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B¨acklund transformations for nonlinear evolution equations:

arXiv:hep-th/9212031v1 4 Dec 1992B¨acklund transformations for nonlinear evolution equations:Hilbert space approachKrzysztof KowalskiDepartment of Biophysics, Institute of Physiology and Biochemistry, Medical Schoolof L´od´z, 3 Lindley St., 90-131 L´od´z, PolandAbstractA new method of determining B¨acklund transformations for nonlinearpartial differential equations of the evolution type is introduced. Using theHilbert space approach the problem of finding B¨acklund transformations isbrought down to the solution of an abstract equation in Hilbert space.Key words:nonlinear dynamical systems, partial differential equations, quan-tum field theory, coherent statesPACS numbers:0.90, 03.40K, 03.50, 03.70

21. IntroductionIn recent years, the interest in the B¨acklund transformations is steadily increasingin connection with a general increase in the understanding of methods for solutionof nonlinear partial differential equations.

Let us only recall the rˆole played by theMiura transformation in discovery of the method of inverse scattering [1].In the recent paper [2] we introduced a new method for finding linearizationtransformations for nonlinear partial differential equations of the evolution typebased on the Hilbert space approach to the theory of nonlinear dynamical systemsdeveloped by the author [2-7]. The theory was illustrated by an example of theBurgers equation (we obtained in a simple way the celebrated Hopf-Cole transfor-mation).

The present work is devoted to a generalization of the treatment to thecase involving general B¨acklund transformations. Proceeding analogously as in thepaper [2] we demonstrate that the problem of finding B¨acklund transformations forevolution equations can be reduced to the solution of an abstract equation in Hilbertspace.

We illustrate the algorithm on the example of the Miura transformation.2. B¨acklund transformationsWe begin with a brief account of the Hilbert space description of nonlinear partialdifferential equations of the evolution type [4].

Consider the equation∂tu(x, t) = F(u, Dαu),u(x, 0) = u0(x),(1)where u : Rs × R →R, Dβ = ∂|β|/∂xβ11 . .

. ∂xβss , |β| =sPi=1 βi, F is analytic in u,Dαu and u0 ∈L2R(Rs, dsx) (real Hilbert space of square-integrable functions).

3Let |u⟩be a normalized functional coherent state (see appendix), where u satisfies(1) and we assume that u is square-integrable. We define the vectors |u, t⟩as follows|u, t⟩= exp12Zdsxu2 −Zdsxu20|u⟩.

(2)Suppose now we are given a boson operator of the formM =Zdsxa†(x)F(a(x), Dαa(x)),(3)where a†(x) and a(x) are the standard Bose field operators.An easy differentiation shows that the vectors (2) satisfy the following linear evolu-tion equation in Hilbert spaceddt|u, t⟩= M|u, t⟩,|u, 0⟩= |u0⟩. (4)Taking into account (2) we find that the following eigenvalue equation holds truea(x)|u0, t⟩= u[u0|x, t]|u0, t⟩,(5)where |u0, t⟩is the solution of (4) and u[u0|x, t] is the solution of (1) (the squarebrackets designate the functional dependence of u on u0).It thus appears that the nonlinear equation (1) can be brought down to the linearabstract Schr¨odinger-like equation (4).

The restriction to square integrable data isnot too serious. Indeed, the approach was shown to work also in the case when theinitial data were not square integrable [2,4].

The postulate that the solutions aresquare integrable at any time is rather restrictive one. It should be noted, however,that there exist numerous equations of classical and of current interest satisfying

4this requirement. Examples include Korteweg-de Vries equation, Burgers equation,nonlinear Schr¨odinger equation and Kadomtsev-Petviashvili equation.

We shouldalso mention that the treatment can be immediately extended to the case of complexmultidimensional systems of partial differential equations (1) with a right-hand sidedependent on x, t [4].We now discuss the transformation of variables within the Hilbert space ap-proach. Consider the following transformationu′ = φ[u|x],(6)where φ is analytic in u.Taking into account (2) we find that under (6) the “Hamiltonian” M transforms asM′ =Zdsxa†(x)[φ[a|x], M].

(7)Therefore, whenever the transformation (6) converts the equation (1) into the equa-tion∂tu′ = F ′(u′, Dβu′)(8)then the following commutation relation holds[φ[a|x], M] = F ′(φ[a|x], Dβφ[a|x]). (9)On taking the Hermitian conjugate of (9) and using (B.7) we arrive at the followingequationM†|φ(x)⟩= F ′(φ[a†|x], Dβφ[a†|x])|0⟩,(10)where |φ(x)⟩= φ[a†|x]|0⟩.

5The vector |φ(x)⟩is related to the B¨acklund transformation (6) byφ[u|x] = ⟨u|φ(x)⟩exp12Zdsxu2. (11)It thus appears that the problem of determining B¨acklund transformation (6) isequivalent to solving Hilbert space equation (10).

The particular caseF ′(u′, Dβu′) = Lu′,(12)where L is a linear differential operator,when (6) is the linearization transformation and (10) takes the formM†|φ(x)⟩= L|φ(x)⟩(13)was discussed in ref. 2.

Solving (13) we obtained in a simple way the celebratedHopf-Cole transformation reducing the Burgers equation to the heat equation. Wenote that the case of the linearization transformation is the only one when (10) islinear.

Nevertheless, it appears that there exist nontrivial cases when the B¨acklundtransformations can be determined easily by solving (10). We now illustrate thisobservation by the example of the Miura transformation.

6Example.Consider the modified Korteweg-de Vries equation∂tu = −∂3xu + 6u2∂xu(14)and the Korteweg-de Vries equation∂tu′ = −∂3xu′ + 6u′∂xu′. (15)We seek for the B¨acklund transformation φ such thatu′ = φ[u|x].(16)Eq.

[10] corresponding to (16) can be written asM†|φ(x)⟩= −∂3x|φ(x)⟩+ 6φ[a†|x]∂x|φ(x)⟩,(17)where the conjugation M† of the “Hamiltonian” corresponding to (14) isM† =Zdx(−a†′′′(x) + 6a†2(x)a†′(x))a(x). (18)On writing (18) in the coordinate representation (see appendix A) we obtain∂3x1φ1(x; x1) = −∂3xφ1(x; x1),(19a)(∂3x1 + ∂3x2)φ2(x; x1, x2) = −∂3xφ2(x; x1, x2)+ 6φ1(x; x2)∂xφ1(x; x1) + 6φ1(x; x1)∂xφ1(x; x2),(19b)

7n+2Xi=1∂3xiφn+2(x; x1, . .

. , xn+2)−12n+2Xi=1Xr,s̸=ir>s∂xi[δ(xi −xr)δ(xi −xs)φn(x; x1, .

. .

, ˇxr, . .

. , ˇxs, .

. .

, xn+2)]= −∂3xφn+2(x; x1, . .

. , xn+2)+ 6n+1Xr=11r!n+2Xi1=1Xi2̸=i1.

. .Xir̸=ir−1φn+2−r(x; x1, .

. .

, ˇxi1, . .

. , ˇxir, .

. .

, xn+2)×∂xφr(x; xi1, . .

. , xir),n = 1, 2, .

. .

, ∞,(19c)where φn(x; x1, . .

. , xn) = ⟨x1, .

. .

, xn|φ(x)⟩and the reversed hat over xr, xs andxi1, xir denotes that these variables should be omitted from the set {x1, . .

. , xn+2}.Hence, passing to the Fourier transformation we get(k3 + k31)˜φ1(k; k1) = 0,(20a)(k3 + k31 + k32)˜φ2(k; k1, k2) = −6kZdk′ ˜φ1(k −k′; k1)˜φ1(k′, k2),(20b) k3 +n+2Xi=1k3i!˜φn+2(k; k1, .

. .

, kn+2)+ 12n+2Xi=1Xr,s̸=ir>ski ˜φn(k; k1, . .

. , ˇkr, .

. .

, ˇks, . .

. , kn+2)|ki→ki+kr+ks= −6n+1Xr=11r!n+2Xi1=1Xi2̸=i1.

. .Xir̸=ir−1Zdk′ ˜φn+2−r(k −k′; k1, .

. .

, ˇki1, . .

. , ˇkir, .

. .

, kn+2)×k′ ˜φr(k′; ki1, . .

. , kir),n = 1, 2, .

. .

, ∞. (20c)

8Making use of the identities(k3 + k31)δ(k + k1) = 0,(21a)nXi=0k3i δ nXi=0ki!= 3Xq>r>skqkrksδ nXi=0ki!,(21b)where q, r, s ∈{0, 1, . .

. , n}, n ≥2 and we set k0 = k, one finds easily the followingsolution to (20)˜φ1 = −ikδ(k + k1),˜φ2 = 2δ(k + k1 + k2),˜φn = 0,n ≥3.

(22)On performing Fourier’s inverse transformations to (22) and using|φ(x)⟩=∞Xn=11n!Zdx1 . .

. dxnφn(x; x1, .

. .

, xn)|x1, . .

. , xn⟩,(23)we obtain the solution to (17) of the form|φ(x)⟩= |xx⟩+ ∂x|x⟩.

(24)Hence taking into account (11) and (B.5) we finally arrive at the desired B¨acklundtransformation (16) such thatu′ = u2 + ∂xu. (25)The mapping (25) coincides with the celebrated Miura transformation relating so-lutions of the modified Korteweg-de Vries equation to solutions of the Korteweg-deVries equation.

9ConclusionApplying the Hilbert space appproach to the theory of nonlinear dynamical sys-tems developed by the author a new method is introduced in this work of findingB¨acklund transformations for nonlinear evolution equations.It should be notedthat regardless of the form of eqs. (20b) and (20c) we have rederived the Miuratransformation from (20) in purely algebraic mannner (we need not have solved anyintegral equation).

The algorithm described herein is an example of the followinggeneral technique of the study of nonlinear partial differential equations based onthe Hilbert space formalism. Namely, using the Hilbert space approach we first de-rive an abstract equation in Hilbert space corresponding to the considered nonlinearevolution problem.

Then writing this equation in the coordinate representation andperforming a Fourier transformation we obtain a system of algebraic equations re-lated to the original problem. This technique was succesfully applied for finding firstintegrals [4,6] and linearization transformations [2] for nonlinear partial differentialequations of the evolution type.

The simplicity of the algorithm for determiningB¨acklund transformations described in this work suggests that it would also be auseful tool in the study of nonlinear evolution equations.AcknowledgementsThis work was supported by KBN grant 2 0903 91 01.

10Appendix A. Coordinate representationWe first recall the basic properties of the coordinate representation.

The Bosecreation (a†(x)) and annihilation (a(x)) operators obey the canonical commutationrelations[a(x), a†(x′)] = δ(x −x′),(A.1)[a(x), a(x′)] = [a†(x), a†(x′)] = 0,x, x′ ∈Rs.Let us assume that there exists in a Hilbert space of states H where act Boseoperators a unique normalized vector |0⟩(vacuum vector) such thata(x)|0⟩= 0,for every x ∈Rs. (A.2)We also assume that there is no nontrivial closed subspace of H which is invariantunder the action of the operators a(x), a†(x′).

The state vectors defined as|x1, . .

. , xn⟩= nYi=1a†(xi)!|0⟩,xi ∈Rs(A.3)satisfy the following orthogonality relation⟨x1, .

. .

, xn|x′1, . .

. , x′m⟩= δnmXσnYi=1δ(xi −x′σ(i)),(A.4)where σ is a permutation of the set {1, .

. .

, n}, and completeness relationXn1n!Zdsx1 . .

. dsxn|x1, .

. .

, xn⟩⟨x1, . .

. , xn| = I.

(A.5)The vectors |x1, . .

. , xn⟩form the basis of the coordinate representation.

The Boseoperators act on the basis vectors as follows

11a(x)|x1, . .

. , xn⟩=nXi=1δ(x −xi)|x1, .

. .

, ˇxi, . .

. , xn⟩,(A.6)a†(x)|x1, .

. .

, xn⟩= |x1, . .

. , xn, x⟩,where the reversed hat over xi denotes that this variable should be omitted fromthe set {x1, .

. .

, xn}.Appendix B. Functional coherent states representationWe now recall the basic properties of the functional coherent states. Considerthe functional coherent states |u⟩, where u ∈L2(Rs, dsx) (the complex Hilbert spaceof square-integrable functions).

The functional coherent states can be defined as theeigenvectors of the Bose annihilation operatorsa(x)|u⟩= u(x)|u⟩. (B.1)The normalized functional coherent states can be defined as|u⟩= exp−12Zdsx|u|2expZdsxu(x)a†(x)|0⟩.

(B.2)These states are not orthogonal. We find⟨u|v⟩= exp−12Zdsx(|u|2 + |v|2 −2u∗v).

(B.3)The coherent states form the complete (overcomplete) set. The formal resolution ofthe identity can be written asZΩ2D2u|u⟩⟨u| = I,(B.4)

12where Ωis the real space D′(Rs) of Schwartz distributions or the real space S′(Rs) oftempered distributions, D2u = D(Reu)D(Imu) and the symbol exp (−R dsxv2) Dv,where v ∈L2R(Rs, dsx) (the real Hilbert space of square-integrable functions) des-ignates the Gaussian measure on Ω.The passage from the coordinate representation to the functional coherent statesrepresentation is given by⟨x1, . .

. , xn|u⟩= nYi=1u(xi)!exp−12Zdsx|u|2.

(B.5)Suppose now that we are given an arbitrary state |φ⟩. It follows immediately from(A.5) and (B.5) that the functional φ[u∗] = ⟨u|φ⟩is of the formφ[u∗] = ˜φ[u∗] exp−12Zdsx|u|2,(B.6)where the functional ˜φ[u∗] is analytic.An easy calculation based on (B.1), (B.2) and (B.6) shows that (B.6) can be writtenin the following abstract basis independent form|φ⟩= ˜φ[a†]|0⟩.

(B.7)

13Taking into account (B.4) and (B.6) we find⟨φ|ψ⟩=ZΩ2D2u exp(−∫dsx|u|2)˜φ∗[u∗] ˜ψ[u∗]. (B.8)The representation (B.8) is the functional Bargmann representation.The Boseoperators act in this representation as followsa(x)˜φ[u∗] =δδu∗(x)˜φ[u∗],(B.9)a†(x)˜φ[u∗] = u∗(x)˜φ[u∗].

14References[1]A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985).[2]K. Kowalski, Physica A 180 (1992) 156.[3]K.

Kowalski, Physica A 145 (1987) 408.[4]K. Kowalski, Physica A 152 (1988) 98.[5]K.

Kowalski and W.-H. Steeb, Progr. Theor.

Phys. 85 (1991) 713.[6]K.

Kowalski and W.-H. Steeb, Progr. Theor.

Phys. 85 (1991) 975.[7]K.

Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and CarlemanLinearization (World Scientific, Singapore, 1991).

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