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Bäcklund transformations as integrable discretization. The geometric approach

Adam Doliwa

Abstract

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by Bäcklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that Bäcklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

Bäcklund transformations as integrable discretization. The geometric approach

Abstract

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by Bäcklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that Bäcklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.
Paper Structure (13 sections, 13 theorems, 83 equations, 4 figures)

This paper contains 13 sections, 13 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discretization of asymptotic nets, and the Moutard transformation
  3. Hyperbolic surfaces, the Moutard equation, and the Lelieuvre formulas
  4. The Moutard transformation and its permutability property
  5. The Moutard transformation as integrable discretization
  6. Discrete BKP equation
  7. Multidimensional conjugate nets, their fundamental transformation, and multidimensional lattices of planar quadrilaterals
  8. Conjugate coordinates on a surface, and the Lévy transformation
  9. Multidimensional conjugate nets and the Darboux equations
  10. The vectorial fundamental (binary Darboux) transformation of conjugate nets
  11. Superpositions of fundamental transformations, and multidimensional lattices of planar quadrilaterals
  12. Curvature coordinates, the Ribaucour transformation, and circular lattices
  13. Conclusion and open problems

Key Result

Theorem 2.1

Moutard Given vector-valued solution $\boldsymbol{\nu}$ of the Moutard equation corresponding to a given potential $f$, and given scalar solution $\theta$ of the same equation, then the vector-valued function $\hat{\boldsymbol{\nu}}$ defined by the compatible equations satisfies the Moutard equation with the potential

Figures (4)

  • Figure 1: Discrete asymptotic nets
  • Figure 2: The fundamental transform $\hat{\boldsymbol{r}}$, the Lévy transforms $\boldsymbol{r}^{(u)}$, $\boldsymbol{r}^{(v)}$, the adjoint Lévy transforms $\boldsymbol{r}_{(u)}$, $\boldsymbol{r}_{(v)}$, and the Laplace transforms $\boldsymbol{r}^{(u)}_{(v)}$, $\boldsymbol{r}^{(v)}_{(u)}$ of two-dimensional conjugate net $\boldsymbol{r}$
  • Figure 3: Elementary quadrilateral of superposition of two fundamental transformations
  • Figure 4: The superposition of two scalar Ribaucour transforms of an orthogonal conjugate net. The vertices $\boldsymbol{r}$, $\boldsymbol{r}^{\{a\}}$, $\boldsymbol{r}^{\{b\}}$, and $\boldsymbol{r}^{\{a,b\}}$ of the elementary quadrilateral are concircular. For circular quadrilaterals opposite angles sum up to $\pi$. The center of the circle is the intersection point of bisectors of all the four sides of the quadrilateral

Theorems & Definitions (26)

  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Remark
  • Theorem 2.4: Permutability of the Moutard transformations
  • Remark
  • Corollary 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Remark
  • ...and 16 more