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  • 1. Basarab-Horwath, P.
    et al.
    Euler, Norbert
    Technische Hochschule Darmstadt.
    Euler, Marianna
    Amplitude-phase representation for solutions of nonlinear d'Alembert equations1995In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 28, no 21, p. 6193-6201Article in journal (Refereed)
    Abstract [en]

    We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.

  • 2.
    Calogero, F.
    et al.
    Dipartimento di Fisica, Università di Roma “La Sapienza”, Rome.
    Euler, Marianna
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    New evolution PDEs with many isochronous solutions2009In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 353, no 2, p. 481-488Article in journal (Refereed)
    Abstract [en]

    Certain nonlinear evolution PDEs in 1 + 1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T = 2 π / ω (or perhaps over(T, ̃) = p T with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited.

  • 3. Duarte, L.G.S.
    et al.
    Euler, Norbert
    Moreira, I.C.
    Steeb, Willi-Hans
    Invertible point transformations, Painlevé analysis and anharmonic oscillators1990In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 23, p. 1457-1463Article in journal (Refereed)
    Abstract [en]

    The techniques of an invertible point transformation and the Painleve analysis can be used to construct integrable ordinary differential equations. The authors compare both techniques for anharmonic oscillators.

  • 4. Duarte, L.G.S.
    et al.
    Moreira, N.
    Euler, Norbert
    Steeb, Willi-Hans
    Invertible point transformations, Lie symmetries and the Painlevé test for the equation d2x/dt2 + f1(t)dx/dt + f2(t)x + f3(t)xn = 01991In: Physica Scripta, ISSN 0031-8949, E-ISSN 1402-4896, Vol. 43, p. 449-451Article in journal (Refereed)
    Abstract [en]

    The anharmonic oscillator d2x/dt2 + f1(t)dx/dt + f2(t)x + f3(t)xn = 0 (n ≠ 0, 1) is investigated applying the invertible point transformation, Lie symmetries and the Painlevé test.

  • 5.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics.
    A class of semilinear fifth-order evolution equations: recursion operators and multipotentialisations2011In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 18, no Suppl. 1, p. 61-75Article in journal (Refereed)
    Abstract [en]

    We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.

  • 6.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Aufgaben, Theorie und Lösungen zur Linearen Algebra, Teil 1: Der Euklidische Raum2017Book (Refereed)
    Abstract [de]

    Dieses Buch ist der erste Teil einer dreiteiligen Serie mit dem Titel Aufgaben, Theorieund L¨osungen zur Linearen Algebra. Dieser erste Teil behandelt Vektoren des euklidischen Raumes sowie Matrizen, Matrixalgebra und Systeme von linearen Gleichungen. Wir l¨osen lineare System mit Hilfe des Gaußschen Eliminationsverfahrens und auf anderem Wege und untersuchen die Eigenschaften dieser Systeme in Bezug auf Vektoren und Matrizen. Dar¨uber hinaus betrachten wir lineare Abbildungen und berechnendie Standardmatrizen dieser Abbildungen.

  • 7.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Integrating factors and conservation laws for some Camassa-Holm type equations2012In: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 11, no 4, p. 1421-1430Article in journal (Refereed)
    Abstract [en]

    We classify all 1st-order integrating factors and the correspondingconservation laws for a class of Camassa-Holm type equations.

  • 8.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Invariance of the Kaup-Kupershmidt equation and triangular auto-Bäcklund transformations2012In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 19, no 3, p. 1220001/1-1220001/7Article in journal (Refereed)
    Abstract [en]

    We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup-Kupershmidt equation derived by E. G. Reyes in his paper titled ``Nonlocal symmetries and the Kaup-Kupershmidt equation'' [ Math. Phys., 46, 073507, 19 pp., 2005]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup-Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialisation of the Kupershmidt equation.

  • 9.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    n-Dimensional real wave equations and the D’Alembert-Hamilton system2001In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 47, no 8, p. 5125-5133Article in journal (Refereed)
    Abstract [en]

    We reduce the nonlinear wave equation □nu = αF[exp(βu)] to ordinary differential equtions and construct exact solutions, by the use of a compatible d'Alembert-Hamilton system. The solutions of these ordinary differential equations, together with the solutions of the corresponding d'Alembert-Hamilton equations, provide a rich class of exact solutions of the multidimensional wave equations. The wave equations are studied in n-dimensional Minkowski space.

  • 10.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    On Möbius‐invariant and symmetry‐integrable evolution equations and the Schwarzian derivative2019In: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 143, no 2, p. 139-156Article in journal (Refereed)
    Abstract [en]

    We consider symmetry‐integrable evolution equations in 1 + 1 dimensions of order 3 and order 5. We show that there exist only three equations in this class that are invariant under the Möbius transformation, and we name those Schwarzian equations. We report an interesting relation between the recursion operators of the Schwarzian equations and the corresponding adjoint operators that generate hierarchies of Schwarzian systems in terms of the Schwarzian derivative. This indicates a deep relation between the Schwarzian equations and the Schwarzian derivative. A classification of the fully nonlinear third‐order Schwarzian equations is also reported.

  • 11.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Problemas, Teoria y Soluciones en Algebra Lineal: Parte 1 Espacio Euclideo2016 (ed. 1)Book (Refereed)
    Abstract [en]

    This book is the first part of a three-part series titled Problems, Theory and Solutions in Linear Algebra. This first part contains over 100 solved problems and 100 exercises on vectors, matrices, linear systems, as well as linear transformations in Euclidean space. It is intended as a supplement to a textbook in Linear Algebra and the aim of the series it to provide the student with a well-structured and carefully selected set of solved problems as well as a thorough revision of the material taught in a course on this subject for undergraduate engineering and science students.

  • 12.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Problems, Theory and Solutions in Linear Algebra: Part 1: Euclidean Space2015 (ed. 2)Book (Refereed)
    Abstract [en]

    This book is the first part of a three-part series titled Problems, Theory and Solutions in Linear Algebra. This first part contains over 100 solved problems and 100 exercises on vectors, matrices, linear systems, as well as linear transformations in Euclidean space. It is intended as a supplement to a textbook in Linear Algebra and the aim of the series it to provide the student with a well-structured and carefully selected set of solved problems as well as a thorough revision of the material taught in a course on this subject for undergraduate engineering and science students.

  • 13.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Symmetries for a class of explicitly space- and time-dependent (1+1)-dimensional wave equations1997In: Proceedings of the second international conference: Memorial Prof. W. Fushchych conference, July 7 - 13, 1997, Kyiv, Ukraine / [ed] Mykola Shkil, Kyev: Institute of Mathematics of the National Academy of Sciences of Ukraine , 1997, Vol. 1, p. 70-78Conference paper (Refereed)
    Abstract [en]

    In this paper the nonlinear wave equation $\partial^2 u/\partial x_0^2-\partial^2 u/\partial x_1^2+f(x_0,x_1,u)=0$, where $f$ is an arbitrary smooth function of its arguments, is considered from the symmetry standpoint. The form of the most general Lie point symmetry generator of this equation is obtained. The classes of functions $f$, for which the equation in question admits a one-parameter Lie point symmetry group, are constructed. Then, the authors investigate the possible form of generators of conformal transformations, assuming the usual form of generators of Lorentz and scaling transformations, and study the wave equations invariant under such operators. The symmetry groups of obtained equations are used for the construction of ansätze and reductions of these equations to ordinary differential equations. $Q$-conditional (nonclassical) symmetries of the wave equation are also considered. Namely, the determining equations for the coefficients of a $Q$-conditional symmetry operator are found and their compatibility is investigated.

  • 14.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Théorie et Problémes Résolus d'Algèbre Linéaire: Volume 1: Espaces Euclidiens2017 (ed. 1)Book (Refereed)
    Abstract [fr]

    Ce livre est le premier d'une série de trois ouvrages intitulés Théorie et Problèmes Résolus d'algèbre linéaire. Cette première partie contient plus de 100 problèmes résolus et plus de 100 exercices sur les vecteurs en espaces euclidiens, les matrices, les systèmes linéaires ainsi que les applications linéaires entre espaces euclidiens. Le but de cette série est de fournir aux étudiants de filières scientifiques et techniques un ensemble structuré de problèmes soigneusement choisis ainsi qu'une opportunité d'approfondir leurs connaissances acquises en cours d'algèbre linéaire.

  • 15. Euler, Marianna
    et al.
    Euler, Norbert
    Köhler, A.
    On the construction of approximate solutions for a multidimensional nonlinear heat equation1994In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 27, no 6, p. 2083-2092Article in journal (Refereed)
    Abstract [en]

    Summary: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation $\partial u/\partial x_0+ \Delta u= au^n+ \varepsilon f(u)$, where $a$ and $n$ are arbitrary real constants, $f$ is a smooth function, and $0< \varepsilon\ll 1$.

  • 16.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics.
    Leach, Peter
    University of KwaZulu-Natal.
    Properties of the Calogero-Degasperis-Ibragimov-Shabat differential sequence2011In: Lobachevskii Journal of Mathematics, ISSN 1995-0802, E-ISSN 1818-9962, Vol. 32, no 1, p. 61-70Article in journal (Refereed)
    Abstract [en]

    We present a differential sequence based upon the Calogero-Degasperis-Ibragimov-Shabat Equation and determine first integrals and the general solution. Under suitable transformations eachmember of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form.

  • 17.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Lindblom, Ove
    Luleå tekniska universitet.
    Explicitly space- and time-dependent d'Alembert equations with symmetries1999In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 14, no 26, p. 4189-4200Article in journal (Refereed)
    Abstract [en]

    The general d'Alembert equation $\square u+ f(x_0, x_1,u)= 0$ is considered, where $\square$ is the two-dimensional d'Alembert operator. We classify the equation for functions $f$ by which it admits several Lie symmetry algebras, which include the Lorentz symmetry generator. The corresponding symmetry reductions are listed.

  • 18.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Lundberg, Staffan
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Invertible Bäcklund transformations for autonomous evolution equations2009In: Theoretical and mathematical physics, ISSN 0040-5779, E-ISSN 1573-9333, Vol. 159, no 3, p. 418-427Article in journal (Refereed)
  • 19.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Lundberg, Staffan
    Reciprocal Bäcklund transformations of autonomous evolution equations2009In: Theoretical and mathematical physics, ISSN 0040-5779, E-ISSN 1573-9333, Vol. 159, no 3, p. 770-778Article in journal (Refereed)
    Abstract [en]

    We discuss the construction of reciprocal Bäcklund transformations for evolution equations using integrating factors of zeroth and higher orders with their corresponding conservation laws. As an example, we consider the Harry Dym equation and the Schwarzian KdV equation.

  • 20.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Nucci, M.C.
    Dipartimento di Matematica e Informatica Universita di Perugia.
    On nonlocal symmetries generated by recursion operators : second-order evolution equations2017In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 37, no 8, p. 4239-4247Article in journal (Refereed)
    Abstract [en]

    We introduce a new type of recursion operator to generate a class of nonlocal symmetries for second-order evolution equations in 1+1 dimensions, namely those evolution equations which allow the complete integration of their stationary equations. We show that this class of evolution equations is C-integrable (linearizable by a point transformation). We also discuss some applications.  

  • 21.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Petersson, Niclas
    Luleå tekniska universitet.
    Linearizable hierarchies of evolution equations in (1+1) dimensions2003In: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 111, no 3, p. 315-337Article in journal (Refereed)
    Abstract [en]

    In our article [1], "A tree of linearisable second-order evolution equations by generalised hodograph transformations" we present a class of linearizable (C-integrable) second-order evolution equations in (1+1) dimensions, using a generalized hodograph transformation. We report here the complete set of recursion operators for this class and present the resulting linearizable (C-integrable) hierarchies in (1+1) dimensions. The autonomous class of linearizable hierarchies are extended further by considering the equations in potential form followed by the pure hodograph transformation.

  • 22.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Reyes, Enrique G.
    Departamento de Matematica y Ciencia de la Computacion, Universidad de Santiago de Chile, Chile.
    Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations2017In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 24, no 3, p. 303-314Article in journal (Refereed)
    Abstract [en]

    In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetriesand that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can be recovered using potentializations.

  • 23.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Strömberg, Anders
    Luleå tekniska universitet.
    Åström, Erik
    Luleå tekniska universitet.
    Transformation between a generalized Emden-Fowler equation and the first Painlevé transcendent2007In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 30, no 16, p. 2121-2124Article in journal (Refereed)
    Abstract [en]

    We use the generalized Sundman transformation to obtain a relation between a generalized Emden-Fowler equation and the first Painlevé transcendent.

  • 24.
    Euler, Marianna
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Wolf, Thomas
    Brock University.
    The two-component Camassa-Holm equations CH(2,1) and CH(2,2): First-order integrating factors and conservation laws2012In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 19, no supplement 1, p. 1240002-Article in journal (Refereed)
    Abstract [en]

    Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, [J. Phys A: Math. Theor, vol. 43, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).

  • 25. Euler, Marianna
    et al.
    Euler, Norbert
    Mathematical Institute of the Ukrainian Academy of Sciences.
    Zachary, W.W
    Computational Science and Engineering Research Center, Howard University.
    Mahmood, M.F.
    Computational Science and Engineering Research Center, Howard University.
    Gill, T.L.
    Computational Science and Engineering Research Center, Howard University.
    Symmetry classification for a coupled nonlinear Schrödinger equation1994In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 1, no 4, p. 358-379Article in journal (Refereed)
    Abstract [en]

    We do a Lie symmetry classification for a system of two nonlinear coupled Schrödinger equations. Our system under consideration is a generalization of the equations which follow from the analysis of optical fibres. Reductions of some special equations are given.

  • 26.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    A First Course in Ordinary Differential Equations2015 (ed. 1)Book (Refereed)
    Abstract [en]

    The book consists of lecture notes intended for engineering and science students who are reading a first course in ordinary differential equations and who have already read a course on linear algebra, including general vector spaces and integral calculus for functions of one variable. No knowledge of differential equations is required to read and understand this material. Many examples have been included and most statements are proved in full detail. The aim is to provide the student with a thorough understanding of the methods to obtain solutions of certain classes of mainly linear scalar differential equations.

  • 27.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Aktivitet: Journal of Nonlinear Mathematical Physics1997Other (Other (popular science, discussion, etc.))
    Abstract [en]

    The Journal of Nonlinear Mathematical Physics (JNMP) publishes research papers on fundamental mathematical and computational methods in mathematical physics in the form of Letters, Articles, and Review Articles.  The latest ISI Impact factor (2005) is: 0.508. Bibliografisk note: http://www.atlantis-press.com/publications/jnmp/

  • 28.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Ecuaciones diferenciales ordinarias: Introducción a las ecuaciones lineales2016 (ed. 1 edición)Book (Refereed)
    Abstract [en]

    The book consists of lecture notes intended for engineering and science students who are reading a first course in ordinary differential equations and who have already read a course on linear algebra, including general vector spaces and integral calculus for functions of one variable. No prior knowledge of differential equations is required to read and understand this material. Many examples have been included and most statements are proved in full detail. The aim is to provide the student with a thorough understanding of methods to obtain solutions of certain classes of mainly linear scalar differential equations.

  • 29.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Gewöhnliche Differentialgleichungen: Eine Einführung2016 (ed. 1)Book (Refereed)
    Abstract [en]

    The book consists of lecture notes intended for engineering and science students who are reading a first course in ordinary differential equations and who have already read a course on linear algebra, including general vector spaces and integral calculus for functions of one variable. No knowledge of differential equations is required to read and understand this material. Many examples have been included and most statements are proved in full detail. The aim is to provide the student with a thorough understanding of the methods to obtain solutions of certain classes of mainly linear scalar differential equations.

  • 30.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Linear operators and the general solution of elementary linear ordinary differential equations2012In: CODEE Journal, ISSN 2160-5211, E-ISSN 2160-5211, Vol. CJ12-1802Article in journal (Refereed)
    Abstract [en]

    We make use of linear operators to derive the formulae for the generalsolution of elementary linear scalar ordinary differential equations of order n.The key lies in the factorization of the linear operators in terms of first-orderoperators. These first-order operators are then integrated by applying theircorresponding integral operators.This leads to the solution formulae for bothhomogeneous- and nonhomogeneous linear differential equations in a naturalway without the need for any ansatz (or “educated guess”). For second-orderlinear equations with nonconstant coefficients, the condition of the factorizationis given in terms of Riccati equations.

  • 31. Euler, Norbert
    Nonlinear field equations and Painleve test1988Book (Other academic)
  • 32.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Nonlinear Systems and Their Remarkable Mathematical Structures2018Collection (editor) (Refereed)
    Abstract [en]

    The book aims to provide a comprehensive account of the state of the art on the mathematical description of nonlinear systems. The book consists of 20 invited contributions written by leading experts in different aspects of nonlinear systems that include ordinary and partial differential equations, difference equations and q-difference equations, discrete or lattice equations, non-commutative and matrix equations, as well as supersymmetric equations.

  • 33. Euler, Norbert
    Painlevé analysis and conditional auto-Bäcklund transformations for a two-dimensional Boltzmann model1994In: Doklady Akademii Nauk Ukrainy, no 8, p. 42-48Article in journal (Refereed)
    Abstract [en]

    Summary (translated from the Russian): Using Painlevé analysis we construct a conditional auto-Bäcklund transformation for a two-dimensional four-velocity Boltzmann equation with a quadratic nonlinearity. We construct exact multiparameter solutions of this model.

  • 34.
    Euler, Norbert
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Transformation properties of x''+f1(t)x + f2(t)x + f3(t)xn = 01997In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 4, no 3-4, p. 310-337Article in journal (Refereed)
    Abstract [en]

    The author studies the integrability of the nonlinear second-order ordinary differential equation described in the title. Using the invertible point transformation technique, Lie point symmetry technique and the Painlevé analysis, he classifies all integrable cases. The Lie algebra structure of the Lie point symmetries is also discussed. Finally, the three different methods are compared.

  • 35.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    A tree of linearisable second-order evolution equations by generalised hodograph transformations2001In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 8, no 3, p. 342-362Article in journal (Refereed)
    Abstract [en]

    We present a list of (1 + 1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations

  • 36.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    An alternate view on symmetries of second-order linearisable ordinary differential equations2012In: Lobachevskii Journal of Mathematics, ISSN 1995-0802, E-ISSN 1818-9962, Vol. 33, no 2, p. 191-194Article in journal (Refereed)
    Abstract [en]

    We discuss the Sundman symmetries and point symmetries of linearisable second-order ordinary differential equations, which include discrete and nonlocal symmetries. The method of construction of such symmetries is also introduced for general n-th order equations.

  • 37. Euler, Norbert
    et al.
    Euler, Marianna
    Madelung representation for complex nonlinear d'Alembert equations in n-dimensional Minkowski space1995In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 2, no 3-4, p. 292-300Article in journal (Refereed)
    Abstract [en]

    The Madelung representation $\psi = u\exp(iv)$ is considered for the d'Alembert equation $\square_n\psi - F(|\psi|)\psi = 0$ to develop a technique for finding exact solutions. The authors classify the nonlinear function $F$ for which the amplitude and phase of the d'Alembert equation are related to the solutions of the compatible d'Alembert-Hamiltonian system. The equations are studied in the $n$-dimensional Minkowski space.[A.~Ju.~Obolenskij (Kyïv)]

  • 38.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    Multipotentialisations and iterating-solution formulae: the Krichever-Novikov equation2009In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 16, no Suppl. 1, p. 93-106Article in journal (Refereed)
    Abstract [en]

    We derive solution-formulae for the Krichever-Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever-Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.

  • 39.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies2018In: Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1 / [ed] Norbert Euler, Boca Raton, USA: CRC Press, 2018, 1, p. 317-351Chapter in book (Refereed)
    Abstract [en]

    The term multipotentialisation of evolution equations in 1+1 dimensions refers to the process of potentialising a given evolution equation, followed by at least one further potentialisation of the resulting potential equation. For certain equations this process can be applied several times to result in a finite chain of potential equations, where each equation in the chain is a potential equation of the previous equation. By a potentialisation of an equation with dependent variable u to an equation with dependent variable v, we mean a differential substitution v_x=\Phi^t, where \Phi^t is a conserved current of the equation in u. The process of multipotentialisation may lead to interesting nonlocal transformations between the equations. Remarkably, this can, in some cases, result in nonlocal invariance transformations for the equations, which then serve as iteration formulas by which solutions can be generated for all the equations in the chain.

    In the current paper we give a comprehensive introduction to this subject and report new nonlocal invariance transformations that result from the multipotentialisation of the Kupershmidt equation and its higher-order hierarchies. The recursion operators that define the hierarchies are given explicitly.

  • 40.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations: two linearisable hierarchies2009In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 16, no 4, p. 489-504Article in journal (Refereed)
    Abstract [en]

    We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries.We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach and derive second degree potential symmetries for the Burgers' hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy.

  • 41. Euler, Norbert
    et al.
    Euler, Marianna
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Sundman symmetries of nonlinear second-order and third-order ordinary differential equations2004In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 11, no 3, p. 399-421Article in journal (Refereed)
    Abstract [en]

    Sundman symmetries arise from more general transformations than do point or contact symmetries. This paper first shows how to systematically calculate Sundman symmetries of second- and third-order nonlinear ordinary differential equations. Secondly, the authors illustrate the application of these symmetries by computing first integrals of the corresponding equations.

  • 42. Euler, Norbert
    et al.
    Euler, Marianna
    Mathematical Institute of the Ukrainian Academy of Sciences.
    Symmetry properties of the approximations of multidimensional generalized van der Pol equations1994In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 1, no 1, p. 41-59Article in journal (Refereed)
    Abstract [en]

    The subject of the paper are symmetries of the nonlinear hyperbolic equation, $$\frac{\partial^2u}{\partial t^2} - \sum_{n=1}^N \frac{\partial^2u}{\partial^2x_n} +m^2u - \varepsilon f(u)\left(\lambda_0 \frac{\partial u}{\partial t} + \sum_{n=1}^N\lambda_n \frac{\partial u}{\partial x_n}\right) =0. $$ The case $f(u)=1-u^2$ corresponds to the generalized van der Pol equation. The equation is expanded in powers of the parameter $\varepsilon$, which stands in front of the nonlinear term, and then symmetries of the resulting chain of approximate equations are studied by means of the Lie-group technique. Emphasis is made on a special type of the function $f(u)$ which admits conformal invariance of the equation.

  • 43.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    The converse problem for the multipotentialisation of evolution equations and systems2011In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 18, no Suppl. 1, p. 77-105Article in journal (Refereed)
    Abstract [en]

    We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-Bäcklund transformations for the equations/systems. In this respect we define the triangular-auto-Bäcklund transformation and derive its connections to the converse problem. Several explicit examples are given. In particular, we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.

  • 44. Euler, Norbert
    et al.
    Euler, Marianna
    Köhler, A.
    Conditional and approximate symmetries for a generalized van der Pol equation1994In: Lie Groups and their Applications, Vol. 1, no 1, p. 79-94Article in journal (Refereed)
    Abstract [en]

    Non-Lie ansätze and exact solutions are constructed by means of conditional symmetries for a class of nonlinear heat equations in one space dimension. We study approximate symmetries by which one can obtain approximate solutions for multidimensional partial differential equations. The method of approximate symmetries is applied to an $n$-dimensional generalized van der Pol equation. Approximate conditions symmetries are introduced to construct approximate solutions for the latter equation.

  • 45.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Euler, Marianna
    Leach, Peter
    School of Mathematical Sciences, Howard College, University of KwaZulu-Natal, Durban.
    The Riccati and Ermakov-Pinney hierarchies2007In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 14, no 2, p. 290-302Article in journal (Refereed)
    Abstract [en]

    The concept and use of recursion operators is well-established in the study of evolution, in particular nonlinear, equations. We demonstrate the application of the idea of recursion operators to ordinary differential equations. For the purposes of our demonstration we use two equations, one chosen from the class of linearisable hierarchies of evolution equations studied by Euler et al (Stud Appl Math 111 (2003) 315-337) and the other from the class of integrable but nonlinearisible equations studied by Petersson et al (Stud Appl Math 112 (2004) 201-225). We construct the hierarchies for each equation. The symmetry properties of the first hierarchy are considered in some detail. For both hierarchies we apply the singularity analysis. For both we observe intersting behaviour of the resonances for the different possible leading order behaviours. In particular we note the proliferation of subsidiary solutions as one ascends the hierarchy.

  • 46.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Gandarias, Maria Luz
    Cadiz University.
    Euler, Marianna
    Lindblom, Ove
    Luleå tekniska universitet.
    Auto-hodograph transformations for a hierarchy of nonlinear evolution equations2001In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 257, no 1, p. 21-28Article in journal (Refereed)
    Abstract [en]

    We introduce nonlocal auto-hodograph transformations for a hierarchy of nonlinear evolution equations. This is accomplished by composing nonlocal transformations (one of which is a hodograph transformation) which linearize the given equations. This enables one to construct sequences of exact solutions for any equation belonging to the hierarchy.

  • 47. Euler, Norbert
    et al.
    Köhler, A.
    Fushchich, W.I.
    Q-symmetry generators and exact solutions for nonlinear heat conduction1994In: Physica Scripta, ISSN 0031-8949, E-ISSN 1402-4896, Vol. 49, no 5, p. 518-524Article in journal (Refereed)
    Abstract [en]

    Summary: We investigate conditional invariance by considering $Q$-symmetry generators of the nonlinear heat equation $\partial u/\partial x\sb 0-\lambda \partial\sp 2u/\partial x\sp 2\sb 1=f(u)$, where $\lambda$ is a real constant and $f$ an arbitrary differentiable function. With the obtained $Q$-generators we construct exact solutions by the use of similarity ansatze and reductions to ordinary differential equations. A generalization to $m$ space dimensions is performed.

  • 48.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Leach, P. G. L.
    School of Mathematical and Statistical Sciences, University of Natal, Durban.
    First integrals and reduction of a class of nonlinear higher order ordinary differential equations2003In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 287, no 2, p. 473-486Article in journal (Refereed)
    Abstract [en]

    We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.

  • 49. Euler, Norbert
    et al.
    Leach, P. G. L.
    Mahomed, F. M.
    Steeb, W-H
    Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a maxwell distribution1988In: International journal of theoretical physics, ISSN 0020-7748, E-ISSN 1572-9575, Vol. 27, no 6, p. 717-723Article in journal (Refereed)
  • 50.
    Euler, Norbert
    et al.
    Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
    Leach, Peter
    University of KwaZulu-Natal.
    Aspects of proper differential sequences of ordinary differential equations2009In: Theoretical and mathematical physics, ISSN 0040-5779, E-ISSN 1573-9333, Vol. 159, no 1, p. 474-487Article in journal (Refereed)
    Abstract [en]

    We define a proper differential sequence of ordinary differential equations and introduce a method for deriving an alternate sequence of integrals for such a sequence. We describe some general properties, illustrated by several examples.

12 1 - 50 of 89
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