Flat space curved space11/6/2022 ![]() ![]() the robust inverse-scattering transform method recently introduced by Bilman and Miller. The resulting functions are encoded in a Riemann-Hilbert problem using. Starting from the zero background, we generate multiple-pole solitons by $n$-fold application of Darboux transformations. We analyze the large-$n$ behavior of soliton solutions of the integrable focusing nonlinear Schr\"odinger equation with associated spectral data consisting of a single pair of conjugate poles of order $2n$. We also discuss actions of recursion operators on shadows (in the sense of ) of nonlocal symmetries. For all the for pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not the same) structures: the are (semi) direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras all of them contain a component of finite grading. #Flat space curved space fullequation, using the know Lax pairs and expanding the latter in formal series in spectral parameter, we construct two infinite-dimensional differential coverings and give a full description of nonlocal symmetry algebras associated to these coverings. We consider four three-dimensional equations: (1) the rdDym equation $u_$. We continue here the study of Lax integrable equations. This gives a space of solutions which has the functional dimension of finitely. Introducing the necessary loop algebras in Section 6 we then apply the finite type integration scheme to construct solutions to the curved flat equations from a hierarchy of commuting algebraically integrable ODE in Lax form. Therefore one can write the curved flat condition as a zero-curvature equation on a Lie algebra valued 1-form involving the deformation - spectral - parameter linearly. Besides their geometric relevance curved flats are also interesting from the integrable systems point of view: scaling the derivative of a curved flat gives rise to a non-trivial deformation which obviously preserves the curved flat condition (Section 5). Of course, every curve in a symmetric space is a curved flat. Further examples of curved flats come from isothermic surfaces and conformally flat 3-folds in the 4-sphere. The roots of the Cartan subalgebra are mapped via this isometry into principal curvature coordinates of the corresponding isometric immersion (Section 4). The developing isometry identifies (locally) the curved flat with a Cartan subalgebra of the Grassmanian (in which the Gauss map takes values in). Important examples of curved flats arise from isometric immersions of space forms into space forms via their Gauss maps (Section 3). Since curved flats are intrinsically flat (Theorem 2) they may also be viewed as natural analogues of developable surfaces in 3-space. A curved flat is simply a map into a symmetric space which is tangent at each point to a flat of the symmetric space, i.e. This paper we introduce a natural class of maps into symmetric spaces, curved flats, which, in a certain sense, contains all the above examples as special cases. ![]()
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