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Schiebold, Cornelia
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Publications (10 of 32) Show all publications
Carillo, S., Schiavo, M. L. & Schiebold, C. (2019). Abelian versus non-Abelian Bäcklund charts: Some remarks. Evolution Equations and Control Theory, 8(1), 43-55
Open this publication in new window or tab >>Abelian versus non-Abelian Bäcklund charts: Some remarks
2019 (English)In: Evolution Equations and Control Theory, ISSN 2163-2472, Vol. 8, no 1, p. 43-55Article in journal (Refereed) Published
Abstract [en]

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. 

Keywords
Burgers equations, Bäcklund transformations, Cole-Hopf Transformations, Invariances, Korteweg deVries-type equations, Nonlinear evolution equations, Recursion operators
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-35834 (URN)10.3934/eect.2019003 (DOI)000462022800004 ()2-s2.0-85061866201 (Scopus ID)
Available from: 2019-03-20 Created: 2019-03-20 Last updated: 2019-05-20Bibliographically approved
Carillo, S., Lo Schiavo, M., Porten, E. & Schiebold, C. (2018). A novel noncommutative KdV-type equation, its recursion operator, and solitons. Journal of Mathematical Physics, 59(4), Article ID 043501.
Open this publication in new window or tab >>A novel noncommutative KdV-type equation, its recursion operator, and solitons
2018 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, no 4, article id 043501Article in journal (Refereed) Published
Abstract [en]

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. 

National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-33634 (URN)10.1063/1.5027481 (DOI)000431271800031 ()2-s2.0-85045104787 (Scopus ID)
Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-05-30Bibliographically approved
Carillo, S., Lo Schiavo, M. & Schiebold, C. (2018). Recursion operators admitted by non-Abelian Burgers equations: some remarks. Mathematics and Computers in Simulation, 147(SI), 40-51
Open this publication in new window or tab >>Recursion operators admitted by non-Abelian Burgers equations: some remarks
2018 (English)In: Mathematics and Computers in Simulation, ISSN 0378-4754, E-ISSN 1872-7166, Vol. 147, no SI, p. 40-51Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Recursion operator, Bäcklund transformations, Non-Abelian nonlinear evolution equations, Burgers equation
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-29469 (URN)10.1016/j.matcom.2017.02.001 (DOI)000425315100004 ()2-s2.0-85014578707 (Scopus ID)
Note

Available online 24 February 2017

https://arxiv.org/pdf/1606.07270.pdf

Available from: 2016-12-07 Created: 2016-12-07 Last updated: 2018-04-25Bibliographically approved
Nilson, T. & Schiebold, C. (2018). Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour.
Open this publication in new window or tab >>Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour
2018 (English)Report (Other academic)
Abstract [en]

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. 

Publisher
p. 35
Series
Mid Sweden Mathematical Reports ; 2
Keywords
2d Toda lattice, asymptotic behaviour, operator identities, multiple pole solutions
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-35283 (URN)978-91-88527-86-8 (ISBN)
Available from: 2018-12-18 Created: 2018-12-18 Last updated: 2018-12-19Bibliographically approved
Schiebold, C. (2017). Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Nonlinearity, 30(7), 2930-2981
Open this publication in new window or tab >>Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation
2017 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 7, p. 2930-2981Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
nonlinear Schrodinger equation, multiple pole solutions, asymptotic behaviour, Cauchy-type determinants, inverse scattering method
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-31347 (URN)10.1088/1361-6544/aa6d9a (DOI)000403576400002 ()2-s2.0-85021249183 (Scopus ID)
Available from: 2017-08-08 Created: 2017-08-08 Last updated: 2017-08-10Bibliographically approved
Li, S., Biondini, G. & Schiebold, C. (2017). On the degenerate soliton solutions of the focusing nonlinear Schrödinger equation. Journal of Mathematical Physics, 58(3), Article ID 033507.
Open this publication in new window or tab >>On the degenerate soliton solutions of the focusing nonlinear Schrödinger equation
2017 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 3, article id 033507Article in journal (Other academic) Published
Abstract [en]

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.

Keywords
Solitons, Nonlinear Schrodinger equation, Inverse scattering, Long-time asymptotics
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-29468 (URN)10.1063/1.4977984 (DOI)000397873800036 ()2-s2.0-85016255505 (Scopus ID)
Available from: 2016-12-07 Created: 2016-12-07 Last updated: 2017-04-25Bibliographically approved
Carillo, S., Lo Schiavo, M. & Schiebold, C. (2016). Bäcklund transformations and non-abelian nonlinear evolution equations: A novel bäcklund chart. SIGMA. Symmetry, Integrability and Geometry, 12, Article ID 087.
Open this publication in new window or tab >>Bäcklund transformations and non-abelian nonlinear evolution equations: A novel bäcklund chart
2016 (English)In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 12, article id 087Article in journal (Refereed) Published
Abstract [en]

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.

National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-28915 (URN)10.3842/SIGMA.2016.087 (DOI)000383277400001 ()2-s2.0-84986576759 (Scopus ID)
Available from: 2016-09-26 Created: 2016-09-26 Last updated: 2017-11-21Bibliographically approved
Schiebold, C. (2014). Asymptotics for the multiple pole solutions of the Nonlinear Schrödinger equation.
Open this publication in new window or tab >>Asymptotics for the multiple pole solutions of the Nonlinear Schrödinger equation
2014 (English)Report (Other academic)
Publisher
p. 51
Series
Mid Sweden Mathematical Reports ; 1
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-22109 (URN)
Available from: 2014-06-05 Created: 2014-06-05 Last updated: 2014-06-11Bibliographically approved
Carillo, S. & Schiebold, C. (2012). On the recursion operator for the noncommutative Burgers hierarchy. Journal of Nonlinear Mathematical Physics, 19(1), Art. no. 1250003
Open this publication in new window or tab >>On the recursion operator for the noncommutative Burgers hierarchy
2012 (English)In: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 19, no 1, p. Art. no. 1250003-Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
Bäcklund transformations; Burgers equation; hereditariness; noncommutativity; recursion operators
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-15343 (URN)10.1142/S1402925112500039 (DOI)000302022500003 ()2-s2.0-84858822229 (Scopus ID)
Available from: 2011-12-16 Created: 2011-12-16 Last updated: 2017-12-08Bibliographically approved
Carillo, S. & Schiebold, C. (2011). Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions. Journal of Mathematical Physics, 52(5), Art. no. 053507
Open this publication in new window or tab >>Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: Noncommutative soliton solutions
2011 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, no 5, p. Art. no. 053507-Article in journal (Refereed) Published
Abstract [en]

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]

Keywords
OPERATOR IDEALS; NONLINEAR EQUATIONS; MKDV-EQUATION; KDV EQUATION; TRACE
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-14178 (URN)10.1063/1.3576185 (DOI)000291106000032 ()2-s2.0-79957899011 (Scopus ID)
Available from: 2011-07-19 Created: 2011-07-19 Last updated: 2017-12-08Bibliographically approved
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