Symmetry Reduction and Exact Solutions of Nonlinear Partial Differential Equations

Clarkson, Peter A.; Winternitz, P.

doi:10.1007/978-1-4612-1532-5_10

Cited by 32 publications

(38 citation statements)

References 137 publications

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“…To compute the solutions of this system, we have to satisfy the compatibility conditions provided by the algorithm presented in ref. [5]. Thus the solution will be a nonreducible PIS because L 1 does not contain any two dimensional subalgebra.…”

Section: Examplementioning

confidence: 99%

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Some spherical solutions of ideal magnetohydrodynamic equations

Picard

2007

JNMP

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Some spherical solutions of the ideal magnetohydrodynamic (MHD) equations are obtained from the method of the weak transversality method (WTM), which is based on Lie group theory. This analytical method makes use of the symmetry group of the MHD system in situations where the "classical" Lie approach of symmetry reductions is no longer applicable. Also, a brief physical interpretation of these solutions is given.

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Section: Examplementioning

confidence: 99%

“…[1]- [5], which also provide numerous examples of such calculations. Now, let us introduce the ideal magnetohydrodynamic (MHD) equations which describe nonrelativistic isentropic flows of an ideal (dissipative effects are negligible) and conductive (an infinite electrical conductivity is supposed) fluid, they have the following form [8]:…”

mentioning

confidence: 99%

Some spherical solutions of ideal magnetohydrodynamic equations

Picard

2007

JNMP

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“…[12,14,25,26,27,28]. Nonclassical (conditional) symmetries of nonlinear heat equations of the general form…”

Section: On Symmetries and Superposition Principles Of Nonlinear Heatmentioning

confidence: 99%

“…were completely described in [14]. In [12] nonclassical partial symmetries of equations from the family…”

Section: On Symmetries and Superposition Principles Of Nonlinear Heatmentioning

confidence: 99%

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Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Tychynin¹,

Petrova²,

Tertyshnyk³

2007

SIGMA

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Abstract. We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal symmetries and formulae of nonlocal nonlinear superposition of solutions of these equations were used then for construction of chains of exact solutions. Linearization by means of the Legendre transformations of a second-order PDE with three independent variables allowed to obtain nonlocal superposition formulae for solutions of this equation, and to generate new solutions from group invariant solutions of a linear equation.

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Partial Differential Equations: Painlevé Test

Conte

Musette

2020

The Painlevé Handbook

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Symmetry Reduction and Exact Solutions of Nonlinear Partial Differential Equations

Cited by 32 publications

References 137 publications

Some spherical solutions of ideal magnetohydrodynamic equations

Some spherical solutions of ideal magnetohydrodynamic equations

Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Partial Differential Equations: Painlevé Test

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