Pseudospherical Surfaces and Evolution Equations

Abstract

We consider evolution equations, mainly of type ut = F(u, ux,..., ∂ku/∂xk), which describe pseudo‐spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.

Geometric Integrability of the Camassa–Holm Equation. II

“…Explicitly, if X is given by (12), the symmetry conditions (10), on solutions to the augmenten system (11), become…”

“…On the other hand, a Darboux transform for ACH was considered before by Schiff in his paper [49]. The novelty of our approach rests on the fact that, while Schiff's transformation was originally obtained with the help of loop groups, we find it simply by using discrete symmetries of an equation associated with a quadratic pseudo-potential of ACH, as in [12]. With respect to the CH recursion operator, it is well known that it can be constructed starting from the CH bihamiltonian structure [10,11], or from the recursion operator for the KdV equation [20].…”

“…It was shown in that paper that the geometric approach introduced by Chern and Tenenblat in 1986 (see [12]) can be profitably applied to the study of Equation (1): The CH equation describes pseudo-spherical surfaces, and the analysis of their geodesics allows us to define a "Miura transform" and a "modified CH equation", which has been recently discussed in [22]. As a straightforward consequence, we can obtain infinite sequences of (local and nonlocal) conservation laws for (1).…”

“…The pseudo-potential y can be understood geometrically in terms of geodesics of the pseudo-spherical surfaces associated with the CH equation (23); see [12] and the later work [22]. A geometric derivation of (24) appears in [43], but of course Theorem 1 can be checked directly.…”

Bäcklund Transformations for Integrable Geometric Curve Flows

“…These solutions can be obtained using the Bianchi's permutability formula through purely algebraic means [2]. In [3], Chern and Tenenblat performed a complete classification to a class of nonlinear evolution equations which describe pseudospherical surfaces. It is noted that a nonlinear PDE describes pseudospherical surface if it admits sl(2) prolongation structure.…”