A unified approach to Bäcklund type theorems for surfaces in 3-dimensional pseudo-euclidean space

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

“…r = 1). In [16], we introduced the following definition of a Bäcklund-type line congruence in R 3 s for space-like or time-like surfaces in R 3 s . Let M 2 r , M 2 r → R 3 s be two surfaces that are isometrically immersed in R 3 s with 0 ≤ r, r ≤ s ≤ 1.…”

“…−1). One can determine an angle between a pair of independent vectors in R 3 s (see [16]). By definition, six cases may occur, according to the values of s, r, r and ϵ.…”

“…In [16], we provided a unified geometric proof for the six cases of Bäcklund type theorem and integrability theorem for surfaces in 3-dimensional pseudo-Euclidean space, R 3 s , s = 0, 1. These cases include the classical results (s = 0) and also those in the Lorentz-Minkowski 3-space (s = 1), previously obtained by McNertney [17], Palmer [18], Tian [24] and Gu-Hu-Inoguchi [13].…”

“…In this paper, we provide the analytic superposition formulae for the Bäcklund transformations discussed in [16]. The geometric permutability property of the Bäcklund theorem for distinct parameters has an analytic version as a superposition formula, which gives new solutions by algebraic expressions.…”

“…This paper is organized as follows. In Section 2, we recall from [16], the six cases of Bäcklundtype theorem (Theorem 2.1) and the integrability theorem (Theorem 2.2) for surfaces in the 3-dimensional pseudo-Euclidean space R 3 s . Moreover, by considering the partial differential equations whose solutions correspond locally to space-like or time-like surfaces with non-zero constant Gaussian curvature (Theorem 2.4), we provide the analytic interpretation of the geometric results in terms of solutions of these partial differential equations (Theorem 2.5).…”