Recursion operator for the stationary Nizhnik-Veselov-Novikov equation

M Marvan and A Sergyeyev

Mathematical Institute, Silesian University in Opava, Bezrucovo nám. 13, 746 01 Opava, Czech Republic

Email: Michal.Marvan@math.slu.cz and Artur.Sergyeyev@math.slu.cz

Received 15 October 2002, in final form 18 December 2002
Published 22 January 2003

PACS numbers: 02.30.Ik, 02.30.Jr, 02.40.Hw, 02.40.Ky, 02.70.Wz

Mathematics Subject Classification: 37K10, 35A30, 35Q58

In this letter, we suggest a new method for the construction of a recursion operator using the zero-curvature representation. Unlike the majority of the hitherto known methods (see e.g. [1-4] and references therein), ours gives not only the recursion operator, but also its inverse, leading to the `negative' part of the hierarchy of the system in question. What is more, it is immediately applicable to both evolutionary and non-evolutionary systems. We apply the method to the stationary Nizhnik-Veselov-Novikov (NVN) equation for which no recursion operator has been found so far.

Let F =  0 be a system of PDEs in two independent variables x, y for the unknown n-component vector function u =  (u¹,  ..., uⁿ)^T, where the superscript `T' denotes matrix transposition. Let this system have a zero-curvature representation D_y(A)  - D_x(B)  +  [A, B]  =  0, where A and B take values in a (matrix) Lie algebra and depend on λ, x, y, and u and its derivatives. Here D_x, D_y are the operators of total x- and y-derivatives (see e.g. ch 2 of [5], and [6]). Note that A and B may involve an essential (spectral) parameter λ.

Consider a function P (possibly vector- or matrix-valued) of x, y, u and its derivatives. Then the directional derivative of P along an n-component vector U =  (U¹,  ..., Uⁿ)^T is given by P '[ U]  = ∑ⁿ_α = 1∑^∞_i,j = 0 (∂P/∂u^α_ij)Dⁱ_x D^j_y(U^α), where u^α₀₀≡u^α, u^α_ij = ∂^i + ju^α/∂xⁱ∂y^j (cf e.g. [7]). In [6], P '[ U] is called a linearization and denoted by ℓ_P U.

Let U be a symmetry of the system F =  0, that is, let U satisfy F '[ U]  =  0 on the solution manifold of F =  0 [5, 6]. Consider a -valued solution S of the system

Assume that we have found n linear combinations of entries S_ij of S, α =  1,  ..., n, with the property that is another symmetry of F =  0. Then the linear operator defined by is a recursion operator for the system F =  0 in Guthrie's [8] sense. The coefficients a^α,ij may depend on λ, x, y and u and its derivatives.

However, testing the above scheme on a number of known examples such as the KdV or the Harry Dym equation shows that generates the nonlocal symmetries that belong to the `negative' part of the hierarchy of F =  0. Then we should, if possible, invert in order to obtain a `conventional' recursion operator which will generate the `positive' local part of the hierarchy in question. The inversion is an algorithmic process described in [8]. Note [9] that if the coefficients of the recursion operator are local, then so are the coefficients of its inverse.

Let us now apply this procedure to the stationary NVN equation

recently studied by Ferapontov [10] (see also Rogers and Schief [11]) in connection with isothermally asymptotic surfaces in projective differential geometry.

The stationary NVN equation is a reduction of the NVN equation [12, 13]

obtained upon assuming that u, v, w are independent of t. The latter is well known to be integrable via the inverse scattering transform, as it has the Lax pair

The NVN equation (3) is the first member of the hierarchy describing the deformations preserving the zero-energy level of the two-dimensional Schrödinger operator [13]. It also naturally arises in the theory of surfaces (see [11] and references therein) and its modified version appears in string theory [14, 15].

Upon setting [11] , where λ is a constant, the Lax pair (4) can be transformed into a zero-curvature representation for (2) of the form D_y (A)  - D_x(B)  +  [A, B]  =  0. This representation involves an essential parameter λ, and the matrices A and B belong to the semisimple Lie algebra sl₆ of traceless 6 × 6 matrices. They read

where A₄₁ =  - u_yy +  3wu, B₄₁ =  - u_xx +  3vu.

Let U =  (U, V, W)^T be a symmetry of (2), i.e. let U, V, W satisfy

Consider a traceless 6 × 6 matrix S that solves (1), where A, B are given by (5), and

are the directional derivatives of the matrices A and B (5) along the vector , .

The next step is to find linear combinations of entries of S that solve (6). A straightforward but tiresome computation shows that for A and B (5) these are , , , i.e. if U =  (U, V, W)^T is a symmetry of (2), then so is . Hence, the linear operator mapping U to is a recursion operator for (2).

However, the application of to the simplest symmetries of (2), e.g., to the zero symmetry, yields nonlocal symmetries of (2), so we should invert in order to obtain a recursion operator generating hierarchies of local symmetries for (2).

It turns out that our is invertible only for λ≠ 0. Inverting involves solving a system of algebraic and differential equations for the components of , which is a fairly tiresome but algorithmic process. For the sake of simplicity, we set all the integration constants to zero. Then , where id is the identity operator, is independent of λ and provides a conventional recursion operator for (2).

The action of on a symmetry U =  (U, V, W)^T of (2) is given by . Here is a general solution of the system

Note that this system is compatible if and only if U solves (6).

The operators and are of the form

Note that the above formula defines a recursion operator in the sense of Guthrie [8], and the system (7) defines a covering [6] over (6). Formally, we could express Z_i from (7) as Z₁ = D ^- 1_x U, Z₂ = D ^- 1_x V, etc, and thus write as an integro-differential operator as has become a tradition in the literature (see e.g. [3, 5, 7]). However, if we drop the y-part of (7), we encounter certain difficulties in constructing new symmetries (cf e.g. [8, 16, 17]).

As integrating (7) involves arbitrary constants, we have , where c_i are constants, and S₁,  ..., S₅ are symmetries of (2) of the following form:

The repeated application of to S₁,  ..., S₅ produces five hierarchies of symmetries of the stationary NVN equation (2), which can be visualized as follows (numbers in the top line denote the orders of symmetries):

We conjecture that all these symmetries are local and commute, as is the case for the symmetries of orders 1, 3, 5,  ..., 11.

Note that (2) has a scaling symmetry S = x u_x + y u_y +  2 u. The application of to S yields a nonlocal symmetry of seventh order, which we conjecture to be a master symmetry for (2), cf [18], meaning that commuting with any symmetry belonging to one of the five hierarchies, described above, yields (up to a constant multiplier) the next member of the same hierarchy. The repeated application of to S yields an infinite hierarchy of nonlocal symmetries for (2).

We believe that is hereditary in the sense of [19], but we have not yet checked this because of the huge amount of computations involved.

As a final remark, let us mention the nonstandard structure of nonlocal terms of in (7): they involve the derivatives of components of the symmetry, which is quite unusual (cf e.g. [20] for another example of this kind and [21] for a comprehensive list of integrable systems in (1  +  1) dimensions and their recursion operators known today).

Acknowledgments

We are grateful to Dr E V Ferapontov and Dr M V Pavlov for stimulating discussions. We are also pleased to thank the anonymous referee and Dr Tomáš Kopf for providing valuable advice on improving the presentation.

This research was supported by the Ministry of Education, Youth and Sports of the Czech Republic under grant MSM:J10/98:192400002 (MM and AS), and by the Czech Grant Agency under grant no 201/00/0724 (AS).

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