Surfaces and curves induced by nonlinear schrödinger-type equations and their spin systems
Myrzakul A. Nugmanova G. Serikbayev N. Myrzakulov R.
October 2021MDPI
Symmetry
2021#13Issue 10
In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.
Chen–Lee–Liu equation , Derivative spin system , Heisenberg ferromagnet equation , Isomorphism of Lie algebras , Nonlinear Schrödinger equation , Nonlocal integrable equations , Soliton solution , Soliton surfaces , Symmetry in nonlinear integrable equation
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Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics, Nur-Sultan, 010009, Kazakhstan
Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana, 010008, Kazakhstan
Ratbay Myrzakulov Eurasian International Centre for Theoretical Physics
Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics
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