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• 1.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Correctors for some nonlinear monotone operators2001In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 8, no 1, p. 8-30Article in journal (Refereed)

In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/εh,Duh)) = fh on Ω with Dirichlet boundary conditions. Here the sequence (εh) tends to 0 as h→∞ and the map a (x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions.W e prove that uh → u weakly in W1,p 0 (Ω) as h → ∞, where u is the solution of a homogenized problem of the form -div (b (x,Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (Ph) such that Duh -Ph (Du) → 0 in Lp (Ω,Rn). () 0 in (Ω).

• 2.
Luleå tekniska universitet.
Viewing the efficiency of chaos control1999In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 6, no 3, p. 314-331Article in journal (Refereed)

This paper aims to cast some new light on controlling chaos using the OGY- and the Zero-Spectral-Radius methods. In deriving those methods we use a generalized procedure differing from the usual ones. This procedure allows us to conveniently treat maps to be controlled bringing the orbit to both various saddles and to sources with both real and complex eigenvalues. We demonstrate the procedure and the subsequent control on a variety of maps. We evaluate the control by examining the basins of attraction of the relevant controlled systems graphically and in some cases analytically

• 3.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Fourth-order recursion operators for third-order evolution equations2008In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 15, no 2, p. 147-151Article in journal (Refereed)

We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit fourth-order recursion operators. Under the given assumptions we obtain the complete list of equations, one of which is the well-known Krichever-Novikov equation.

• 4.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Second-order recursion operators of third-order evolution equations with fourth-order integrating factors2007In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 14, p. 313-315Article in journal (Refereed)

We report the recursion operators for a class of symmetry integrable evolution equa- tions of third order which admit a fourth-order integrating factor. Under some as- sumptions we obtain the complete list of equations, one of which is a special case of the Schwarzian Korteweg-de Vries equation.

• 5.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.

• 7.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. 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)

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.

• 8.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. 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)

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).

• 9. Euler, Marianna
Mathematical Institute of the Ukrainian Academy of Sciences. Computational Science and Engineering Research Center, Howard University. Computational Science and Engineering Research Center, Howard University. 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)

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.

• 10.
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)

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.

• 11.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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

• 12. Euler, Norbert
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)

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)]

• 13.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.

• 14.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.

• 15. Euler, Norbert
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)

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.

• 16. Euler, Norbert
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)

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.

• 17.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.

• 18.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
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)

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.

• 19.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
University of Santriago, Chile.
Local and Nonlocal Symmetries in Mathematical Physics2017In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 24, no Suppl. 1, p. 1-2Article in journal (Refereed)
• 20.
University of KwaZulu-Natal.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A novel Riccati sequence2009In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 16, no Suppl. 1, p. 157-164Article in journal (Refereed)

Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity, symmetry and integrability properties. We examine a variation of this sequence and find that there are some remarkable changes in properties consequential upon this variation.

• 21. Lukkassen, Dag
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On weak convergence of locally periodic functions2002In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 9, no 1, p. 42-57Article in journal (Refereed)

We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boandary value problems are also considered.

• 22.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some homogenization and corrector results for nonlinear monotone operators1998In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 5, no 2, p. 331-348Article in journal (Refereed)

This paper deals with the limit behaviour of the solutions of quasi-linear equations of the form - div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. The sequence (h) tends to 0 and the map a(x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions. It is proved that uh u weakly in H1,2 0 (), where u is the solution of a homogenized problem - div(b(x, Du)) = f on . We also prove some corrector results, i.e. we find (Ph) such that Duh - Ph(Du) 0 in L2 (, Rn ).

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