The Diffiety Institute Preprint Series

Preprint DIPS–4 March 21, 2002

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

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Scalar second order evolution equations possessing an irreducible sl_{2}-valued zero

curvature representation by

Michal Marvan

Scalar second order evolution equations possessing an irreducible sl_{2}-valued zero curvature representation

Michal Marvan

March 21, 2002

Mathematical Institute, Silesian University in Opava, Bezručovo nám. 13, 746 01 Opava, Czech Republic. E-mail: Michal.Marvan@math.slu.cz

Abstract

We find all scalar second order evolution equations possessing an sl_{2}-valued zero curvature representation that is not reducible to a proper subalgebra of sl_{2}. None of these zero-curvature representations admits a parameter.

For more than twenty years, researchers are being attracted by the problem of classification of nonlinear systems possessing a zero-curvature representation (ZCR). Efforts are focused on ZCR’s taking values in a non-solvable Lie algebra 𝔤 and depending on a nonremovable parameter, in expectation that they will be suitable for the Zakharov and Shabat [16] formulation of integrability (S-integrability) and hence admit soliton solutions. But even without parameter, a ZCR may be useful for construction of Bäcklund transformations, nonlocal symmetries and nonlocal conservation laws. The problem of finding a ZCR is then equivalent to that of computing finite-dimensional linear coverings in the sense of Krasil’shchik and Vinogradov [3], which are very often just realizations of the Wahlquist–Estabrook prolongation structures [15]. However, existing computational procedures are insufficient for solving general classification problems, unless in combination with methods based on different criteria of integrability. The most complete lists of integrable systems obtained so far resulted from the formal symmetry approach [6, 7, 10].

In this paper we apply the method of [4, 5] and find all second order scalar evolution equations

u_{t} = F (t, x, u, u_{x}, u_{xx})  (1)

possessing an sl_{2}-valued zero curvature representation that is irreducible in the sense of being not reducible to a proper subalgebra of sl_{2}. We arrive at a single previously unnoticed class of equations parametrized by a single function of the coordinates t, x. We also distinguish a particular subclass of equations that admit a single conservation law. None of the corresponding ZCR’s admits a substantial parameter, which is in accordance with the general belief that no second order scalar evolution equation is S-integrable. All the previously known ZCR’s [1, 8] for second order scalar evolution equations turn out to be reducible to a solvable subalgebra and as such fall outside our classification.

1 Preliminaries

Let E be a nonlinear partial differential equation on a number of functions in two independent variables t, x. Let 𝔤 be a non-solvable matrix Lie algebra. By a 𝔤-valued zero-curvature representation (ZCR) for E we mean a 𝔤-valued one-form α = A dx+B dt such that

dα = \frac{1}{2} [α, α]  (2)

holds as a consequence of E.

Let G be the connected and simply connected matrix Lie group associated with 𝔤. Then for an arbitrary G-valued function S, the form α^{S} = dSS^{−1} + SαS^{−1} is another ZCR, which is said to be gauge equivalent to the former. Gauge equivalent ZCR’s may be viewed as identical geometric objects (connections). A 𝔤-valued ZCR is said to be reducible if it is gauge equivalent to a ZCR taking values in a proper subalgebra of 𝔤; otherwise it is said to be irreducible.

Let us proceed to a description of the general algorithm of computing ZCR’s [4, 5] as we use it here. For simplicity we restrict ourselves to a single non-linear nth order evolution equation

u_{t} = F (t, x, u, u_{1}, …, u_{n}) = 0.  (3)

Here t, x are coordinates, u is a single field variable, and u_{1} = u_{x}, u_{2} = u_{xx}, etc., represent the derivatives. Let us consider the corresponding equation manifold E, that is, the infinite-dimensional space ℝ^{∞} endowed with coordinates t, x, u and u_{k}, 1 ≤ k. The vector fields D_{x} = ∂/∂x + u_{1}∂/∂u + ⋯ + u_{k+1} ∂/∂u_{k} + ⋯, D_{t} = ∂/∂t + F ∂/∂u + ⋯ + D_{x}^{k}F ∂/∂u_{k} + ⋯ defined on E generate a diffiety structure in the sense of [3] and encode all essential geometric information about the equation. In these terms, a ZCR for eq. (3) is a pair of 𝔤-valued functions A, B on E satisfying eq. (2), which may be written as

D_{t}A − D_{x}B + [A, B] = 0.  (4)

Let us introduce operators D̂_{I} acting on an arbitrary 𝔤-valued function C on E as follows:

D̂_{x}C = D_{x}C − [A, C], D̂_{t}C = D_{t}C − [B, C].  (5)

Operators D̂_{x}, D̂_{t} commute whenever (A, B) is a ZCR. We also set D̂_{i} = D̂_{x} ∘ ⋯ ∘ D̂_{x} (i times).

By [4] for every ZCR there exists a characteristic matrix R, which is a 𝔤-valued function defined on E (see also the independent work by Sakovich [11]). The following proposition is proved in [4, Prop. 2.7 and Prop. 3.9], see also [5, Prop. 2]

Proposition 1 (1) The characteristic matrix R for a ZCR of the evolution equation (3) satisfies

−D̂_{t}R = ∑_{i}(−D̂)_{i}(\frac{∂F}{∂u_{i}} R).  (6)

(2) Gauge-equivalent ZCR’s have conjugate characteristic matrices.

Using Proposition 1(2), one can restrict the gauge freedom by requiring that the characteristic matrix be in the Jordan normal form. The remaining gauge freedom then may be fixed by transforming A into its normal form with respect to the gauge action by the stabilizer S ⊂ G of R.

In the sequel we consider a ZCR A dx + B dt taking values in sl_{2}. We shall write the two sl_{2}-matrices as

A = (\begin{matrix}a_{1} & a_{2} \\ a_{3} & −a_{1} \end{matrix}), B = (\begin{matrix}b_{1} & b_{2} \\ b_{3} & −b_{1} \end{matrix}).  (7)

Reducible ZCR’s will be excluded from our classification. A subalgebra of sl_{2} to which the ZCR may be reduced is either an abelian algebra or the two-dimensional solvable subalgebra representable by lower triangular matrices. Obviously, a ZCR taking values in an abelian algebra is equivalent to one or more conservation laws. What concerns solvable algebras, the situation is not much different.

Definition 2 An sl_{2}-valued ZCR satisfying the condition a_{2} = b_{2} = 0 is said to be lower triangular.

For a lower triangular ZCR, it follows from eq. (4) that ϕ = a_{1} dx + b_{1} dt is a conservation law. Let h be the potential of ϕ; then, by the same eq. (4), ϕ′ = (a_{3} dx + b_{3} dt)e^{2h} is a conservation law nonlocal over the potential h. This situation will be referred to as a chain of conservation laws. Clearly, one can reconstruct the reducible ZCR from the corresponding chain (ϕ, ϕ′). In this way, reducible sl_{2}-valued ZCR’s are equivalent to certain chains of conservation laws. It also follows that methods to find them are to be sought among methods to compute nonlocal conservation laws.

Proposition 3 Let the matrices (7) form a ZCR for for the evolution equation (3). Suppose that a_{2} = 0. Then also b_{2} = 0 or the ZCR is gauge equivalent to zero.

Proof Let us denote by C the matrix (4) evaluated at a_{2} = 0. By assumption, C is zero on the equation manifold E. If b_{2} ≠ 0, then from the condition 0 = c_{2} = −D_{x}b_{2} + 2a_{1}b_{2} we compute that a_{1} = \frac{1}{2}D_{x}b_{2}/b_{2} on E, and then from the condition 0 = c_{1} = D_{t}a_{1} −D_{x}b_{1} −a_{3}b_{2} we compute that a_{3} = \frac{1}{2}D_{tx}b_{2}/b_{2}^{2} − \frac{1}{2}D_{t}b_{2}D_{x}b_{2}/b_{2}^{3} −D_{x}b_{1}/b_{2} on E. Let us introduce a function G by the requirement b_{3} = −b_{1}^{2}/b_{2} − D_{t}b_{1}/b_{2} + b_{1}D_{t}b_{2}/b_{2}^{2} + \frac{1}{2}D_{tt}b_{2}/b_{2}^{2} − \frac{3}{4}(D_{t}b_{2})^{2}/b_{2}^{3} +G/b_{2}. Then 0 = c_{3} = −D_{x}G/b_{2}, which in the case of an evolution equation implies that G is a function of t only. Then, under the above substitutions for a_{1}, a_{3}, b_{3}, the gauge matrix

(\begin{matrix}b_{2}^{−1/2} & 0 \\ − \frac{1}{2} D_{t}b_{2}b_{2}^{−3/2} + b_{1}b_{2}^{−1/2} & b_{2}^{1/2} \end{matrix})

sends A to zero and B to

(\begin{matrix}0 & 1 \\ G & 0 \end{matrix}),

which depends on t at most. The last matrix is sent to zero by gauge transformation with the gauge matrix composed of independent solutions of the equation s_{tt} = Gs.

2 The classification

Let us consider a second order evolution equation (1) along with the sl_{2}-matrices A, B satisfying eq. (4) but not reducible to a solvable subalgebra. We also assume that ∂F /∂u_{xx} ≠ 0. Following [5], we consider the two cases distinguished by their Segre characteristics separately.

2.1 The nilpotent case

Under the notation (7), the Jordan form for R corresponds to r_{1} = 0, r_{2} = 0, r_{3} = 1. The normal form for A, obtained in [5], is given by the single requirement a_{1} = 0. Indeed, one can set a_{1} = 0 in a general metrix A by means of the gauge matrix

(\begin{matrix}1 & 0 \\ a_{1}/a_{2} & 1 \end{matrix})

whenever a_{2} ≠ 0; otherwise the ZCR is either lower triangular or trivial by Proposition 3. The equation (6) then reduces to the system T_{i} = 0, i = 1, 2, 3, where

\begin{aligned}T_{1} & := & 2D_{x}(\frac{∂F}{∂u_{xx}} )a_{2} + \frac{∂F}{∂u_{xx}} D_{x}a_{2} − \frac{∂F}{∂u_{x}} a_{2} + b_{2}, \\ T_{2} & := & 2\frac{∂F}{∂u_{xx}} a_{2}^{2}, \\ T_{3} & := & −D_{xx} \frac{∂F}{∂u_{xx}} + D_{x} \frac{∂F}{∂u_{x}} − 2\frac{∂F}{∂u_{xx}} a_{2}a_{3} − \frac{∂F}{∂u} − 2b_{1}. \end{aligned}

Then a_{2} = 0 by the second equation and b_{2} = 0 by the first equation, whence the ZCR is lower triangular. Consequently, this case is void in our classification.

2.2 The semisimple case

It will be convenient to change the notation for sl_{2}-matrices to

A = (\begin{matrix}a_{1} & a_{2} + a_{3} \\ a_{2} − a_{3} & −a_{1} \end{matrix}).

The Jordan form for R has r_{2} = r_{3} = 0 with r := r_{1} arbitrary. Unlike in [5], we choose the normal form for A characterized by the single requirement a_{3} = 0. And indeed, whenever a_{2} + a_{3} ≠ 0, which is irrestrictive by Proposition 3, one can set a_{3} to zero by gauge transformation. The relevant gauge matrix is diagonal with the diagonal entries h and 1/h, where h = ((a_{2} − a_{3})/(a_{2} + a_{3}))^{1/4}.

Eq. (4) and (6) then assume the form S_{i} = 0 = T_{i}, i = 1, 2, 3, with

\begin{matrix}S_{1} = −D_{t}a_{1} + D_{x}b_{1} + 2a_{2}b_{3}, \\ S_{2} = −D_{t}a_{2} + D_{x}b_{2} − 2a_{1}b_{3}, \\ S_{3} = D_{x}b_{3} + 2a_{2}b_{1} − 2a_{1}b_{2}, \\ T_{1} = −D_{t}r − \frac{∂F}{∂u} r + D_{x}(\frac{∂F}{∂u_{x}} r) − D_{xx}(\frac{∂F}{∂u_{xx}} r) − 4\frac{∂F}{∂u_{xx}} ra_{2}^{2}, \\ T_{2} = −b_{3} + 2\frac{∂F}{∂u_{xx}} a_{1}a_{2}, \\ T_{3} = −b_{2} + \frac{∂F}{∂u_{x}} a_{2} − 2D_{x}(\frac{∂F}{∂u_{xx}} )a_{2} − 2\frac{D_{x}r}{r} \frac{∂F}{∂u_{xx}} a_{2} − \frac{∂F}{∂u_{xx}} D_{x}a_{2}. \end{matrix}  (8)

If a_{2} = 0, then we have b_{2} = b_{3} = 0 by the last two equations and the ZCR reduces to a single conservation law. Therefore, we assume that a_{2} ≠ 0 in the sequel.

Proposition 4 As solutions to eq. (8), functions r, a_{1}, a_{2}, b_{3} cannot depend on jet coordinates other than t, x, u, u_{x}, u_{xx}, whereas functions b_{1}, b_{2} cannot depend on jet coordinates other than t, x, u, u_{x}, u_{xx}, u_{xxx}.

Proof We may assume, without loss of generality, that the functions r, a_{1}, a_{2}, b_{3} depend on t, x, u, …, u_{k} and the functions b_{1}, b_{2} depend on t, x, u, …, u_{k}, u_{k+1} for some k ≥ 2. We perform a downward induction, each step of which consists in deriving appropriate differential consequences of the system (8). Thus, let k > 2. Then we have

0 = \frac{∂T_{1}}{∂u_{k+2}} = −2\frac{∂F}{∂u_{xx}} \frac{∂r}{∂u_{k}} ,

but ∂F /∂u_{xx} ≠ 0, whence r does not depend on u_{k}. Then, similarly,

\begin{matrix}0 = \frac{∂S_{2}}{∂u_{k+2}} − \frac{∂T_{3}}{∂u_{k+1}} = 2\frac{∂b_{2}}{∂u_{k+1}} \\ 0 = \frac{∂S_{2}}{∂u_{k+2}} + \frac{∂T_{3}}{∂u_{k+1}} = −2\frac{∂F}{∂u_{xx}} \frac{∂a_{2}}{∂u_{k}} , \end{matrix}

whence b_{2} does not depend on u_{k+1} and a_{2} does not depend on u_{k}. Finally,

0 = −2a_{2} \frac{∂S_{1}}{∂u_{k+2}} + \frac{∂S_{3}}{∂u_{k+1}} + \frac{∂T_{2}}{∂u_{k}} = 4a_{2} \frac{∂F}{∂u_{xx}} \frac{∂a_{1}}{∂u_{k}} ,

0 = 2a_{2} \frac{∂S_{1}}{∂u_{k+2}} + \frac{∂S_{3}}{∂u_{k+1}} + \frac{∂T_{2}}{∂u_{k}} = 4a_{2} \frac{∂b_{1}}{∂u_{k+1}} ,

0 = −2a_{2} \frac{∂S_{1}}{∂u_{k+2}} + \frac{∂S_{3}}{∂u_{k+1}} − \frac{∂T_{2}}{∂u_{k}} = 2\frac{∂b_{3}}{∂u_{k}} ,

whence a_{1}, b_{3} do not depend on u_{k} and b_{1} does not depend on u_{k+1} (recall that a_{2} ≠ 0). This completes the induction step.

Under the restrictions established in Proposition 4, the determining system (8) becomes an overdetermined system of partial differential equations. As such, it can be solved routinely, but its solution is troublesome even with the employment of software capable of automated deriving of differential consequences. The reason is that the class of second order evolution equations is invariant with respect to a large group of contact transformations x̄ x = x̄ x(t, x, u, u_{x}), ū u = ū u(t, x, u, u_{x}), t̄ t = t̄ t(t). Below we shall apply a series of suitably chosen contact transformations to achieve substantial reduction of the matrix A.

Proposition 5 For every second order evolution equation (1) possessing an irreducible sl_{2}-valued ZCR there exists a contact transformation such that the transformed a_{2} depends on t, x, u, u_{x} at most.

Proof Let functions r, a_{i}, b_{i} depend on t, x, u, u_{x}, u_{xx}, u_{xxx} as in Proposition 4. Taking successively the derivatives ∂S_{1}/∂u_{xxxx}, ∂S_{2}/∂u_{xxxx}, T_{2}, ∂T_{3}/∂u_{xxx}, ∂T_{1}/∂u_{xxxx}, ∂S_{3}/∂u_{xxx}, ∂T_{3}/∂u_{xxx}, ∂^{2}S_{1}/∂u_{xxx}^{2}, ∂^{2}S_{2}/∂u_{xxx}^{2} one may check routinely that

\frac{∂^{2}a_{2}}{∂u_{xx}^{2}} = 0 and a_{1} \frac{∂a_{2}}{∂u_{xx}} − \frac{∂a_{1}}{∂u_{xx}} a_{2} = 0,

are among differential consequences of the system (8). Hence,

(a) a_{2} is linear in u_{xx}, i.e., a_{2} = a_{21}(t, x, u, u_{x})u_{xx} + a_{20}(t, x, u, u_{x});

(b) the ratio a_{1}/a_{2} does not depend on u_{xx}.

Now, if a_{21} = 0, then the statement is proved. Otherwise, let f_{1}, f_{2} be two functionally independent solutions of the linear equation

−\frac{a_{20}}{a_{21}} \frac{∂f}{∂u_{x}} + u_{x} \frac{∂f}{∂u} + \frac{∂f}{∂x} = 0.  (9)

In particular, both f_{1} and f_{2} do depend on u_{x}. Then t̄ t = t, x̄ x = f_{1}, ū u = f_{2} and ū ux̄ ; x = (∂f_{2}/∂u_{x})/(∂f_{1}/∂u_{x}) satisfy the well-known necessary conditions of being a contact transformation:

\frac{\frac{∂ū u}{∂u_{x}}}{\frac{∂x̄ x}{∂u_{x}}} = ū ux̄ ; x = \frac{\frac{∂ū u}{∂x} + u_{x} \frac{∂ū u}{∂u}}{\frac{∂x̄ x}{∂x} + u_{x} \frac{∂x̄ x}{∂u}} .

Under this transformation, A dx+B dt becomes – A A dx̄ x+ – B B dt̄ t with dx̄ x = D_{x}x̄ x dx+D_{t}x̄ x dt, dt̄ t = dt, so that

A = – A A D_{x}x̄ x = (\frac{∂f_{2}}{∂x} + u_{x} \frac{∂f_{2}}{∂u} + u_{xx} \frac{∂f_{2}}{∂u_{x}} ) – A A = \frac{a_{2}}{a_{21}} \frac{∂f_{2}}{∂u_{x}} – A A,

where we have used eq. (9). Hence

– A A = \frac{a_{21}}{∂f_{2}/∂u_{x}} (\begin{matrix}a_{1}/a_{2} & 1 \\ 1 & −a_{1}/a_{2} \end{matrix}),

which is independent of u_{xx}, hence of ū u_{xx}, by virtue of statement (b) above.

Theorem 6 Every second order scalar evolution equation (1) possessing an irreducible sl_{2}-valued ZCR is transformable to an equation of the form

u_{t} = \frac{∂β}{∂x}u^{2}u_{xx} + 2\frac{∂^{2}β}{∂x^{2}} u^{2}u_{x} + 4βu_{x} + (\frac{∂^{3}β}{∂x^{3}} − 4\frac{∂β}{∂x})u^{3} − 4\frac{∂β}{∂x}u  (10)

through a contact transformation. Here β is an arbitrary function of t, x with ∂β/∂x ≠ 0. The ZCR is then A dx + B dt with

\begin{matrix}A = (\begin{matrix}\frac{1}{u} & 1 \\ 1 & −\frac{1}{u} \end{matrix}), \\ B = (\begin{matrix}−\frac{∂β}{∂x}u_{x} + 4\frac{β}{u} − \frac{∂^{2}β}{∂x^{2}} u & 4β + 2\frac{∂β}{∂x}u \\ 4β − 2\frac{∂β}{∂x}u & \frac{∂β}{∂x}u_{x} − 4\frac{β}{u} + \frac{∂^{2}β}{∂x^{2}} u \end{matrix}). \end{matrix}  (11)

Proof Following Proposition 5, we assume that the matrix A depends on t, x, u, u_{x} at most. One may check routinely that

\frac{∂^{2}a_{2}}{∂u_{x}^{2}} = 0 and \frac{∂^{2}a_{2}}{∂x ∂u_{x}} + u_{x} \frac{∂^{2}a_{2}}{∂u ∂u_{x}} − \frac{∂a_{2}}{∂u} = 0

are among differential consequences of the system (8). The general solution is a_{2} = ∂h/∂x + u_{x} ∂h/∂u = D_{x}h for a suitable function h(t, x, u). If a_{2} does depend on u_{x}, then ∂h/∂u ≠ 0, whence t̄ t = t, x̄ x = h, ū u = x is a point transformation. If a_{2} does not depend on u_{x}, then h does not depend on u, but does depend on x (otherwise a_{2} = 0), and t̄ t = t, x̄ x = h, ū u = u is a point transformation. In both cases A = – A A D_{x}h = – A Aa_{2}, whence ā a_{2} = 1 in the transformed matrix – A A.

With a_{2} = 1, one can check routinely that ∂a_{1}/∂u_{x} = 0 is among differential consequences of system (8). If moreover ∂a_{1}/∂u = 0, then A is completely independent of u and its derivatives, and then so is B, whence the ZCR is gauge equivalent to zero. Therefore, we shall continue with ∂a_{1}/∂u ≠ 0. Then we can apply a point transformation x̄ x̄ x = x̄ x, ū ū u = 1/a_{1}, which sends a_{1} to 1/ū ū u (this choice prevents terms quadratic in u_{x} from appearing on the right-hand side of eq. (1)). It is then a matter of routine to compute all possible forms of the right-hand side F of eq. (1) and also the corresponding matrices B.

There seem to be no earlier appearance of the class (10) in the literature, let alone its ‘simplest’ member u_{t} = u^{2}u_{xx} + 4xu_{x} − 4u^{3} − 4u.

The results would be incomplete if we do not establish irreducibility of the ZCR (11). Since reducibility implies existence of at least one local conservation law, we shall start the following result.

Proposition 7 Within the class (10), the only equations to possess a conservation law are those with

β = \frac{1}{8} \frac{p_{t}e^{2x} + q_{t}e^{−2x}}{pe^{2x} − qe^{−2x}} ,  (12)

where p, q are arbitrary functions of t such that (pq)_{t} ≠ 0. In all these cases the equation has a single conservation law

D_{t} \frac{pe^{2x} + qe^{−2x}}{u} = D_{x}(\frac{1}{2} \frac{(pq)_{t}(pe^{2x} + qe^{−2x})}{(pe^{2x} − qe^{−2x})^{2}}u_{x} + \frac{1}{2} \frac{(p_{t}e^{2x} + q_{t}e^{−2x})(pe^{2x} + qe^{−2x})}{(pe^{2x} − qe^{−2x})} \frac{1}{u} − \frac{(pq)_{t}(3p^{2}e^{4x} + 2pq + 3q^{2}e^{−4x})}{(pe^{2x} − qe^{−2x})^{3}}u).  (13)

Proof A routine computation shows that any characteristics ψ of a conservation law depends on t, x, u at most and satisfies the equations

\frac{∂^{2}ψ}{∂x^{2}} − 4ψ = 0, \frac{∂ψ}{∂u} + 2\frac{ψ}{u} = 0, \frac{∂ψ}{∂t} − 4β \frac{∂ψ}{∂x} = 0.

The rest is easy.

Another computation shows that for none of the equations of the class (12) the corresponding ZCR (11) can be reduced to the lower triangular form with multiples of (13) on the diagonal. Thus, the ZCR’s (11) are indeed irreducible.

Finally, a remark on equations determining pseudospherical surfaces (PSS equations) is due. In anticipation of finding new S-integrable nonlinear systems, a number of attempts have been made to classify equations describing pseudospherical surfaces (PSS equations), see [14] and references therein. Even though being a PSS equation is equivalent to possessing an sl_{2}-valued ZCR, the classification of second order scalar evolution PSS equations as obtained by Reyes [8] (see also [9, 2]) has no intersection with ours. This seeming paradox is easily resolved. In fact, each of the ZCR’s found by Reyes is reducible to the lower triangular form (the generalized Burgers equation) or even to a single conservation law (the other equations), which are disregarded in our classification. On the other side, equations (10) are not integrable, hence do not enter the classification of integrable equations by Svinolupov and Sokolov [12, 13], which was the starting point of the Reyes work.

Acknowledgements

The support from the grants MSM:J10/98:192400002 and GAČR 201/00/0724 is gratefully acknowledged.

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