Local isometric immersions of pseudospherical surfaces described by evolution equations in conservation law form

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• 作者：D. Catalano Ferraioli^a ; ^{diego.catalano@ufba.br" class="auth_mail" title="E-mail the corresponding author} ; L.A. de Oliveira Silva^b ; ^{lulaluiz@ufrb.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Equations describing pseudospherical surfaces ; Second order evolution equations ; Pseudospherical surfaces ; Local isometric immersions ; Nonlinear partial differential equations
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1606-1631
• 全文大小：532 K

文摘

For the pseudospherical surfaces described by a class of second order evolution equations, of the form z_t=A(x,t,z)z₂+B(x,t,z,z₁)$z_t=A(x,t,z)z_2+B(x,t,z,z_1)$, we consider the problem of local isometric immersion into the 3-dimensional Euclidean space E³${\mathbf{E}}^3$ with a second fundamental form depending on finite-order jets of solutions z of the considered equations. We also provide an extension of our analysis to the case of k-th order evolution equations in conservation law form. Examples of equations admitting such local isometric immersions, are provided by equations like Burgers, Murray, Svinolupov–Sokolov, Kuramoto–Sivashinsky, Sawada–Kotera, Kaup–Kupershmidt, as well as hierarchies of evolution equations in conservation law form like Burgers, mKdV and KdV.