2007/12/31 by Roman O. Popovych · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #Molecular spectroscopy and chirality #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP #msc:35A30 #msc:35C05 #msc:35K05 #msc:35K10
paper · pdf · doi:10.1088/1751-8113/41/18/185202
published as J. Phys. A: Math. Theor., 2008, V. 41, 185202, 31 pp · 31 pages, minor misprints are corrected
arxiv created 2008/04/16 · openalex publication_date 2008/04/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
The reduction operators, i.e. the operators of nonclassical (conditional) symmetry, of (1 + 1)-dimensional second-order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving initial equations. The 'no-go' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.