Differential geometric approach of Betchov-Da Rios soliton equation. (English) Zbl 1524.35583
Summary: 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.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 53A05 | Surfaces in Euclidean and related spaces |
Keywords:
Betchov-Da Rios equation; localized induction equation (Lie); smoke ring equation; vortex filament equation; nonlinear Schrodinger (NLS) equationReferences:
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