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Integrable systems from inelastic curve flows in 2- and 3- dimensional Minkowski space

Integrable systems are derived from inelastic flows of timelike, spacelike, and null curves in 2- and 3- dimensional Minkowski space. The derivation uses a Lorentzian version of a geometrical moving frame method which is known to yield the modified Korteveg-de Vries (mKdV) equation and the nonlinear Schrodinger (NLS) equation in 2- and 3- dimensional Euclidean space, respectively. In 2-dimensional Minkowski space, timelike/spacelike inelastic curve flows are shown to yield the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while inelastic null curve flows are shown to give rise to Burgers' equation and its symmetry integrability structure. In 3-dimensional Minkowski space, the complex defocusing mKdV equation and the NLS equation along with their bi-Hamiltonian integrability structures are obtained from timelike inelastic curve flows, whereas spacelike inelastic curve flows yield an interesting variant of these two integrable equations in which complex numbers are replaced by hyperbolic (split-complex) numbers.


Introduction
There is an interesting geometric relationship between integrable systems and non-stretching (inelastic) geometrical flows of curves in various geometric spaces. For example, in the Euclidean plane, inelastic curve flows yield [12] the focusing mKdV equation u t = u xxx + 3 2 u 2 u x where x is the arclength of the curve and u is the curvature invariant of the curve, with the geometrical motion of the curve being given by the vortex patch equation. In Euclidean space, inelastic curve flows whose geometrical motion is given by the vortex filament equation [15] and its axial generalization [11] respectively yield [4,11,15,16,18] the focusing NLS equation −iu t = u xx + 1 2 |u| 2 u and the focusing complex mKdV equation u t = u xxx + 3 2 |u| 2 u x . Here u = κ exp(i τdx) is the Hasimoto variable [15] which is a covariant of the curve defined in terms of the curvature and torsion invariants κ and τ up to arbitrary (constant) phase rotations u → e iφ u.
The derivation of these integrable systems from the underlying curve flows is based on a moving frame method that uses a Frenet frame [13] in the case of curves in the Euclidean plane and a parallel frame [5] in the case of curves in Euclidean space. Parallel frames differ from a Frenet frame by a gauge transformation given by a rotation of the two normal vectors (in the normal plane of the curve) such that, at each point, their derivative along the curve is purely tangential. In both the case of the Euclidean plane and Euclidean space, the components of the Cartan matrix of the frame define the flow variable u, and the Cartan structure equations of the frame yield a Lax pair as well as a pair of Hamiltonian operators, which provide the integrability structure of the flows on u. In addition, in the case of space curves, the U(1) phase-rotation invariance of the flows on u geometrically corresponds to the fact that a parallel frame is determined by a curve only up to the action of arbitrary rigid SO(2) rotations on the normal vectors in the frame, whereas a Frenet frame is determined uniquely by a curve.
In this paper, integrable systems are derived in an analogous way from inelastic curve flows in the Minkowski plane and in Minkowski space [21]. Several new results are obtained by considering timelike curves, spacelike curves, and null curves. The paper is organized as follows.
In Sec. 2, a Lorentzian version of a Frenet frame is applied to inelastic curve flows in the Minkowski plane. Geometrical inelastic flows of timelike and spacelike curves are found to yield the defocusing mKdV equation and its bi-Hamiltonian structure. In contrast, for null curves, a Frenet frame does not exist and instead a null frame is introduced. Geometrical inelastic flows of null curves are shown to yield Burger's equation and the Cole-Hopf transformation under which Burger's equation is mapped into the heat equation. This geometric realization of Burger's equation is new.
(See Ref. [10] for related work.) Frame formulations of inelastic flows of timelike, spacelike, and null curves has appeared previously in Ref. [14].
In Sec. 3, starting from a Frenet frame, a Lorentzian counterpart of a parallel frame is introduced for timelike curves in 3-dimensional Minkowski space. The normal plane of a timelike curve is spacelike, and so a Lorentzian parallel frame is determined by a curve up to the action of rigid SO (2) rotations, similarly to the case of Euclidean space. Consequently, the Cartan matrix of this frame yields the same complex-valued Hasimoto variable as in the case of Euclidean space. For timelike curves undergoing an inelastic geometrical flow given by a timelike version of the vortex filament equation and its axial generalization, the Cartan structure equations of the Lorentzian parallel frame are shown to yield the defocusing NLS equation and the defocusing complex mKdV equation along with their bi-Hamiltonian integrability structure. A similar derivation of the defocusing NLS equation has appeared in Ref. [8] without, however, deriving the bi-Hamiltonian structure of this equation or considering the defocusing complex mKdV equation. Parallel frames for non-null curves have been considered previously in Ref. [22].
In Sec. 4, spacelike curves in Minkowski space are considered. Because the normal plane of a spacelike curve is timelike, two different cases arise depending on whether the principal normal vector of the curve is non-null or null.
When the principal normal vector of a spacelike curve is non-null, a Frenet frame exists which is used to define a Lorentzian parallel frame by a gauge transformation given by a boost (hyperbolic rotation) of the two normal vectors. Consequently, the resulting frame is determined by the curve only up to the action of rigid SO(1, 1) boosts. These boosts comprise a group of hyperbolic rotations, and the Hasimoto variable arising from the Cartan matrix of this frame is no longer a complex variable but instead is a hyperbolic generalization [7] based on the split-complex numbers defined by j 2 = 1 andj = − j. For spacelike curves undergoing an inelastic geometrical flow given by a spacelike generalization of the vortex filament equation and its axial generalization, the Cartan structure equations of a non-null parallel frame are shown to yield variants of the defocusing NLS equation and the defocusing complex mKdV equation, along with their bi-Hamiltonian integrability structure, in which i is replaced by j. Such integrable systems have been derived previously in the literature [6] by purely algebraic methods. Our results provide an interesting geometric realization for these systems. A special case of the SO(1, 1)-invariant complex mKdV equation has been derived in Ref. [9] using similar geometrical methods, and a system equivalent to the SO(1, 1)-invariant NLS equation appears in Ref. [8] in a less geometric form, without any results on the bi-Hamiltonian structure of these equations.
When a spacelike curve has a null principal normal vector, a complete Frenet frame does not exist. Instead a null frame is introduced for the normal plane. Then a Lorentzian parallel frame is defined by a gauge transformation such that the derivative of the pair of null normal vectors in the frame is purely tangential to the curve. The gauge transformation acts as a scaling on these null normal vectors, and consequently the resulting frame is determined by the curve only up to the action of rigid scalings (where the two null vectors scale reciprocally to each other). Because the principal normal vector of the curve is constrained to be null, the Cartan matrix has one fewer component than in the case when the principal normal vector is non-null. Interestingly, the Cartan structure equations are found to yield Burgers' equation and the KdV equation, as well as the heat equation and the Airy equation. These results are new. The corresponding geometrical inelastic curve flows are shown to be given by variants of a heat map equation and an Airy map equation.
Some concluding remarks are made in Sec. 5.

Curve flows and integrable systems in R 1,1
The Minkowski plane is a 2-dimensional vector space R 1,1 equipped with a Lorentz-signature metric η and a compatible volume form ε. A vector v ∈ R 1,1 is respectively timelike, spacelike, or null if its Minkowski norm η( v, v) is negative, positive, or zero. The set of all null vectors spans a pair of 1-dimensional null lines through the origin in R 1,1 , called the lightcone. Up to a sign, the volume form is determined from the metric by the relation holding for any vectors v, w ∈ R 1,1 . Associated to the metric and the volume form is the Hodge dual operator * defined by and holding for any vectors v, w ∈ R 1,1 .
In inertial coordinates X i = (X 0 , X 1 ), the Minkowski metric has the components η i j = −1 0 0 1 and the volume tensor has the components ε i j = 0 1 −1 0 , where X 0 is a timelike coordinate and X 1 is a spacelike coordinate. The Hodge dual operator * is given by the associated tensor ε i j = ε ik η jk . The isometries of the Minkowski plane consist of time and space translations and boosts. These transformations comprise the 3-dimensional Poincare group ISO(1, 1) SO(1, 1) R 2 (also called the inhomogeneous Lorentz group).

Timelike/spacelike curves and Frenet frames
Let γ(x) = (X 0 (x), X 1 (x)) be a curve in R 1,1 such that In both cases we hereafter choose x to be the proper-time or proper-distance arclength parameter defined by From relation (2.3), we note that the vectors γ x and * γ x are orthogonal in the Minkowski metric. Since γ x is assumed to be non-null, then this implies * γ x is also non-null. Hence, for a timelike/spacelike curve γ(x) in R 1,1 , this pair of vectors can be used to define a Lorentzian version of a Frenet frame [14] e = γ x = T, unit timelike/spacelike tangent vector (2.7a) e ⊥ = * γ x = * T = N, unit spacelike/timelike normal vector (2.7b) satisfying η(e , e ) = ∓1, η(e ⊥ , e ⊥ ) = ±1, The Frenet equations for this frame (2.7) are easily derived as follows. First, from the xderivative of equation (2.8), we have η(e x , e ) = 0 and η(e ⊥x , e ⊥ ) = 0, which implies e x = u 1 e ⊥ and e ⊥x = u 2 e for some functions u 1 (x) and u 2 (x). Next, from the x-derivative of equation (2.9), we have η(e x , e ⊥ ) + η(e ⊥x , e ) = 0. Now substituting the previous expressions for the x-derivatives of e and e ⊥ , and then using the norm relations (2.8), we find 0 = u 1 η(e ⊥ , e ⊥ ) + u 2 η(e , e ) = ±(u 1 − u 2 ). This yields u 1 = u 2 = u in both cases. Thus the Frenet equations are given by where u geometrically represents the Lorentzian curvature invariant of the curve γ(x), and where the Cartan matrix 0 u u 0 belongs to the Lie algebra so(1, 1) of the SO(1, 1) group of boost isometries in R 1,1 .
We now consider curve flows γ(t, x) that locally preserve both the timelike/spacelike signature (2.5) of the curve and the proper time/distance normalization (2.6) of the arclength parameter. Such flows are called inelastic and are specified by a flow vector expressed in terms of tangential and normal components in the Frenet frame (2.7). The Frenet frame itself will be carried by the flow, according to the equations which arise from equations (2.8)-(2.9) similarly to the derivation of the Frenet equations (2.10).
Here ω geometrically represents a Lorentzian invariant of the curve flow, and the Cartan matrix 0 ω ω 0 belongs to the Lie algebra so(1, 1).
The Frenet equations (2.10) and the flow equations (2.12) are related by the compatibility condition that the mixed t, x-derivatives of the Frenet frame (2.7) are equal. Since the Cartan matrices commute, this yields (2.13) Likewise, the tangent vector (2.7a) and the flow vector (2.11) of the curve are related by the compatibility condition that the mixed t, x-derivatives of γ(t, x) are equal, which gives (2.14) After substituting the Frenet equations (2.10) and the flow equations (2.12) for the derivatives of e and e ⊥ , we obtain Hence, from the vanishing of the coefficients of the frame vectors e and e ⊥ , we find Theorem 1. For timelike or spacelike inelastic curve flows in R 1,1 , the curve invariant u satisfies the system where H and J are a pair of compatible Hamiltonian and symplectic operators. Composition of these operators yields the evolution equation for u in terms of the normal component h ⊥ of the flow, where is a hereditary recursion operator.
This theorem is a counterpart of a similar result (see e.g. [3,23]) known for inelastic curve flows in the Euclidean plane and can be proved by the same methods. In particular, compared to the Euclidean case, the Hamiltonian operator (2.20) is exactly the same while the symplectic operator (2.21) differs only by the sign of the nonlocal term.
As a consequence of the compatibility of this pair of operators (2.20) and (2.21), their obvious invariance under x-translations can be used to derive a hierarchy of flows starting from a root flow u t = u x corresponding to the generator of x-translations on u, due to a general theorem of Magri [17]. This leads to the following main result.
There is a hierarchy of integrable bi-Hamiltonian flows on u(t, x) given by are compatible Hamiltonian operators, and where are local Hamiltonian densities.
The expression for the Hamiltonians (2.26) in this theorem arises from a scaling formula derived in Ref. [1].
The +1 flow in the hierarchy (2.24) is explicitly given by which is the defocusing mKdV equation. It has the explicit bi-Hamiltonian structure in terms of the Hamiltonian densities (modulo trivial densities given by total x-derivatives).
The entire hierarchy (2.24) of flows corresponds to a hierarchy of commuting vector fields  We have and hence We can simplify this equation of motion further by using the frame expansion holding for any vector v ∈ R 1,1 . This yields which is a Lorentzian version of the non-stretching mKdV map equation [2], in the Minkowski plane.

Null curves and null frames
Let γ(x) = (X 0 (x), X 1 (x)) be a null curve in R 1,1 , satisfying at every point x on the curve. Here, in contrast to the case of non-null curves, the parameter x is arbitrary and cannot be normalized solely in terms of γ x . Moreover, the vectors γ x and * γ x are now parallel, since we have 0 = η( γ x , γ x ) = −η( * γ x , * γ x ) = ε( * γ x , γ x ), from the relations (2.2) and (2.4), where antisymmetry of ε directly implies * γ x = a γ x for some function a(x). Similarly, the vectors γ x and γ xx are parallel, since the x-derivative of η( γ x , γ x ) = 0 yields η( γ x , γ xx ) = 0 from which we obtain ε( * γ x , γ xx ) = −η( γ x , γ xx ) = 0 through the relations (2.2) and (2.4). Antisymmetry of ε then implies γ xx = b * γ x for some function b(x), and thus γ xx is parallel to * γ x which itself is parallel to γ x .
Hence, for defining a frame, we need another vector, linearly independent of γ x . One natural choice is to use a null vector on the opposite side of the lightcone. If v is a null vector in R 1,1 , let N be a linear map that produces a null vector Hence the null vectors v and N ( v) are spatial reflections of each other with respect to this timelike vector in R 1,1 , with the reflection being given by v − 1  However, if the null parameter x is changed by reparameterizing the curve, then the equations (2.44) undergo a gauge transformation. In particular, for x →x(x), withx = 0, we have u →ũ = (1/x )u + (1/x ) . By restricting such reparameterizations to affine transformationsx = αx + β , where α = 0 and β are constants, we see that u only undergoes a scaling by α. Therefore, u geometrically represents a relative affine invariant of the parameterized null curve. We now consider inelastic null curve flows γ(t, x), in which both the null signature η( γ x , γ x ) = 0 and the null parameter x of the curve are locally preserved by the flow. Such flows are specified by a flow vector expressed in terms of the null frame (2.41). The null frame will be carried by the flow, such that where the Cartan matrix belongs to the Lie algebra of the abelian group of null boosts in R 1,1 . These equations (2.46) arise from the null frame equations (2.42)-(2.43) similarly to the derivation of the Frenet equations (2.44) and are therefore preserved if the normalization (4.90) of the null frame is changed. Consequently, ω represents a relative affine invariant of the time-parameterized curve flow. The Frenet equations (2.44) and the null flow equations (2.46) are related by the compatibility condition that the mixed t, x-derivatives of the null frame (2.41) are equal. Likewise, the tangent vector (2.41a) and the flow vector (2.45) of the curve are related by the compatibility condition that the mixed t, x-derivatives of γ(t, x) are equal. By the same steps used in the spacelike/timelike case, these conditions yield which are the Cartan structure equations of the null frame. From equation (2.49), we naturally obtain a potential variable related to the variable u by after a suitable gauge transformation is used to absorb an arbitrary integration constant (which is a function of t). Note, since u has the geometrical meaning of a relative affine invariant of the curve under reparameterizations x → αx + β , where α = 0 and β are constants, we see that v = udx is invariant up to an additive constant. Therefore v represents an affine covariant of the curve.
The system (2.50)-(2.51) has an interesting operator structure, involving a recursion operator, as follows. This recursion operator R can be used to derive a hierarchy of flows starting from a root flow v t = v x corresponding to the generator of x-translations on v. The following main result shows that the odd flows have a gradient-energy structure, while the even flows have a Hamiltonian structure.
is a Hamiltonian operator, and where are local densities.
The +1 flow in this hierarchy (2.53) is explicitly given by which is Burgers' equation in potential form. It has the gradient structure in terms of the energy density We remark that this structure implies The odd part of the hierarchy (2.53) has a gradient-energy structure similar to this +1 flow. In contrast, the even part of the hierarchy is quite different.
The +2 flow is given by which has the Hamiltonian structure where the Hamiltonian is the same expression H (1) = 1 2 exp(2v)v x 2 dx as the energy integral appearing in the +1 flow. There is a similar Hamiltonian structure for the other even flows.
The entire hierarchy (2.53) of flows on v(t, x) corresponds to a hierarchy of commuting vector fields with the root vector field being the generator of x-translations, In terms of u and h − , the vector fields (2.63) become are given by the recursion operators using equation (2.50). The odd and even vector fields in this hierarchy (2.64) will respectively inherit a gradient structure and a Hamiltonian structure through the variational relation This leads to the following result.
are local densities. Under the Hopf-Cole transformation (2.50), the hierarchy (2.68) is mapped to where H (l) = H (l) dx is given by the densities and where are a positive symmetric operator and a Hamiltonian operator, respectively.
Corresponding to the potential form of Burgers' equation (2.57), the +1 flow in these two hierarchies (2.71) and (2.68) is given by the heat equation and Burgers' equation (up to a scaling of u) Both of these equations are integrable and have a gradient structure in terms of the energy density All of the odd flows have a similar structure. The +2 flow in the hierarchies (2.71) and (2.68) is given by and which is the Airy equation. These two integrable equations have the Hamiltonian structure where the Hamiltonian is the same expression as the energy integral appearing in the +1 flow. There is a similar structure for all of the even flows. (2.84) The expression shows that u and its x-derivatives are invariant under translations and boosts applied to the curve γ in the Minkowski plane, so thus the equations of motion (2.82) are invariant under the isometry group ISO(1, 1) of the Minkowski plane.
These equations of motion (2.82) describe geometrical non-stretching motions of the curve. An interesting geometrical aspect of the motion emerges when the flow vector γ t at each point x on the curve is decomposed into complementary null components with respect to the light cone, and where ( γ t ) ± is a null vector parallel to e ± . Now, from the frame structure equations (2.44), (2.46), in terms of the tangent vector of the curve All of these curve flows preserve the null condition η( γ x , γ x ) = 0 and exhibit invariance under the isometry group ISO(1, 1). Thus, each equation of motion (2.88) describes a geometric nonstretching motion of the curve. The n = 1 curve flow (2.88) corresponding to the heat equation (2.75), and equivalently to Burgers' equation (2.76) or its potential form (2.57), is given by Through the frame structure equation (2.44), we can express this equation of motion (2.91) in the simple form which is a non-stretching heat map equation in the Minkowski plane. Similarly, the n = 2 curve flow (2.88) corresponding to the Airy equation (2.80), and equivalently to equations (2.79) and (2.61) obtained through the Hopf-Cole transformation, is given by This equation of motion (2.93) can be expressed entirely in terms of x-derivatives of γ x by use of the frame structure equation (2.44), as follows. We have u This curve flow is a non-stretching Airy map equation in the Minkowski plane.
3. Timelike curve flows and integrable systems in R 2,1 Three-dimensional Minkowski space is a vector space R 2,1 equipped with a Lorentz-signature (−1, 1, 1) metric η and a compatible volume form ε. A vector v ∈ R 2,1 is respectively timelike, spacelike, or null if its Minkowski norm η( v, v) is negative, positive, or zero. The set of all null vectors spans a 2-dimensional null surface through the origin in R 2,1 , called the lightcone.
Up to a sign, the volume form is determined from the metric by the relation holding for any vectors v, w, x ∈ R 2,1 . Associated to the metric and the volume form is the Hodge dual operator * defined by This operator maps pairs of vectors into vectors, and thus it can be alternatively formulated as a Lorentzian version of the cross product defined by holding for any vectors v, w, x ∈ R 2,1 .
In inertial coordinates X i = (X 0 , X 1 , X 2 ), the Minkowski metric and the volume tensor respectively have the components while the Hodge dual operator * and the cross product × are both given by the associated tensor ε i jk = ε l jk η il . Here X 0 is a timelike coordinate and X 1 , X 2 are spacelike coordinates. The isometries of 3-dimensional Minkowski space are given by the Poincare group ISO(2, 1) SO(2, 1) R 3 , which comprises time and space translations, rotations, and boosts.

Timelike curves and parallel frames
at every point x on the curve. Hereafter we choose x to be the proper-time arclength parameter defined by Hence the normal plane orthogonal to T at each point on the curve is a spacelike (Euclidean) plane, We begin by introducing the Lorentzian analog of a Frenet frame for such curves [22]. The xderivative of T defines the principal normal vector T x , which is spacelike since it lies in the normal plane, due to η(T, T x ) = 0. Therefore where the function κ(x) is given by which is the Lorentzian curvature invariant of the curve. The cross product of T and T x yields a vector which is orthogonal to both T and T x from property (3.5). Since this vector T × T x lies in the normal plane, it is spacelike. Hence The triple of mutually orthogonal vectors defines a Frenet frame for a timelike curve γ(x) in R 2,1 . Note that the orientation of this frame is ε(T, N, B) = 1 (3.18) in terms of the volume form.
The Frenet equations of this frame (3.17) are straightforward to derive. First, from equation (3.12), we have for some function τ(x). To obtain an expression for τ in terms of T and its x-derivatives, we substitute equations (3.12) and (3.15) into equation (3.20) and take its Minkowski inner product with B. This yields which is the Lorentzian torsion invariant of the curve. Last, from the x-derivative of equation (3.15), (3.19) and (3.20). Then using property (3.4), we have Therefore, the Frenet equations are given by is the Cartan matrix, which belongs to the Lie algebra so(2, 1) of the SO(2, 1) group of rotation and boost isometries in R 2,1 . A general frame for a timelike curve in R 2,1 is related to this Frenet frame by the action of arbitrary x-dependent SO(2, 1) rotations and boosts applied to the vectors (3.17). If the tangent vector T is preserved as one of the frame vectors, then the resulting frame is given by applying a general x-dependent SO (2)  We now use this gauge freedom to define a Lorentzian parallel frame by the geometrical condition that the x-derivative of both normal vectors e 1 and e 2 is parallel to the tangent vector e 0 . Thus we require e 1x = u 1 e 0 , e 2x = u 2 e 0 (3.28) for some functions u 1 (x), u 2 (x). This determines part of the Cartan matrix. To derive the remaining part, we need to work out e 0x . By substituting equation (3.28) into the x-derivative of equations (3.25) and (3.26), we get η(e 0x , e 1 ) = u 1 , η(e 0x , e 2 ) = u 2 , and η(e 0x , e 0 ) = 0. These relations determine e 0x = u 1 e 1 + u 2 e 2 .
(3.29) Therefore, the Cartan matrix of a Lorentzian parallel frame for a timelike curve in R 2,1 is given by This matrix belongs to the perp space of the stabilizer subalgebra so(2) ⊂ so(2, 1) of the frame vector e 0 . In particular, there is a decomposition so(2, 1) = so(2) ⊕ so(2) ⊥ as a symmetric Lie algebra, where so(2) is the subalgebra of rotations and so(2) ⊥ R 2 is its perp space given by the span of the 1-dimensional subalgebras of boosts which act in two orthogonal planes containing e 0 .
To determine the rotation angle θ (x) under which a Frenet frame is transformed to a parallel frame, we first need the inverse transformation (3.32) Now, we take the x-derivative of the frame (3.24) and substitute the Frenet equations (3.23) followed by the inverse transformation (3.32), which yields A parallel frame is unique up to x-independent (rigid) rotations where φ is constant. Under these rotations, the tangent vector e 0 is preserved, while the normal vectors e 1 and e 2 are rigidly rotated (3.37) Any two parallel frames related by this transformation are gauge equivalent. The SO(2) group of rigid rotations thereby defines the gauge (equivalence) group for parallel frames. Stated another way, a parallel frame is determined by a timelike curve only up to the action of this gauge group.

Inelastic Flow Equations
We now consider curve flows γ(t, x) that locally preserve both the timelike signature We can write these equations in the form which belongs to the Lie algebra of the SO(2, 1) group of rotation and boost isometries in R 2,1 . The flow equations (3.41) and the Frenet equations (3.30) of the parallel frame are related by the compatibility condition ∂ t (E x ) = ∂ x (E t ). This condition is equivalent to a zero curvature equation relating the Cartan matrices W and U. After substituting these matrices (3.31) and (3.42) into equation (3.43), we obtain ) Likewise, the flow vector (3.38) and the tangent vector (3.10) of the curve are related by the compatibility condition ∂ x ( γ t ) = ∂ t ( γ x ). By writing we have γ x = A T E and γ t = H T E. Then the compatibility condition becomes )

U(1)-invariant formalism
The gauge group for parallel frames consists of rigid SO(2) rotations (3.37) acting on the normal vectors. Under the action of this group, the components of the Cartan matrix (3.31) along the curve are transformed by (3.50) Similarly, the components of the Cartan matrix (3.42) along the flow are transformed by (3.52) Since the flow vector (3.38) is gauge invariant, its normal and tangential components in a parallel frame are transformed by Hence it is natural to introduce a complex formalism in which the rigid rotations (3.50), (3.52), (3.54) become phase rotations Since u is determined by the curve only up to rigid U(1) phase rotations (3.50), this complex variable u is a Hasimoto variable which geometrically represents a U(1)-covariant of the curve, in contrast to the invariants κ, τ which are determined uniquely by the curve. This system (3.57)-(3.60) has the following operator formulation which encodes a triple of U(1)-invariant Hamiltonian operators.
Theorem 6. For timelike inelastic curve flows in R 2,1 , the curve covariant u satisfies the U(1)invariant system where I is a Hamiltonian operator compatible with H and J . Composition of these operators yields the U(1)-invariant evolution equation for u in terms of the normal component h ⊥ of the flow, where is a hereditary recursion operator.
This theorem is a counterpart of a similar result [4,18] for inelastic curve flows in R 3 and can be proved by the same methods. In particular, compared to the Euclidean case, there is only a change in the sign of the nonlocal term in both operators H and J .
are compatible Hamiltonian operators, and where are local Hamiltonian densities.
The expression for the Hamiltonians (3.71) in this theorem arises from a scaling formula derived in Ref. [1].

Spacelike curve flows and integrable systems in R 2,1
In 3-dimensional Minkowski space R 2,1 , let γ(x) = (X 0 (x), X 1 (x), X 2 (x)) be a spacelike curve, with the tangent vector γ x satisfying at every point x on the curve. Hereafter we choose x to be the proper-distance arclength parameter defined by Hence the normal plane orthogonal to T at each point on the curve is a Minkowski plane, R 1,1 . The x-derivative of T defines the principal normal vector T x which lies in this plane, due to η(T, T x ) = 0.
Thus T x can be either timelike, spacelike, or null. The case when T x is non-null is similar to the cases of timelike/spacelike curve flows in 2-dimensional Minkowski space discussed in Sec. 2.1. In contrast, the case when T x is null has no counterpart in 2-dimensional Minkowski space. We will consider the non-null case first and discuss the null case afterward.

Spacelike curve flows with a non-null normal vector
We begin by introducing the Lorentzian analog of a Frenet frame for spacelike curves whose principal normal vector T x is assumed to be non-null at every point on the curve. In this case, where the function κ(x) is given by which is the Lorentzian curvature invariant of the curve. As shown by property (3.5), the cross product of T and T x yields a vector which is orthogonal to both T and T x and has the norm η(T × is the same. Next, from the x-derivative of equation (4.7), we have N x = ±κT + τB (4.14) for some function τ(x). An expression for τ can be obtained in terms of T and its x-derivatives by use of equations (4.6) and (4.9). This yields which is the Lorentzian torsion invariant of the curve. Last, from the x-derivative of equation (4.9) combined with equations (4.13), (4.14), and property (3.4), we have which has changed in sign. Therefore, the Frenet equations are given by 1) is the Cartan matrix, which belongs to the Lie algebra so(2, 1) of the SO(2, 1) group of rotation and boost isometries in R 2,1 . A general frame for a spacelike curve in R 2,1 is related to this Frenet frame by the action of arbitrary x-dependent SO(2, 1) rotations and boosts applied to the vectors (4.11). If the tangent vector T is preserved as one of the frame vectors, then the resulting frame is given by applying a general x-dependent SO(1, 1) boost on the normal vectors N and B in the R 1,1 normal plane of the curve. Geometrically, this transformation on the Frenet frame is a hyperbolic rotation, yielding Therefore, the Cartan matrix of a Lorentzian parallel frame for a spacelike curve with a non-null principal normal in R 2,1 is given by This matrix belongs to the perp space of the stabilizer subalgebra so(1, 1) ⊂ so(2, 1) of the frame vector e . The perp space so(1, 1) ⊥ R 1,1 is given by the span of the rotation subalgebra so (2) and another boost subalgebra so(1, 1) which act in two orthogonal planes (one spacelike and one timelike) containing e . In particular, so(1, 1) ⊕ so(1, 1) ⊥ = so(2, 1) is a symmetric Lie algebra decomposition.
To determine the hyperbolic rotation angle θ (x) under which a Frenet frame is transformed to a parallel frame, we first need the inverse transformation (4.26) Now, we take the x-derivative of the frame (4.18) and substitute the Frenet equations (4.17) followed by the inverse transformation (4.26), which yields The resulting frame given by equations (4.18) and (4.28) thereby defines a parallel frame, where the components of its Cartan matrix are related to the curvature and torsion invariants κ, τ by These expressions are a Lorentzian counterpart of the well-known Hasimoto transformation [15] in Euclidean space. A parallel frame is unique up to x-independent (rigid) hyperbolic rotations where φ is constant. Under these transformations, the tangent vector e is preserved, while the normal vectors e 1 and e 2 are rigidly rotated Any two parallel frames related by this hyperbolic rotation are gauge equivalent. The SO(1, 1) group of rigid hyperbolic rotations thereby defines the gauge (equivalence) group for parallel frames. Stated another way, a parallel frame for a spacelike curve with a non-null principal normal is determined only up to the action of this gauge group.

Inelastic Flow Equations
We now consider spacelike inelastic curve flows γ(t, x) that locally preserve the proper distance normalization (3.9) of the arclength parameter and the non-null signature The parallel frame will be carried by the flow, such that the orthogonality relations (4.19)-(4.20) are preserved. This implies that the t-derivative of the frame vectors e , e 1 , e 2 is given by e t = ±(ω 1 e 1 − ω 2 e 2 ), (4.33) e 1t = ω 1 e + ω 0 e 2 , e 2t = ω 2 e + ω 0 e 1 . (4.34) We can write these equations in the form with the Cartan matrix which belongs to the Lie algebra of the SO(2, 1) group of rotation and boost isometries in R 2,1 .
The flow equations (4.35) and the Frenet equations (4.24) of the parallel frame are related by the compatibility condition ∂ t (E x ) = ∂ x (E t ), which is equivalent to a zero curvature equation (3.43) relating the Cartan matrices W and U. Substituting these matrices (4.25) and (4.36) into equation (3.43), we obtain There is a similar compatibility condition ∂ x ( γ t ) = ∂ t ( γ x ) relating the flow vector (4.32) and the tangent vector (4.3) of the curve, which is given by equation (3.47) using the notation (3.46). After substituting the matrices (4.25) and (4.36) along with the vectors (3.46) into this equation, we find that its tangential and normal components yield ) These compatibility equations (4.37), (4.38), (4.39), (4.40) are the Cartan structure equations of the parallel frame, describing all inelastic spacelike curve flows γ(t, x) with a non-null principal normal in R 2,1 .

SO(1,1)-invariant formalism
The gauge group for parallel frames consists of rigid SO(1, 1) hyperbolic rotations (4.31) acting on the normal vectors. Under the action of this group, the components of the Cartan matrix (4.25) along the curve are transformed by (4.41) Similarly, the components of the Cartan matrix (4.36) along the flow are transformed by (4.43) Since the flow vector (4.32) is gauge invariant, its normal and tangential components in a parallel frame are transformed by We can now write the evolution equations in a SO(1, 1)-invariant form: Since u is determined by the curve only up to rigid hyperbolic phase rotations (4.41), this splitcomplex variable u is a Hasimoto variable which geometrically represents a SO(1, 1)-covariant of the curve, in contrast to the invariants κ, τ which are determined uniquely by the curve. This system (4.51)-(4.54) has an operator formulation, encoding a triple of SO(1, 1)-invariant Hamiltonian operators, as follows.
Theorem 8. For inelastic flows of spacelike curves with a non-null principal normal in R 2,1 , the curve covariant u satisfies the SO(1, 1)-invariant system where I is is a Hamiltonian operator compatible with H and J . Composition of these operators yields the SO(1, 1)-invariant evolution equation for u in terms of the normal component h ⊥ of the flow, where is a hereditary recursion operator.
This theorem is analogous to the similar result for timelike inelastic curve flows in Sec. 3.3 and has a similar proof by replacing i with j.
An operator D here is Hamiltonian iff it defines an associated Poisson bracket (3.68) having the same structure as in the case of timelike inelastic curve flows, with j in place of i. Similarly, based on Magri's general theorem [17], the SO(1, 1) invariance of the pair of compatible Hamiltonian operators H and I can be used here to derive a hierarchy of flows starting from a root flow u t = ju corresponding to the generator of SO(1, 1) hyperbolic phase-rotations on u. This leads to the following main result.   are local Hamiltonian densities.
The expression for the Hamiltonians (4.64) in this theorem arises from a scaling formula derived in Ref. [1].
The +1 flow in the hierarchy (4.62) is explicitly given by which is a variant of the defocusing NLS equation with j in place of i. This equation (4.65) has the explicit tri-Hamiltonian structure in terms of the Hamiltonian densities (modulo trivial densities given by total x-derivatives). Similarly, the +2 flow in the hierarchy (4.62) is given by which is a variant of the complex defocusing mKdV equation, where j replaces i. It has the tri-Hamiltonian structure where H (1) and H (2) are given by the Hamiltonian densities (4.67), while H (3) is given by the Hamiltonian density (modulo trivial densities given by total x-derivatives).

Spacelike curve flows with a null normal vector
We now consider spacelike curves whose principal normal vector T x is assumed to be null at every point on the curve. In contrast to the case when the principal normal is non-null, here the vectors T x and T × T x are parallel, by the following argument. We note ε( (3.3), and (3.5). Antisymmetry of ε thereby implies T × T x = aT + bT x for some functions a(x), b(x). Then, since T × T x is orthogonal to T from property (3.5), we find 0 = η(T, T × T x ) = a due to η(T, T x ) = 0, and hence we have T × T x = bT x .
To define a frame, we therefore need another vector, linearly independent of T and T x . Similarly to the case of null curves in 2 dimensions, it is natural to use a null vector on the opposite side of the lightcone in the normal plane R 1,1 (which is orthogonal to T ). If v is a null vector in the plane R 1,1 , let N be a linear map that produces a null vector N ( v) such that η (N ( v), v) = −1. A general frame for a spacelike curve in R 2,1 is related to this Frenet frame by the action of arbitrary x-dependent SO(2, 1) rotations and boosts applied to the vectors (4.88). If the tangent vector T is preserved as one of the frame vectors, then the resulting frame is given by applying a general x-dependent SO(1, 1) boost on the null normal vectors e ± in the R 1,1 normal plane of the curve. This yields the boosted frame where the boost acts as a scaling, parameterize by θ (x), and where the frame vectors satisfy the orthogonality relations η(e ,ẽ ± ) = η(ẽ ± ,ẽ ± ) = 0 (4.98) η(e , e ) = −η(ẽ + ,ẽ − ) = 1 (4.99) and the cross product relationsẽ To derive the Cartan matrix of the boosted frame (4.97), we first take the x-derivative of the frame vectors, and then we substitute the Frenet equations (4.96) followed by We now see that, similarly to the case of a spacelike curve with a non-null principal normal, here the gauge freedom in equation (4.102) can be used to define a Lorentzian parallel frame by the geometrical condition that the x-derivative of each null normal vector e ± is parallel to the tangent vector e . This condition can be achieved if (and only if) The resulting frame given by equations (4.97) and (4.103) is a parallel frame whose Cartan matrix is given by  Note that the Cartan matrix (4.105) has one fewer component than in the case of a spacelike curve with a non-null principal normal, since here the principal normal is constrained to lie on the lightcone. In particular, this Cartan matrix belongs to a 1-dimensional subalgebra consisting of a boost combined with a rotation, which lies in a subspace of the perp space so(1, 1) ⊥ R 1,1 of the stabilizer subalgebra so(1, 1) ⊂ so(2, 1) of the frame vector e . A parallel frame is unique up to x-independent (rigid) boosts given by where φ is constant. Under these transformations, the tangent vector e is preserved, while the null normal vectorsẽ ± are rigidly scaled Any two parallel frames related by this transformation are gauge equivalent, and thus the SO(1, 1) group of these rigid boosts thereby defines the gauge (equivalence) group for parallel frames. Stated another way, a parallel frame for a spacelike curve with a null principal normal is determined only up to the action of this gauge group.

Inelastic Flow Equations
Let γ(t, x) be a spacelike inelastic curve flow that locally preserves the proper distance normalization (3.9) of the arclength parameter and the null signature (4.87) of the principal normal. Such flows are specified by a flow vector expressed in terms of a tangential component h and a pair of normal components h ± with respect to the frame vectors e ,ẽ ± . The parallel frame will be carried by the flow, such that the orthogonality relations (4.89)-(4.91) are preserved. This implies that the t-derivative of the frame vectors e ,ẽ ± is given by We can write these equations in the form with the Cartan matrix which belongs to the Lie algebra of the SO(2, 1) group of rotation and boost isometries in R 2,1 .
Under the action of the gauge group of the parallel frame, the Cartan matrix (4.105) along the curve is scaled by while the Cartan matrix (4.113) along the flow is similarly scaled by Since the flow vector (4.109) is gauge invariant, its normal and tangential components also scale by The flow equations (4.112) and the Frenet equations (4.104) of the parallel frame are related by the compatibility condition ∂ t (E x ) = ∂ x (E t ), which is equivalent to a zero curvature equation (3.43) relating the Cartan matrices W and U. Substituting these matrices (4.105) and (4.113) into equation (3.43), we obtain A similar compatibility condition ∂ x ( γ t ) = ∂ t ( γ x ) relates the flow vector (4.109) and the tangent vector (4.3) of the curve, as formulated by equation (3.47) using the notation (3.46). After substituting the matrices (4.105) and (4.113) along with the vectors (3.46) into this equation, we find that its tangential and normal components yield These compatibility equations (4.119)-(4.121) and (4.122)-(4.124) are the Cartan structure equations of the parallel frame, describing all inelastic spacelike curve flows γ(t, x) with a null principal normal in R 2,1 . Compared to all of the previous cases, here the system (4.119)-(4.124) has a quite different operator structure, as follows.
From equations (4.119) and (4.123), we have Finally, from equation (4.121), we obtain which directly implies for some functions c(t), d(t). Hence we have derived the following result.
Theorem 10. For inelastic flows of spacelike curves with a null principal normal in R 2,1 , the curve invariant u satisfies the system for w in terms of the normal component h + of the flow, where a(t), b(t), c(t), d(t) are arbitrary functions.
We will now restrict attention to the class of flows that exhibit invariance under x-translations. This implies a = c = 0, so thus the system (4.131) reduces to  is the energy density. Next, consider the flow generated by where the Hamiltonian H (1) is the same expression as the energy integral appearing in the +1 flow, and where H is the Hamiltonian operator (4.147). There is a similar structure for all of the even flows in the hierarchy. To conclude, we now work out the underlying equations of motion for the curve flows determined by the preceding flows. We begin by using the Frenet relations (4.88) to express the frame vectors (4.97) in terms of the tangent vector γ x : We next derive an expression for the curve invariant u in terms of x-derivatives of these frame vectors. From the Frenet equations (4.104) of the frame, we have e xx = u xẽ+ = u x u −1 e x . Then, since e x is a null vector orthogonal to e , we obtain (ln u) x = −η(e xx , N (e x )) = −η( γ xxx , N ( γ xx )) (4.165) and hence u = exp(−η( γ xxx , N ( γ xx )))dx. where u is the curve covariant (4.166) and v is its potential.
The resulting curve flows are explicitly given by

Conclusion
There are several interesting directions in which the methods and results in this paper can be extended.
Firstly, one can consider inelastic curve flows in n-dimensional Minkowski space R n,1 for n ≥ 4. Inelastic flows of timelike curves will yield an integrable multi-component SO(n − 1)-invariant vector version of the defocusing mKdV equation and its bi-Hamiltonian integrability structure, while spacelike curves will similarly yield a SO(n − 2, 1)-invariant variant of that equation. Inelastic flows of null curves will yield multi-component Burgers' equations and their symmetry-integrability structures.
Secondly, one can consider inelastic curve flows in Lorentzian symmetry spaces M = G/SO(n− 1, 1), with G = SO(n, 1) and G = SU(n − 1, 1) for n ≥ 3. These two spaces are curved generalizations of Minkowski space in which their frame bundles have the same SO(n − 1, 1) gauge group of rotations and boosts as in the flat space R n,1 . The space M = SO(n, 1)/SO(n − 1, 1) will yield the same multi-component vector defocusing mKdV equations obtained from R n,1 , whereas the space M = SU(n−1, 1)/SO(n−1, 1) will yield different multi-component vector mKdV equations, analogously to the geometric origin of the two known multi-component vector focusing mKdV equations from [2] the Riemannian symmetric spaces M = G/SO(n), with G = SO(n + 1) and G = SU(n) for n ≥ 3. For n = 3, the Lorentzian symmetric space M = SO(3, 1)/SO(2, 1) will yield the same two integrable defocusing NLS equations as found in this paper, while the space M = SU(2, 1)/SO(2, 1) will yield two other integrable defocusing NLS equations (and their bi-Hamiltonian integrability structures).