In this talk, we  study a new  non-linear integrable equation, $u_{t}+2/frac{u_{x}u_{xx}}{u}=/epsilon u_{xxx}$, which is invariant under scaling of dependent variable and was called the SIdV equation, see Commun. Nonlinear Sci. Numeric. Simulat. 17 (2012) 4155. The order-n kink solution $u^{[n]}$ of the SIdV equation, which is associated with the n-soliton solution of the Korteweg-de Vries equation, is constructed by using the n-fold Darboux transformation (DT) from zero “seed” solution. Moreover, we also provide the evolution scenarios of surfaces of revolution associated with the kink-type solutions of the SIdV, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.

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