arXiv subjects
Publications and source records attributed to Tarcísio Castro Silva.
In this paper, we show how factorisation with respect to nonlocal pseudosymmetries allows one to obtain Bäcklund transformations, interpreted as nonlocal $\mathcal{C}$-morphisms of differential equations. According to this approach, which is illustrated through several examples, the Bäcklund transformations are determined by basic invariants of the exploited nonlocal pseudosymmetries.
Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the form \[ ν\,z_{t}-λ\,z_{xxt}=A(z,z_{x},z_{xx})\,z_{xxx}+B(z,z_{x},z_{xx}) \] with $ν$, $λ$ $\in$ $\mathbb{R}$, $ν^2+λ^2\neq 0$, are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K=1$ or $K=-1$, respectively. Alternatively they can be seen as the compatibility condition of an associated $\mathfrak{su}(2)$-valued or $\mathfrak{sl}(2,\mathbb{R})$-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we obtain an explicit classification for equations of the considered form that describe ss or pss, in terms of some arbitrary differentiable functions. Several examples of such equations, which describe also a number of already known equations, are provided by suitably choosing the arbitrary functions.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the short pulse equation and some generalizations.
Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K = 1$ or $K = -1$, respectively, and they can be seen as the compatibility condition of an associated su(2)-valued or sl(2, R)-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we give a complete and explicit classiffcation of equations of the form $z_{tt} = A(z, z_x , z_t) z_{xx} + B(z, z_x , z_t ) z_{xt} + C(z, z_x , z_t)$ describing pss or ss, in terms of some arbitrary differentiable functions. Several examples of such equations are provided by choosing the arbitrary functions. In particular, well known equations which describe pseudospherical surfaces, such as the short pulse and the constant astigmatism equations, as well as their generalizations and their spherical analogues are included in the paper.
The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in ${\bf E}^{3}$ of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the pde. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In the present paper, we consider third-order equations of the form $u_{t}-u_{xxt}=λu u_{xxx} + G(u,u_x,u_{xx}),\, λ\in \mathbb{R}, $ which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in ${\bf E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, then these coefficients are universal in the sense of being independent on the choice of solution $u$.