The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

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1 Nonlinear Mathematical Physics 997, V.4, N 3 4, The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation Juha POHJANPELTO Department of Mathematics, Oregon State University, Corvallis, Oregon , USA Abstract The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by its symmetry algebra originally posed by David, Levi, and Winternitz.. Introduction. Let E = (t, x, y, u) (t, x, y) be the bundle of the independent and dependent variables for the potential Kadomtsev-Petviashvili (PKP) equation u tx + 3 u xu xx + u xxxx s u yy = 0, where s = ±. In this paper, we compute the cohomology of the variational bicomplex and the associated Euler-Lagrange complex on E invariant under the Lie algebra Γ of point symmetries of the PKP equation. The algebra Γ is an infinite-dimensional infinitesimal Lie pseudo-group spanned by the vector fields X f = f t + 3 yf y + ( 3 xf 9 s y f ) x + ( 3 uf + 9 x f 4 7 s xy f + 4 ) 43 y4 f u, Y g = g y 3 s yg x + ( 4 9 s xyg y3 g ) u, (.) Z h = h x + ( 3 xh 4 9 s y h ) u, W k = yk u, and U l = l u, where f = f(t), g = g(t), h = h(t), k = k(t) and l = l(t) are arbitrary smooth functions of t. See [5]. Copyright c 997 by Mathematical Ukraina Publisher. All rights of reproduction in form reserved.

2 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 365 Our work is motivated by the following two considerations. Firstly, the cohomology of variational bicomplexes and the associated Euler-Lagrange complexes invariant under some transformation group naturally arise in a host of problems in differential geometry and mathematical physics; see, for example, [], []. It is therefore important to develop effective methods for computing the cohomology spaces of these complexes and to find new and useful applications for the cohomology classes. In [], Anderson and Pohjanpelto find that, under some mild regularity conditions the local cohomology of the Euler-Lagrange complex invariant under a finite-dimensional transformation group Γ can be computed from the Lie algebra cohomology of Γ. In [], it is discovered that this relationship continues to hold in a variety of examples involving infinite-dimensional transformation groups Γ provided that the Lie algebra cohomology is replaced by the Gelfand-Fuks cohomology of Γ relative to some subalgebra of Γ. Accordingly, the local cohomology of the Γ invariant Euler-Lagrange complex is isomorphic with the Gelfand-Fuks cohomology of Γ ; see [3]. In contrast, as the results of this paper show, the Euler-Lagrange complex of Γ invariant everywhere smooth forms provides an example where this simple correspondence fails to hold. In addition, it is presently not clear what the precise relationship, if any, between the two cohomology spaces is in this case. Secondly, the symmetry algebras of several other integrable equations in + dimensions possess a Kac-Moody-Virasoro structure similar to that of Γ ; see, for example, [4], [7], [8], [9]. With these examples at hand, it is reasonable to speculate that the symmetry algebras of integrable equations in + dimensions characterize the equations in some fashion. In fact, the formulation of virtually all physical models is based on symmetry considerations. In [6], David, Levi, and Winternitz study the characterization problem for the KP equation u tx + 3 uu xx + 3 u x + 4 u xxxx s u yy = 0 in terms of the differential invariants of its symmetry algebra Γ KP specifically, they classify all fourth-order equations involving only a first or second-order time derivative that are invariant under Γ KP. However, David, Levi, and Winternitz find that the algebra Γ KP admits 0 functionally independent differential invariants of the specified type rendering the sought after characterization unfeasible. One possible way to shorten the list of equations is to repeat the analysis in [6] for the potential Kadomtsev-Petviashvili equation and only consider those differential invariants of the symmetry algebra Γ that are the Euler-Lagrange expressions of some Lagrangian on E. Surprisingly, the source form = (u tx + 3 u xu xx + u xxxx + 34 ) s u yy dt dx dy du associated with the PKP equation is invariant under Γ, that is, the algebra Γ consists solely of distinguished symmetries of the PKP equation. However, even though the source form is the Euler-Lagrange expression of the Lagrangian ( λ = u tu x 4 u3 x + 8 u xx 3 ) 8 s u y dt dx dy,

3 366 J. POHJANPELTO one can prove without too much trouble that it is not the Euler-Lagrange expression of any Γ invariant Lagrangian. Thus, the source form generates nontrivial cohomology in the Γ invariant Euler-Lagrange complex. The results of this paper show that the PKP equation is essentially uniquely characterized by this property, that is, as an obstruction to constructing Γ invariant Lagrangians for Γ invariant differential equations satisfying the Helmholtz conditions. Our main result is the following. Theorem. i) The interior rows (Ω,s (J (E)), ), s, of the augmented Γ variational bicomplex are exact. invariant ii) The dimensions of the vertical cohomology spaces H r,s (Ω (J (E)), d V ) of the Γ invariant variational bicomplex are s s = s = 0 4 s = 0 0 s = r = 0 r = r = r = 3 iii) The dimensions of the cohomology spaces H r (E PKP (J (E))) of the Γ invariant Euler-Lagrange complex are r = r 7 dimh r (E PKP ) We will obtain explicit generators for all the cohomology classes in the course of the proof of Theorem.. Let uxxxx be the source form uxxxx = u xxxx dt dx dy du. As a Corollary, Theorem. yields the following characterization of the PKP equation. Corollary.. Let be a Γ invariant source form on E satisfying the Helmholtz conditions. Then there are constants c, c and a Γ invariant Lagrangian form λ such that = c + c uxxxx + E(λ). The rest of the paper is dedicated to the proof of Theorem.. In section, we review some basic definitions and results from the theory of variational bicomplexes and transformation groups needed in the proof. In section 3, we first describe a basis for the

4 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 367 Γ invariant variational bicomplex. Then with this description at hand, it is an easy task to complete the proof of Theorem... Preliminaries. In this section, we collect together some definitions and results from calculus of variations needed in the sequel. For more details and proofs, we refer to [], [0], [], [3]. Let π E = (t, x, y, u) (t, x, y) be the bundle of the independent and dependent variables for the PKP equation. Often we will also write x = t, x = x, x 3 = y. Let J (E) be the infinite jet bundle of local sections of E with coordinates (t, x, y, u, u t, u x,...). Let Ω (J (E)) be the de Rham complex on J (E), and let Ω p (J (E)) = Ω r,s (J (E)) p=r+s be the decomposition of the de Rham complex into forms of horizontal degree r and vertical degree s. A type (r, s) form ω Ω r,s (J (E)) is a sum of terms f(x i, u [k] )dx i dx ir θ x j x j t θ x j s x j sts, where f(x i, u [k] ) is a smooth function on J (E) and the forms θ x k x kv = du x k x kv u x k x kv x ldxl are the contact one forms on J (E). We write d = + d V for the induced decomposition of the exterior derivative into the horizontal and vertical derivatives. Let I : Ω 3,s (J (E)) Ω 3,s (J (E)), s, be the interior Euler operator given by I(ω) = [( ) ( s θ ω L Di u u x i Here L Di ) ( ) ] ω + L Di L Dj ω. (.) u x i x j stands for the Lie derivative with respect to the total vector field D i = x i + u x i u + u x i x j +. u x j Recall that I = 0. (.) The spaces F s (J (E)), s, of functional s forms are, by definition, the images of the spaces Ω 3,s (J (E)) under the mapping I. The vertical derivative d V induces differentials E Ω 3,0 (J (E)) F (J (E)) and δ V F s (J (E)) F s+ (J (E)), s, by E = I d V and δ V = I d V.

5 368 J. POHJANPELTO The elements of Ω 3,0 (J (E)) can be regarded as Lagrangians for variational problems on E and source forms, the elements of F (J (E)), as partial differential equations on E. The mapping E Ω 3,0 (J (E)) F (J (E)) is just a usual Euler-Lagrange operator while the differential δ V F (J (E)) F (J (E)) agrees with the Helmholtz operator of the inverse problem of calculus of variations. The infinitesimal transformation group Γ acts projectably on E; consequently, its prolongation prγ preserves the spaces Ω r,s (J (E)) and commutes with the operators, d V, and I. Recall that the prolongation of a vector field X = ξ i (x j ) x i + φ(xj, u) u on E is given by prx = totx + prxev, (.3) where totx = ξ i D i and prxev = D i D ip (φev), u p 0 x i x ip and where φev = φ ξ i u i. Also recall the commutation formula [D i, pry ] = 0, i =,, 3, (.4) where Y = φ(x j, u) / u is any evolutionary vector field. Let Ω r,s (J (E)) consist of all Γ invariant type (r, s) forms on J (E), that is, Ω r,s (J (E)) = { ω Ω r,s (J (E)) L prxω = 0 for all X Γ }. We define the spaces FP s KP (J (E)) of Γ invariant functional s forms in a similar fashion. We thus obtain the augmented Γ invariant variational bicomplex (Ω r,s (J (E)),, d V ) 0 0, 0 0, 0 R 0, d V d V d V d V δ V,, 3, I F 0 d V d V d V d V δ V,, 3, d V d V d V d V,0,0 3,0 I F 0

6 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 369 and the Γ invariant Euler-Lagrange complex E (J (E)) 0 R 0,0,0,0 3,0 E F δ V F δ V F 3 δ V. The vertical cohomology spaces H r,s (Ω (J (E)), d V ), r, s 0, of the Γ invariant variational bicomplex are, by definition, the quotient spaces H r,s (Ω (J (E)), d V ) = Ker {d V Ω r,s (J (E)) Ω r,s+ (J (E))} Im {d V Ω r,s (J (E)) Ω r,s (J (E))}. Here Ω 0, = {0}. The horizontal cohomology spaces Hr,s (Ω (J (E)), ) of the Γ invariant variational bicomplex and the cohomology spaces H r (E (J (E))) of the Γ invariant Euler-Lagrange complex are defined similarly. Note, in particular, that the elements of H 4 (E (J (E))) characterize those Γ invariant source forms on E which satisfy the Helmholtz conditions and consequently are the Euler-Lagrange expression of some Lagrangian form on E but which are not the Euler-Lagrange expression of any Γ invariant Lagrangian form on E. 3. Proof of Theorem. As is easily checked by a direct computation, generators (.) of the Lie algebra Γ satisfy the bracket relations X ˆf Yĝ Zĥ Wˆk Uˆl X f X (f ˆf f ˆf) Y (fĝ 3 f ĝ) Z (f ĥ 3 f ĥ) W (f ˆk +f ˆk) U (fˆl + 3 f ˆl) Y g 3 s Z (gĝ g ĝ) 4 9 s W (g ĥ +g ĥ g ĥ) U gˆk 0 Z h 3 U (hĥ h ĥ) 0 0 W k 0 0 U l 0 (3.) Let H Γ be the subalgebra generated by the vector fields H = {X f f = f(t) is a smooth function}. Note that, by virtue of the bracket relations (3.), a form ω Ω r,s (J (E)) that is invariant under the algebra H and under the single vector field Y = / y is necessarily invariant under the full algebra Γ. Next let R t, R x, and R y be the total differential operators R t = D t + 3 u xd x, R x = D x, R y = D y, (3.) and let S stand for the operator S = R t R x s R y R y. (3.3)

7 370 J. POHJANPELTO Proposition 3.. The total differential operators R t, R x, and R y and the operator S satisfy [ prx f, R t ] = f (t)r t 3 yf (t)r y, [ prx f, R x ] = 3 f (t)r x, [ prx f, R y ] = 3 f (t)r y s yf (t)r x, [ prx f, S] = 4 3 f (t)s (3.4) for all smooth f = f(t). Proof. By (3.), we have that [ prx f, R t ] = 3 prx f (u x )D x + [ prx f, D t ] + 3 u x[ prx f, D x ]. (3.5) We compute [ prx f, D t ] = [ totx f, D t ] = f (t)d t ( 3 yf (t)d y 3 xf (t) ) 9 s y f (t) D x,(3.6) where we used the prolongation formula (.3) and the commutation formula (.4). Also by (.) and (.3), Hence, prx f (u x ) = 3 f (t)u x + 9 xf (t) 4 7 s y f (t). 3 prx f (u x )D x + [ prx f, D t ] = f (t)(d t + u x D x ) 3 yf (t)d y. (3.7) We also have that [ prx f, D x ] = 3 f (t)d x. (3.8) Now substitute (3.7), (3.8) into (3.5) and simplify to arrive at the first equation in (3.4). The proof of the remaining identities in (3.4) is based on a similar computation and will be omitted. We let v tx = Su, v txx = R x v tx, v ty = R t R y u, v x k y l = u x k y l, k, l 0. (3.9) Let {σ t, σ x, σ y } be the horizontal frame dual to the total differential operators {R t, R x, R y }, σ t = dt, σ x = dx 3 v x dt, σ y = dy, (3.0) and define new contact forms by Θ t = L Rt θ + 3 v xxθ, Θ tx l = d V v tx l, l =,, Θ ty = d V v ty + 9 v xxθ y, Θ x l yy = θ x l yy s v xx θ x l+, l 0, (3.)

8 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 37 and Θ x l y m = θ x l ym, l, m 0; m. By (.) and (.3), we have that prx f (v xx ) = f (t)v xx + 9 f (t). (3.) Now a straightforward but tedious computation employing the commutation relations (3.4) establishes the following Lie derivative formulas: prx f v tx l = l f (t)v tx l, l =,, + m + prx f v x l ym = l f (t)v 3 x l y m + 4m 9 s yf (t)v x l+ y m, m = 0, ; l + m 3. (3.3) Furthermore, L Xf σ t = f (t)σ t, L Xf σ x = 3 f (t)σ x 4 9 s yf (t)σ y, L Xf σ y = 3 f (t)σ y + 3 yf (t)σ t, (3.4) and, on account of (3.), L Xf Θ t = 4 3 f (t)θ t 3 yf (t)θ y, L Xf Θ tx l = l f (t)θ tx l, l =,, ( ) 4 L Xf Θ ty = f (t)θ ty + yf (t) 9 s Θ tx Θ yy, + m + L Xf Θ x l ym = l f (t)θ 3 x l y m + 4m 9 s yf (t)θ x l+ y m, l 0, m = 0,,. (3.5) Note that all the Lie derivative formulas above only involve the terms f (t) and yf (t) and no other terms in f(t) or its derivatives. We say that a form ω Ω r,s (J (E)) has a weight w if L X3t ω = w ω. In this case, write W (ω) = w. For example, the scalar form ω = u tx has the weight W (ω ) = 5 while the type (, ) form ω = dx Θ t has the weight W (ω ) = 3. Lemma 3.. Let ω Ω r,s (J (E)) be Γ invariant. Then ω can be expressed in terms of exterior products of the forms σ t, σ x, σ y and Θ t k x l y m (3k + l + m 5) with coefficients which are polynomial in the variables v tx l (l =, ), v x l (l = 3, 4, 5), and v x l y (l =, 3).

9 37 J. POHJANPELTO Proof. Let ω Ω r,s (J (E)) be Γ invariant. Our first goal is to show that the components ω I I s of ω in the basis {dt, dx, dy, θ, θ t,...} are polynomials in the variables u t k x l y m. For this, let Φ ɛ J (E) J (E), ( ) Φ ɛ (t, x, y, u t k x l y m) = e 3ɛ t, e ɛ x, e ɛ y, e (3k+l+m+)ɛ u t k x l y m, (3.6) be the one-parameter group of transformations generated by the vector field prx 3t. Since the action of Φ ɛ leaves ω invariant, it follows from (3.6) that the components ω I I s satisfy ω I I s = e pɛ ω I I s Φ ɛ, (3.7) for some p R. Let k, l, m be some nonnegative integers, and apply a q-fold derivative with respect to the variable u t k x l y m to equation (3.7) to obtain q ω I I s u = q ω I Is t k x l y mq e[p q(3k+l+m+)]ɛ u Φ ɛ. (3.8) t k x l y mq Suppose that q > p, and let ɛ in (3.8). Recall that by the Γ invariance of ω the component functions ω I I s do not involve the independent variables t, x, y. Thus, we see that q ω I I s = 0. u t k x l ymq Hence, any q-fold, q > p, derivative with respect to the variable u t k x l y m vanishes identically. Since k, l, m are arbitrary nonnegative integers, it follows that the ω I I s must be polynomial in the derivative variables u t k x l y m. Next write ω = ω i i r dx i dx ir, where ω i i r Ω 0,s (J (E)). Each monomial dx i dx ir has a weight 0 W (dx i dx ir ) 6. Hence, we have that 6 W (ω i i r ) 0 for all i,..., i r. By (3.6), the weight of the derivative variable u t k x l y m is W (u t k x l ym) = 3k l m. Note, moreover, that the invariance of the form ω under the vector fields prw k and pru l simply means that the components of ω do not depend on the variables u t k or u t k y, where k 0. Therefore, by (3.9) (3.), we are able to express ω as a combination of the exterior products of the adapted horizontal forms σ t, σ x, σ y, the adapted contact forms Θ t k x l y m (3k + l + m 5), the variables v tx l (l =, ), the variables v x l+ ym (l + m 4), and the variable v yy.

10 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 373 We still need to show that the coefficients ˆω I I s of ω in the adapted frame σ t, σ x, σ y and Θ t k x l y m do not involve the variables v x, v xx, v xy, v yy, and v xyy. Write X f = 4 f (i) (t)x i, i=0 where X i, i = 0,..., 4, do not involve the function f(t) or its derivatives. By the prolongation formula (.3), prx f (v x l y m) = 4 i=0 f (i) prx i (v x l ym), l, m 0. Now prx 3 = 4 (xy 7 s v ) + y vx + xy vy + y vxy + x vyy + vxyy. (3.9) By (3.3) (3.5), the Lie derivatives of the forms σ t, σ x, σ y, Θ t k x l ym (3k + l + m 5), and the variables v tx l (l =, ) only involve the term f (t) or the term yf (t) and no other terms in f(t) or its derivatives. Since f(t) is an arbitrary function, the coefficient of f (t) in the expression for L prx f ω must vanish. It therefore follows that the components ˆω I I s are invariant under prx 3. Recall that the ˆω I I s do not involve the variables x, y. Hence, on account of the formula (3.9) for prx 3, we see that the ˆω I I s can not depend on the variables v x, v xy, v yy, and v xyy. We can similarly use (3.) to show that the ˆω I I s can not depend on v xx. This concludes the proof of the Lemma. We let = σ t Θ xx, = σ t Θ y 3 s σ y Θ x, α, = σ t Θ Θ x, α,0 = v tx σ t σ y, α,0 = v xxxx σ t σ y, α,0 3 = v xxx σ t σ x + v xxy σ t σ y, α, = v xxx σ t σ y Θ, α 3, = v tx σ t σ x σ y Θ, α 3, = v xxxx σ t σ x σ y Θ, α, = (σ t σ y Θ t 3 ) s σ t σ x Θ y σ x σ y Θ x Θ. (3.0) One can verify without difficulty that, on account of the Lie derivative formulas (3.3) (3.5) and the remark following the bracket relations (3.), the forms α i,j k above are all Γ invariant. Proposition 3.3. The spaces Ω r,s (J (E)) of Γ spanned by the following forms. invariant type (r, s) forms are

11 374 J. POHJANPELTO s s = s = 0 α, s = 0 α,, α,, a, d V α,, α, α,,, d V α,0, d V α,0 α, α, α,, (a, ) d V α,, ( ) α 3,, α3,, d V α,0 α,, d V α,0 s = 0 0 α,0, α,0, α,0 3 α,0, α,0 r = 0 r = r = r = 3 Proof. Due to Lemma 3. and the Lie derivative formulas (3.3) (3.5), the proof reduces to routine computations. We illustrate the details by finding a basis for Ω, (J (E)) assuming that bases for Ω, (J (E)) and Ω,0 (J (E)) have already been found. Let ω Ω, (J (E)), and write ω = ω σ t σ x + ω σ t σ y + ω 3 σ x σ y, (3.) where ω i Ω 0, (J (E)), i =,, 3. By the Γ invariance of ω, W (ω ) = 4, W (ω ) = 5, W (ω 3 ) = 3. It now follows from Lemma 3. that ω must be a linear combination of the forms Θ t, Θ xy, Θ xxx, that ω must be a linear combination of the forms v xxx Θ, Θ tx, Θ yy, Θ xxy, Θ xxxx, and that ω 3 must be a linear combination of the forms Θ y, Θ xx. Next note that = σ t (σ x Θ xxx + σ y Θ xxy ), = σ t σ x Θ xy 3 s σ t σ y d V α,0 = σ t σ y Θ tx, d V α,0 = σ t σ y Θ xxxx. (Θ tx + 34 s Θ yy ) 3 s σ x σ y Θ xx, (3.) are all Γ invariant type (, ) forms. Thus, by subtracting a suitable linear combination of the form α, and the forms in (3.) from ω, we can assume that there are constants so that b i j ω = b Θ t + b Θ xy, ω = b Θ yy + b Θ xxy, ω 3 = b 3Θ y.

12 THE COHOMOLOGY OF A VARIATIONAL BICOMPLEX 375 It remains to show that the invariance of ω in (3.) with ω, ω, ω 3 as above forces b j i = 0 for all i, j. But this is an immediate consequence of the Lie derivative formulas (3.3) (3.5). Proof of Theorem.. One can verify without difficulty that, on account of (.), each of the forms α 3,, α3,, α, α, and α, is fixed by the interior Euler operator I. Thus, F (J (E)) = R < α 3,, α3, >, FP 4 KP (J (E)) = R < α,, α, α, >, F s (J (E)) = < 0 >, s, 4. (3.3) Now the exactness of the augmented interior rows (Ω,s (J (E)), ), s, of the Γ invariant variational bicomplex immediately follows from Proposition 3.3. This proves (i). To prove (ii), we only need to verify that d V i = 0, i =,, d V α, = 0, d V α,0 3 =, d V α 3, = α,, d V α 3, = d V α, 3 s ( ). Then we can conclude with the help of Proposition 3.3 that the vertical cohomology spaces H r,s (Ω (J (E)), d V ) are generated by the following forms. s s = s = 0 α, α,, α, α,, α, α, α, ( ) s = 0, α, 0 s = r = 0 r = r = r = 3 Now (ii) follows. Finally, by (3.3), the differential δ reduces to the zero map when restricted to the spaces F s (J (E)), s. Thus, by Proposition 3.3 and by (3.3), H (E (J (E))) = R < [α,0 3 ] >, H 4 (E (J (E))) = R < [α 3, ], [α3, ] >, H 6 (E (J (E))) = R < [ α, ], [α, H s (E (J (E))) = < 0 >, s, 4, 6. α, ] >, (3.4) Now (iii) follows.

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