Differential geometric approach of Betchov-Da Rios soliton equation

Abstract

In the present paper, we investigate differential geometric properties the soliton surface $M$ associated with Betchov-Da Rios equation. Then, we give derivative formulas of Frenet frame of unit speed curve $\Phi=\Phi(s,t)$ for all $t$. Also, we discuss the linear map of Weingarten type in the tangent space of the surface that generates two invariants: $k$ and $h$. Moreover, we obtain the necessary and sufficient conditions for the soliton surface associated with Betchov-Da Rios equation to be a minimal surface. Finally, we examine a soliton surface associated with Betchov-Da Rios equation as an application.