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The aim of this work is to study affine translation surfaces in the Euclidean 3-space with density. We completely classify affine translation surfaces with zero weighted mean curvature.

In this paper, we obtain the parametric representation for a family of surfaces through a given asymptotic curve by using the Frenet frame in the Galilean space G 3 . Necessary and sufficient conditions are given for that curve to be an isoasymptotic curve on the parametric surfaces. We also provide an example in support of our results.

Abstract:In this paper, we investigate the parametric representation for a family of surfaces through a given geodesic curve G 3 . We provide necessary and sufficient conditions for this curve to be an isogeodesic curve on the parametric surfaces using Frenet frame in Galilean space. Also, for the sake of visualizing of this study, we plot an example for this surfaces family.

In this work, the new coupled non-linear partial differential equations (CNLPDE) getting the time evolution of the curvatures of the evolving curve are derived in the Galilean space. Exact solutions for these new CNLPDE are obtained. Finally, Lie symmetry analysis is performed on these new CNLPDE and the algebra of Lie point symmetries of these new equations is found.