Darboux-Bäcklund Transformations
for Spin-Valued Linear Problems
Jan L. Cie´ sli ´
nski
This paper is dedicated to Prof. Decio Levi on the occasion of
his 70th birthday.
Abstract Motivated by geometry of submanifolds we develop an algebraic construction of Darboux transformations using Clifford numbers and Spin groups.
Eigenvalues parameterizing solitons, usually computed as zeros of determinants,
are identified as zeros of the spinor norm. Reduction groups (loop groups) for Spinvalued linear problems are identified with involutions in Clifford algebras.
Keywords Darboux matrix · Spin groups · Clifford algebras · Soliton surfaces
1 Introduction
This paper is a continuation of my long research on Darboux transformations, which
started more than 30 years ago and the interaction with Decio Levi has been crucial
in this context. My PhD thesis was focused on the following problems, all of which
stemmed from the cooperation between Antoni Sym (my supervisor) and Decio
Levi:
• Group interpretation of the spectral parameter [1–6].
• Darboux transformations (non-isospectral and non-canonical cases) [1, 7–9].
• Symmetric formulas for multi-soliton surfaces [10–12].
My several visits to Rome were very stimulating for this research. Recently, I
returned to these topics, see [13, 14] and the formula (24) in the present paper,
with many hopes for further progress in the near future. In the meantime I devoted
J. L. Cie´ sli´ nski ()
University of Bialystok, Faculty of Physics, Białystok, Poland
e-mail: j.cieslinski@uwb.edu.pl
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_3
37
for Spin-Valued Linear Problems
Jan L. Cie´ sli ´
nski
This paper is dedicated to Prof. Decio Levi on the occasion of
his 70th birthday.
Abstract Motivated by geometry of submanifolds we develop an algebraic construction of Darboux transformations using Clifford numbers and Spin groups.
Eigenvalues parameterizing solitons, usually computed as zeros of determinants,
are identified as zeros of the spinor norm. Reduction groups (loop groups) for Spinvalued linear problems are identified with involutions in Clifford algebras.
Keywords Darboux matrix · Spin groups · Clifford algebras · Soliton surfaces
1 Introduction
This paper is a continuation of my long research on Darboux transformations, which
started more than 30 years ago and the interaction with Decio Levi has been crucial
in this context. My PhD thesis was focused on the following problems, all of which
stemmed from the cooperation between Antoni Sym (my supervisor) and Decio
Levi:
• Group interpretation of the spectral parameter [1–6].
• Darboux transformations (non-isospectral and non-canonical cases) [1, 7–9].
• Symmetric formulas for multi-soliton surfaces [10–12].
My several visits to Rome were very stimulating for this research. Recently, I
returned to these topics, see [13, 14] and the formula (24) in the present paper,
with many hopes for further progress in the near future. In the meantime I devoted
J. L. Cie´ sli´ nski ()
University of Bialystok, Faculty of Physics, Białystok, Poland
e-mail: j.cieslinski@uwb.edu.pl
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_3
37