Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

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.

Exploring the Dynamics of Dark and Singular Solitons in Optical Fibers Using Extended Rational Sinh–Cosh and Sine–Cosine Methods

“…The intensity profile of the singular soliton, described by solution (28), is generated through plotting. This profile is obtained with the following parameters: p 1 = 0.5, ν = 0.05, k = 0.005, α 2 = 1, and α 3 = 0.5, as depicted in Figure 3a.…”

“…Through this investigation, we aim to analyze the notable characteristics associated with these soliton solutions by adjusting the system parameters. Exploring solitary wave solutions for nonlinear equations is pivotal in unraveling various nonlinear physical phenomena [28][29][30][31][32][33][34][35]. Nonlinear wave phenomena manifest across a spectrum of engineering and scientific domains, encompassing fluid dynamics, plasma physics, optics, biology, condensed matter physics, and beyond [1,9].…”