Functional Analysis and Its Applications, Vol. 38, No. 2, pp. 138–148, 2004 Translated from Funktsional nyi Analiz i Ego Prilozheniya, Vol. 38, No. 2, pp. 71–83, 2004 c by I. T. Khabibullin and E. V. Gudkova Original Russian Text Copyright
Boundary Conditions for Multidimensional Integrable Equations ∗ I. T. Khabibullin and E. V. Gudkova Received November 11, 2002
Abstract. We suggest an efficient method for finding boundary conditions compatible with integrability for multidimensional integrable equations of Kadomtsev–Petviashvili type. It is observed in all known examples that imposing an integrable boundary condition at a point results in an additional involution for the t-operator of the Lax pair. The converse is also likely to be true: if constraints imposed on the coefficients of the t-operator of the L-A pair result in a broader group of involutions of the t-operator, then these constraints determine integrable boundary conditions. New examples of boundary conditions are found for the Kadomtsev–Petviashvili and modified Kadomtsev–Petviashvili equations. Key words: integrable equation, Hamiltonian structure, Kadomtsev–Petviashvili equation, Lax pair.
Sklyanin’s paper [1] has greatly stimulated the interest in boundary conditions compatible with integrability for nonlinear integrable equations. In the last fifteen years, efficient algorithms have been discovered for finding such boundary conditions for 1+1-dimensional equations (i.e., for equations with one spatial and one time variable) on the basis of conservation laws and the Hamiltonian structure [1], higher symmetries of the equation [2], or the Lax pair itself [3–5]. Although integrable boundary conditions are a rather narrow subclass of all admissible boundary conditions, they can be used to analyze the influence of arbitrary boundary conditions, say, by methods of perturbation theory. Note that the issue of boundary conditions compatible with integrability have been little explored for equations with two or more spatial variables. By way of example, one can mention only several types of boundary conditions found in [6] for the Ishimori equation and also the so-called generalized Toda lattices, that is, reductions of the infinite two-dimensionalized Toda lattice that are related to classical series of finite-growth Lie algebras (e.g., see the monograph [7]). The generalized lattices are obtained from the infinite lattice if one imposes truncation conditions (boundary conditions with respect to the discrete variable) compatible with integrability. The present paper has been inspired by a curious observation that if one subjects the infinite Toda lattice uxt (n) = exp{u(n − 1) − u(n)} − exp{u(n) − u(n + 1)}
(1)
to an integrable truncation condition of the form F (u(0), u(−1)) = 0, then the second equation of the Lax pair φ(n + 1) = (Dx + ux (n))φ(n),
(2)
φxt (n) = −ux (n)φt (n) − exp{u(n − 1) − u(n)}φ(n)
(3)
of this lattice acquires an additional involution at the point n = 0. Here Dx stands for the operator of differentiation with respect to x. Prior to giving the precise statement, note that Eq. (1) is invariant with respect to the transformation x ↔ t; hence not only system (2), (3) but also the ∗ This work was supported by the RFBR (under project No. 04-01-00190), by the Education Ministry of the Russian Federation (under grant No. E02-1.0-77), and by the Program of Supporting Leading Scientific Schools (under grant NSh-1716.2003.1).
138
0016–2663/04/3802–0138
c 2004 Plenum Publishing Corporation
system ψ(n + 1) = (Dt + ut (n))ψ(n),
(4)
ψxt (n) = −ut (n)ψx (n) − exp{u(n − 1) − u(n)}ψ(n)
(5)
is a Lax pair for the lattice (1). Moreover, these two pairs are not related by a similarity transformation. However, these pairs become similar after imposing additional constraints relating the coefficients of the operators at some given point. More precisely, the following assertion holds. Proposition 1. Suppose that there exists a differential operator of the form M = aDx2 +bDx +c such that, for n = 0, the function φ = M ψ is a solution of Eq. (3) whenever ψ is a solution of Eq. (5). Then the variables u(0) and u(−1) satisfy one of the following relations: eu(−1) = 0,
(6.1)
u(−1) = 0,
(6.2)
u(−1) = −u(0),
(6.3) −u(0)−u(−1)
ux (−1) = −ut (0)e
.
(6.4)
The corresponding operator M has the form M1 = a0 eu Dx2 + (b0 eu + a0 ux eu )Dx , u
M2 = a0 (e
u
Dx2
u
+ ux e Dx ),
(7.1) (7.2)
M3 = a0 e Dx ,
(7.3)
M4 = a0 (eu Dx2 + ux eu Dx + e−u ),
(7.4)
respectively, where a0 and b0 are arbitrary numerical parameters and u = u(0). Note that M1 is a linear combination of M2 and M3 . Scheme of the proof of Proposition 1. By differentiating the relation φ = M ψ with respect to x and t, we obtain (8) φxt = Mxt ψ + Mx ψt + Mt ψx + M ψxt . Let us express the variables φ, φx , φt , φxt , . . . in (8) in terms of the dynamical variables ψ, ψx , ψt , ψxx , ψtt , . . . with the use of (3) and the relation φ = M ψ. Now by matching the coefficients of these dynamical variables in (8) we obtain an overdetermined system of equations for the coefficients a, b, and c of the desired operator M . The solvability conditions for the resulting system are just the differential constraints between u(−1) and u(0) given in Proposition 1. One can readily see that relations (6) are none other than the well-known integrable truncation conditions for the Toda lattice. (The truncation condition corresponding to the Dn series is missing from the list, since it is associated with a higher-order operator M .) Hence Proposition 1 can serve as a basis for a preliminary test on the integrability of the boundary condition. Namely, we say that a boundary condition satisfies the test on the compatibility with an L-A pair if the respective operators of the nonsimilar Lax pairs become similar once the boundary condition is imposed. An analogous test for 1+1-dimensional integrable equations has a simple explanation: the presence of an additional discrete symmetry at the boundary point permits one to solve the Degasperis–Manakov–Santini equation [8] describing the time dynamics of the scattering matrix by reducing this equation to a matrix Riemann problem. For 1+2-dimensional equations, our approach enables one to classify integrable boundary conditions simply and efficiently by using only the LA pair of the equation. In what follows, we use this operator test to find boundary conditions for several popular multidimensional equations. In particular, we present new types of boundary conditions for the Kadomtsev–Petviashvili equation (KP), the modified Kadomtsev–Petviashvili equation (mKP), and the Veselov–Novikov equation in Sections 1, 5, and 6. We show that each of the three boundary conditions w|y=0 = 0 and (vx ± αw)|y=0 = 0 is compatible with the KP equation vτ + vxxx − 6vvx = −3α2 wy , wx = vy , where α has one of the two values α = i and α = 1. We show in Sec. 2 that these boundary conditions are compatible with all odd-order homogeneous 139
higher symmetries of the KP equation (see Proposition 3). In Secs. 3 and 4, we briefly discuss how the boundary conditions are related to higher symmetries and to the B¨ acklund transformation. 1. Boundary Conditions for the KP Equation The main model in this section is the KP-I equation vτ + vxxx − 6vvx = 3wy , w x = vy ,
(9)
which has applications in a number of physical problems, e.g., in plasma physics and in the theory of ocean waves [9]. (All formulas below can readily be recomputed for the KP-II equation, which is related to KP-I by the change of variables y → iy, w → −iw.) The Lax pair for Eq. (9) has the form [10–12] φxx = iφy + vφ,
(10)
φτ = −4φxxx + 6vφx + 3(vx + iw)φ.
(11)
The form of Eq. (9) is preserved under a simultaneous change of sign of the variables y and w. Hence the system ψxx = −iψy + vψ,
(12)
ψτ = −4ψxxx + 6vψx + 3(vx − iw)ψ
(13)
determines a Lax pair (for the KP equation) dual to (10), (11). One can readily see that Eqs. (11) and (13) are not related by a similarity transformation in general; in other words, one cannot relate these two equations by a substitution of the form φ = M ψ or ψ = M φ, where M is a differential operator. At the same time, the equations become similar if additional constraints are imposed on the coefficients. The following assertion holds. Proposition 2. Suppose that there exists a differential operator of the form M = aDx2 +bDx +c such that, for y = 0, the function φ = M ψ is a solution of Eq. (11) whenever ψ is a solution of Eq. (13). Then one of the following relations holds: w|y=0 = 0,
(14.1)
(vx − iw)|y=0 = 0,
(14.2)
(wτ − 2vxxy + 6ivyyx + 6vx w − 6vwx − 6iw2 − 12cvy )|y=0 = 0,
(14.3)
where the function c = c(x, τ ) is determined by the equation cx = (−vx + 2i w)|y=0 . The corresponding operator M has the form M = 1,
(15.1)
M = Dx ,
(15.2)
M = Dx2 + c,
(15.3)
respectively. Proof. We set A1 = −4Dx3 + 6vDx + 3(vx + iw) and A2 = −4Dx3 + 6vDx + 3(vx − iw) and rewrite Eqs. (11) and (13) in the form φτ = A1 φ,
ψτ = A2 ψ.
(16)
Next, we differentiate the relation φ = M ψ with respect to τ and use Eqs. (16) to rewrite the result in the form A1 M ψ = (Mτ + M A2 )ψ. Since ψ is arbitrary, we have the operator equation Mτ = A1 M − M A2 . 140
(17)
By matching the coefficients of like powers of Dx in (17), we obtain a strongly overdetermined system of equations for the unknown functions a, b, and c: Dx4 : ax = 0, Dx3 : bx = 0, Dx2 : aτ + 12cx + 6a(2vx − iw) = 0,
(18)
Dx : bτ + 12cxx + 6a(2vxx − iwx ) + 6b(vx − iw) = 0, Dx0 : cτ + 4cxxx − 6vcx + 3a(vxxx − iwxx ) + 3b(vxx − iwx ) − 6icw = 0. Let us differentiate the third equation in (18) with respect to x and subtract the result from the τ fourth equation. We obtain bτ + 6b(vx − iw) = 0. Now since b(τ ) = b(0) exp(−6 0 (vx − iw) dτ ), we have either (i) b ≡ 0 or (ii) b(τ ) = 0 for all τ . Moreover, i (19) cxx = −a vxx − wx . 2 Suppose that case (i) holds. Then the fifth equation in (18) becomes cτ + 4cxxx − 6vcx + 3a(vxxx − iwxx ) − 6icw = 0.
(20)
The function c = c(x, τ ) satisfies both equations (19) and (20) simultaneously. When verifying the compatibility of these equations, we should consider two cases, (j) a = 0 and (jj) a = 0. In the first case one has cx = 0 and cτ = 6iwc, and hence wx = 0. Here we arrive at the boundary condition w|y=0 = w0 (τ ), where w0 (τ ) is an arbitrary function, but the ambiguity can be eliminated by the transformation w → w − w0 (τ ), which does not change the KP equation. Hence the boundary condition can be reduced to the form (14.1). In the second case, a = 0, one can set a = 1, and then the boundary condition becomes wτ = 2vxxy − 6ivyyx − 6vx w + 6vwx + 6iw2 + 12cvy , where c is determined by the equation cx = −vx + 2i w; i.e., the boundary condition coincides with (14.3). In case (ii), we have 6(vxx − iwx ) = −(ln b)xτ = 0. Then the third and fourth equations in (18) acquire the form aτ = −12cx − 6avx and bτ = −12cxx − 6avxx , respectively. It follows that bτ = axτ = 0, but then the condition bτ + 6b(vx − iw) = 0 implies vx − iw = 0, which coincides with (14.2). Moreover, Eqs. (19) and (20) are compatible only if a = 0. Hence we have cx = 0 and cτ = 6iwc, where wx = 0 in general. Consequently, c = 0, and the operator M acquires the form (15.2). By interchanging Eqs. (11) and (13) (i.e., the functions ψ and φ), we obtain two more boundary conditions, (vx + iw)|y=0 = 0,
(14.4)
(wτ − 2vxxy − 6ivyyx − 6vx w + 6vwx − 6iw − 12cvy )|y=0 = 0, 2
(14.5)
where cx = vx + 2i w. Proposition 2 shows that the boundary conditions (14.1)–(14.5) imposed on the KP equation pass the preliminary operator test on the compatibility with integrability. Moreover, each of the boundary conditions (14.1), (14.2), and (14.4) has a simple statement, but the other two are cumbersome. It may happen that they are simplified under further reductions. In what follows, we restrict ourselves to conditions (14.1), (14.2), and (14.4) and show that they are compatible with ingredients of integrability like higher symmetries and the B¨ acklund transformation. The inverse scattering method (e.g., see [13]) can possibly be transferred to initial-boundary value problems with boundary conditions of the form (14.1)–(14.5); however, in this paper we do not consider the construction of solutions of the boundary value problem. 141
2. Master Symmetry and Boundary Conditions To describe the hierarchy of higher symmetries of an equation, it is convenient to use the notion of master symmetry. A master symmetry of an equation uτ = f (u, ux , uxx , . . . ) is an equation of the form ut = g(u, ux , uxx , . . . ) such that, for each symmetry uτ1 = f1 (u, ux , uxx , . . . ) of the former, ∂ f1 − ∂τ∂1 g, is also a symmetry. In particular, the equation uτ2 = f2 (u, ux , uxx , . . . ), where f2 = ∂t the equation (e.g., see [14]) vt1 = 12 yvy − 2yvvy − ywvx − 12 yDx−2 vyyy + 18 xv3
− 34 xvvx − 38 xDx−1 vyy + 14 ivy − 34 Dx−2 vyy − 14 vx Dx−1 v − v 2
(21)
is a master symmetry of the KP equation. The t1 -evolution of the eigenfunctions φ and ψ of Eqs. (10) and (12) is determined by the conditions φt1 = B1 φ,
ψt1 = B2 ψ,
(22)
where B1 = −iyDx4 + 12 xDx3 +(1+2iyv)Dx2 +(2iyv1 − 34 xv − 14 Dx−1 v −yDx−1 vy )Dx +iyv2 − 38 xv1 − 34 v − 1 3 1 1 x 3 1 −1 −2 −2 2 4 2 2 yvy − 8 ixDx vy − 4 iDx vy − 2 iyDx vyy − iyv and B2 = iyDx + 2 Dx + ( 2 − 2iyv)Dx + (−2iyv1 − 3 1 −1 3 1 1 3 1 1 −1 −1 −2 −2 2 4 xv − 4 Dx v − yDx vy )Dx − iyv2 − 8 xv1 − 4 v − 2 yvy + 8 ixDx vy + 4 iDx vy + 2 iyDx vyy + iyv . We introduce a dependence on higher times τn by the rule (n)
(n)
φτn = A1 φ, (0)
(0)
ψτn = A2 ψ,
(23)
(n)
where A1 = A1 , A2 = A2 , and the operators Aj , j = 1, 2, are defined for n > 0 recursively: (n+1)
Aj
=
∂ (n) ∂ (n) Aj − Bj + [Aj , Bj ]. ∂t1 ∂τn
(24)
One can readily see that the condition of compatibility of the first equation in (23) with Eq. (10) and the condition of compatibility of the second equation in (23) with Eq. (12) determine the same higher symmetry of odd order 2n + 3. Let us show that this symmetry is compatible with the boundary conditions (14.1), (14.2), and (14.4). Proposition 3. Let one of the following boundary conditions hold : w|y=0 = 0,
(25.1)
vx − iw|y=0 = 0,
(25.2)
vx + iw|y=0 = 0.
(25.3)
Then for y = 0 system (23) admits the respective reduction condition φ = ψ,
(26.1)
φ = Dx ψ,
(26.2)
ψ = Dx φ
(26.3)
for all n 0. In other words, the reduction condition φ = M ψ (or ψ = M φ) imposed at y = 0 is compatible with (23) for all n 0 if the corresponding boundary condition is satisfied (see Proposition 2). This means that the boundary conditions are compatible with all homogeneous odd-order higher symmetries. Proof of Proposition 3. To be definite, we consider the boundary condition (25.2). (The other cases can be treated in a similar way.) Then M = Dx . Moreover, it is easily seen that the relation φ = Dx ψ is compatible with system (22) for y = 0 and vx = iw. Indeed, B1 |y=0 = 12 xDx3 +Dx2 +Dx h and B2 |y=0 = 12 xDx3 + 12 Dx2 +hDx , where h = − 14 (3xv+D−1 v). It readily follows that B1 Dx = Dx B2 for y = 0. 142
Now let us prove Proposition 3 by induction on n. For n = 0, the assertion follows from (n) (n) Proposition 2. Suppose that the assertion is valid for some n, i.e., A1 Dx = Dx A2 . It follows from the recursion formula (24) that (n+1)
A1
(n+1)
Dx − Dx A2
=
∂ (n) ∂ (n) (n) (n) (A1 Dx − Dx A2 ) − (B1 Dx − Dx B2 ) + [A1 , B1 ]Dx − Dx [A2 , B2 ]. ∂t ∂τn (n)
(n)
By setting y = 0 and vx = iw in the resulting equation and by using the fact that A1 Dx = Dx A2 (n+1) (n+1) and B1 Dx = Dx B2 , we arrive at the relation A1 Dx = Dx A2 . The proof of Proposition 3 is complete. 3. The KP Equation and Truncations of the Toda Lattice Let a symmetry of the KP equation be given as the condition of compatibility of the linear equations (3) and (11). The variable n is a parameter in these equations; hence we can assume that n = 0. We introduce the new variables r = e−u(0) and s = eu(−1) to make formulas shorter. In terms of these variables, the desired symmetry has the form vt = −2(rs)x ,
iwt = 2(sxx r − srxx ).
(27)
Here the nonlocal variables r and s are determined by the conditions rτ = −4rxxx + 6vrx + 3(vx + iw)r,
(28)
sτ = −4sxxx + 6vsx + 3(vx − iw)s.
(29)
Note that the last two equations coincide with (11) and (13). Moreover, here one observes some kind of duality between the KP equation and the Toda lattice. Indeed, the same system (27)– (29) specifies a symmetry of the Toda lattice. More precisely, the symmetry itself is given by the pair of equations (28) and (29), while relations (27) are viewed as a method for introducing the nonlocal variables v and w (see [15]). This duality manifests itself amazingly at the level of boundary conditions. Namely, an integrable boundary condition for the KP equation is compatible with the symmetry (27) only if the lattice (1) is subjected to the corresponding (dual) truncation condition at the point n = 0. Let us discuss this fact in more detail by using the example of the boundary condition (14.2), for which the dual truncation condition for the Toda lattice has the form u(−1) = 0. We essentially claim that the phase flows given by (9) and (27) commute on the manifold S determined by the equation (vx − iw)|y=0 = 0 and its differential corollaries with respect to x and τ only if the additional condition s = eu(−1) = const is imposed. This additional condition is equivalent, modulo translations of the variable u(n) (which obviously do not affect the lattice itself), to the condition s = 1 or the condition u(−1) = 0. Indeed, by differentiating the boundary condition (vx − iw)|y=0 = 0 with respect to t, we obtain, in view of the symmetry (27), d (vx − iw)|y=0 = −4(rsx )x , (30) dt which shows that the condition (rsx )x = 0 is necessary for the flows to commute on S . The last relation is compatible with the τ -dynamics only if rsx = 0. If we additionally assume that r = 0, then from (27) we readily obtain superfluous constraints like vt = 0, wt = 0, etc. Hence we set s = 1. The differentiation of this equation with respect to t in accordance with (27) does not result in new constraints; in other words, the flows commute on S for s = 1. Likewise, one can readily see that the condition vx +iw|y=0 = 0 is compatible with the symmetry (27) only if u(0) = 0. Let us find the truncation condition dual to the boundary condition w|y=0 = 0. By differentiating the last formula with regard to (27), we see that wt = −2i(sxx r − srxx ) = 0. Now elementary computations give (r/s)|y=0 = const, which is equivalent to u(0) = −u(−1) modulo translations with respect to u. 143
4. The B¨ acklund Transformation and Boundary Conditions Recall that the KP equation admits a dressing transformation, which is defined as the linear transformation acting on the class of solutions of Eq. (10) by the rule φ1 = (Dx + β)φ,
(31)
where β = 12 Dx−1 (v1 − v) and
φ1xx = iφ1y + v1 φ1 . (32) The corresponding B¨ acklund transformation has the form of the nonlocal partial differential equation vx + v1x = (v − v1 )Dx−1 (v − v1 ) + iθDx−1 (vy − v1y ), (33) where v = v(x, y, τ ) and v1 = v1 (x, y, τ ) are solutions of the KP equation and θ = 1. The B¨ acklund transformation (33) with θ = −1 is determined by Eq. (31) with φ1 and φ replaced by the solutions ψ1 and ψ of the adjoint equations ψ1xx = −iψ1y + v1 ψ1 and ψxx = −iψy + vψ, respectively. The dressing transformation permits one to construct a new solution of the KP equation from a given solution v. Indeed, let φ be a solution of Eq. (10). We take β = −Dx ln φ and use the formula v1 = v + 2Dx β = v − 2Dx2 ln φ. acklund transformaNow suppose that solutions v and v1 of the KP equation are related by the B¨ tion (33) and satisfy the additional involution condition v1 (x, y, τ ) = v(x, −y, τ ). One can prove the existence of such solutions by using the fact that both the KP equation itself and its B¨ acklund transformation do not change if the sign of y is changed. By differentiating the involution condition, we arrive at the equations v1x (x, y, τ ) = vx (x, −y, τ ) and v1y (x, y, τ ) = −vy (x, −y, τ ). With regard to the relations obtained above, it follows from Eq.(33) for y = 0 that vx (x, 0, τ ) = −iθDx−1 vy (x, 0, τ ); this coincides with the boundary conditions (14.2) and (14.4), since θ = ±1. 5. The Modified KP Equation Let us find boundary conditions for the modified Kadomtsev–Petviashvili equation (mKP) vt + vxxx − 32 v 2 vx + 3σ 2 wy − 3σvx w = 0, w x = vy ,
(34)
where σ 2 = ±1. This equation appeared for the first time in [16, 17]. Equation (34) is a condition for the existence of a common solution of the following pair of linear equations: σφy + φxx + vφx = 0, φτ + 4φxxx + 6vφxx + (3vx − 3σw +
3 2 2 v )φx
(35) + δφ = 0,
(36)
where δ is an arbitrary constant. According to the test suggested in Sec. 1, one should find point discrete symmetries of Eq. (34). One can readily see that Eq. (34) is preserved under the substitution y → −y, v → −v. This substitution reduces the Lax pair (35), (36) to the form σψy − ψxx + vψx = 0, ψτ + 4ψxxx − 6vψxx − (3vx + 3σw −
3 2 2 v )ψx
(37) + δψ = 0.
(38)
Proposition 5. Suppose that there exists a differential operator M = αDx + β such that, for y = 0, the function φ = M ψ is a solution of Eq. (36) for every solution ψ of Eq. (38). Then the variables v and w are related by one of the formulas v|y=0 = 0,
(39.1)
(σwy + vvy + vxy )|y=0 = 0.
(39.2)
The corresponding operator M has the form M = 1, M = αDx , 144
(40.1) where αx = −vα,
(40.2)
respectively. If we interchange φ and ψ in Proposition 5, then we obtain another boundary condition (σwy + vvy − vxy )|y=0 = 0.
(39.3)
The relation ψ = αDx φ specifying the corresponding operator holds only if αx = vα. Proof of Proposition 5. Just as in the proof of Proposition 2, we introduce the differential operators A1 = −4Dx3 − 6vDx2 − bDx − δ and A2 = −4Dx3 + 6vDx2 − kDx − δ, where b = 3vx − 3σw + 32 v 2 and k = −3vx − 3σw + 32 v 2 . Then Eqs. (36) and (38) become φτ = A1 φ and ψτ = A2 ψ, respectively. The desired operator satisfies the relation Mτ = A1 M − M A2 , which is equivalent to the overdetermined system of differential equations αx + αv = 0, αxx + αx v + αvx + βx + βv = 0, αxxx + 32 vαxx + 14 bαx + 3βxx + 3vβx + 32 βvx − 14 kx α = − 14 αt ,
(41)
βxxx + 32 vβxx + 14 bβx = − 14 βt . It follows from the first two equations that βx + βv = 0. Hence one can eliminate the higher derivatives of α and β from the next two equations and rewrite them in the form βt = W β,
(42)
αt = (W − kx )α + 6vx β,
(43)
where W = 4vxx − 6vvx − 2v 3 + vb. The condition of compatibility of Eq. (42) with the equation βx + βv = 0 has the form (Wx + vt )β = 0. Suppose that the first factor is zero: vt = −Wx . By comparing this formula with the mKP equation, we see that v = 0. Now we have two equations, αx = 0 and αt = −3σvy α, for the function α, whence it follows that α = 0, since vxy = 0. Let the second factor be zero: β = 0. Then the condition of compatibility of Eq. (43) with the equation αx + αv = 0 has the form (Wx − kxx + vt )α = 0. The case α = 0 results in the trivial answer M = 0. Hence we equate the first factor with zero: Wx − kxx + vt = 0. This equation can be rewritten as 3 (−vt − vxxx + v 2 vx + 3σvvy + 3σvx w + 3σvxy )|y=0 = 0. (44) 2 By comparing Eq. (44) with the original equation (34), we see that condition (39.2) is valid. 6. The Veselov–Novikov Equation To illustrate the test, we consider yet another example, the Veselov–Novikov equation, which arises as a 1+2-dimensional generalization of the Korteweg–de Vries equation [18]. It has the form ut = uxxx + (cu)x ,
cy = −3ux
(45)
and is a compatibility condition for the system of linear differential equations (e.g., see [14]) φy = ψ, φt = φxxx + cφx ,
ψx = uφ,
ψt = uφxx − ux φx + (uxx + cu)φ.
(46) (47)
The involution y → −y, u → −u does not change the Veselov–Novikov equation but affects the Lax pair: ¯ ¯ ψ¯x = −uφ, (48) φ¯y = −ψ, ¯ φ¯t = φ¯xxx + cφ¯x , ψ¯t = −uφ¯xx + ux φ¯x − (uxx + cu)φ. (49) Suppose that Eqs. (47) and (49) are related for y = 0 by a linear transformation of the form φ φ¯ =M ¯ , (50) ψ ψ 145
where M = αDx + β is a first-order differential operator and the coefficients α and β are matrix functions of x and t. Then the condition Mt = V M − M V¯ should be satisfied, where 0 Dx3 + cDx , V = uDx2 − ux Dx + uxx + cu 0
V¯ =
(51)
0 Dx3 + cDx . −uDx2 + ux Dx − uxx − cu 0
An elementary analysis of Eq. (51) shows that, first, the equation has the two-parameter family of solutions 0 0 0 β11 Dx + M= α22 u −β11 0 α22 without any restrictions on the coefficients of the operators V and V¯ . Second, a necessary and sufficient condition for the existence of a three-parameter family of solutions β11 0 0 0 Dx + M= 0 β22 0 α22 is the constraint u = 0. Here α22 , β11 , and β22 are arbitrary constants. Consequently, the boundary condition u|y=0 = 0 is compatible with the L-A pair of the Veselov–Novikov equation. 7. Conclusion In the present paper, we do not analyze the question as to how the resulting boundary value problems can be treated in the framework of the inverse scattering method. In what follows, we give some reasoning suggesting that the inverse scattering method can be adapted to these problems in principle. First, all known integrable truncation conditions, and only these conditions, pass the operator test in the case of the two-dimensionalized Toda lattice; these conditions are a fortiori compatible with the inverse scattering method. Second, the boundary conditions found above for the KP equation pass also the well-approved symmetry test (see Proposition 3 and also Section 3), which is already less formal. The compatibility of a boundary condition with a higher symmetry essentially means that the corresponding boundary value problem admits closed-form particular solutions that are simultaneously stationary solutions of the symmetry. Indeed, the class of soliton-like solutions of the KP-II equation contains a wide subclass of solutions satisfying the condition (vx ± w)|y=0 = 0. Third, the operator test is a straightforward generalization of the results concerning 1+1dimensional equations in [4, 5]. It was discovered in [5] at the level of examples involving the Korteweg–de Vries equation, the sine-Gordon equation, and the Harry Dym equation that, once an equation is supplemented by an integrable boundary condition, the other equation of the Lax pair acquires an additional discrete symmetry. This fact was used as a basis of the integrability test. Let us briefly explain the essence of the test for several examples. We start from the Korteweg–de Vries equation ut = uxxx − 6uux , which is compatible with the boundary conditions u|x=0 = a and uxx |x=0 = b, where a and b are arbitrary real numbers. The Korteweg–de Vries equation is a condition for the compatibility of two families of linear differential equations “indexed” by a spectral parameter λ ∈ C: −4λ − 2u 0 1 ux Y. Y, Yt = Yx = −ux uxx − (4λ + 2u)(u − λ) u−λ 0 Along the line x = 0, the second system becomes ux −4λ − 2a Y (λ, t). Yt (λ, t) = b − (4λ + 2a)(a − λ) −ux 146
(52)
It admits the involution λ → h(λ), Y (λ, t) → F (λ, t)Y (h(λ), t). In other √ words, the function Y1 (λ, t) = F (λ, t)Y (h(λ), t) is also a solution of system (52) if h(λ) = (−λ+ c − 3λ2 )/2, c = 3a2 −b (see [5]), and 2λ + a 0 F (λ, t) ≡ F (λ) = . 0 2h + a The appearance of an additional involution permits one to linearize the evolution equation for the scattering matrix (see [19]) and hence “integrate” the corresponding initial-boundary value problem. Now consider the Harry Dym equation ut + u3 uxxx = 0 with Lax pair 0 1 Y (x, t, λ), Yx (x, t, λ) = −λ/u2 0 −2u ux Y (x, t, λ). Yt (x, t, λ) = −2λ uxx + 2λ/u −ux The involution corresponding to the integrable boundary conditions u(0, t) = 0, ux (0, t) = b, b = const, has the form λ → h(λ), Y (0, t, λ) → F (λ, t)Y (0, t, h(λ)), where h(λ) = λ and the conjugation matrix is 0 0 , α = const . F = I + αe4λbt 1 0 For another type of integrable boundary conditions, d a2 u+ , a, d ∈ R, 2 u the corresponding involution has the form given above with parameters √ −d − 2λ ± d2 − 4λd − 12λ2 λ 0 , h= F = 1 . 4 2 a(λ − h(λ)) h(λ) ux (0, t) = au,
uxx (0, t) =
For the zero (a fortiori integrable) boundary conditions u(0, t) = 0, v(0, t) = 0 for the system iut = −uxx + 2vu2 ,
−ivt = −vxx + 2uv 2 ,
which can be reduced to a nonlinear Schr¨ odinger equation, the second equation of the Lax pair iλ u Yx (x, t, λ) = Y (x, t, λ), v −iλ −2iλ2 − iuv −2λu + iux Y (x, t, λ) Yt (x, t, λ) = −2λv − ivx 2iλ2 + iuv acquires the form
Yt (x, t, λ) =
−2iλ2 iux Y (x, t, λ) −ivx 2iλ2
and obviously admits the involution λ → −λ. The Lax pair for 2+1-dimensional equations does not contain a spectral parameter. More precisely, one can assume that the spectral parameter takes finitely many values. Indeed, each of the models considered above has exactly two nonequivalent (not related by a gauge transformation) Lax pairs. Hence the involution λ → h(λ) in the 2+1-dimensional case is none other than the passage to a nonequivalent L-A pair. The conjugation matrix F (λ) is replaced by a differential operator M = M (Dx ). We have restricted ourselves to conjugation operators M = M (Dx ) of order 2 in the examples. The computational difficulties rapidly increase with the order of the operator, and nonlocal variables occur in the boundary conditions. The authors are grateful to R. I. Yamilov for useful discussions. 147
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[email protected] Ufa State Oil Technical University
Translated by V. E. Nazaikinskii 148
