An algebraic formula for superposition and the completeness of the Bäcklund transformations of (2+1)-dimensional integrable systems

Abstract

In a crystal containing paramagnetic impurities the parameters that specify mean squares and products of the intrinsic strain components at the impurity sites can in principle be obtained from EPR linewidths provided that the spin-strain coupling tensor is known. It is shown that if the strain-inducing defects are distributed with the symmetry of the crystal lattice the number of such parameters is restricted by the relation ( . E~, . E~~) = & u 6 , , (~) J~k )where &jJi is the ith component of the rth repetition of the irreducible representation A of the crystal's point group. Some strain-broadening measurements in MgO are considered. Ruby linewidths have been measured and an expression containing four such strain parameters, a mean-square crystallite misalignment, and an expression for the linewidth arising from Cr3' interaction with A1 nuclei fitted to them to obtain meaningful strain parameters.

Two-dimensional rational solitons and their blowup via the moutard transformation

“…In this case an algebraic superposition formula exists for three initial solutions (cf. [9] for a detailed discussion of this phenomenon). Namely, as one may check, the following statement, obtained by L. Bianchi [4] for the case of the Moutard equation ϕ xy = u(x, y)ϕ, is also valid in our case:…”

The vectorial Ribaucour transformation for submanifolds and applications

“…For this case, the independence assumption is equivalent to the condition that none of f 1 , f 2 or f 3 belong to the associated family determined by the other two. Then, uniqueness was proved in [11] by using a nice elementary argument relying on the version of Miquel's Theorem for four circumferences.…”

“…The following Bianchi k-cube theorem was proved in [11] for k = 3 in the context of triply orthogonal systems of Euclidean space. A nice proof in the setup of Lie sphere geometry was recently given in [2], where an indication was also provided of how the general case can be settled by using results of [12] for discrete orthogonal nets together with an induction argument.…”

Classification of Three-Dimensional Integrable Scalar Discrete Equations

“…On the other hand, starting from the classical theorems on non-linear superposition principles and permutability of the aforementioned transformations between smooth surfaces of one of these types we obtain precisely the underlying discrete system. One of the cornerstones of the discrete differential geometry (the idea to look for cubic nonlinear superposition formulas of Bäcklund transformations of nonlinear integrable PDEs) was laid down in [5]. The duality between the smooth objects in any of the geometric classes of integrable smooth surfaces mentioned above and their Bäcklund-type transformations is therefore…”

“…The dBKP-system has many equivalent forms and appears in very different contexts. In addition to the known geometric interpretations and a reformulation as Yang-Baiter system ( [8]), the dBKP-system may be considered as a nonlinear superposition principle for the classical 2-dimensional Moutard transformations ( [5]). The expression (5) enjoys an extra symmetry property: the equation Q (−−−) = 0 is invariant under the action of the SL 2 (C) group of fractional-linear transformations…”