Deformation of surfaces in three-dimensional space induced by means of integrable systems

Bracken, Paul

doi:10.1016/j.nonrwa.2012.09.021

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…Most works deal with applications where the image of F is a two-dimensional surface. Less commonly applications deal with an F whose image is an immersed submanifold in a Lie algebra [6]. All such results can be found in a simpler and more general way, e.g.…”

Section: Immersion Formulasmentioning

confidence: 99%

“…Our procedure provides simple geometric proofs for the theoretical results in the previous literature on the topic, simplifies expressions for immersion formulas via geometric structures, and extends the formalism of immersion formulas to create a theory of immersion formulas for soliton submanifolds and generalized Lax pairs with many potential applications (see e.g. [1,6,44,67,68] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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A cohomological approach to immersed submanifolds via integrable systems

Grundland,

de Lucas

2017

Preprint

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A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of g-valued differential forms. This leads us to introduce Poincaré-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings also allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through CP N −1 sigma models.

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Section: Immersion Formulasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A cohomological approach to immersed submanifolds via integrable systems

Grundland,

de Lucas

2017

Preprint

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Lax Triples for Integrable Surfaces in Three-Dimensional Space

Cieśliński

Kobus²

2016

Advances in Mathematical Physics

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We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean spaceE3. We begin with a natural integrable deformation of the principal chiral model. Then, we show that all deformations linear in the spectral parameterλare trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the groupSpin(6). Finally, the obtained results are interpreted in the context of the soliton surfaces approach.

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Deformation of surfaces in three-dimensional space induced by means of integrable systems

Cited by 2 publications

References 19 publications

A cohomological approach to immersed submanifolds via integrable systems

A cohomological approach to immersed submanifolds via integrable systems

Lax Triples for Integrable Surfaces in Three-Dimensional Space

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