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Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

Filipe Kelmer, Keti Tenenblat

Abstract

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

Abstract

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.
Paper Structure (5 sections, 15 theorems, 68 equations, 4 figures, 1 table)

This paper contains 5 sections, 15 theorems, 68 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Superposition formulae
  4. Six cases
  5. Examples

Key Result

Theorem 2.1

Let $M^2_{r},\bar{M}^2_{\bar{r}}\hookrightarrow{\mathbb{R}}^3_{s}$, with $0\leq r,\bar{r}\leq s\leq 1$, be surfaces contained in the pseudo-Euclidean space ${\mathbb{R}}^3_s$. Suppose that $M$ and $\bar{M}$ are related by a Bäcklund type line congruence in ${\mathbb{R}}^3_{s}$. Then both surfaces ha where $\lambda$ is the distance of the line congruence, $\phi$ is the angle between the normals and

Figures (4)

  • Figure 1: Solutions of the sinh-Gordon equation: $\alpha_1$, $\alpha_2$ and $\alpha_3$ are obtained from the null solution by the Bäcklund transformation with parameters $\phi_1=\pi/2$, $\phi_2=\pi/3$ and $\phi_3=\pi/8$ respectively. $\alpha_{12}$ is a solution obtained from $\alpha_1$ and $\alpha_2$ by the superposition formula. $\alpha_{23}$ is a solution obtained from $\alpha_2$ and $\alpha_3$ by the superposition formula and $\alpha_{123}$ is obtained from $\alpha_{12}$ and $\alpha_{23}$ by the superposition formula.
  • Figure 2: $X_1$ is a time-like surface with Gaussian curvature $K=1$ contained in ${\mathbb{R}}^3_1$ corresponding to the solution $\alpha_1$. $X_{12}$ is a time-like surface obtained from the surface $X_1$ by a Bäcklund transformation, with parameter $\phi_2=\pi/3$ and it corresponds to the solution $\alpha_{12}$. $X_{123}$ is the surface obtained from $X_{12}$ by a Bäcklund transformation, with parameter $\phi_3=\pi/8$ and it corresponds to $\alpha_{123}$.
  • Figure 3: $\alpha_1$ and $\alpha_2$ are solutions of the elliptic sine-Gordon equation, obtained from the null solution of the elliptic sinh-Gordon equation, by Bäcklund transformations with parameters $\phi_1=0$ and $\phi_2=\ln( (1+\sqrt{5})/2)$ respectively. $\alpha_{12}$ is a solution of the elliptic sinh-Gordon equation defined on the complement of two curves, obtained by using the superposition formula.
  • Figure 4: $\alpha$ is a solution of ESGE. $\alpha_1$ and $\alpha_2$ are solutions of the ESHGE obtained from $\alpha$ by the Bäcklund transformations with parameter $\phi_1=0$ and $\phi_2=\ln(\sqrt{2}+1)$ respectively. $\alpha_{12}$ is a solution of the ESGE obtained from $\alpha_1$ by using the superposition formula with parameters $\phi_1$ and $\phi_2$.

Theorems & Definitions (21)

  • Theorem 2.1: Bäcklund-type theorem KRT2022
  • Theorem 2.2: integrability theorem KRT2022
  • Remark 2.3
  • Theorem 2.4: KRT2022
  • Theorem 2.5: KRT2022
  • Remark 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Theorem 3.1
  • Theorem 3.2
  • ...and 11 more