Abstract
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.
Highlights
The paper proposes an algebraic-geometric approach, which enables a universal description of symmetric nonlinear integrable equations
As a second example of the application of the approach described in Section 2 to other nonlinear integrable equations, we demonstrate it to the derivative nonlinear Schrödinger type equations (NLSE) [4,5]
Let us we construct the soliton surface corresponding to the 1-soliton solution of the derivative HFE (dHFE) (72) which we presented in the previous section
Summary
The paper proposes an algebraic-geometric approach, which enables a universal description of symmetric nonlinear integrable equations. The solutions of these two equations (NLSE and HFE) are related by the Hasimota transformation κ R (5). As a second example of the application of the approach described in Section 2 to other nonlinear integrable equations, we demonstrate it to the derivative NLSE [4,5]. The last equation is known as the derivative HFE (dHFE). From (42), taking into account the last relation, we can always obtain the following generalized HFE in the form e3t + e3 × e3xx + 2κe3x = 0. The specific form of the spin system depends on the accepted value κ
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