A class of third order partial differential equations describing spherical or pseudospherical surfaces

2023/03/23 by Diego Catalano Ferraioli, Ferraioli, Diego Catalano, Tarcísio Castro Silva +1
Mathematics · Computer Science · #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2303.13715

Abstract

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 ∈ ℝ, ν²+λ²≠ 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 \mathfraksu(2)-valued or \mathfraksl(2,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.

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