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.

Keywords

• Betchov-Da Rios equation
• localized induction equation (LIE)
• smoke ring equation
• vortex filament equation
• nonlinear Schrodinger (NLS) equation

Supporting Institution

National Natural Science

Project Number

12101168

Thanks

This work was funded by the National Natural Science Foundation of China (Grant No. 12101168).

References

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