Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation

The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin vortex filament) and the nonlinear Schr\"odinger equation (NLS) has proven very useful in the past, since it allows one to construct new solutions to the LIA once a solution to the NLS is known. In the present paper, the quantum form of the LIA (which includes mutual friction effects) is put into correspondence with a type of complex nonlinear dispersive partial differential equation (PDE) with cubic nonlinearity (similar in form to a Ginsburg-Landau equation, with additional nonlinear terms). Transforming the quantum LIA in such a way enables one to obtain quantum vortex filament solutions once solutions to this dispersive PDE are known. From our quantum Hasimoto transformation, we determine the form and behavior of Stokes waves and a standing 1-soliton solution under normal and binormal friction effects. The soliton solution on a quantum vortex filament is a natural generalization of the classical 1-soliton solution constructed mathematically by Hasimoto (which motivated subsequent real-world experiments). The quantum Hasimoto transformation is useful when normal fluid velocity is relatively weak, so for the case where the normal fluid velocity is dominant we resort to other approaches. We consider the dynamics of the tangent vector to the vortex filament directly from the quantum LIA, and this approach, while less elegant than the quantum Hasimoto transformation, enables us to study waves primarily driven by the normal fluid velocity.

The quantum form of the local induction approximation (LIA, a model approximating the motion of a thin vortex filament in superfluid) including superfluid friction effects is put into correspondence with a type of cubic complex Ginsburg-Landau equation, in a manner analogous to the Hasimoto map taking the classical LIA into the cubic nonlinear Schrödinger equation. From this formulation, we determine the form and behavior of Stokes waves, 1-solitons, and other traveling wave solutions under normal and binormal friction. The most important of these solutions is the soliton on a quantum vortex filament, which is a natural generalization of the 1-soliton solution constructed mathematically by Hasimoto which motivated subsequent real-world experiments. We also conjecture on the possibility of chaos in such systems, and on the existence more complicated solitons such as breathers. The self-induced motion of a vortex filament in a superfluid is approximated by the HVBK model [1]. Applying the local induction approximation (LIA) to the non-local term, Schwarz [2] obtained a kind of quantum LIA which is given in non-dimensional form by v = γκt×n+αt×(U−γκt×n)−α ′ t×(t×(U−γκt×n)) , (1) where U is the dimensionless normal fluid velocity, t and n are the unit tangent and unit normal vectors to the vortex filament, κ is the dimensionless average curvature, γ = Γ ln(c/κa 0 ) is a dimensionless composite parameter (Γ is the dimensionless quantum of circulation, c is a scaling factor of order unity, a 0 ≈ 1.3 × 10 −8 cm is the effective core radius of the vortex), α and α ′ are dimensionless friction coefficients which are small (except near the λ-point; for reference, the λ-point is the temperature (≈ 2.17K, at atmospheric pressure) below which normal fluid Helium transitions to superfluid Helium [3]). Table  1 of Schwarz [2] shows that at temperature T = 1K we have α = 0.005 and α ′ = 0.003, while at temperature T = 1.5K we have α = 0.073 and α ′ = 0.018. In the limit (α, α ′ ) → (0, 0), we recover the classical Da Rios equations for the motion of a vortex filament in a classical fluid [4].
A number of studies exist on the solutions to the quantum LIA. In the α, α ′ → 0 limit, these solutions should collapse into solutions of the classical LIA. One highly important class of solutions to the classical LIA would be the 1-soliton solution found by Hasimoto [5], by way of what is now referred to as the Hasimoto transformation, which puts the classical LIA into correspondence with the cubic NLS. While a number of solutions to the quantum LIA have been studied either numerically or analytically, the Hasimoto 1-soliton have never been extended to the quantum LIA. The purpose of this paper is to fill this important gap. Applying a method analogous to that of Hasimoto, we are able to put the quantum LIA (1) into correspondence with a type of complex Ginzburg-Landau equation (a natural complex-coefficient generalization of NLS). From this, we study Stokes waves, 1-solitons, and other traveling wave solutions. Each of these solutions generalizes known results for the classical LIA. We also conjecture on the possibility of chaos in such systems.
Differentiating with respect to the arclength variable s, and performing several vector manipulations, we have that the quantum LIA (1) becomeṡ Taking t to be a unit vector, the equation simplifies slightly tȯ (3) This puts the quantum LIA (1) into the form of a vector conservation law.
In what follows, we shall take U = 0, for brevity of the calculations. Many studies on specific structures in the quantum LIA model have taken the normal fluid velocity to zero, as it permits one to study such structures without the influence of drift or other distorting effects on the filaments [6]. The physical applicability of such a scenario is limited to the very low temperature regime in superfluid Helium 4. On the other hand, in the case of superfluid Helium 3, the normal fluid velocity U is zero (because Helium 3 is very viscous, unlike Helium 4, so it is always at rest or in solid body rotation, but α and α ′ are not zero [7]). Similar results were recently attempted in the case of U = 0 [8], however the system was not solved and only the limiting reduction to α, α ′ = 0 was given. Some qualitative observations were also given at lowest order.
Let b denote the binormal vector, and take κ and τ to be the curvature and torsion, respectively. Recall that

)dŝ) and also a new vectorvalued function by
We seek to derive an equation for ψ in analogy to that which was obtained by Hasimoto in the case of a standard fluid (i.e., α = α ′ = 0). On the one hand, note thaṫ On the other hand, assume that we have a representation forṁ of the formṁ First, observe that therefore a must take the form a = iφ(s, t), for some real-valued function φ. By a similar process, b ≡ 0. We should also find that Therefore, we have the representatioṅ Differentiation of this representation with respect to arclength giveṡ Clearly, the coefficients of t, m and m * in equations (5) and (10) should match exactly. The m * coefficients already match exactly. Setting the m coefficients equal, we obtain where A(t) is an arbitrary function of time. Despite the appearance of i, this representation is real-valued, since ψ 2 − ψ * 2 is purely imaginary. Matching the coefficients of t, we obtaiṅ Using (12), we obtain an evolution equation for the function ψ: Evidently, for the solutions we take interest in, the term ψ 2 − ψ * 2 will be small (negligible), so we remove it. This term would need to be considered in the the case of higher-order perturbations to the system (at order α 2 ). Making this reasonable reduction, we obtaiṅ where ǫ = α/(1 − α ′ ) << 1. Eq. (16) is a type of complex Ginzburg-Landau equation. If we take α, α ′ = 0 (which corresponds to a standard fluid), then ǫ = 0, and (16) reduces to the cubic NLS, and therefore these results are completely consistent with those of Hasimoto for the standard fluid LIA.
A Stokes wave solution exists for the classical LIA. To recover a Stokes wave along a quantum vortex filament, we assume a solution of the form Ψ(s, T ) = P (T ), so that Writing P (T ) = R(T ) exp(iΘ(T )), we find R T = −2ǫR 3 and Θ T = R 2 , which gives R(T ) = (1 + 4ǫT ) −1/2 and Θ(T ) = (4ǫ) −1 ln(1 + 4ǫT ). P (T ) then takes the form Taking ψ(s, t) = P (T ) exp(i t 0 A(t)dt) gives us the general form of a Stokes wave. In the ǫ = 0 limit, we obtain the standard Stokes wave of constant modulus.
The most interesting solution associated with the Hasimoto transformation of the LIA is likely the soliton on a vortex filament. It is natural to wonder if such a soliton solution is possible for the quantum LIA. In order to obtain a soliton, we shall consider a stationary solution Ψ(s, T ) = √ 2ωq(S) exp(−iωt), with S = √ ωs. This puts (16) into the form or, equivalently, (20) Since ǫ = O(α) and α << 1, we may neglect terms of order ǫ 2 and higher, to obtain When ǫ = 0 (corresponding to the standard fluid case), we find q(S) = sech(S), so any solution for ǫ > 0 should reduce to this case in the ǫ → 0 limit. We should therefore consider a solution of the form q(S) = ρ(S) exp(iǫθ(S)). This has the interpretation that curvature is determined by the ǫ = 0 case, while ǫ > 0 influence the torsion of the filament. Making the relevant transformation, discarding contributions of order ǫ 2 or higher, and splitting (21) into real and imaginary parts, we obtain Clearly, ρ(S) = sech(S), which is just the soliton from the standard fluid case. We then find that θ ′ (S) = (C 1 + 5 tanh(S) − 2 tanh 3 (S)) cosh 2 (S) . (23) This derivative blows up rapidly for all values of C 1 except for C 1 = 3. When C 1 = 3, θ ′ (S) → 1/2 as S gets large. This in turn implies that θ(S) would grow linearly as S gets large. Therefore, we pick C 1 = 3, and upon integrating (23) we find (24) There is a far simpler, yet still rather accurate, way to represent θ by way of an asymptotic expansion. We find that This solution gives a linear growth in S, for large enough S. We therefore have that q(S) is accurately approximate by up to order ǫ 2 . Putting this solution back into the natural variables s and t, we obtain a soliton on a quantum vortex filament: The solution (28)-(29) constitutes a soliton along a vortex filament. The solution is stationary, with only the phase depending on time. It is, however, possible to consider a traveling wave along a vortex filament. In the case, both the phase and the amplitude of ψ would vary in time. Let us define Ψ(s, T ) =Ψ(ξ), where ξ = s − ηT . Ignoring terms of order ǫ 2 and higher, we obtain , we obtain the system which is effectively a third-order dynamical system. The system (31) has the interesting property that it has either one or infinitely many equilibrium points, depending on the value of the wave speed, η. If η = 3/2, there exist infinitely many equilibria of the form (F , G) = (F , −2F 2 ), where F ∈ R. On the other hand, when η = 3/2, the only equilibrium is the zero equilibrium (F , G) = (0, 0). From numerical simulations, we find that there is an interesting bursting pattern associated with the solutions to (31). For large negative values of ξ, the phase and amplitude functions are reasonably well-behaved. Then, near some finite value ξ = ξ 0 (which in general depends on both ǫ and η), there is a bursting behavior to the phase G(ξ), near where the amplitude F (ξ) reaches its minimal value. Past ξ 0 , the phase switches signs and gradually the dynamics become more tame. This behavior is demonstrated in Fig. 1. This behavior becomes more clear when we view the system in phase space. In Fig. 2, we plot the solution to (31) in the phase space (F, F ′ , G). The solution corresponds to a trajectory which originates infinitely far from the origin as ξ → −∞, then approaches the origin, goes through a bursting pattern, and then leaves the origin in a similar manner to which it came.
With this, we have successfully transformed the quan- tum LIA into a type of complex Ginzburg-Landau equation (16), by use of a method analogous to the Hasimoto transformation for a standard fluid vortex filament. Doing so, we are able to reduce the quantum LIA (3) (a vector conservation law) into a complex scalar PDE (16), which makes the system far more amenable to mathematical analysis. Such a mapping between the quantum LIA and this PDE is also desirable from a physical point of view, since it allows for greater qualitative comparison of the quantum and standard fluid LIA solutions. Upon transforming the quantum LIA into a complex Ginzburg-Landau equation (16), we were able to study a number of solutions. First we obtained Stokes wave type solutions. In the case of a standard fluid, these solutions takes the form of oscillating waves with constant ampli-tudes. However, we were able to demonstrate that for a quantum fluid modeled under LIA, such solutions have an algebraic decay rate and therefore dissipate as time becomes large. The period of oscillation for such solutions is variable, as well, as gradually increases in time.
Since the function ψ used in this paper is a composite function of curvature and torsion, the physical interpretation for these solutions to the quantum LIA is that the curvature of the filament decreases in time, while the torsion increases in time, in contrast to the standard fluid solutions, where curvature is constant.
A second and rather fundamental solution is that of the soliton on a vortex filament. Hasimoto originally employed the aforementioned transform in order to demonstrate the existence of a soliton on a vortex filament under the standard LIA. In the present paper, we have been able to demonstrate analogously that such a soliton also exists under the quantum LIA. The soliton takes the form of a sech function (i.e., a bright soliton), which is what one finds for the standard fluid case as well. However, the inclusion of superfluid friction parameters results in the appearance of an additional phase factor that depends on arclength. Therefore, the curvature of the filament solution corresponding to a 1-soliton does not change when we go from the classical to the quantum LIA, while the torsion is modified -by a factor that scales as α -when we go from the classical to the quantum LIA. The Hasimoto formulation has proven useful in experiments [9], and we expect the present results should be similarly useful for experiments in superfluid vortex dynamics. Breather solitons have been found on the classical LIA [10] (with no superluid friction parameters present), and one extension of the present paper would be to consider breather solitons for the quantum LIA.
We considered a family of traveling waves solutions. The phase of the waves undergo a type of bursting behavior, during which they change sign (going from positive to negative). However, we did not find more complicated dynamics, such as chaos. Still, there are other possible solutions to the PDE (16), so more complex dynamics are certainly possible. Indeed, chaos has been shown to arise from related models [11]. Chaos in the quantum LIA was previously conjectured [12], but as of yet has not been shown. Note that our derivations exclude any strong effects from the normal fluid velocity vector, U. It is possible to include the effects of the normal fluid, although the derivations will be much more complicated and lengthy. Due to the added complexity of the resulting equation, it may be possible to demonstrate chaotic behavior in the analogous equations which account for the normal fluid flow. It is also possible that the inclusion of the term ψ 2 − ψ * 2 will give more complicated dynamics in some instances. R.A.V. supported in part by NSF grant # 1144246.