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• 1.
Sapienza Università di Roma, Rome, Italy.
Sapienza Università di Roma, Rome, Italy. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego W Kielcach, Kielce, Poland.
A novel noncommutative KdV-type equation, its recursion operator, and solitons2018In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, no 4, article id 043501Article in journal (Refereed)

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in Carillo et al. [Symmetry Integrability Geom.: Methods Appl. 12, 087 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in Olver and Sokolov [Commun. Math. Phys. 193, 245 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies, and an explicit solution class are derived.

• 2.
Dipartimento “Scienze di Base e Applicate per l’Ingegneria”, Sapienza - Università di Roma, 16, Via A. Scarpa, Rome, Italy; I.N.F.N. - Sez. Roma1, Gr. IV - Mathematical Methods in NonLinear Physics, Rome, Italy .
Dipartimento “Scienze di Base e Applicate per l’Ingegneria”, Sapienza - Università di Roma, 16, Via A. Scarpa, Rome, Italy. Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland.
Bäcklund transformations and non-abelian nonlinear evolution equations: A novel bäcklund chart2016In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 12, article id 087Article in journal (Refereed)

Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

• 3.
Dipartimento “Scienze di Base e Applicate per l’Ingegneria”, Sapienza - Universita di Roma, 16, Via A. Scarpa, 00161 Rome, Italy.
Dipartimento “Scienze di Base e Applicate per l’Ingegneria”, Sapienza - Universita di Roma, 16, Via A. Scarpa, 00161 Rome, Italy. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland.
Recursion operators admitted by non-Abelian Burgers equations: some remarks2018In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 147, no SI, p. 40-51Article in journal (Refereed)

The recursion operators admitted by different operator Burgers equations, in the framework of the study of nonlinear evolution equations, are here considered. Specifically, evolution equations wherein the unknown is an operator acting on a Banach space are investigated. Here, the mirror   non-Abelian Burgers equation is considered: it can be written as rt=rxx+2rxrrt=rxx+2rxr. The structural properties of the admitted recursion operator are studied; thus, it is proved to be a strong symmetry for the mirror non-Abelian Burgers equation as well as to be the hereditary. These results are proved via direct computations as well as via computer assisted manipulations; ad hoc routines are needed to treat non-Abelian quantities and relations among them. The obtained recursion operator generates the mirror   non-Abelian Burgers hierarchy. The latter, when the unknown operator rr is replaced by a real valued function reduces to the usual (commutative) Burgers hierarchy. Accordingly, also the recursion operator reduces to the usual Burgers one.

• 4.
Università di Roma La Sapienza, Scarpa, Italy; I.N.F.N.-Sez. Roma1, Gr.IV: Mathematical Methods in NonLinear Physics Rome, Italy.
Università di Roma La Sapienza, Rome, Italy. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego w Kielcach, Poland.
Abelian versus non-Abelian Bäcklund charts: Some remarks2019In: Evolution Equations and Control Theory, ISSN 2163-2472, Vol. 8, no 1, p. 43-55Article in journal (Refereed)

Connections via Bäcklund transformations among different nonlinear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Bäcklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Bäcklund charts in the case of the Burgers-heat equation, on one side, and KdV-type equations, on the other, are considered. The Bäcklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.

• 5.
Universita Roma La Sapienza.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
A non-commutative operator-hierarchy of Burgers equations and Bäcklund transformations2009In: Series on Advances in Mathematics for Applied Sciences, ISSN 1793-0901, Vol. 82Article in journal (Refereed)
• 6.
Universita Roma La Sapienza.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Matrix KdV and mKdV hierarchies: Noncommutative soliton solutions and explicit formulae2009Report (Other academic)
• 7.
Univ Roma La Sapienza, Dipartimento Sci Base & Applicate Ingn, Sez Matemat, Rome, Italy .
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 5, p. Art. no. 053507-Article in journal (Refereed)

The present work continues work on KdV-type hierarchies presented by S. Carillo and C. Schiebold ["Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods," J. Math. Phys. 50, 073510 (2009)]. General solution formulas for the KdV and mKdV hierarchies are derived by means of Banach space techniques both in the scalar and matrix case. A detailed analysis is given of solitons, breathers, their countable superpositions as well as of multisoliton solutions for the matrix hierarchies. (C) 2011 American Institute of Physics. [doi:10.1063/1.3576185]

• 8.
Universita Roma La Sapienza.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods2009In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 50, no 7, article id 073510Article in journal (Refereed)

Here, noncommutative hierarchies of nonlinear equations are studied. They represent a generalization to the operator level of corresponding hierarchies of scalar equations, which can be obtained from the operator ones via a suitable projection. A key tool is the application of Baumlcklund transformations to relate different operator-valued hierarchies. Indeed, in the case when hierarchies in 1+1-dimensions are considered, a "Baumlcklund chart" depicts links relating, in particular, the Korteweg-de Vries (KdV) to the modified KdV (mKdV) hierarchy. Notably, analogous links connect the hierarchies of operator equations. The main result is the construction of an operator soliton solution depending on an infinite-dimensional parameter. To start with, the potential KdV hierarchy is considered. Then Baumlcklund transformations are exploited to derive solution formulas in the case of KdV and mKdV hierarchies. It is remarked that hierarchies of matrix equations, of any dimension, are also incorporated in the present framework.

• 9.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria Sez. Matematica, SAPIENZA Università di Roma, Rome, Italy.
Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
On the recursion operator for the noncommutative Burgers hierarchy2012In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 19, no 1, p. Art. no. 1250003-Article in journal (Refereed)

The noncommutative Burgers recursion operator is constructed via the ColeHopf transformation, and its structural properties are studied. In particular, a direct proof of its hereditary property is given. © S. Carillo and C. Schiebold.

• 10.
Universita Roma I La Sapienza.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Recursion techniques and explicit solutions of integrable noncommutative hierarchies2010In: Proceedings WASCOM 2009 / [ed] A. Greco, S. Rionero, T. Ruggeri, 2010, p. 74-80Conference paper (Refereed)
• 11.
Universität Jena.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Ein direkter Ansatz zur Untersuchung von Solitonengleichungen2000In: Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner), ISSN 0012-0456, E-ISSN 1869-7135, Vol. 102, no 3, p. 102-148Article in journal (Refereed)
• 12.
Universität Jena.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Nonlinear equations in soliton physics and operator ideals1999In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 12, no 2, p. 333-364Article in journal (Refereed)

An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively. Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper. Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation. Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows.

• 13.
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260 – USA.
Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260 – USA. Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielcach, Poland.
On the degenerate soliton solutions of the focusing nonlinear Schrödinger equation2017In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 3, article id 033507Article in journal (Other academic)

We characterize the N-soliton solutions of the focusing nonlinear Schrodinger (NLS) equation with degenerate velocities, i.e., solutions in which two or more soliton velocities are the same, which are obtained when two or more discrete eigenvalues of the scattering problem have the same real parts. We do so by employing the operator formalism developed by one of the authors to express the N-soliton solution of the NLS equation in a convenient form. First we analyze soliton solutions with fully degenerate velocities (a so-called multi-soliton group), clarifying their dependence on the soliton parameters. We then consider the dynamics of soliton groups interaction in a general N-soliton solution. We compute the long-time asymptotics of the solution and we quantify the interaction-induced position and phase shifts of each non-degenerate soliton as well as the interaction-induced changes in the center of mass and soliton parameters of each soliton group.

• 14.
Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
On the noncommutative two-dimensional Toda latticeManuscript (preprint) (Other academic)
• 15.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education. Uniwersytet Jana Kochanowskiego w Kielcach, Poland.
Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour2018Report (Other academic)

The first main aim of this article is to derive an explicit solution formula for the scalar 2d-Toda lattice depending on three independent operator parameters, ameliorating work in [29]. This is achieved by studying a noncommutative version of the two-dimensional Toda lattice, generalizing its soliton solution to the noncommutative setting.

The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

• 16.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Cauchy-type determinants and integrable systems2010In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 433, no 2, p. 447-475Article in journal (Refereed)

It is well known that the Sylvester matrix equation AX + XB = C has a unique solution X if and only if 0 ∉ spec(A) + spec(B). The main result of the present article are explicit formulas for the determinant of X in the case that C is one-dimensional. For diagonal matrices A, B, we reobtain a classical result by Cauchy as a special case. The formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine-Gordon equation and the Toda lattice. We will provide a concise introduction to the background from soliton theory, an operator theoretic approach originating from work of Marchenko and Carl, and discuss examples for the application of the main results.

• 17.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
A non-abelian Nonlinear Schrödinger equation and countable superposition2008In: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, no 3, p. 5p. 245-250Article in journal (Refereed)
• 18.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
An operator-theoretic approach to the Toda lattice equation1998In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 122, no 1-4, p. 37-61Article in journal (Refereed)

We treat the Toda lattice equation with operator methods and derive an explicit solution formula in terms of determinants. As an application, we investigate solutions which are given by special settings. In the finite-dimensional case matrices in Jordan canonical form give rise to a new class of solutions. Within this class the well-known N-soliton solutions can be recovered by the special choice of diagonal matrices. Moreover, using diagonal operators we get solutions depending on an infinite number of parameters. We comprehensively discuss the case involving diagonal operators and show that it can be reduced to a very particular situation.

• 19.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics. Uniwersytet Jana Kochanowskiego Kielcach, Inst Matemat, Kielce, Poland.
Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation2017In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 7, p. 2930-2981Article in journal (Refereed)

Multiple pole solutions consist of groups of weakly bound solitons. For the (focusing) nonlinear Schrodinger equation the double pole solution was constructed by Zakharov and Shabat. In the sequel particular cases have been discussed in the literature, but it has remained an open problem to understand multiple pole solutions in their full complexity.

In the present work this problem is solved, in the sense that a rigorous and complete asymptotic description of the multiple pole solutions is given. More precisely, the asymptotic paths of the solitons are determined and their position-and phase-shifts are computed explicitly. As a corollary we generalize the conservation law known for the N-solitons. In the special case of one wave packet, our result confirms a conjecture of Olmedilla.

Our method stems from an operator theoretic approach to integrable systems. To facilitate comparison with the literature, we also establish the link to the construction of multiple pole solutions via the inverse scattering method. The work is rounded off by many examples and MATHEMATICA plots and a detailed discussion of the transition to the next level of degeneracy.

• 20.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Science Education and Mathematics.
Asymptotics for the multiple pole solutions of the Nonlinear Schrödinger equation2014Report (Other academic)
• 21.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Cauchy-type determinants and integrable systems2009Report (Other academic)
• 22.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Die klassische Charaktertheorie der Mathieugruppen1993Licentiate thesis, monograph (Other academic)
• 23.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Explicit solution formulas for the matrix-KP2009In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 51A, p. 147-155Article in journal (Refereed)

We study a non-commutative version of the Kadomtsev-Petviashvili equations and construct a family of solutions generalizing naturally the soliton to the non-commutative setting. From this we derive explicit solution formulas as well for the scalar as for the matrix-Kadomtsev-Petviashvili equation which still depend on operator parameters.

• 24.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
From the non-abelian to the scalar two-dimensional Toda lattice2005In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 47, no A, p. 177-189Article in journal (Refereed)

We extend a solution method used for the one-dimensional Toda lattice in [1], [2] to the two-dimensional Toda lattice. The idea is to study the lattice not with values in $\mathbb{C}$ but in the Banach algebra ${\cal L}$ of bounded operators and to derive solutions of the original lattice ($\mathbb{C}$-solutions) by applying a functional $\tau$ to the ${\cal L}$-solutions constructed in 1. The main advantage of this process is that the derived solution still contains an element of $\cal L$ as parameter that may be chosen arbitrarily. Therefore, plugging in different types of operators, we can systematically construct a huge variety of solutions. In the second part we focus on applications. We start by rederiving line-solitons and briefly discuss discrete resonance phenomena. Moreover, we are able to find conditions under which it is possible to superpose even countably many line-solitons.

• 25.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Funktionalanalytische Methoden bei der Behandlung von Solitonengleichungen1997Doctoral thesis, monograph (Other scientific)
• 26.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
Integrable Systems and Operator Equations2004Doctoral thesis, monograph (Other scientific)
• 27.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Integrable Systems and Operator Equations: Habilitationsschrift2004Book (Refereed)
• 28.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, no Suppl, p. 678-690Article in journal (Refereed)

The main result is a very general solution formula for the noncommutative AKNS system, extending work by Bauhardt and P¨oppe. As anapplication, we construct for the matrix sine-Gordon equation N-soliton solutions analogous to the multisoliton solutions for the KdV equation due toGoncharenko.

• 29.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
On negatons of the Toda lattice2003In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 10, no suppl.2, p. 181-193Article in journal (Refereed)
• 30.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Solitons of the sine-Gordon equation coming in clusters2002In: Revista Matemática Complutense, ISSN 1139-1138, Vol. 15, no 1, p. 265-325Article in journal (Refereed)
• 31.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Structural properties of the noncommutative KdV recursion operator2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 11, p. 16p. Art. no. 113504-Article in journal (Refereed)

The present work studies structural properties of the recursion operator of the noncommutative KdV equation. As the main result, it is proved that this operator is hereditary. The notion of hereditary operators was introduced by Fuchssteiner for infinite-dimensional integrable systems, building on classical concepts from differential topology. As an illustration for the consequences of this property, it is deduced that the flows of the noncommutative KdV hierarchy mutually commute.

• 32.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
The noncommutative AKNS system: projection to matrix systems, countable superposition and soliton-like solutions2010In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 43, no 43, p. 434030-Article in journal (Refereed)

Starting from the recent work on noncommutative AKNS systems for functions with values in the bounded operators on a Banach space, it is shown how their formal 1-soliton solution (depending on operator parameters) can be mapped to solutions of matrix AKNS systems. The main result is rather general solution formulas for matrix AKNS systems. The most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton-like solutions of the classical AKNS system.

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