Space Curves from Nonlinear Schr\"odinger Solutions: A Direct Approach

TL;DR

This paper introduces a direct method to construct space curves from any nonlinear Schrödinger (NLS) solution, enabling new ways to analyze vortex filament evolution with potential applications in physical systems.

Contribution

It presents a novel direct construction approach linking NLS solutions to space curves using ordered integrals and Magnus expansion, simplifying the mapping process.

Findings

Provides a new explicit mapping from NLS solutions to space curves.

Utilizes ordered integrals and Magnus expansion for the construction.

Highlights potential applications in physical systems analysis.

Abstract

The connection between vortex filament evolution in the local induction approximation and non-linear Schr\"odinger (NLS) equation by Hasimoto [H. Hasimoto, J. Fluid Mechanics 51, (1972) 477] has led to space curves corresponding to NLS solitons in the past. Utilizing this map, we propose a direct construction of parametric curve evolution from any NLS solution. It includes ordered (or nested) integrals of products of local matrices akin to the causal evolution of quantum theory, necessitating the implementation of the Magnus expansion. Such a straightforward mapping may be a simple tool to study the evolution of various systems of physical concern, although the actual computation can be a challenge for most NLS solutions.

Figures

Equations

$${\kappa }_t+2\tau {\kappa }_s+{\tau }_s\kappa =0$$

$$\mathrm{and}$$

$$q_t-i{q}_{ss}-2i\eta |q{|}^{2}q=0,$$

$$q(s,t)=\kappa (s,t)\exp \left\{i{\int }_0^s\tau (s^{\prime },t)d{s}^{\prime }\right\}.$$

$${\mathbf{W}}_s=\mathcal{A}(s,t)\mathbf{W},$$

$${\cal A}(s,t)=\left[{\begin{array}[]{ccc}0&\kappa(s,t)&0\\ -\kappa(s,t)&0&\tau(s,t)\\ 0&-\tau(s,t)&0\\ \end{array}}\right].$$

$$\mathbf{W}(s,t)=\mathfrak{S}\left(\exp \left[{\int }_0^s\mathcal{A}(s^{\prime },t)d{s}^{\prime }\right]\right){\mathbf{W}}_0(t),$$

$$\mathfrak{S}\left(\mathcal{A}(s_1,t)\mathcal{A}(s_2,t)\right)$$

$$:=\Theta (s_1-s_2)\mathcal{A}(s_1,t)\mathcal{A}(s_2,t)+\Theta (s_2-s_1)\mathcal{A}(s_2,t)\mathcal{A}(s_1,t);$$

$$\text{where}\Theta (s_1-s_2)=\{\begin{array}{l}1\text{for}{s}_1>s_2\\ 0\text{otherwise}\end{array},$$

$$\frac{1}{n!}\mathfrak{S}\left({\int }_0^{s}d{s}_1{\mathcal{A}}_1{\int }_0^{s}d{s}_2{\mathcal{A}}_2\cdots {\int }_0^{s}d{s}_n{\mathcal{A}}_n\right)$$

$$={\int }_0^{s}d{s}_1{\mathcal{A}}_1{\int }_0^{s_1}d{s}_2{\mathcal{A}}_2\cdots {\int }_0^{s_{n-1}}d{s}_n{\mathcal{A}}_n,$$

$$\mathrm{where}{\mathcal{A}}_j:=\mathcal{A}(s_j,t).$$

$$\mathbf{W}(s,t)=\mathbf{W}(0,t)\exp \mathbf{\Omega }(s,t),\mathbf{W}(0,t)=\mathbb{I},$$

$$\mathbf{\Omega }(s,t)=\sum _{k=1}^{\infty }{\mathbf{\Omega }}_k(s,t).$$

$${\mathbf{\Omega }}_1(s,t)={\int }_0^{s}d{s}_1{\mathcal{A}}_1,$$

$${\mathbf{\Omega }}_2(s,t)=\frac{1}{2}{\int }_0^{s}d{s}_1{\int }_0^{s_1}d{s}_2\left[{\mathcal{A}}_1,{\mathcal{A}}_2\right],$$

$$\displaystyle\bm{\Omega}_{3}(s,t)=\frac{1}{6}\int_{0}^{s}ds_{1}\int_{0}^{s_{1}}ds_{2}\int_{0}^{s_{2}}ds_{3}\,\Bigg{(}\left[{\cal A}_{1},\,\left[{\cal A}_{2},\,{\cal A}_{3}\right]\right]+\left[\left[{\cal A}_{1},\,{\cal A}_{2}\right],\,{\cal A}_{3}\right]\Bigg{)},$$

$$\displaystyle\bm{\Omega}_{4}(s,t)=\frac{1}{12}\int_{0}^{s}ds_{1}\int_{0}^{s_{1}}ds_{2}\int_{0}^{s_{2}}ds_{3}\int_{0}^{s_{3}}ds_{4}\,\Bigg{(}\left[\left[\left[{\cal A}_{1},\,{\cal A}_{2}\right],\,{\cal A}_{3}\right],\,{\cal A}_{4}\right]+\left[{\cal A}_{1},\,\left[\left[{\cal A}_{2},\,{\cal A}_{3}\right],\,{\cal A}_{4}\right]\right]$$

$$\displaystyle\qquad\qquad\qquad\qquad\qquad+\left[{\cal A}_{1},\,\left[{\cal A}_{2},\,\left[{\cal A}_{3},\,{\cal A}_{4}\right]\right]\right]+\left[{\cal A}_{2},\,\left[{\cal A}_{3},\,\left[{\cal A}_{4},\,{\cal A}_{1}\right]\right]\right]\Bigg{)},$$

$$⋮$$

$$x^i(s,t)={\int }_0^s{W}_{1i}(s^{\prime },t)d{s}^{\prime }+x_0^i(t),$$

$$\kappa (s,t)=|q(s,t)|$$

$$\mathrm{and}$$

$$q(s,t)=\frac{1}{\sqrt{t}}\exp i\left(\frac{x^2}{4t}+2\ln t\right),$$

$$\kappa (s,t)=\frac{1}{\sqrt{t}}\mathrm{and}\tau (s,t)=\frac{x}{2t},$$

$$\mathbf{\Omega }(s,t)=\left(\begin{array}{ccc}0& y-\frac{y^5}{960}& \frac{y^3}{24}+\frac{y^5}{1440}\left(1+\frac{3y^2}{56}\right)\\ -y+\frac{y^5}{960}& 0& \frac{y^2}{4}\\ -\frac{y^3}{24}-\frac{y^5}{1440}\left(1+\frac{3y^2}{56}\right)& \frac{y^2}{4}& 0\end{array}\right),y:=\frac{s}{\sqrt{t}}.$$

Full text

Nonlinear Schrödinger equation, Inverse Hasimoto Map, Quasi-integrable deformation, Non-holonomic deformation, Solitons.

Evolution of certain systems like Vortex filaments in perfect fluids and classical one-dimensional continuum Heisenberg spin chain [1] may mathematically be modeled as a moving space curve parameterized as $\gamma(s,t)\in{\mathbb{R}}^{3}$. Da Rios [2] derived a set of two coupled equations governing the in-extensional motion of a vortex filament within an irrotational fluid in terms of the time evolution of geometric parameters intrinsic to the filament: curvature ($\kappa(s,t)$) and torsion ($\tau(s,t)$).

[TABLE]

These equations were rediscovered by Betchov decades later [3].

Hasimoto showed that these Da Rios–Betchov equations can elegantly be combined to give a focusing-type nonlinear Schrödinger (NLS) equation [4],

[TABLE]

using the so-called Hasimoto map [5]:

[TABLE]

This map relates the Frenet-Serret space-time parameters $\tau(s,t)$ and $\kappa(s,t)$ to the NLS amplitude $q(s,t)$. Subsequently, physical characteristics of the NLS system like densities of energy ($\kappa^{2}$) and momentum ($\kappa^{2}\tau$) can directly be constructed from these geometric parameters. In contrast, the non-linear coupling $\eta$ reflects the on-site interaction strength of the physical system. These equations prescribe (up to a rigid motion) the evolution of a vortex filament in an infinite domain of ${\mathbb{R}}^{3}$ for given initial conditions $\kappa(s,0)$ and $\tau(s,0)$. In particular, the single-soliton solution of this equation describes a isolated loop of helical twisting motion along the vortex line [6].

Extending Hasimoto’s work, Lamb [7] initiated the study on the dynamics of space curves corresponding to soliton solutions of a class (sine-Gordon, MKdV, and NLS) of nonlinear differential equations. Later, in addition to the Hasimoto map, more ways of connecting nonlinear solutions to Frenet-Serret curves were obtained for the same class of systems [8]. However, the explicit construction of the space curve corresponding to a solution of a given differential equation (the reconstruction problem) was still lacking [9, 10]. The initial computation of an inverse Hasimoto transformation to reformulate the Da Rios–Betchov type equations had been performed by Sym [10] and Aref-Linchem [11].

In this letter we obtain a direct inversion yielding the Frenet-Serret curve dynamics from the NLS solution. It is achieved by solving the matrix-valued Frenet-Serret equations that necessarily introduce $s$-integrals of ordered products curvature and torsion; quantities that can be read off of a given solution to the NLS equation. The novelty of the present work is the straightforward approach, that obtains the exact space-curve coordinates from a given NLS solution, which has not been achieved till now as per the best of our knowledge. In addition to the formal interest, this can yield a crucial understanding of various physical systems (e.g. spin configuration, filament dynamics in fluids, etc.) which are effectively represented through localized and other specific types of NLS solutions. It can further shed light on the stability, or lack thereof, of a particular physical state based on the properties of the corresponding nonlinear solution. Such correspondence between physical systems and NLS systems [12, 13] with subsequent deformations [14] have been studied before.

The Frenet-Serret equations can be cast into the matrix form [15]:

[TABLE]

where $\bm{W}=\left[{\bf t}~{}{\bf n}~{}{\bf b}\right]^{T}$ and

[TABLE]

Here the parametric space derivatives of the tangent ($\bm{t}$), normal ($\bm{n}$) and binormal ($\bm{b}$) of a space-time curve are interrelated through corresponding $\kappa$ and $\tau$. As the Hasimoto map relates $\kappa$ and $\tau$ to the NLS amplitude $\psi(s,t)$ the IHM amounts to solve for $\bm{W}$ that in turn should yield the $\bm{t}=\bm{x}_{s}$. Then, on integrating the components of torsion, the three-dimensional coordinates $\{x_{i}\}$ of the curve can be obtained as a function of $s$ and $t$.

Although of first-order, Eq. 4 is matrix-valued and thus cannot be integrated directly. This situation is identical to the time evolution of a quantum state under a time-dependent Hamiltonian invoking time-ordering of the unitary evolution [16]. In the same way, respecting the matrix order, the general solution of Eq. 4 takes the form:

[TABLE]

with $\bm{W}_{0}(t)=\bm{W}(s=0,t)$. The symbol $\mathfrak{S}$ marks ordered product:

[TABLE]

among the products of the integrands in the exponent. These integrals of ordered products can equivalently be written in terms of nested integrals as,

[TABLE]

Such results appear often in physical systems, tempting possible interpretations of the vortex filament evolution, more so as the matrix ${\cal A}$ is anti-Hermitian making the ordered exponent in Eq. 6 unitary.

The Magnus expansion [] is employed to approximate $\bm{W}(s,t)$ in the form:

[TABLE]

The choice of $\bm{W}(0,t)$ is totally arbitrary and so it can be chosen as the identity matrix for brevity. The first few terms of the expansion are,

[TABLE]

By definition the first row of the matrix $\bm{W}$ represents the components of the tangent, the identification $dx_{i}/ds=W_{1i}(s,t)$ immediately provides the space coordinates of the curve,

[TABLE]

subjected to arbitrary initial conditions $x^{i}_{0}(t)$. From Eq. 3, the curvature and torsion are expressed in terms of the NLS amplitude and phase as,

[TABLE]

A case of particular interest is that of the focusing-type NLS system of Eq. 2 related to the spin orientation of 1-D Heisenberg XXX model in the continuum limit through the Hasimoto map [17]. The spin orientation itself is identified with the tangent to the Frenet-Serret curve, implying a direct physical interpretation of the curve dynamics. We demonstrate this process for the particular periodic solution to the NLSE obtained by Sall’ [18]:

[TABLE]

plotted in Fig. 1. The solution is well-behaved and quite amiable to the calculation of the matrix exponent. Notably, for other standard solutions of the NLS system, even obtaining a closed form of $\bm{\Omega}_{4}(s,t)$ is not possible. Then the curvature and torsion takes the particular forms,

[TABLE]

on substituting which the Magnus exponent is can be obtained as,

[TABLE]

up to the 4th order. Following the exponentiation of $\bm{\Omega}(s,t)$, Eq. 11 then leads to the parametric vortex filament in the $\{x_{i}\}$ space, time-snaps of which with respect to the NLS variable $t$ are shown in Fig. 2. We have used Mathematica 12 to obtain them numerically.

In conclusion, we have obtained a direct approach to construct the evolving vortex filament corresponding to solutions of the NLS system. It includes evaluation of integrals containing ordered products of geometric parameters pertaining to the curve expressed in terms of the amplitude and phase of the particular solution. Such nested integrals are not very easy to evaluate but could be approximated under particular conditions. For the given solution of Eq. 13 we could obtain the exact forms of $\exp\bm{\Omega}(s,t)$ up to the 4th order which is the usual extent in most of the literature that deals with the Magnus expansion.

Acknowledgement: Kumar Abhinav’s research is supported by Mahidol University, Thailand under the Grant Number MRC-MGR 04/2565. Work of Partha Guha was supported by the Khalifa University of Science and Technology, United Arab Emirates under Grant Number FSU-2021-014.