DIFFERENTIAL GEOMETRIC ASPECTS OF NONLINEAR SCHRÖDINGER EQUATION

The main scope of this paper is to examine the smoke ring (or vortex lament) equation which can be viewed as a dynamical system on the space curve in E3: The di¤erential geometric properties the soliton surface associated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures kn; kg and g . Then, we give a di¤erent proof of that the s parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame. 1. Introduction Investigation of motion of a vortex lament provides the crucial problems of mathematical physics and di¤erential geometry. The work of Hasimoto in 1972 is one of the leading studies that guide us in solving these extremely important problems. It was concerned with an approximation to the self-induced motion of a thin isolated vortex lament traveling without stretching in an incompressible uid. If the position vector of vortex lament is denoted by r = r(s; t), then the equation rt = rs rss is hold [7]. This equation is called the smoke ring or vortex lament equation. It can be considered that these vortex motions, which involve no change of form, correspond to traveling wave solutions of the Nonlinear Schrödinger (NLS) equation, [14]. These kind of soliton surfaces that are associated with the NLS equation are called NLS surfaces or Hasimoto surfaces. Hasimoto stated the complex function 2020 Mathematics Subject Classication. 14J25, 53Z05. Keywords and phrases. Smoke ring equation, vortex lament equation, NLS surface, darboux frame. ayasar@erbakan.edu.tr-Corresponding author; merdogdu@erbakan.edu.tr 0000-0002-0469-3786; 0000-0001-9610-6229. c 2021 Ankara University Communications Facu lty of Sciences University of Ankara-Series A1 Mathematics and Statistics


Introduction
Investigation of motion of a vortex …lament provides the crucial problems of mathematical physics and di¤erential geometry. The work of Hasimoto in 1972 is one of the leading studies that guide us in solving these extremely important problems. It was concerned with an approximation to the self-induced motion of a thin isolated vortex …lament traveling without stretching in an incompressible ‡uid. If the position vector of vortex …lament is denoted by r = r(s; t), then the equation r t = r s r ss is hold [7]. This equation is called the smoke ring or vortex …lament equation. It can be considered that these vortex motions, which involve no change of form, correspond to traveling wave solutions of the Nonlinear Schrödinger (NLS) equation, [14]. These kind of soliton surfaces that are associated with the NLS equation are called NLS surfaces or Hasimoto surfaces. Hasimoto  iq t + q ss + vq jqj 2 = 0 [7]. In the work [15], the binormal motion of curves of constant curvature is shown to lead to integrable extensions of the Dym equations. Moreover, the binormal motion of curves of constant torsion is shown to lead to integrable extensions of classical sine-Gordon equations. A reciprocal invariance is used to establish the existence of novel dual-soliton surfaces associated with a given soliton surface in the case of the extended Dym equation. Then a cc-ideal formulation is adduced to obtain a matrix Darboux invariance for the extended Dym and reciprocally linked m 2 KdV equations. At the same time, a Bäcklund transformation is introduced which allows us to generate the associated soliton surfaces. Similarities of both Bäcklund's and Bianchi's classical transformations are derived for the extended sine-Gordon system. In the study [13], NLS equation is examined in a general intrinsic geometric setting as introduced earlier in a kinematic analysis of certain hydrodynamic motions by [16], and subsequently applied in magneto-hydrodynamics by [17]. Furthermore, di¤erential geometric properties of the soliton surfaces associated with NLS equation are obtained. Furthermore, the connection between Hasimoto derivation is stated. An intrinsic decomposition for the auto-Backlund transformation is obtained at level of the soliton surfaces. Finally, the connection between decomposition and a linear representation for NLS equation is investigated.
Emerging problems revealed that studies should be conducted for non-Euclidean geometries. The NLS equation of repulsive type for timelike curves and nonlinear heat system were examined in a general intrinsic geometric setting including a normal congruence in 3-dimensional Minkowski space in the study [5]. Additionally, the motion of timelike surfaces correspond to the repulsive type NLS equation in timelike geodesic coordinates was studied in [6].
Erdo¼ gdu and Özdemir investigated the Hasimoto surfaces in Minkowski 3-space and obtained the geometric properties of these surfaces [2]. Kelleci and others examined the curvatures of Hasimoto surface according to Bishop frame and give some characterization of parameter curves of these surfaces [10]. Grbović and Nešović investigated vortex …lament equation for pseudo null curves in Minkowski 3-space by using Bäcklund transformation in Minkowski 3-space [4].
In this paper, we investigate di¤erential geometric properties of the soliton surface associated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using the Darboux frame. Firstly, a brief summary is presented to provide the necessary background. We give a proof of that the s parameter curves of NLS surface are the geodesics of the soliton surface and that this surface provides the NLS equation. Then, we discuss the geometric properties of NLS surface. We …nd Gaussian and mean curvature of NLS surface. 512 M . ERDO ¼ GDU, A.YAVUZ Also, we obtain new results and the necessary and su¢ cient conditions for NLS surfaces to be ‡at or minimal surfaces. Finally, we investigate two di¤erent NLS surfaces as applications.

Preliminaries
In this section, we give the necessary information about the regular curves on surfaces to understand the main subject of the study. Let : I ! M be a regular unit speed curve on the orientiable surface M: We may de…ne Frenet frame fT; N; Bg at each points of the curve where T is unit tangent vector, N is principal normal vector and B is binormal vector. The Serret-Frenet formulas of the curve are given by where the functions and are called the curvature and the torsion of the curve ; respectively. On the other hand, we may also de…ne another orthonormal frame …elds on the curve ; which is called Darboux frame, since the curve lies on the orientiable surface M: Darboux frame of ( ; M ) curve-surface couple includes unit tangent vector …eld T of and unit normal vector …eld of the surface n = N on the curve : To describe an orthonormal frame including these vector …elds, there is only one way to choose last frame …eld as g = n T: This implies the following relations between Frenet and Darboux frames: where is the angle between the vector …elds N and n: The derivative formulas of Darboux frame can be given as follows: d ds where k n (s) = (s) cos (s); The functions k g is called geodesic curvature, k n is called normal curvature and t r is called geodesic torsion of : For a curve (s) lying on a surface, the following cases are satis…ed: The …rst fundamental form characterizes the interior geometry of the surface in a neighborhood of a given point. This means that measurements on the surface can be carried out by means of it. The …rst fundamental form is given by At the same time, the second fundamental form Gaussian curvature is given as K = det S and the mean curvature is related to the trace as follows H = 1 2 trS: Gaussian and mean curvatures of a surface are given by respectively.

Nonlinear Schrödinger Surfaces
In this section, NLS surface or Hasimoto surface are investigated by using the Darboux frame and discuss the geometric properties of NLS surface. We …nd the Gaussian and mean curvatures of this surface. Also, we obtain some of the results and some necessary conditions for surfaces to be ‡at or minimal surfaces. We give a proof of that s parameter curves of NLS surface are geodesics of the soliton surface and that this surface provides the NLS equation. The movement of a thin vortex in a thin vicous ‡uid by the motion of a curve propagating in E 3 is described by the following equation: This is called the vortex …lament or smoke ring equation, and can be viewed as a dynamically system on the space of curves.
Theorem 1. Suppose r = r(s; t) is a NLS surface such that r = r(s; t) is a unit speed curve with normal vector …eld for all t: Then the following is satis…ed: where fT; n; gg is the Darboux frame, ; ; are smooth functions given by following equalities = k gs k n t r ; (2) = k ns k g t r ; (3) Proof. We need to …nd these functions in terms of the curvatures k g ; k n ; t r of the solution curve r = r(s; t) of smoke ring equation for all t: Using compatibility conditions t ts = t st , we get s = k nt + k g t r ; s = k gt k n + t r ; and using equality of n ts = n st ; we have s = k nt + k g t r ; s = k n k g + t rt : Similarly, using equality of g ts = g st ; we get following equalities s = k n k g + t rt ; s = k gt k n + t r : Thus, by the above relations, the following equations are obtained Again by compatibility condition of r st = r ts , we …nd the following equalities = k gs k n t r ; = k ns t r k g :

DIFFERENTIAL GEOM ETRIC ASPECTS OF NLS EQUATION 515
Now, we assume that the velocity of the curve is of the form n (k g k gs + k n k ns ) s (k ns + k gs t r ) 2 (k gs k n t r ) 2 + k gt k n k nt k g : For a solution of smoke ring equation, the velocity vector is given by r t = r s r ss = t (k n n+k g g) = k g n k n g: Theorem 2. If r = r(s; t) is a NLS surface where r = r(s; t) is a unit speed curve with normal vector …eld for all t; then the Gaussian curvature K and mean curvature H of r = r(s; t) are respectively. Here, is the curvature function, and is the torsion function of the curve r = r(s; t) for all t: Proof. We have found the coe¢ cients of …rst fundamental forms of the r(s; t) as E = 1; F = 0; and G = k 2 g + k 2 n : Thus, we say that EG F 2 = k 2 g + k 2 n : Normal vector …eld of the NLS surface is given by N = r s r t kr s r t k = T (k g n k n g) kT (k g n k n g)k = k g g k n n q k 2 g + k 2 n : After some computations, one can easily obtain the coe¢ cients of the second fundamental form as Thus, the Gaussian curvature K and mean curvature H of the surface r(s; t) are respectively.
Theorem 3. Suppose r = r(s; t) is a NLS surface in E 3 ; the s-parameter curves of the surface r = r(s; t) are geodesics.
Proof. Suppose r = r(s; t) is a NLS surface such that r = r(s; t) is unit speed curve for all t: We know that r ss = k n n+k g g: And the surface normal of r = r(s; t) is Thus, r ss is parallel to surface normal which means s-parameter curves of the surface are geodesics.

Remark 4.
Since the s-parameter curves of NLS surface are geodesics, this implies that k g = 0: Theorem 5. Suppose r = r(s; t) is a NLS surface in E 3 : t-parameter curves of the surface r = r(s; t) are geodesics if and only if k ns = 0.
Proof. By above remark, we have r t = k g n k n g = k n g: Thus, we obtain the tangent vector …eld of t -parameter curve as t = r t kr t k = g: Then, the third vector …eld of Darboux frame of t-parameter curve is found g = n t = n ( g) = T: The geodesic curvature of the t-parameter curve is obtained as k g = t t ; g = h T n; T i = where = k ns k g t r = k ns : Therefore, t-parameter curves of the surface r = r(s; t) are geodesics if and only if k ns = 0: is provide with Hasimoto transformation q = k n e i Proof. We know that Hasimoto transformation with Frenet frame is obtained as follows q = e i such that = R ds: Since the following equations are satis…ed 2 = k 2 n + k 2 g ; t r (s) = 0 (s) (s); and k g = 0, we obtain that Hasimoto transformation with Darboux frame as follows q = k n e i R trds : By taking derivative of q according to t parameter, we get We also know that k nt = s + t r ; t rt = s k n ; such that = k n t r ; = k ns ; by equations (5) and (6). Therefore, we obtain k nt = ( k n t r ) s k ns t r = 2k ns t r k n t rs and t rt = 1 k n k nss k n t 2 r s + k ns k n = k nss k n t 2 r + k 2 n 2 s : If we substitute above equations into Equation (3.8), then we obtain q t = 2k ns t r k n t rs + ik n k nss k n t 2 r + k 2 n 2 e i R trds : Thus, we also get q ss = k nss + 2ik ns t r + ik n t rs k n t 2 r e i R trds : According to above …ndings, it is seen that the following NLS equation is satis…ed iq t + q ss + vq jqj 2 = 0:  (k n k ns ) s = 1 + k 2 ns + k 2 n t 2 r :

Declaration of Competing Interests
The author declares no con ‡icts of interest in this paper.