A certain class of solutions of the nonlinear wave equation

Cieciura, Grzegorz; Grundland, A. M.

doi:10.1063/1.526102

Cited by 28 publications

(12 citation statements)

References 3 publications

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“…arises in the study of relativistic field equations [7] and as a constraint in reducing the nonlinear wave equation to an ODE [12,13]. One of its symmetries is generated by the vector field v = ϕ(u)∂ u where the function ϕ has to satisfy F ϕ u − ϕF u + Gϕ uu = 0 2Gϕ u − ϕG u = 0 .…”

Section: Examplementioning

confidence: 99%

Group Analysis of Differential Equations and Generalized Functions

Kunzinger¹,

Oberguggenberger²

2000

SIAM J. Math. Anal.

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Abstract. We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.

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Section: Examplementioning

confidence: 99%

Group Analysis of Differential Equations and Generalized Functions

Kunzinger¹,

Oberguggenberger²

2000

SIAM J. Math. Anal.

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“…x3 − u 2 x4 = 0, which is the eikonal equation in the five-dimensional space, will be investigated. As shown in [7,8], the system obtained is compatible if and only if χ = 0. We will construct general solutions of multi-dimensional systems of partial differential equations (PDE) 2 n u = 0, u µ u µ = 0 (2) in the four-and five-dimensional complex pseudo-Euclidean spaces.…”

Section: Introductionmentioning

confidence: 85%

On the general solution of the d’Alembert equation with a nonlinear eikonal constraint and its applications

Zhdanov¹,

Revenko

Fushchych

1995

Journal of Mathematical Physics

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“…Systems of PDEs similar to the type in (7.3) in n-dimensional real and complex vector spaces, with scalar product (∇ρ|∇ρ) = a(ρ) with arbitrary signature (p = 0, n − p), were investigated by geometrical methods for some classes of functions a and b [38][39][40][41]. Note that system (7.3) is invariant under the conformal group Conf (2, C) in general [39].…”

Section: )mentioning

confidence: 99%

Symmetry properties and explicit solutions of the generalized Weierstrass system

Bracken

Grundland

2001

Journal of Mathematical Physics

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The method of symmetry reduction is systematically applied to derive several classes of invariant solutions for the generalized Weierstrass system inducing constant mean curvature surfaces and to the associated two-dimensional nonlinear sigma model. A classification of subgroups with generic orbits of codimension one of the Lie point symmetry group for these systems provides a tool for introducing symmetry variables and reduces the initial systems to different nonequivalent systems of ordinary differential equations. We perform a singularity analysis for them in order to establish whether these ordinary differential equations have the Painlevé property. These ordinary differential equations can then be transformed to standard forms and next solved in terms of elementary and Jacobi elliptic functions. This results in a large number of new solutions and in some cases new interesting constant mean curvature surfaces are found. Furthermore, this symmetry analysis is extended to include conditional symmetries by subjecting the original systems to certain differential constraints. In this case, several types of nonsplitting algebraic, trigonometric and hyperbolic multi-soliton solutions have been obtained in explicit form. A new procedure for constructing solutions of the overdetermined system which is composed of the generalized Weierstrass system and the complex eikonal equations is studied. Finally, an approach to the classical configurations of strings in three-dimensional Euclidean space based on the obtained solutions of the generalized Weierstrass system is presented. RésuméLa méthode de réduction par symétries est appliquée systématiquement pour dériver plusieurs classes de solutions invariantes du système de Weierstrass généralisé induisant des surfaces de courbure moyenne constante et du modèle sigma euclidien bidimensionnel associéà ce système. Une classification des sous-groupes avec orbites génériques de codimension un des groupes de Lie de symétrie ponctuels pour ces systèmes fournit un moyen d'introduire des variables de symétrie et réduit le système initialà différents systèmes inéquivalents d'équations différentielles ordinaires. Nous effectuons une analyse des singularités afin d'établir si ceséquations différentielles ordinaires possèdent la propriété de Painlevé. Ceséquations différentielles ordinaires peuvent alorsêtre transformées dans des formes standards et ensuite résolues en termes de fonctionsélémentaires et de fonctions elliptiques de Jacobi. Cela résulte en un grand nombre de nouvelles solutions et, dans certains cas, de nouvelles surfacesà courbure moyenne constante sont trouvées. De plus, cette analyse des symétries estétendue au cas des symétries conditionnelles en soumettant les systèmes initiauxà certaines contraintes différentielles. Dans ce cas, plusieurs solutions multi-solitoniques non-séparantes de type algébrique, trigonomé-trique et elliptique sont obtenues sous une forme explicite. Une nouvelle procédure pour construire des solutions du système surdéterminé composé du ...

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A certain class of solutions of the nonlinear wave equation

Cited by 28 publications

References 3 publications

Group Analysis of Differential Equations and Generalized Functions

Group Analysis of Differential Equations and Generalized Functions

On the general solution of the d’Alembert equation with a nonlinear eikonal constraint and its applications

Symmetry properties and explicit solutions of the generalized Weierstrass system

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