[Paper Review] On one-parametric families of Backlund transformations

Sergei Igonin, Joseph Krasil’shchik|ArXiv.org|Oct 25, 2000

Nonlinear Waves and Solitons8 references6 citations

TL;DR

This paper provides a cohomological description of one-parameter families of Bäcklund transformations within the framework of the geometrical theory of nonlinear PDEs. It shows that such families evolve in the direction of a nonlocal symmetry shadow, generalizing classical results on Bäcklund transformations for equations like sine-Gordon and potential KdV through the use of covering theory and deformation cohomology.

ABSTRACT

In the context of the cohomological deformation theory, infinitesimal description of one-parametric families of Backlund transformations of special type including classical examples is given. It is shown that any family of such a kind evolves in the direction of a nonlocal symmetry shadow.

Motivation & Objective

• To develop an infinitesimal description of one-parameter families of Bäcklund transformations using cohomological deformation theory.
• To clarify the geometric and cohomological structure underlying classical Bäcklund transformations, such as those for the sine-Gordon and potential KdV equations.
• To establish a connection between smooth families of coverings and nonlocal symmetries via the concept of shadows in the sense of [10].
• To characterize deformations of covering structures that preserve the equation's underlying geometry.
• To provide a systematic framework for understanding the nonlinear superposition principle (Bianchi permutability) through cohomological invariants.

Proposed method

• Utilizes the theory of jet bundles and infinite prolongations to model nonlinear PDEs and their symmetries.
• Applies the Cartan connection and Cartan distribution on the infinite jet space to define differential operators and integrability conditions.
• Introduces the cohomology complex associated with a covering $\tau$ of a PDE $\mathcal{E}$, with three components: $H_g^\bullet$, $H_s^\bullet$, and $H_C^\bullet$.
• Employs the exact sequence $H_g^\bullet \to H_C^\bullet \to H_s^\bullet$ to relate gauge symmetries, Cartan symmetries, and shadows.
• Uses the Frölicher–Nijenhuis bracket to describe the infinitesimal action of a one-parameter family of diffeomorphisms on the covering structure.
• Identifies the infinitesimal generator of a smooth family of coverings with a $\tau$-shadow, which lies in $H_s^0(\mathcal{E};\tau)$, and shows it generates the deformation via the Lie bracket $[\![U_\tau, X]\!]$.

Experimental results

Research questions

• RQ1 How can one-parameter families of Bäcklund transformations be described infinitesimally using cohomological methods?
• RQ2 What is the role of nonlocal symmetries and their shadows in generating smooth families of coverings?
• RQ3 How do deformations of covering structures relate to the underlying equation's geometry and integrability?
• RQ4 In what way do classical Bäcklund transformations (e.g., for sine-Gordon) fit into this cohomological framework?
• RQ5 What is the precise cohomological interpretation of the Bianchi permutability theorem in terms of shadow symmetries?

Key findings

• Any smooth one-parameter family of coverings $\tau_\lambda$ induces a deformation of the covering structure $U_{\tau_\lambda}$ that is linear in $\lambda$, with the infinitesimal generator given by $[\![U_\tau, X]\!]$, where $X$ is a $\tau$-shadow.
• The infinitesimal part of such a family lies in the image of the connecting homomorphism $\phi: H_s^0(\mathcal{E};\tau) \to H_g^1(\mathcal{E};\tau)$, linking shadows to deformations of the covering structure.
• The space of all such families of coverings is tangent at $\tau$ to the quotient space $\mathrm{shad}_\tau\mathcal{E}/\overline{\operatorname{sym}}_\tau\mathcal{E}$, where $\mathrm{shad}_\tau\mathcal{E} = H_s^0(\mathcal{E};\tau)$ and $\overline{\operatorname{sym}}_\tau\mathcal{E}$ is the quotient of $\tau$-symmetries modulo gauge symmetries.
• The classical infinitesimal symmetries of the sine-Gordon and potential KdV equations arise as shadows of specific coverings, confirming their role in generating Bäcklund families.
• The construction provides a cohomological justification for the nonlinear superposition principle (Bianchi permutability) via the existence of compatible shadows and deformations.
• The framework generalizes classical Bäcklund transformations by showing they are generated by the action of shadows in the covering space, thus unifying their infinitesimal description.