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Euler, M. & Euler, N. (2019). On Möbius‐invariant and symmetry‐integrable evolution equations and the Schwarzian derivative. Studies in applied mathematics (Cambridge), 143(2), 139-156
Open this publication in new window or tab >>On Möbius‐invariant and symmetry‐integrable evolution equations and the Schwarzian derivative
2019 (English)In: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 143, no 2, p. 139-156Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
John Wiley & Sons, 2019
Keywords
dynamical systems, mathematical physics, partial differential equations
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-74953 (URN)10.1111/sapm.12268 (DOI)000476678700002 ()2-s2.0-85065337676 (Scopus ID)
Note

Validerad;2019;Nivå 2;2019-08-06 (johcin)

Available from: 2019-06-25 Created: 2019-06-25 Last updated: 2019-08-06Bibliographically approved
Euler, N. (Ed.). (2018). Nonlinear Systems and Their Remarkable Mathematical Structures (1ed.). Boca Raton, USA: CRC Press
Open this publication in new window or tab >>Nonlinear Systems and Their Remarkable Mathematical Structures
2018 (English)Collection (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.

Place, publisher, year, edition, pages
Boca Raton, USA: CRC Press, 2018. p. 582 Edition: 1
Keywords
Nonlinear Evolution Equations, Nonlinear Dynamical Systems, Integrable Systems
National Category
Natural Sciences Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-70946 (URN)9781138601000 (ISBN)
Available from: 2018-09-22 Created: 2018-09-22 Last updated: 2019-02-11Bibliographically approved
Euler, N. & Euler, M. (2018). Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies (1ed.). In: Norbert Euler (Ed.), Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1 (pp. 317-351). Boca Raton, USA: CRC Press
Open this publication in new window or tab >>Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies
2018 (English)In: 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.

Place, publisher, year, edition, pages
Boca Raton, USA: CRC Press, 2018 Edition: 1
Keywords
Nonlinear Partial Differential Equations, Integrable Equations, Invariance
National Category
Natural Sciences Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-70947 (URN)9781138601000 (ISBN)
Available from: 2018-09-22 Created: 2018-09-22 Last updated: 2019-07-09
Euler, M. & Euler, N. (2017). Aufgaben, Theorie und Lösungen zur Linearen Algebra, Teil 1: Der Euklidische Raum. London: BookBooN.com
Open this publication in new window or tab >>Aufgaben, Theorie und Lösungen zur Linearen Algebra, Teil 1: Der Euklidische Raum
2017 (German)Book (Refereed)
Alternative title[en]
Problems, Theory and Solutions in Linear Algebra. Part 1 : Euclidean Space
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.

Place, publisher, year, edition, pages
London: BookBooN.com, 2017. p. 241
Keywords
Aufgaben und Lösungen zur Linearen Algebra, Theorie zur Linearen Algebra, Der Euklidische Raum
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-61295 (URN)978-87-403-1568-4 (ISBN)
Available from: 2017-01-01 Created: 2017-01-01 Last updated: 2018-03-08Bibliographically approved
Euler, M., Euler, N. & Reyes, E. G. (2017). Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations [Letter to the editor]. Journal of Nonlinear Mathematical Physics, 24(3), 303-314
Open this publication in new window or tab >>Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations
2017 (English)In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 24, no 3, p. 303-314Article in journal, Letter (Refereed) Published
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.

Place, publisher, year, edition, pages
Taylor & Francis, 2017
Keywords
Nonlinear partial differential equations, Nonlocal symmetries, Potentialization
National Category
Natural Sciences Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-64359 (URN)10.1080/14029251.2017.1341694 (DOI)000414996700001 ()2-s2.0-85021139923 (Scopus ID)
Note

Validerad;2017;Nivå 2;2017-08-14 (kribac)

Available from: 2017-06-21 Created: 2017-06-21 Last updated: 2018-12-14Bibliographically approved
Euler, M., Euler, N. & Nucci, M. (2017). On nonlocal symmetries generated by recursion operators : second-order evolution equations. Discrete and Continuous Dynamical Systems, 37(8), 4239-4247
Open this publication in new window or tab >>On nonlocal symmetries generated by recursion operators : second-order evolution equations
2017 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 37, no 8, p. 4239-4247Article in journal (Refereed) Published
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.  

Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2017
Keywords
Evolution equations, nonlocal symmetries, nonlinear PDEs, recursion operators
National Category
Natural Sciences Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-63177 (URN)10.3934/dcds.2017181 (DOI)000401074500005 ()2-s2.0-85038083356 (Scopus ID)
Note

Validerad; 2017; Nivå 2; 2017-05-11 (andbra)

Available from: 2017-04-27 Created: 2017-04-27 Last updated: 2018-11-20Bibliographically approved
Euler, M. & Euler, N. (2017). Théorie et Problémes Résolus d'Algèbre Linéaire: Volume 1: Espaces Euclidiens (1ed.). London: BookBooN.com
Open this publication in new window or tab >>Théorie et Problémes Résolus d'Algèbre Linéaire: Volume 1: Espaces Euclidiens
2017 (French)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.

Place, publisher, year, edition, pages
London: BookBooN.com, 2017. p. 214 Edition: 1
Keywords
Linear Algebra Course Literature, Problems and Solutions, Linear Algebra Theory
National Category
Natural Sciences Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-65190 (URN)978-87-403-1867-8 (ISBN)
Available from: 2017-08-18 Created: 2017-08-18 Last updated: 2018-03-08Bibliographically approved
Euler, N. (2016). Ecuaciones diferenciales ordinarias: Introducción a las ecuaciones lineales (1 edicióned.). Paper presented at . London: BookBooN.com
Open this publication in new window or tab >>Ecuaciones diferenciales ordinarias: Introducción a las ecuaciones lineales
2016 (Spanish)Book (Refereed)
Alternative title[en]
Ordinary differential equations: An introduction to linear equations
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.

Place, publisher, year, edition, pages
London: BookBooN.com, 2016. p. 233 Edition: 1 edición
Keywords
Ordinary differential equations, linear differential equations, methods of solutions, Mathematics, Matematik
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-16586 (URN)87a7befc-d453-4a11-8487-aa103c2cebc5 (Local ID)978-87-403-1213-3 (ISBN)87a7befc-d453-4a11-8487-aa103c2cebc5 (Archive number)87a7befc-d453-4a11-8487-aa103c2cebc5 (OAI)
Note
Godkänd; 2016; 20160122 (norbert)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-03-08Bibliographically approved
Euler, N. (2016). Gewöhnliche Differentialgleichungen: Eine Einführung (1ed.). Paper presented at . London: BookBooN.com
Open this publication in new window or tab >>Gewöhnliche Differentialgleichungen: Eine Einführung
2016 (German)Book (Refereed)
Alternative title[en]
A First Course in Ordinary Differentrial Equations
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.

Place, publisher, year, edition, pages
London: BookBooN.com, 2016. p. 237 Edition: 1
Keywords
ordinary differential equations, Linear and elementary, textbook, Mathematics, Matematik
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-16393 (URN)096e35e4-a60f-4c0e-8de8-082ce70be150 (Local ID)978-87-403-1290-4 (ISBN)096e35e4-a60f-4c0e-8de8-082ce70be150 (Archive number)096e35e4-a60f-4c0e-8de8-082ce70be150 (OAI)
Note
Godkänd; 2016; 20160411 (norbert)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-03-08Bibliographically approved
Euler, M. & Euler, N. (2016). Problemas, Teoria y Soluciones en Algebra Lineal: Parte 1 Espacio Euclideo (1ed.). Paper presented at . London: BookBooN.com
Open this publication in new window or tab >>Problemas, Teoria y Soluciones en Algebra Lineal: Parte 1 Espacio Euclideo
2016 (Spanish)Book (Refereed)
Alternative title[en]
Problems, Theory and Solutions in Linear Algebra: Part 1. Euclidean Space
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.

Place, publisher, year, edition, pages
London: BookBooN.com, 2016. p. 232 Edition: 1
Keywords
Linear Algebra, Solved probelms and exercises, theory, Euclidean space, Mathematics, Matematik
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:ltu:diva-16388 (URN)0725e5a5-d70c-42ce-b1f9-099a423bcec4 (Local ID)978-87-403-1415-1 (ISBN)0725e5a5-d70c-42ce-b1f9-099a423bcec4 (Archive number)0725e5a5-d70c-42ce-b1f9-099a423bcec4 (OAI)
Note
Godkänd; 2016; 20160812 (norbert)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-03-08Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-0370-7274

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