×

Infinite hierarchies of nonlocal symmetries of the Chen-Kontsevich-Schwarz type for the oriented associativity equations. (English) Zbl 1186.37036

The author of this interesting paper constructs infinite hierarchies of nonlocal higher symmetries for the oriented associativity equations using for this purpose the solutions of auxiliary spectral problems. The structure constants \(c_{\alpha\beta }^{\delta }(x^1,...,x^n)\) of a commutative algebra satisfy the relations \[ c_{\alpha\rho }^{\nu }c_{\beta\gamma }^{\rho }=c_{\rho \gamma }^{\nu }c_{\alpha\beta }^{\rho }, \;\;{\partial c_{\beta\gamma }^{\alpha }\over \partial x^{\rho }}={\partial c_{\rho\gamma }^{\alpha }\over \partial x^{\beta }}. \] The symmetries mentioned above generalize those found by Chen, Kontsevich and Schwarz for the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. An interesting result that the author obtains is a Darboux-type transformation and a conditional Bäcklund transformation for the oriented associativity equation. As a conclusion one could find some open problems arising in the theory.

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDFBibTeX XMLCite
Full Text: DOI arXiv