
Keyword(s):  
Lie Algebra ◽  
Zeta Functions ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Petviashvili Equation ◽  
Symmetry Method ◽  
Similarity Reductions

Based on Lie symmetry analysis, we study nonlinear waves in fluid mechanics with strong spatial dispersion. The similarity reductions and exact solutions are obtained based on the optimal system and power series method. We obtain the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reductions for the [Formula: see text]-dimensional Kadomtsev–Petviashvili equation. For different Lie algebra, Lie symmetry method reduces Kadomtsev–Petviashvili equation into various ordinary differential equations (ODEs). Some of the solutions of [Formula: see text]-dimensional Kadomtsev–Petviashvili equation are of the forms — traveling waves, Weierstrass’s elliptic and Zeta functions and exponential functions.

Exact solutions of Zabolotskaya–Khokhlov equation using optimal system

2018 ◽  
Vol 15 (07) ◽  
pp. 1850111
Author(s):  
Mohsin Umair ◽  
Tooba Feroze
Keyword(s):  
Exact Solutions ◽  
Lie Symmetry ◽  
Optimal System ◽  
Lie Symmetry Method ◽  
Symmetry Method

In this paper, using the Lie symmetry method, we obtain optimal system, group invariants and exact solutions of [Formula: see text] dimensional Zabolotskaya–Khokhlov equation.

Classical Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave system

2020 ◽  
pp. 2150052
Author(s):  
Preeti Devi ◽  
K. Singh
Keyword(s):  
Wave Equation ◽  
Symmetry Algebra ◽  
Lie Symmetry ◽  
Lie Symmetries ◽  
Wave System ◽  
Lie Symmetry Method ◽  
Long Wave ◽  
Virasoro Type ◽  
Symmetry Method ◽  
Similarity Reductions

In the present work, classical Lie symmetry method is adopted to obtain the Lie symmetries and similarity reductions of the (2 + 1)-dimensional dispersive long wave equation. We also demonstrate that the symmetry algebra of this equation is an infinte dimensional, together with Kac–Moody–Virasoro type subalgebras.

Numerical analytic method for solving a large class of generalized non-homogeneous variable coefficients KdV problems based on lie symmetries

2020 ◽  
pp. 2150115
Author(s):  
Azadeh R. Moghaddam ◽  
Mortaza Gachpazan
Keyword(s):  
Numerical Methods ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Analytic Method ◽  
Initial Value ◽  
Lie Symmetry Method ◽  
Generalized Kdv Equation ◽  
Symmetry Method ◽  
Reduced Problem

In this study, we use Lie symmetry method to reduce a generalized KdV equation with initial and boundary conditions with non-homogeneous variable coefficients to initial value problem. In particular, we concentrate on the cases that the reduced IVP cannot be solved analytically. We also compare the approximated solutions of IVP using numerical methods with IBVP, and note that they are more efficient than doing the same procedure for IBVP. In fact, it is shown that reducing IBVP to IVP and solving the reduced problem numerically will lead to more accurate solutions.

scholarly journals Lie Symmetry Analysis, Exact Solutions, and Conservation Laws for the Generalized Time-Fractional KdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Maria Ihsane El Bahi ◽  
Khalid Hilal
Keyword(s):  
Differential Equation ◽  
Conservation Laws ◽  
Nonlinear Partial Differential Equation ◽  
Lie Symmetry ◽  
Power Series Method ◽  
Lie Symmetry Method ◽  
Symmetry Operators ◽  
Generalized Time ◽  
Symmetry Method ◽  
Conservation Theorem

In this paper, the problem of constructing the Lie point symmetries group of the nonlinear partial differential equation appeared in mathematical physics known as the generalized KdV-Like equation is discussed. By using the Lie symmetry method for the generalized KdV-Like equation, the point symmetry operators are constructed and are used to reduce the equation to another fractional ordinary differential equation based on Erdélyi-Kober differential operator. The symmetries of this equation are also used to construct the conservation Laws by applying the new conservation theorem introduced by Ibragimov. Furthermore, another type of solutions is given by means of power series method and the convergence of the solutions is provided; also, some graphics of solutions are plotted in 3D.

New soliton solutions of the CH–DP equation using Lie symmetry method

2018 ◽  
Vol 32 (20) ◽  
pp. 1850234 ◽  
Author(s):  
A. H. Abdel Kader ◽  
M. S. Abdel Latif
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Dark Soliton ◽  
Lie Symmetry ◽  
Jacobi Elliptic Functions ◽  
Lie Symmetry Method ◽  
Wave Solutions ◽  
Symmetry Method

In this paper, using Lie symmetry method, we obtain some new exact traveling wave solutions of the Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Some new bright and dark soliton solutions are obtained. Also, some new doubly periodic solutions in the form of Jacobi elliptic functions and Weierstrass elliptic functions are obtained.

scholarly journals Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Lamine Thiam ◽  
Xi-zhong Liu
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Symmetry Reduction ◽  
Lie Symmetry ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Lie Symmetry Method ◽  
Consistent Riccati Expansion ◽  
Symmetry Method

The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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