Conditional symmetry of the equations of nonlinear mathematical physics

Fushchich, V. I.

doi:10.1007/bf01067273

Cited by 26 publications

(31 citation statements)

References 8 publications

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“…We illustrate the possibility of generation of solutions for equations (12) by the following example. The transformation (14) maps the known traveling wave solution…”

Section: Exact Solutionsmentioning

confidence: 99%

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva

Boyko

Zhalij

et al. 2019

Journal of Mathematical Analysis and Applications

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A class of the Newell-Whitehead-Segel equations (also known as generalized Fisher equations and Newell-Whitehead equations) is studied with Lie and "nonclassical" symmetry points of view. The classifications of Lie reduction operators and of regular nonclassical reduction operators are performed. The set of admissible transformations (the equivalence groupoid) of the class is described exhaustively. The criterion of reducibility of variable coefficient Newell-Whitehead-Segel equations to their constant coefficient counterparts is derived. Wide families of exact solutions for such variable coefficient equations are constructed.

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“…We illustrate the possibility of generation of solutions for equations (12) by the following example. The transformation (14) maps the known traveling wave solution…”

Section: Exact Solutionsmentioning

confidence: 99%

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

Vaneeva

Boyko

Zhalij

et al. 2019

Journal of Mathematical Analysis and Applications

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“…We are particularly interested in combining singularity structure analysis and Lie point symmetries in order to recover the Auto-Bäcklund transformation (Auto-BT) and Darboux transformation (DT) for PDEs if such exist. In the literature, several attempts at treating this subject can be found, for example [5][6][7][8], and references therein. More recently, this subject has been studied for first order systems of PDEs, by one of the authors, leading to the development of a new version of the conditional symmetry method [10,11].…”

Section: Conditional Symmetriesmentioning

confidence: 99%

On Complete Integrability of the Generalized Weierstrass System

Bracken

Grundland

2002

JNMP

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In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsingular 2 × 2 matrices based on an sl(2, C) representation. We derive the explicit form of the Darboux transformation for the Weierstrass system. New classes of multisoliton solutions of the Generalized Weierstrass system are obtained through the use of the Bäcklund transformation and some physical applications of these results in the area of classical string theory are presented.

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“…al. [13,22,25,26,29,34,35,42] while in the second one are the conditional symmetries [7,17,18,27,30]. In this paper we will be interested in showing that one can construct equations having given conditional symmetries.…”

Section: Introductionmentioning

confidence: 99%