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  • 1.
    Basarab-Horwath, Peter
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lie algebras and classification of partial differential equations2004In: Symmetry in Nonlinear Mathematical Physics,2004, 2004Conference paper (Other academic)
  • 2.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Gungor, F
    Dogus University, Turkey .
    Lahno, V
    Pedag University, Ukraine .
    Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras2013In: 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)
    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.

  • 3.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Güngör2, F.
    Istanbul Technical University, Turkey.
    Özemir, C.
    Istanbul Technical University, Turkey.
    Infinite-dimensional symmetries of a general class of variable coefficient evolution equations in 2+1 dimensions2013In: ISQS21, Institute of Physics Publishing (IOPP), 2013, Vol. 474Conference 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.

  • 4.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V
    Group classification of the general quasi-linear wave equation: invance under low-dimensional Lie algebras2004Report (Other academic)
  • 5.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V.
    Magda, O.
    The structure of Lie algebras and the classification problem for partial differential equations2004In: Proceedings of Institute of Mathematics of NAS of Ukraine.Mathematics and its Applications, Vol. 50, p. 40-46Article in journal (Refereed)
  • 6.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V.
    Zhdanov, R.
    Classifying evolution equations2001In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 47, no 8, p. 5135-5144Article in journal (Refereed)
    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.

  • 7.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lahno, V
    Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine.
    Zhdanov, R
    Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine.
    The structure of lie algebras and the classification problem for partial differential equations2001In: 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)
    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.

  • 8.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Zhdanov, R.Z.
    Institute of Mathematics, 3 Tereshchenkivska Street, 252004 Kyiv, Ukraine.
    Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries2001In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 42, no 1, p. 376-389Article in journal (Refereed)
    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.

  • 9.
    Kurujyibwami, Célestin
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Basarab-Horwath, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Popovych, Roman O.
    Institute of Mathematics of NAS of Ukraine, Ukraine / Wolfgang Pauli Institute, Wien, Austria.
    Algebraic method for group classification of (1+1)-dimensional linear Schrödinger equationsManuscript (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.

  • 10.
    Kurujyibwami, Célestin
    et al.
    Linköping University, Department of Mathematics. Linköping University, Faculty of Science & Engineering.
    Basarab-Horwath, Peter
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Popovych, Roman O.
    Institute of Mathematics of NAS of Ukraine, Ukraine / Wolfgang Pauli Institute, Wien, Austria.
    Group classification of multidimensional linear Schrödinger equations with algebraic methodManuscript (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.

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