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Basarab-Horwath, Peter
Publications (10 of 10) Show all publications
Basarab-Horwath, P., Güngör2, F. & Özemir, C. (2013). Infinite-dimensional symmetries of a general class of variable coefficient evolution equations in 2+1 dimensions. In: ISQS21: . Paper presented at XXIst International Conference on Integrable Systems and Quantum Symmetries (ISQS21),. Institute of Physics Publishing (IOPP), 474
Open this publication in new window or tab >>Infinite-dimensional symmetries of a general class of variable coefficient evolution equations in 2+1 dimensions
2013 (English)In: ISQS21, Institute of Physics Publishing (IOPP), 2013, Vol. 474Conference paper, Published paper (Refereed)
Abstract [en]

We consider generalized KP-Burgers equations and attempt to identify subclasses admitting Virasoro or Kac-Moody type algebras as their symmetries. We give reductions to ODEs constructed from invariance requirement under these infinite-dimensional Lie symmetry algebras and integrate them in cases where it is possible. We also look at the conditions under which the equation passes the Painleve test and construct some exact solutions by truncation.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2013
Series
Journal of Physics: Conference Series, ISSN 1742-6588, E-ISSN 1742-6596 ; 474
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-103260 (URN)10.1088/1742-6596/474/1/012010 (DOI)000327951000010 ()
Conference
XXIst International Conference on Integrable Systems and Quantum Symmetries (ISQS21),
Available from: 2014-01-17 Created: 2014-01-16 Last updated: 2018-01-26Bibliographically approved
Basarab-Horwath, P., Gungor, F. & Lahno, V. (2013). Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras. Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, 124(1), 123-170
Open this publication in new window or tab >>Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras
2013 (English)In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 124, no 1, p. 123-170Article in journal (Refereed) Published
Abstract [en]

We give a complete point-symmetry classification of all third-order evolution equations of the form u (t) =F(t,x,u,u (x) ,u (xx) )u (xxx) +G(t,x,u,u (x) ,u (xx) ) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2013
Keywords
Symmetries, Lie algebras, Equivalence group, Semi-simple Lie algebras, Solvable Lie algebras
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-90747 (URN)10.1007/s10440-012-9773-4 (DOI)000316009000007 ()
Note

Funding Agencies|Swedish Research Council|624-2004-1073|Research Council of Turkey (TUBITAK)||

Available from: 2013-04-05 Created: 2013-04-05 Last updated: 2017-12-06
Basarab-Horwath, P. & Lahno, V. (2004). Group classification of the general quasi-linear wave equation: invance under low-dimensional Lie algebras. Linköping: Linköpings universitet
Open this publication in new window or tab >>Group classification of the general quasi-linear wave equation: invance under low-dimensional Lie algebras
2004 (English)Report (Other academic)
Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004
Series
LiTH-MAT-R ; 18
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22997 (URN)2372 (Local ID)2372 (Archive number)2372 (OAI)
Available from: 2009-10-07 Created: 2009-10-07
Basarab-Horwath, P. (2004). Lie algebras and classification of partial differential equations. In: Symmetry in Nonlinear Mathematical Physics,2004.
Open this publication in new window or tab >>Lie algebras and classification of partial differential equations
2004 (English)In: Symmetry in Nonlinear Mathematical Physics,2004, 2004Conference paper, Published paper (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-23261 (URN)2685 (Local ID)2685 (Archive number)2685 (OAI)
Available from: 2009-10-07 Created: 2009-10-07
Basarab-Horwath, P., Lahno, V. & Magda, O. (2004). The structure of Lie algebras and the classification problem for partial differential equations. Proceedings of Institute of Mathematics of NAS of Ukraine.Mathematics and its Applications, 50, 40-46
Open this publication in new window or tab >>The structure of Lie algebras and the classification problem for partial differential equations
2004 (English)In: Proceedings of Institute of Mathematics of NAS of Ukraine.Mathematics and its Applications, Vol. 50, p. 40-46Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-23881 (URN)3426 (Local ID)3426 (Archive number)3426 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2011-01-12
Basarab-Horwath, P., Lahno, V. & Zhdanov, R. (2001). Classifying evolution equations. Nonlinear Analysis, 47(8), 5135-5144
Open this publication in new window or tab >>Classifying evolution equations
2001 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 47, no 8, p. 5135-5144Article in journal (Refereed) Published
Abstract [en]

A Lie point symmetry classification of evolution equations in 1+1 time-space dimensions was presented. A combination of the standard Lie algorithm for point symmetry and the equivalence group of the given type of equation was used for the classification. For each canonical evolution the maximal symmetry algebra was calculated and related theorems were proved.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-47298 (URN)10.1016/S0362-546X(01)00623-X (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
Basarab-Horwath, P. & Zhdanov, R. (2001). Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries. Journal of Mathematical Physics, 42(1), 376-389
Open this publication in new window or tab >>Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries
2001 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 42, no 1, p. 376-389Article in journal (Refereed) Published
Abstract [en]

We suggest a new approach to the problem of dimensional reduction of initial/ boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie's classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries. © 2001 American Institute of Physics.

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-47490 (URN)10.1063/1.1330199 (DOI)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13
Basarab-Horwath, P., Lahno, V. & Zhdanov, R. (2001). The structure of lie algebras and the classification problem for partial differential equations. Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, 69(1), 43-94
Open this publication in new window or tab >>The structure of lie algebras and the classification problem for partial differential equations
2001 (English)In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 69, no 1, p. 43-94Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
group classification, equivalence group, symmetry analysis, Lie algebras
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-49058 (URN)
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-12
Kurujyibwami, C., Basarab-Horwath, P. & Popovych, R. O.Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equations.
Open this publication in new window or tab >>Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equations
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We carry out the complete group classification of the class of (1+1)-dimensional linear Schrödinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we compute the equivalence groupoid of the class under study and show that it is uniformly semi-normalized. More specifically, each admissible transformation in the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This allows us to apply the new version of the algebraic method based on uniform semi-normalization and reduce the group classification of the class under study to the classification of low-dimensional appropriate subalgebras of the associated equivalence algebra. The partition into classification cases involves two integers that characterize Lie symmetry extensions and are invariant with respect to equivalence transformations.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-125134 (URN)
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2016-02-15Bibliographically approved
Kurujyibwami, C., Basarab-Horwath, P. & Popovych, R. O.Group classification of multidimensional linear Schrödinger equations with algebraic method.
Open this publication in new window or tab >>Group classification of multidimensional linear Schrödinger equations with algebraic method
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the group classification problem for multidimensional linear Schrödinger equations with complex-valued potentials. Using the algebraic approach, we compute the equivalence groupoid of the class and thus show that this class is uniformly semi-normalized. More specifically, any point transformation connecting two equations from the class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of this class. This is why the algebraic method of group classification is applied, which reduces the group classification to the classification of specific low-dimensional subalgebras of the associated equivalence algebra. Inequivalent Lie symmetry extensions are listed for dimension 1+2. Splitting into different classification cases is based on several integer parameters that are invariant with respect to the adjoint action of equivalence transformations. These parameters characterize the dimensions of parts of the corresponding Lie symmetry algebra that are related to generalized scalings, rotations and generalized Galilean boosts, respectively. As expected, the computation for dimension 1+2 is much trickier and cumbersome than for dimension 1+1 due to two reasons, appearing a new kind of transformations rotations and increasing the range for dimensions of essential subalgebras of Lie symmetry algebras, whose least upper bound is n(n+3)/2+5.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-125135 (URN)
Available from: 2016-02-15 Created: 2016-02-15 Last updated: 2016-02-15Bibliographically approved
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