Some classifications of hyperbolic vector evolution equations

Stephen C. ANCO ^{†} and Thomas WOLF ^{‡}

Department of Mathematics, Brock University, Canada L2S 3A1 E-mail: ^{†}sanco@brocku.ca ,^{‡}twolf@brocku.ca

Abstract

Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations U_{tx} = f(U, U_{t}, U_{x}) for an N-component vector U(t, x) are considered. In each class we find all scalinghomogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.

1 Introduction

Integrability theory of nonlinear partial differential equations (PDEs) has many different aspects which have developed over the years — Bäcklund transformations, Lax pairs, inverse scattering transform and soliton solutions (Calogero’s “S-integrability”), linearizing transformations (“C-integrability”), reduction to Painlevé transcendents, hierarchies of higher symmetries and conservation laws, master symmetries, recursion (Nijenhuis) operators, bi-Hamiltonian structures, and others. To date the most computationally direct and effective test of integrability has proved to be the condition that a PDE system should possess sufficiently many higher symmetries [13, 8] (see also [24, 16, 9, 17]). In particular, for all currently known examples of nonlinear scalar PDEs, the existence of one higher symmetry implies the existence of infinitely many, i.e. a symmetry hierarchy.

A rigorous proof of the observation “one symmetry implies infinitely many” was established in [21, 4] for a wide class of semilinear scalar evolutionary PDEs

u_{t} = u_{nx} + f(u, u_{x}, …, u_{n−1x}), n > 1, (1.1)

under a homogeneity restriction with respect to a scaling symmetry on t, x, u (so-called λ-homogeneous form). The proof uses the symbolic method of Gelfand-Dickey, combined with computer algebra computations. As a main result, it was shown that any such symmetry-integrable scalar polynomial PDE with positive scaling weight (λ > 0) for u belongs to one of the well known hierarchies [12]: n = 2 Burger’s (λ = 1); n = 3 Korteweg de Vries (KdV) (λ = 2), modified Korteweg de Vries (mKdV) (λ = 1), Ibragimov-Shabat (IS) (λ = 1/2); n = 5 Kaup-Kupershmidt (KK) (λ = 2), Sawada-Kotera (SK) (λ = 2), Kupershmidt (K) (λ = 1), potential Kaup-Kupershmidt (λ = 1), potential Sawada-Kotera (λ = 1). The main evolutionary PDEs of semilinear form (1.1) not covered by this classification are the hierarchy of the nonlinear Schrödinger (NLS) equation and its various derivative-type generalizations [12], in which u is complex-valued so its real and imaginary parts satisfy a pair of coupled scalar evolutionary PDEs. There has been much work and interest in finding multi-component generalizations of all these hierarchies, particularly where u is replaced by a vector U or scalar-vector pair (u, U) or matrix U. Recent symmetry-integrability classification results in this direction have been obtained by extensive computer algebra computations [18, 26, 28]. These classifications test for the existence of a higher symmetry, in analogy with the scalar case. ^{1}

Comparatively less work has been devoted to investigating symmetry-integrability of hyperbolic PDE systems [32], apart from the sine-Gordon (SG) equation and its scalar variants related to the S^{2} sigma model, which both have the wave equation form u_{tx} = f(u, u_{t}, u_{x}) and possess zero scaling-weight for u. In this paper we present some symmetryintegrability classifications of hyperbolic vector PDEs. The classifications are motivated by the following three types of examples of scalar wave equations:

(1) the SG equation u_{tx} = sin u, which is distinguished by scaling weight λ = 0 for u; (2a) the variant SG equation u_{tx} = u\sqrt{1 − u_{t}^{2}} coming from the transformation u →

sin^{−1}(u_{t}), which has scaling weight λ = 1 for u and shares the same higher symmetries as the mKdV hierarchy;

(2b) the complex-valued version u_{tx} = u\sqrt{1 − |u_{t}|^{2}}, which shares the higher symmetries of the NLS hierarchy;

(3) the mKdV equation in potential form u_{tx} = u_{4x} + u_{x}^{2}u_{2x} and the NLS equation in potential form iu_{tx} = u_{3x} + |u_{x}|^{2}u_{x}.

Our results give interesting vector counterparts of the first two types of wave equations. We show that these types of hyperbolic vector equations all have a natural geometric origin in terms of integrable nonlinear sigma models and integrable inverse flows of curves into Riemannian manifolds. For the third type of wave equation, we find a null result that no vector counterparts exist.

2 Classification Results

For an N-component vector U(t, x), consider an O(N)-invariant vector PDE of generalized evolutionary form

U_{t mx} = f(U, U_{x}, …, U_{nx}, U_{t}, …, U_{t m−1x}), n, m ≥ 0 (2.1)

namely f is a sum of terms proportional to its vector arguments with scalar coefficients depending on dot products ⟨·, ·⟩ of these vectors. Such a vector PDE is strictly evolutionary if m = 0, n ≥ 1, or is hyperbolic if m = 1, n ≤ 1; we call it a wave equation if m = 1, n ≥ 0. In all cases it is semilinear whenever n ≤ m or if f is linear in U_{nx} when n > m. We say it is of minimal differential order (m, n) provided f is not a total derivative with respect to x. The order of the vector PDE (2.1) in the ordinary sense is max(n, m + 1).

If we view a vector PDE (2.1) as formally defining the generator of a flow D_{t}D_{x}^{m} in the jet space J^{(∞)} = (t, x, U, U_{t}, U_{x}, …, U_{kt lx}, …), then a higher symmetry on the solution jet space R^{(∞)} ⊂ J^{(∞)} (i.e. modulo equation (2.1) and differential consequences) will be a commuting flow D_{τ} whose generator is of the form

U_{τ} = g(U, U_{x}, …, U_{rx}, U_{t}, …, U_{t m−1x}) (2.2) for some order r > max(n, m + 1), where g is a sum of terms proportional to its vector arguments with scalar coefficients depending on dot products ⟨·, ·⟩ of these vectors. Thus the higher symmetry condition is

1It should be noted that there is a special class of Bakirov-type multicomponent evolutionary PDE systems that, surprisingly, admit only finitely many higher symmetries [3]. However, all these systems are of a triangular form in which one PDE is a decoupled linear evolution equation. In contrast, the classifications in [18, 26, 28] in addition to our results deal with fully coupled, nonlinear multicomponent PDE systems, describing a single vector or matrix nonlinear evolution equation for U or U, or a nontriangular scalar-vector pair of nonlinear evolution equations for (u, U).

[D_{τ} , D_{t}D_{x}^{m}] = 0 on R^{(∞)} . (2.3)

A flow (2.1) is said to be λ-homogeneous of weight (λ, μ) if it admits a scaling symmetry group (with parameter ϵ ∈ ℝ)

(t, x, U) → (e^{μϵ}t, e^{ϵ}x, e^{−λϵ}U). (2.4)

Note, with this convention for scaling weights, the order of a semilinear flow (2.1) is equal to the weight μ in the evolutionary case and μ + 1 in the hyperbolic case. Without loss of generality, higher symmetries for such flows can be assumed to be homogeneous under the same scaling symmetry group (since any symmetry necessarily decomposes into a sum of terms with homogeneous forms which themselves will define symmetries). Correspondingly, we will refer to higher symmetries by their weight as defined through the scaling

τ → e^{νϵ}τ (2.5)

induced by the scaling symmetry group (2.4). Note this weight ν is equal to the order r precisely when a higher symmetry is of semilinear form, i.e. linear in U_{rx}.

Throughout, we will assume a non-strict polynomial form for λ-homogeneous vector PDEs (2.1) and higher symmetries (2.2). More specifically, as explained in Sec. 3, f and g will be linear in their vector arguments and polynomial in all nonnegative-weight dot products of these vectors, with undetermined coefficients that either are functions of any zero-weight dot products or otherwise are constants. We refer to such a form as the nonnegatively-weighted polynomial class of vector PDEs and higher symmetries. Hereafter we restrict attention to scaling weights λ = 0, \frac{1}{2}, 1, 2 in analogy with the classes of symmetry-integrable scalar evolutionary PDEs summarized in Sec. 1.

In the case of evolutionary vector PDEs, the following classification was obtained in [26]. Up to scalings of t, x, U, the only vector counterparts of λ-homogeneous evolutionary scalar polynomial PDEs (1.1) with scaling weights λ = \frac{1}{2}, 1, 2 possessing a homogeneous higher symmetry of least possible weight are given by, in the case of real-valued U:

• vector mKdV equations λ = 1, μ = 3,

U_{t} = U_{3x} + ⟨U, U⟩U_{x},  (2.6)
U_{t} = U_{3x} + ⟨U, U⟩U_{x} + ⟨U, U_{x}⟩U;  (2.7)

• vector IS equation λ = \frac{1}{2}, μ = 3,

U_{t} = U_{3x} + 3⟨U, U⟩U_{2x} + 6⟨U, U_{x}⟩U + 3⟨U, U⟩^{2}U_{x} + 3⟨U_{x}, U_{x}⟩U; (2.8)

and in the case of complex-valued U:

• vector NLS equations λ = 1, μ = 2,

iU_{t} = U_{2x} ± ⟨U, Ū⟩U,  (2.9)
iU_{t} = U_{2x} ± 2⟨U, Ū⟩U ∓ ⟨U, U⟩Ū;  (2.10)

• vector derivative-NLS equations λ = \frac{1}{2}, μ = 2,

iU_{t} = U_{2x} + a_{1}⟨U, Ū⟩U_{x} + a_{2}⟨U, U⟩Ū_{x} + a_{3}⟨U, Ū_{x}⟩U + a_{4}⟨Ū, U_{x}⟩U + a_{5}⟨U, U_{x}⟩Ū
+a_{6}⟨U, U⟩⟨Ū, Ū⟩U + a_{7}⟨U, U⟩⟨U, Ū⟩Ū + a_{8}⟨U, Ū⟩^{2}U  (2.11)

whose coefficients a_{1}, …, a_{8} depend on two constant parameters, falling into 6 classes, as listed in [26]. We remark that this classification does not consider the complex-valued case for μ = 3 or μ = 5, i.e. vector analogs of complexly-coupled mKdV, IS, K, KK, SK, and potential KK, SK equations, not in the NLS hierarchy. Hereafter, in the case of complexed-valued U, we allow t → it, τ → iτ, and impose invariance under the phase symmetry U → e^{iϵ}U (ϵ ∈ ℝ), as displayed by NLS type systems.

2.1 Vector-potential wave equations

We now proceed to discuss symmetry-integrability classifications of vector U_{tx} equations. Our results in theorems 1–3 are established by computer algebra computations, as outlined in detail in Sec. 3. In each case we explicitly determine all equations admitting a higher symmetry of least weight.

The first class we will consider consists of potential equations related to the integrable evolutionary vector equations (2.6)–(2.11) via U = V_{x}, with scaling weight λ_{V} = λ − 1. Note that the only ones whose potential form has non-negative scaling weight λ_{V} are the vector mKdV equations and NLS equations with λ_{V} = 0. A natural question is whether they still possess higher symmetries in potential form. We will settle a slightly more general classification problem by considering λ_{V} -homogeneous vector potential wave equations

V_{tx} = f(V, V_{x}, …, V_{nx}), n ≥ 0 (2.12)

where f is a sum of terms proportional to the vectors V_{lx}, 0 ≤ l ≤ n, with coefficients depending on the dot products ⟨V_{jx}, V_{kx}⟩.

Theorem 1. In the nonnegatively-weighted polynomial class for the potential-mKdV weight (λ_{V} , μ) = (0, 3), every λ_{V} -homogeneous vector wave equation (2.12) that possesses a homogeneous higher symmetry of weight ν = 5 containing no t derivatives is linear. The same result is true for the potential-NLS weight (λ_{V} , μ) = (0, 2) with ν = 3.

In contrast to this null result, there are many nonlinear scalar u_{tx} equations possessing a higher symmetry. For example, the potential-mKdV equation u_{tx} = u_{4x} + u^{2}u_{x}, and the 3rd-order Burger’s equation in potential form u_{tx} = u_{4x} + u_{x}u_{3x} + u_{2x}^{2} + \frac{1}{3}u_{x}^{2}u_{2x}.

2.2 Vector sine-Gordon equations

The next class of vector wave equations we investigate will be hyperbolic vector variants of the SG equation u_{tx} = sin u, distinguished by the scaling symmetry (t, x, u) → (e^{−ϵ}t, e^{ϵ}x, u). Although the transcendental form of nonlinearity in the SG equation precludes any obvious vector analog, we note there are other integrable hyperbolic scalar equations with the same scaling symmetry yet compatible with a vector nature for u, such as the equation u_{tx} = −u(1 − u^{2})^{−1}u_{t}u_{x} which is related to the scalar wave equation u_{tx} = 0 by the transformation u → sin u. We now classify analogous hyperbolic vector equations possessing higher symmetries with the same weight as the ones in both the SG hierarchy and the hierarchy connected with the scalar wave equation.

Theorem 2. In the nonnegatively-weighted polynomial class for the SG weight (λ, μ) = (0, −1), every λ-homogeneous nonlinear hyperbolic vector evolution equation (2.1) that possesses a higher symmetry of weight ν = 3 is given by (up to scalings of t, x, U)

U_{tx} = −⟨U_{t}, U_{x}⟩(⟨U, U⟩ + α)^{−1}U, α = −1, 0, 1  (2.13)
U_{τ} = ⟨U, U_{x}⟩^{2}U_{x}.  (2.14)

This result extends at least to weights ν = 4, 5, 6, 7, and in particular

U_{τ} = (⟨U, U_{3x}⟩ + 3⟨U_{x}, U_{2x}⟩)⟨U, U_{x}⟩U_{x} + (⟨U, U_{2x}⟩ + ⟨U_{x}, U_{x}⟩)^{2}U_{x}
+α(⟨U_{x}, U_{3x}⟩ + ⟨U_{2x}, U_{2x}⟩)U_{x},  (2.15)

is an admitted higher symmetry of order 3 for weight ν = 5.

We will give a geometrical interpretation for this wave equation in terms of a Ndimensional sigma model based on a paraboloid Riemannian space in Sec. 4, where we also discuss its the question of its integrability.

2.3 Vector hyperbolic flow equations

The final class of vector wave equations we will consider is motivated by the SG variant equation u_{tx} = u\sqrt{1 − u_{t}^{2}} whose associated flow commutes with the mKdV hierarchy of higher symmetry flows. With respect to this hierarchy, this wave equation represents a −1 flow as it is mapped into the trivial flow U_{τ} = 0 by the mKdV recursion operator. We recall that the recursion operator generates the mKdV hierarchy starting from the x-translation symmetry u_{τ} = u_{x}, referred to as the 0 flow for the hierarchy; the +1 flow is the mKdV equation itself u_{τ} = u_{3x} + u^{2}u_{x}, while its higher symmetries define the +2 flow and so on. We now systematically look for hyperbolic vector equations that describe −1 flows for the hierarchies of each integrable vector evolutionary equation (2.6)–(2.11). In general, the −1 flow corresponding to a λ-homogeneous +1 flow in the vector case will have weight (λ, −1), as we explain next.

Consider first the case of the mKdV and IS hierarchies where the 0 flow is U_{τ} = U_{x} with scaling weight ν_{0} equal to 1. Since the scaling weight of the +1 flow is ν_{1} = 3, the weight difference between successive flows in the hierarchy is w = ν_{1} − ν_{0} = 2. Hence if the −1 flow is regarded as a nonlocal evolutionary equation U_{t} = D_{x}^{−1}f(U, U_{t}, U_{x}) then it will have scaling weight μ = ν_{0} − w = 2ν_{0} − ν_{1} = −1. The remaining case is the NLS and derivative-NLS hierarchies where the 0 flow is U_{τ} = iU with scaling weight ν_{0} equal to 0. The +1 flow is then U_{τ} = U_{x}, and the (derivative-) NLS equation itself corresponds to the +2 flow, with scaling weight ν_{2} = 2. The weight difference between flows in this case is w = ν_{1} − ν_{0} = 1, and so the −1 flow will then have scaling weight μ = 0 − w = 2ν_{0} − ν_{1} = −1. Therefore in all cases the weight μ = −1 of t for the −1 flow is simply the negative of the weight of x. A similar argument clearly applies to the well-known scalar hierarchies.

We now state classifications that determine all −1 flows in the vector case.

Theorem 3. (mKdV and NLS −1 flows) For λ-homogeneous nonlinear hyperbolic vector evolution equations (2.1) with weight (λ, μ) = (1, −1) in the nonnegatively-weighted polynomial class, the ones that possess a higher symmetry of mKdV weight ν = 3 are given by (up to scalings of t, x, U)

U_{tx} = ±\sqrt{1 − α⟨U_{t}, U_{t}⟩}U, α = −1, 1  (2.16)
U_{τ} = U_{3x} + \frac{3}{2}α⟨U, U⟩U_{x},  (2.17)

and

U_{tx} = ⟨U, U_{t}⟩⟨U_{t}, U_{t}⟩^{−1}(1 + A)U_{t} − α(1 − A)U, α = −1, 0, 1  (2.18)
U_{τ} = U_{3x} + 3α⟨U, U⟩U_{x} + 3α⟨U, U_{x}⟩U,  (2.19)
where A = ±\sqrt{1 + α⟨U_{t}, U_{t}⟩},  (2.20)

while the ones that possess a higher symmetry of NLS weight ν = 2 are given by (up to scalings of t, x, U)

U_{tx} = ⟨U, Ū_{t}⟩⟨U_{t}, Ū_{t}⟩^{−1}(α + B)U_{t} − (α − B)U, α = −1, 1  (2.21)
iU_{τ} = U_{2x} + \frac{1}{4}⟨U, Ū⟩U  (2.22)
where B = ±\sqrt{α^{2} − \frac{1}{2}⟨U_{t}, Ū_{t}⟩},  (2.23)

and

U_{tx} = \sqrt{C_{2} + ⟨U_{t}, Ū_{t}⟩ + α} C_{1}^{−1} (⟨U, U_{t}⟩Ū_{t} − ⟨U, Ū_{t}⟩U_{t}) + \sqrt{C_{2} − ⟨U_{t}, Ū_{t}⟩ − α} U, (2.24)
iU_{τ} = U_{2x} + \frac{1}{2}⟨U, Ū⟩U − \frac{1}{4}⟨U, U⟩Ū, α = −1, 0, 1  (2.25)
where C_{1} = \sqrt{⟨U_{t}, U_{t}⟩⟨Ū_{t}, Ū_{t}⟩ − ⟨U_{t}, Ū_{t}⟩^{2}},  (2.26)

C_{2} = ±\sqrt{α^{2} + 2α⟨U_{t}, Ū_{t}⟩ + ⟨U_{t}, U_{t}⟩⟨Ū_{t}, Ū_{t}⟩}.

All four of these −1 flow equations are expected to be integrable, as they possess a +1 flow of the vector mKdV and NLS hierarchies (up to scaling of t, x, U, τ). We will interpret the first three of of them geometrically in terms of flat homogeneous N + 1-dimensional sigma models in Sec. 4.

Finally, we mention that a classification of IS and derivative-NLS −1 flows analogous to theorem 3 in the nonnegatively-weighted polynomial class with weight (λ, μ) = (\frac{1}{2}, −1) yields a null result. However, such flows do exist in a wider polynomial class, which we will present elsewhere.

3 Computational Aspects

Here we describe the algorithmic and implementation issues involved in the computations used to prove the classification theorems in Sec. 2. These computations divide into three parts:

(1) making an ansatz for the PDEs and symmetries with specified homogeneity weights;

(2) computing the symmetry conditions;

(3) solving the system of symmetry conditions.

The first two tasks are handled by the program SVSym (Scalar-Vector-Symmetry) ^{2} and the third computation is performed by the package Crack described in [30].

The class of PDE systems for investigation is allowed to involve any number of scalar functions u^{j} and vector functions U^{k} of t, x. An essential limitation currently in the program is that the left-hand side (l.h.s.) of each PDE must consist of a single derivative u_{pt qx}^{i} or U_{pt qx}^{i} of the same order p, q ≥ 0 for all PDEs in the system.

It is assumed that all derivatives occurring on the right-hand side (r.h.s.) of the PDEs and the symmetries are not equal to any l.h.s. derivatives or their differential consequences. In addition, r.h.s. derivatives must have a lower priority with respect to some total ordering >_{T} of derivatives than all l.h.s. derivatives, to avoid infinite substitution loops in computing the symmetry condition. For example, the equation u_{tx} = u_{tt} + u_{xx} would lead to an infinite loop of substitutions for differential consequences of u_{tx}, as seen by u_{txx} → u_{ttx} + u_{xxx}, u_{ttx} → u_{ttt} + u_{txx}. This problem is avoided if we adopt (without loss of generality) the lexicographical ordering t >_{T} x. Then because we have u_{tt} >_{T} u_{tx}, this equation would be handled by bringing the highest priority derivatives to the l.h.s., u_{tt} = u_{xx} − u_{tx}, which no longer leads to any infinite substitution loops.

3.1 Specifying an Ansatz

Many integrable systems are λ-homogeneous with respect to multiple homogeneity weights. For example, denoting the weight for x as ξ, the vector NLS system [26]

U_{t} = U_{xx} + ⟨U, V ⟩U, V_{t} = −V_{xx} − ⟨U, V ⟩V

and its higher symmetry hierarchy is λ-homogeneous with weights μ, ξ, λ_{U}, λ_{V} equal to 2, 1, 1, 1, as well as 0, 0, 1, −1. The latter weights have been used as an extra filter in our investigation of PDEs in the complex-valued case. To accommodate multiple weightings, the program SVSym allows the specification of a list of weight sets {μ, ξ, ν, λ_{i}}.

The program places two a priori restrictions on possible weights to ensure the number of possible homogeneous terms allowed to appear in the PDEs and symmetries is necessarily finite. A first restriction is that x cannot have zero weight in all weight sets as this would permit an unlimited differential order of x- derivatives (note the differential order of tderivatives is always finite due to the ordering t >_{T} x). A second restriction is that there must be a weight set {μ^{∗}, ξ^{∗}, ν^{∗}, λ_{i}^{∗}} for which the total weight of the l.h.s. of the PDEs and symmetries is non-negative, namely w_{i}^{PDE} := λ_{i}^{∗} + pμ^{∗} + qξ^{∗} ≥ 0 (with ξ^{∗} ≠ 0) and w_{i}^{sym} := λ_{i}^{∗} + ν^{∗} ≥ 0, as then the r.h.s. of the PDEs and symmetries can be generated by the following finite ansatz:

u_{pt qx}^{i} = f^{i}, U_{pt qx}^{i} = ∑_{k,J} F^{ikJ}U_{J}^{k}, and u_{τ}^{i} = g^{i}, U_{τ}^{i} = ∑_{k,J} G^{ikJ}U_{J}^{k}. (3.1)

Here J is a jet index, denoting partial derivatives including those of zeroth order (only tand x-derivatives of u^{k}, U^{k} are used whose total ordering with respect to t >_{T} x is lower than that of the l.h.s. of the PDEs). In (3.1), U_{J}^{k} has ^{∗}-weight ≥ 0 and ≤ w_{i}^{PDE} or ≤ w_{i}^{sym}; f^{i}, g^{i}, F^{ikJ}, G^{ikJ} are the most general polynomials that can be built from the following scalar polynomial variables, all of which are constructed to have positive ^{∗}-weight: u_{J}^{k}, products u_{J_{1}}^{k_{1}}u_{J_{2}}^{k_{2}} where one of the two factors has a negative ^{∗}-weight, and dot products ⟨U_{J_{1}}^{k_{1}}, U_{J_{2}}^{k_{2}}⟩ where at most one of the two factors may have a negative ^{∗}-weight. The undetermined coefficients in all polynomials are arbitrary functions of similarly constructed scalar variables u_{J}^{k}, u_{J_{1}}^{k_{1}}u_{J_{2}}^{k_{2}}, ⟨U_{J_{1}}^{k_{1}}, U_{J_{2}}^{k_{2}}⟩ whose weight is zero in all weight sets. All weight sets other than the ^{∗}-weight set are used as an extra filter on the generated terms.

2SVSym is a completely new program, more general than the program used for the classification of evolutionary PDEs in [26, 28], which was limited to positive weights and a strictly polynomial ansatz.

The restriction of using polynomial variables built from at most two scalar or vector factors could obviously be generalized to allow more factors such that the total weight of the product still satisfies the previous conditions.

It is important to emphasize that the ansatz (3.1) for the PDEs and symmetries need not be strictly polynomial if μ^{∗} < 0 or ξ^{∗} < 0 or any λ_{i}^{∗} ≤ 0.

A few flags give convenient means to cut down the generality of the ansatz, for example by discarding t-derivatives or imposing constant coefficients in the PDEs or symmetries or both. Extra conditions or inequalities on coefficients can be added easily.

The following examples show the ansatz generated for the classifications in theorems 2 and 3. The vector SG ansatz with μ = −1, ν = 3, λ_{U} = 0 is:

U_{tx} = a_{1}U,
U_{τ} = b_{1}U_{3x} + b_{2}U_{2x}⟨U_{x}, U⟩ + b_{3}U_{x}⟨U_{2x}, U⟩ + b_{4}U_{x}⟨U_{x}, U_{x}⟩ + b_{5}U_{x}⟨U, U_{x}⟩^{2} + b_{6}U⟨U_{3x}, U⟩
+b_{7}U⟨U_{2x}, U_{x}⟩ + b_{8}U⟨U_{x}, U_{x}⟩⟨U, U_{x}⟩ + b_{9}U⟨U_{2x}, U⟩⟨U, U_{x}⟩ + b_{10}U⟨U_{x}, U⟩^{3}

where all coefficients a_{i}, b_{j} are undetermined functions of ⟨U, U⟩, ⟨U_{x}, U_{t}⟩. The ansatz for the vector mKdV inverse flow with μ = −1, ν = 3, λ_{U} = 1 is:

U_{tx} = a_{1}U_{t}⟨U, U_{t}⟩ + a_{2}U,
U_{τ} = b_{1}U_{3x} + b_{2}U_{2x}⟨U, U_{t}⟩ + b_{3}U_{x}⟨U_{x}, U_{t}⟩ + b_{4}U_{x}⟨U, U_{t}⟩^{2} + b_{5}U_{x}⟨U, U⟩ + b_{6}U_{t}⟨U_{x}, U_{t}⟩^{2}

+b_{7}U_{t}⟨U_{x}, U_{t}⟩⟨U, U_{t}⟩^{2} + b_{8}U_{t}⟨U_{x}, U_{t}⟩⟨U, U⟩ + b_{9}U_{t}⟨U, U_{t}⟩^{2}⟨U, U⟩ + b_{10}U_{t}⟨U, U_{t}⟩^{4}

+b_{11}U_{t}⟨U, U_{t}⟩⟨U, U_{x}⟩ + b_{12}U_{t}⟨U, U_{2x}⟩ + b_{13}U_{t}⟨U_{x}, U_{x}⟩ + b_{14}U_{t}⟨U, U⟩^{2}
+b_{15}U⟨U_{x}, U_{t}⟩⟨U, U_{t}⟩ + b_{16}U⟨U, U_{t}⟩⟨U, U⟩ + b_{17}U⟨U, U_{t}⟩^{3} + b_{18}U⟨U, U_{x}⟩

where all coefficients a_{i}, b_{j} are undetermined functions of ⟨U_{t}, U_{t}⟩. The ansatz for the vector NLS inverse flow is similar, using a pair of vectors U, V, with μ = −2, ν = 6, λ_{U} = λ_{V} = 1, in addition to the filter μ = ν = 0, λ_{U} = 1, λ_{V} = −1.

3.2 Computing Symmetry Conditions

In the special case of evolutionary PDEs, all substitutions of u_{t}^{i} and u_{τ}^{i} in the symmetry conditions u_{[t,τ]}^{i} := D_{τ} u_{t}^{i} −D_{t}u_{τ}^{i} = 0 are done only once because the r.h.s. of the PDEs and symmetries do not contain t- or τ- derivatives. As a consequence, the symmetry conditions are separately linear in the undetermined coefficients of the PDEs and of the symmetries.

The situation is different when the l.h.s. of the PDEs contain an x-derivative of u. Then any t- or x-derivatives that appear in the r.h.s. of the PDEs and symmetries lead to (repeated) substitutions of u_{pt qx}^{i} through the PDEs. The symmetry conditions become polynomially non-linear in the coefficients of the PDEs but stay linear in the coefficients of the symmetries. Due to this non-linearity, which is further amplified whenever the coefficients are functions of t-derivatives of u^{i}, the symmetry conditions can become extremely large. The size of intermediate expressions can become exceedingly high during the repeated substitution and simplification process. To get over this memory hurdle, the linearity in the coefficients of the symmetry is exploited. If the r.h.s. of all symmetry conditions involve s terms with s undetermined coefficients b_{1}, …, b_{s} then the full symmetry condition is computed via

0 = u_{[t,τ]}^{i} = ∑_{k=1}^{s} u_{[t,τ]}^{i}|_{u_{τ}^{j} =S_{k}^{j}} (3.2)

where S_{k}^{j} denotes the r.h.s. of u_{τ}^{j} = S^{j} such that all coefficients except b_{k} are replaced by zero. This trade-off of computing memory for computing time is also justified by the fact that the time to simplify large expressions grows non-linearly with their size, at least in the computer algebra system Reduce that is used.

3.3 Solving the Symmetry Conditions

Compared to the case of evolutionary PDEs, the symmetry conditions for the general case of PDEs with l.h.s. u_{pt qx}^{i} are not purely algebraic but may involve ordinary and partial differential equations and may be inhomogeneous and non-linear in the undetermined coefficients of the PDEs. The system of symmetry conditions is fed into the computer program Crack, which was used to solve the algebraic symmetry conditions in the classification of evolutionary vector equations [26] and coupled evolutionary scalar-vector systems [28]. Originally, Crack has been developed as a solver for overdetermined PDE systems, and this functionality is essentially required for solving the polynomially nonlinear differential system of symmetry conditions that occur in the classifications of vector hyperbolic equations in Sec. 2. Solutions of non-polynomial form, like square roots in the equations found in theorem 3, may appear and often are in reach of the Reduce ODE solver that is employed by Crack when appropriate.

The strength of Crack compared with other computer algebra packages for solving overdetermined systems lies in the flexibility of its approach, the possibility to run it in a fully automatic or interactive mode, and its support in handling especially large systems (safety, process control, heuristic algorithms to cut down the size of equations, ability to take advantage of the linearity of coefficient functions of the symmetries). In addition, Crack provides a verbose mode reporting all computational steps, which allows in principle the solutions to be checked by (human) inspection (in practise this is hard due to their length).

4 Sigma Model Interpretations

The wave equation u_{tx} = 0 and SG equation u_{tx} = sin u are well-known to be intimately related to two of the simplest sigma models [19], which are described by a geometrical wave equation for maps on 2-dimensional Minkowski space ℝ^{1,1} into target spaces S^{1} and S^{2}, respectively. In general a sigma model on ℝ^{1,1} is given by the nonlinear wave equation

0 = ^{g}∇_{t}∂_{x}u^{A} = ∂_{t}∂_{x}u^{A} + Γ^{A}_{BC}(u)∂_{t}u^{B}∂_{x}u^{C} (4.1)

for scalar functions u^{A}(t, x) representing a map into some Riemannian target space (M, g_{AB}), A = 1, …, dim M. Here ^{g}∇_{t} = ∂_{t} + Γ^{A}_{BC}(u)∂_{t}u^{B} is the pullback to ℝ^{1,1} of the (torsionfree) covariant derivative determined by the metric g_{AB} in local coordinates u^{A} on the manifold M, given in terms of the Christoffel symbol Γ^{A}_{BC} of the metric. This wave equation has the geometrical action principle S[u] = ∫ ⟨∂_{t}u, ∂_{x}u⟩_{g}dtdx where ⟨·, ·⟩_{g} denotes the inner product in T (M) given by the metric g_{AB}(u). Another geometrical meaning for this wave equation arises if we view u^{A}(t, x) for fixed t as a curve embedded into the manifold M. The sigma model thereby describes a flow of this curve, whose arclength is preserved due to the conservation law

∂_{t}(∂_{x}u^{A}∂_{x}u^{B}g_{AB}(u)) = 0. (4.2)

Because of the reflection symmetry t ↔ x in the model, there is also a conservation law

∂_{x}(∂_{t}u^{A}∂_{t}u^{B}g_{AB}(u)) = 0. (4.3)

These conservation laws are connected with invariance of the model under conformal scalings on t, x, namely t → t′ := α(t), x → x′ := β(x). This freedom can be fixed in a natural manner by putting α(t) = ⟨∂_{t}u, ∂_{t}u⟩_{g} and β(x) = ⟨∂_{x}u, ∂_{x}u⟩_{g}. Finally, note that in general the sigma model equation (4.1) is λ-homogeneous with scaling weight λ = 0 on u^{A} while t and x have opposite weights of μ = −1 and +1, in the conventions of Sec. 2.

O(N)-invariant vector wave equations arise from sigma models in various ways and for certain types of target spaces. One direct way is by considering target spaces embedded as an O(N) symmetric hypersurface in Euclidean space ℝ^{N+1} or Minkowski space ℝ^{N,1}. Vector coordinates for the hypersurface given by its embedding lead to natural vector generalizations of the variant scalar wave equation u_{tx} = −u(1 − u^{2})^{−1}u_{t}u_{x} which is the sigma model for S^{1} ⊂ ℝ^{2}. A second way involves introducing a Cartan connection and frame bundle on flat homogeneous target spaces 𝔤/𝔥 ≃ ℝ^{N+1} based on pairs of Lie algebras 𝔤 ⊃ 𝔥. With a suitable choice of frame, the flat connection components yield natural vector generalizations of the variant SG equation u_{tx} = u\sqrt{1 − u_{t}^{2}} which comes from the linear sigma model for 𝔰o(3)/𝔰o(2) ≃ ℝ^{2}. The SG equation itself is related in a similar way to a moving frame formulation of the nonlinear sigma model S^{2} = SO(3)/SO(2) shown in [1].

In Sec. 4.1 we show that the vector wave equation (2.13) in theorem 2 arises from the nonlinear sigma model for an O(N) symmetric paraboloid hypersurface in ℝ^{N+1}. In Sec. 4.2 we derive the first three hyperbolic vector equations in theorem 3 from the Cartan connection of linear sigma models viewed as a flow of curves on flat homogeneous spaces associated with the Lie algebras 𝔰o(N + 1) ⊂ 𝔰o(N + 2) or 𝔰u(N + 1).

4.1 N-sphere and N-dimensional paraboloid sigma models

Consider firstly the N-sphere target space S^{N} as an embedded hypersurface in ℝ^{N+1}. We describe S^{N} by the N + 1-component unit vector u^{i}, i = 1, …, N + 1, constrained to satisfy 1 = u^{i}u^{j}δ_{ij}. If we resolve the constraint by expressing u^{N+1} = ±\sqrt{1 − u^{A}u^{B}δ_{AB}} with u^{i} = (u^{A}, u^{N+1}), A = 1, …, N, then u^{A} represents a set of coordinates on S^{N}. In particular the upper or lower hemisphere of S^{N} is mapped by u^{A} to the open unit ball in ℝ^{N}. In these coordinates, the Riemannian metric on S^{N} as induced by the Euclidean metric δ_{ij} on ℝ^{N+1} is given by

g_{AB}(u) = δ_{AB} + (1 − u^{C}u_{C})^{−1}u_{A}u_{B} (4.4) with the inverse metric g^{AB} = δ^{AB} − u^{A}u^{B}, where u_{A} := δ_{AB}u^{B}. The Christoffel symbol and covariant derivative on S^{N} are readily found from the condition ^{g}∇_{C}g^{AB}(u) = 0, giving

g∇_{C} = \frac{∂}{∂u^{C}} + u^{A}g_{BC} (u), Γ^{A}_{BC}(u) = u^{A}(δ_{BC} + (1 − u^{D}u_{D})^{−1}u_{B}u_{C}). (4.5)

Hence the curvature is given by

[^{g}∇_{A},^{g}∇_{B}] := R_{ABC}^{D} = 2g_{C[A}(u)δ_{B]}^{D}χ, χ = 1 (4.6)

where χ denotes the scalar curvature. Note this indicates S^{N} is an O(N) symmetric space of (positive) constant curvature.

The sigma model equation (4.1) for the target space S^{N} thus has the form

0 = ∂_{t}∂_{x}u^{A} + u^{A}(∂_{t}u^{B}∂_{x}u_{B} + (1 − u^{B}u_{B})^{−1}u_{C}u_{D}∂_{t}u^{C}∂_{x}u^{D}) (4.7)

which is an O(N)-invariant hyperbolic vector equation for u^{A}(t, x). The scalar case N = 1 is seen to reduce to 0 = u_{tx} + u(1 − u^{2})^{−1}u_{t}u_{x}. Integrability of this sigma model for all N ≥ 1 is a well-established result via a Lax pair associated with S^{N} = SO(N +1)/SO(N) as a symmetric space [31, 19], and also more recently via a bi-Hamiltonian structure (higher-symmetry recursion operator) derived from a moving frame formulation of S^{N} as a constant curvature space [22, 1]. Its higher symmetries were first found in [19] using a Bäcklund transformation technique which relies on conformally rescaling t → t′, x → x′ through the conservation laws (4.2) and (4.3). The hierarchy of higher symmetries is homogeneous and has positive scaling weight ν = 3, 5, …, with respect to the scaling group (x′, t′, u^{A}) → (e^{ϵ}x′, e^{−ϵ}t′, u^{A}), ϵ ∈ ℝ. But when expressed in the original variables t, x, the higher symmetries are all homogeneous of weight ν = 0. For this reason the vector wave equation (4.7) does not appear in our classifications.

To account for the vector wave equation (2.13) and its higher symmetry (2.15) with SG scaling weight in theorem 2, we look for a target space with an appropriate Riemannian metric. Working backward from the nonlinear terms in the equation

0 = ∂_{t}∂_{x}u^{A} + u^{A}(1 + u^{C}u_{C})^{−1}∂_{t}u^{B}∂_{x}u_{B} (4.8)

we deduce the Christoffel symbol

Γ^{A}_{BC}(u) = u^{A}δ_{BC} (1 + u^{D}u_{D})^{−1} (4.9)

and hence

g∇_{C} = \frac{∂}{∂u^{C}} + u^{A}δ_{BC} (1 + u^{D}u_{D})^{−1} (4.10)

is the covariant derivative. ¿From the condition ^{g}∇_{C}g^{AB}(u) = 0, it is straightforward to derive g^{AB} = δ^{AB} + (1 + u^{C}u_{C})^{−1}u^{A}u^{B} which then gives the metric

g_{AB}(u) = δ_{AB} − u_{A}u_{B}. (4.11)

Finally we are able to identify this metric as the induced Riemannian metric on the Ndimensional target space given by the paraboloid hypersurface

u^{N+1} = \frac{1}{2}u^{A}u^{B}δ_{AB} in ℝ^{N+1}. (4.12)

This target space is an O(N) symmetric Riemannian manifold, whose scalar curvature is non-constant

χ = (1 + 2u^{N+1})^{−1}, R_{ABC}^{D} = 2δ_{C[A}g_{B]}^{D}(u)χ (4.13)

where g_{A}^{B} := δ_{AC} g^{CB}. (Curiously, the metric tensor g_{A}^{B}(u) on the N-dimensional paraboloid is the same as the inverse metric tensor g^{B}_{A}(u) := g^{BC} (u)δ_{CA} on the Nsphere.)

We note there is a Lorentzian version of these sigma models (4.7) and (4.8) obtained via replacing the Euclidean space ℝ^{N+1} by Minkowski space ℝ^{N,1}. Correspondingly, S^{N} is replaced by its Lorentzian counterpart given by the N-dimensional hyperboloid space u^{N+1} = ±\sqrt{1 + u^{A}u^{B}δ_{AB}} which comes from the spacelike hypersurface 1 = u^{i}u^{j}η_{ij} in ℝ^{N,1}, where η_{ij} is the Minkowski metric and u^{i} = (u^{A}, u^{N+1}). Its induced Riemannian metric is g_{AB}(u) = δ_{AB} + (1 + u^{C}u_{C})^{−1}u_{A}u_{B}, with constant (negative) scalar curvature χ = −1. In contrast, the N-dimensional paraboloid space u^{N+1} = \frac{1}{2}u^{A}u^{B}δ_{AB} is its own Lorentzian counterpart, but with the metric replaced by g_{AB}(u) = δ_{AB} +u_{A}u_{B} as induced by the Minkowski metric η_{ij}, and with non-constant scalar curvature χ = (1 − 2u^{N+1})^{−1}. Note in the Lorentzian case this paraboloid hypersurface has a null tangent direction at points 2u^{N+1} = 1 in ℝ^{N,1} where both the metric and curvature become singular.

Thus the nonlinear sigma model (4.8) for the N-dimensional paraboloid target space in the Euclidean and Lorentzian cases reproduces the vector wave equation (2.13) with the vector U identified as the local coordinates u^{A}. Integrability of this model does not seem to be known in the literature, so its higher symmetry (2.15) stated in theorem 2 is a new result. Unlike the S^{N} model, the existence of a hierarchy of higher symmetries is an open question, apart from the scalar case N = 1 of this model (4.8) which gives the same integrable S^{1} sigma model as the scalar reduction of the S^{N} model (4.7). As noted in the theorem, for N ≥ 1 we checked that there are higher symmetries of all weights ν = 4, 5, 6, 7. This strongly suggests the model (4.8) is indeed integrable. A proof will require exhibiting e.g. a Lax pair, a bi-Hamiltonian structure, or a symmetry-recursion operator.

4.2 Flat homogeneous sigma models

We now turn to the flat homogeneous target spaces M = 𝔤/𝔥 based on Lie algebras 𝔥 ⊂ 𝔤 chosen to be semi-simple and compact, with the structure of a symmetric space 𝔤 = 𝔥 ⊕ 𝔭 such that [𝔥, 𝔭] ⊆ 𝔭 and [𝔭, 𝔭] ⊆ 𝔥 where M is identified with the vector space 𝔭 in this decomposition. These target spaces M = 𝔤/𝔥 ≃ 𝔭 are thus flat Riemannian manifolds whose metric is given by the Cartan-Killing inner product ⟨·, ·⟩_{𝔭} on 𝔤 restricted to M. A few examples are 𝔰o(k + 1)/𝔰o(k), 𝔰u(k)/𝔰o(k), 𝔰u(k + 1)/𝔲(k) (see [11] for a complete list and properties). The sigma model (4.1) for all such flat target spaces is just the linear wave equation

0 = ∂_{t}∂_{x}u^{A} (4.14) for scalar functions u^{A}(t, x). Nevertheless, viewed as a flow of curves in M ≃ 𝔤/𝔥, there is a natural frame bundle formulation for this model using a Cartan connection whose components will satisfy an O(N)-invariant nonlinear vector wave equation [6]. Details of Cartan connections and frame bundles for homogeneous spaces are presented in [23].

Firstly, we consider M ≃ ℝ^{N+1} = 𝔭 as given by 𝔤 = 𝔰o(N + 2) and 𝔥 = 𝔰o(N + 1). Recall, 𝔰o(k) is a real vector space of dimension \frac{k(k−1)}{2} isomorphic to the Lie algebra of k × k skew-symmetric matrices. So M = 𝔰o(N + 2)/𝔰o(N + 1) is isomorphic to 𝔭 = ℝ^{N+1}, as described by the following canonical decomposition

(\begin{matrix}0 & p \\ −p^{T} & h \end{matrix}) ∈ 𝔰o(N + 2), h ∈ 𝔰o(N + 1), p ∈ ℝ^{N+1} (4.15)

parameterized by the N + 1-component vector p^{A}, A = 1, …, N + 1. Note the Cartan-Killing inner product on 𝔤 is (up to a normalization factor) the trace of the product of 𝔰o(N + 2) matrices (4.15) and hence reduces for h = 0 to the ordinary dot product δ_{AB} on vectors p^{A}. On the homogeneous space 𝔭 = 𝔰o(N + 2)/𝔰o(N + 1) we introduce an 𝔰o(N + 2)-valued flat connection ^{𝔤}ω_{A} with the decomposition ^{𝔤}ω_{A} = ^{𝔥}ω_{A} + ^{𝔭}ω_{A}. As a consequence of the Lie algebra structure (4.15), ^{𝔥}ω_{A} provides a Cartan connection to which is associated an ℝ^{N+1}-valued frame e^{A} for the vector space 𝔰o(N + 2)/𝔰o(N + 1) ≃ ℝ^{N+1}. More precisely, the underlying (gauge) structure group here is SO(N + 1), namely the matrix Lie group associated with 𝔥 = 𝔰o(N + 1). Up to the gauge freedom of SO(N + 1) rotations, there is a canonical relation ^{𝔭}ω_{A} = e_{A} between the frame and the connection, where e_{A} is the coframe satisfying ⟨e_{A}, e^{B}⟩_{𝔭} = δ_{A}^{B}.

To derive an O(N)-invariant vector wave equation from the linear sigma model (4.14) for this target space M = 𝔰o(N + 2)/𝔰o(N + 1), we first write ^{𝔥}ω_{t} := ^{𝔥}ω_{A}∂_{t}u^{A}, ^{𝔥}ω_{x} := 𝔥ω_{A}∂_{x}u^{A}, e_{t} := e_{A}∂_{t}u^{A}, e_{x} := e_{A}∂_{x}u^{A}, in terms of u^{A}(t, x). These obey

0 = ∂_{t}e_{x} +^{𝔥}ω_{t}e_{x} = ∂_{x}e_{t} +^{𝔥}ω_{x}e_{t} (4.16)

as a consequence of the wave equation (4.14) and the structure equation of the frame^{𝔭}ω_{A} = e_{A}. Interpreted geometrically, the second equality is the Cartan structure equation for vanishing torsion of the connection ^{𝔤}ω_{A}. The Cartan structure equation for the curvature of this flat connection gives

∂_{x}^{𝔥}ω_{t} − ∂_{t}^{𝔥}ω_{x} + [^{𝔥}ω_{x},^{𝔥}ω_{t}] = [^{𝔭}ω_{t},^{𝔭}ω_{x}] = e_{t} ∧ e_{x} (4.17)

where ∧ denotes the antisymmetric tensor product. Note from equation (4.16) we recover the conservation laws (4.2) and (4.3) for this sigma model, 0 = ∂_{t}⟨e_{x}, e_{x}⟩_{𝔭} = ∂_{x}⟨e_{t}, e_{t}⟩_{𝔭}. By conformally scaling t, x, we normalize ⟨e_{x}, e_{x}⟩_{𝔭} = ⟨e_{t}, e_{t}⟩_{𝔭} = 1. Then we use the inherent gauge freedom on e_{A} using the rotation group SO(N + 1) to put e_{x}^{N+1} = 1, e_{x}^{i} = 0, without loss of generality. Thus one of the conservation laws is satisfied identically; the other one determines

e_{t}^{N+1} = ±\sqrt{1 − e_{t}^{i}e_{t}^{j}δ_{ij}} . (4.18)

Here e_{x} = (e_{x}^{i}, e_{x}^{N+1}) and e_{t} = (e_{t}^{i}, e_{t}^{N+1}), i = 1, …, N, are the N + 1 components of the frames associated with the tangent vector ∂_{x}u^{A} along the curve u^{A}(t, x) and the flow direction vector ∂_{t}u^{A} on M = 𝔰o(N + 2)/𝔰o(N + 1). There now remains SO(N) gauge freedom which preserves the form of e_{x}. We use this freedom to fix the connection^{𝔥}ω_{x} so it corresponds geometrically to a parallel frame [5] along the curve u^{A}(t, x) as follows. Write 𝔥ω_{x} = (^{𝔥}ω_{x}^{ij},^{𝔥}ω_{x}^{iN+1}) and^{𝔥}ω_{t} = (^{𝔥}ω_{t}^{ij},^{𝔥}ω_{t}^{iN+1}) for the components of the (N +1×N +1) connection matrices. The curvature equation (4.17) shows that ^{𝔥}ω_{x}^{ij},^{𝔥}ω_{t}^{ij} are flat 𝔰o(N)- valued connection matrices, and consequently we put ^{𝔥}ω_{x}^{ij} = 0 by the available gauge freedom. This leads to ^{𝔥}ω_{t}^{ij} = 0 without loss of generality. Geometrically speaking, 𝔥ω_{x} then describes the derivative of e_{x} under infinitesimal displacement along the curve u^{A}(t, x) such that the normal vectors in the frame have a purely tangential derivative.

Now, the torsion equation (4.16) can be used to find

𝔥ω_{t}^{iN+1} = 0,  (4.19)
𝔥ω_{x}^{iN+1} = (e_{t}^{N+1})^{−1}∂_{x}e_{t}^{i},  (4.20)

while from the curvature equation (4.17) we have

e_{t}^{i} = −∂_{t}^{𝔥}ω_{x}^{iN+1}. (4.21)

Thus the reduced Cartan equations (4.20) and (4.21) together with the relation (4.18) yield a system of coupled vector equations ∂_{x}e_{t}^{i} = ±(1 − e_{t}^{i}e_{t}^{j}δ_{ij})^{1/2}^{𝔥}ω_{x}^{iN+1} and ∂_{t}^{𝔥}ω_{x}^{iN+1} = −e_{t}^{i}. Viewing ^{𝔥}ω_{x}^{iN+1} as an N-component vector U(t, x), and eliminating e_{t}^{i} from these equations, we obtain precisely the O(N)-invariant nonlinear vector wave equation (2.16) in theorem 3. As shown by the theorem, this vector wave equation possesses a higher symmetry given by the +1 flow in the vector mKdV hierarchy (2.6).

Integrability of the nonlinear vector wave equation (2.16) was first shown via a Lax pair [7, 2]. Related derivations of it using both a zero-curvature and a Lie-algebraic formulation of the Lax pair were obtained in [20, 6, 29]. Our derivative here is more intrinsically geometrical, based on the approach in [15, 22] relying on use of a parallel frame and Cartan connection

e_{x} = (1, 0, …, 0), ^{𝔥}ω_{x} = (\begin{matrix}0 & U \\ −U^{T} & 0 \end{matrix}) (4.22)

that are associated to the sigma model (4.14) interpreted as a flow of curves. In particular, in this frame the connection components U geometrically represent differential invariants of the curve.

The two other O(N)-invariant vector wave equations (2.18) and (2.21) in theorem 3 have a similar derivation, starting from the target space M = 𝔰u(N + 1)/𝔰o(N + 1). One notable difference in this sigma model is that the target space has dimension greater than N + 1 and hence a reduction condition must be imposed on the Cartan connection and its associated frame, allowing them to be expressed in terms of a N + 1-component vector. The higher symmetry structure (2.7) and (2.9) for these vector wave equations obtained by theorem 3, namely +1 flows in the vector mKdV and NLS hierarchies, is a new result. Of course, a proof of their integrability will require e.g. checking that they admit the symmetry-recursion operator of the corresponding hierarchy.

To outline the derivation of these vector wave equations, recall 𝔰u(k) is a complex vector space isomorphic to the Lie algebra of k × k skew-hermitian matrices. The real and imaginary parts of these matrices belong to the real vector space 𝔰o(k) of skew-symmetric matrices and the real vector space 𝔰(k) ≃ 𝔰u(k)/𝔰o(k) defined by k × k symmetric tracefree matrices. Hence 𝔤 = 𝔰u(N + 1) has the decomposition 𝔤 = 𝔥 + i𝔭 where 𝔥 = 𝔰o(N + 1) and 𝔭 = 𝔰(N + 1). There is a natural subset of matrices in 𝔰u(N + 1) given by the special form

\frac{1}{n + 1}p · p 𝕀 − p^{T}p ∈ 𝔰(N + 1), p ∈ ℝ^{N+1} (4.23)

which is parametrized by the arbitrary N +1-component vector p^{A}. This set of symmetric trace-free matrices (4.23), hereafter denoted 𝔰_{p}(N + 1), provides a canonical reduction of 𝔰u(N + 1)/𝔰o(N + 1) to ℝ^{N+1} as a N + 1-dimensional manifold. The restriction of the Cartan-Killing metric of 𝔰u(N + 1) to 𝔰_{p}(N + 1) yields a Riemannian metric ⟨·, ·⟩_{𝔰_{p}} proportional to the trace of the product of matrices (4.23). In particular, the norm of a matrix (4.23) in this metric (with a suitable normalization) is simply p · p. (Note the nonlinearity of the matrix norm in terms of p is due to the lack of a vector space structure for 𝔰_{p}(N + 1).) On this space 𝔰_{p}(N + 1) we introduce an 𝔰u(N + 1)-valued flat connection 𝔤ω_{A} = ^{𝔥}ω_{A} + i^{𝔰}ω_{A}, with ^{𝔥}ω_{A} defining a Cartan connection to which is associated an 𝔰_{p}(N + 1)-valued frame e^{A} and a coframe e_{A} = ^{𝔰}ω_{A} such that ⟨e_{A}, e^{B}⟩_{𝔰_{p}} = δ_{A}^{B}. There is SO(N + 1) gauge freedom inherent in e_{A} and ^{𝔥}ω_{A}.

To proceed we use the Riemannian space M ≃ 𝔰_{p}(N + 1) ⊂ 𝔰u(N + 1)/𝔰o(N + 1) as the reduced target space in the sigma model for u^{A}(t, x). Along the tangent vector ∂_{x}u^{A} and the flow direction vector ∂_{t}u^{A} for the curve u^{A}(t, x), we have the frames e_{x}, e_{t} and the connection matrices ^{𝔥}ω_{x},^{𝔥}ω_{t}. These obey the torsion and curvature equations

0 = ∂_{t}e_{x} +^{𝔥}ω_{t}e_{x} = ∂_{x}e_{t} +^{𝔥}ω_{x}e_{t},  (4.24)
∂_{x}^{𝔥}ω_{t} − ∂_{t}^{𝔥}ω_{x} + [^{𝔥}ω_{x},^{𝔥}ω_{t}] = [^{𝔰}ω_{x},^{𝔰}ω_{t}] = [e_{x}, e_{t}].  (4.25)

Using the sigma model conservation laws (4.2) and (4.3), we put ⟨e_{x}, e_{x}⟩_{𝔰_{p}} = ⟨e_{t}, e_{t}⟩_{𝔰_{p}} = 1. We further fix e_{x},^{𝔥}ω_{x} to be a parallel frame and connection (via the SO(N + 1) gauge freedom). Namely, in matrix notation,

e_{x} = \frac{1}{n + 1}𝕀−(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) ∈ 𝔰_{p}(N +1), ^{𝔥}ω_{x} = (\begin{matrix}0 & U \\ −U^{T} & 0 \end{matrix}) ∈ 𝔰o(N +1), U ∈ ℝ^{N} (4.26)

where 0 ∈ 𝔰o(N), and also

e_{t} = \frac{1}{n + 1}𝕀 − (\begin{matrix}v & V \\ V^{T} & v^{−1}V^{T}V \end{matrix} ) ∈ 𝔰_{p}(N + 1), (V, v) ∈ ℝ^{N+1} (4.27)

where

v^{2} + V · V = v. (4.28)

Note e_{x}, e_{t} are of the form (4.23) with respective parametrizations chosen in vector index notation as 1 = p^{N+1}, 0 = p^{i}, and \sqrt{v} = p^{N+1}, \sqrt{v}^{−1}V^{i} = p^{i}; the norm of e_{t} is 1 = (p^{N+1})^{2} + p^{i}p^{j}δ_{ij} = v + v^{−1}V^{i}V^{j}δ_{ij}. Now, from the 𝔰o(N) part of the curvature equation

(4.25) together with the first half of the torsion equation (4.24), we find ^{𝔥}ω_{t} = (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}).

Then the remaining parts of equations (4.24) and (4.25) reduce to

∂_{x}V = vU − v^{−1}U · V V, ∂_{t}U = −V,  (4.29)
∂_{x}v = −2U · V , ∂_{x}(v^{−1}V^{T} V) = V^{T}U + U^{T}V.  (4.30)

We see equations (4.29) and (4.28) are equivalent to the O(N)-invariant nonlinear vector wave equation (2.18), while the last equation (4.30) is found to hold as an identity.

4.3 Other models and concluding remarks

The last O(N)-invariant nonlinear vector wave equation (2.24) given in theorem 3 stands out from the others in several interesting aspects. First it is not well-defined in the scalar case N = 1 or in the real-valued vector case (precisely when the expression (2.26) vanishes). Thus this wave equation exhibits an intrinsic complex-valued vectorial nature. As a result, it does not appear to be obviously connected with any flat homogeneous sigma models M = 𝔤/𝔥 since they naturally will have well-defined scalar reductions. Some insight on this situation comes from the NLS higher symmetry admitted by this vector wave equation. In [14] the NLS vector hierarchy (2.10) generated by this symmetry is shown to originate from the matrix NLS equation i∂_{t}U = ∂_{x}^{2}U + UU^{†}U where the matrix U = u^{A}γ_{A} is chosen to belong to the span of the gamma matrices γ_{A}, A = 1, …, N, defined to satisfy the Clifford algebra relations γ_{A}γ_{B} + γ_{B}γ_{A} = 2δ_{AB}𝕀. This observation suggests the vector wave equation (2.24) may arise in a similar fashion from a suitable matrix type of sigma model.

Finally, in connection with these remarks, we mention the deep work [25, 10, 27] of Snivolupov and Sokolov on the construction of integrable vector and matrix equations of hyperbolic and evolutionary type, related to special kinds of nonassociative algebras.

Acknowledgements

One author (TW) wants to thank Winfried Neun for discussions, and the Konrad Zuse Institute Berlin for support through its fellowship program. Takayuki Tsuchida is thanked for comments on derivative-NLS −1 flows. Computations were performed partially on computer clusters of the SHARCNET consortium (www.sharcnet.com).

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