Abelian versus non-Abelian Bäcklund charts: Some remarks

Sandra Carillo; Mauro Lo Schiavo; Cornelia Schiebold

doi:10.3934/eect.2019003

Article Contents

2019, Volume 8, Issue 1: 43-55. Doi: 10.3934/eect.2019003

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Abelian versus non-Abelian Bäcklund charts: Some remarks

1.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma La Sapienza, Via A. Scarpa 16, I-00161, Italy
2.
I.N.F.N. - Sez. Roma1, Gr.Ⅳ: Mathematical Methods in NonLinear Physics, Rome, Italy
3.
Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, I-00100, Rome, Italy
4.
Department of Mathematics and Science Education, Mid Sweden University, S-851 70 Sundsvall, Sweden
5.
Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland

Received: April 30, 2018

Revised: April 30, 2018

Early access: December 2018

Published: January 2019

The authors are supported by G.N.F.M.-I.N.d.A.M., I.N.F.N. and La Sapienza Università di Roma, Italy.

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Abstract

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.

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Mathematics Subject Classification: Primary: 58G37, 35Q53, 58F07.

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Figure 1. Burgers and mirror Burgers equations and their Bäcklund links: the non-commutative case.

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Figure 2. Burgers and mirror Burgers hierarchies and their Bäcklund links: the non-commutative case.

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Figure 3. KdV-type Bäcklund chart: the Abelian case

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Figure 4. Abelian KdV-type hierarchies Bäcklund chart: induced invariances

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Figure 5. KdV-type hierarchies Bäcklund chart: the non-Abelian case

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