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The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations
Mathematical Physics · Submitted 12 May 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$.
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The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use, relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. | contribution 0.75 | 0.96 | 0.90 | asserted 1.00 |
| s1 | We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. | result 1.00 | 0.95 | 0.19 | asserted 1.00 |
| s2 | Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. | result 1.00 | 0.96 | 0.26 | asserted 1.00 |
| s3 | Since the procedure which we use, relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. | method 0.77 | 0.47 | 0.02 | asserted 0.97 |
| s4 | This material is dispersed in the literature and is not fully available in English. | gap 0.63 | 0.08 | 0.02 | asserted 1.00 |
| s5 | After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. | method 0.93 | 0.95 | 0.04 | asserted 0.98 |
| s6 | The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study. | result 0.68 | 0.90 | 0.15 | asserted 1.00 |
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Original abstract
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use, relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.
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How the literature uses it
10 arXiv papers cite this in 14 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 6 papers
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The framework of the algebraic method of group classification originated in Lie's classification of second-order ordinary differential equations [ref] but it became a common tool of group analysis of differential equations considerably later, only since the 1990s, although its applications to problems of complete group classification still involved the normalization property implicitly [ref] .
Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations · Introduction · 0.98
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The corresponding theoretical background is presented in detail in [ref] .
Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source · Introduction · 0.99
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Later the practitioners appreciated the method and started to use it intensively in different disguises, cf. [ref] , in particular, to carry out the preliminary group classification [ref] .
Equivalence groupoid and group classification of a class of variable-coefficient Burgers equations · Preliminary group analysis of the imaged regular subclass · 0.80
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