On Third-Order Evolution Systems Describing Pseudo-Spherical or Spherical Surfaces

2024/12/03 by Filipe Kelmer, Kelmer, Filipe
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2412.02657

Abstract

We consider a class of third-order evolution equations of the form \ u_t=F(x,t,u,u_x,u_xx,u_xxx,v,v_x,v_xx,v_xxx), v_t=G(x,t,u,u_x,u_xx,u_xxx,v,v_x,v_xx,v_xxx), . describing pseudos-pherical (pss) or spherical surfaces (ss), meaning that, their generic solutions (u(x,t), v(x,t)) provide metrics, with coordinates (x,t), on open subsets of the plane, with constant curvature K=-1 or K=1. These systems can be described as the integrability conditions of \mathfrakg-valued linear problems, with \mathfrakg=\mathfraksl(2,\R) or \mathfrakg=\mathfraksu(2), when K=-1, K=1, respectively. We obtain characterization and also classification results. Applications of these results provide new examples and new families of such systems, which also contain systems of coupled KdV and mKdV-type equations and nonlinear Schrödinger equations. Additionally, this theory is applied to derive a Bäcklund transformation for the coupled KdV system.

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