On the complete integrability of the Kadomtsev-Petviashvili I equation

Physics LettersA 160(1991) 367—371 North-Holland

PHYSICS LETTERS A

On the complete integrability of the Kadomtsev—Petviashvili I equation V.G. Bakurov Research and Industrial Organization (NPO) ‘Soyuzneftepromchim “, N. Yershov Street 55, Kazan 420045, USSR Received 10 May 1990; accepted for publication 25 September 1991 Communicated by A.P. Fordy

The complete integrability of the Kadomtsev—Petviashivili I equation as a system of Hamiltonian type has been shown. Canonical variables are introduced in terms ofthe scattering data. The trace identities for the integrals of motion and the set of Manakov’s constraints are presented.

1. The Kadomtsev—Petviashvili I equation (KPI) [1]

u,+uu+u

=38

—

u~

(1)

—~

is a Hamiltonian system with Gardner’s Poisson brackets (PB) of the form [2] /

{T, S}=~Jd2r(

—

~T \öu(r)

iF~—F~~+uF=O. (5) The Cauchy problem for KPI (with fast decreasing Cauchy data: u ( r) 0, r cc) was shown to be equivalent to direct and inverse spectral problems for eq. (5). Manakov [4] discovered that the inverse spectral problem for eq. (5) can be reduced to the Riemann—Hilbert problem with a nonlocal boundary condition on the real axis of the complex k-plane:

öS ~u(r)

~u(r) ~ ~u(r))

(2) F~(k)=F-(k)+

and Hamilton function 2+! 2 H~ 1d2r j3 k2Wv ~

J

,~

dsu(s,y).

j

d2ru2, P,=

J d2ruw~.

(Im k=O) (6) Here the F ±(k) are the boundary values on both sides, ±Imk>0, of an eigenfunction F(x, y, k) which is analytic in the upper and lower half complex k-plane, and satisfies the following asymptotic behavior: .

(3)

There is an infinite discrete set of KPI integrals of motion [31.The Hamiltonian (3) and the momentum components P~,P~,are the first representatives of this set: P~=

dzF-~(z)R~(z,k)

3~

—

w=0;’u=

J

—~

(4)

The KPI equation has been related to the nonstationary Schrodinger equation (the so-called auxiliary linear problem)

F ±(k)= [1 +O( Iki _!)] exp(ik2y—ikx) The spectral transform R ±(z, k) of the potential u(x, y) can be expressed in closed form in terms of the Cauchy data u (x, y, 0) and of a special nonanalytic solution ofeq. (5) [5]. Recently an alternative spectral transform b ±1 k, 1) which can be explicitly expressed in terms of u(r) and of the sectionally ho-

0375-9601/9i/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

367

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lomorphic eigenfunction F was proposed [6]. This spectral transform is defined as 2r b ±(k, 1) = 1 d u (r)F ±(r, 1) exp ( ikx— ik2y) 2it (7)

J

—

.

The old spectral transform R ±(z, k) of Manakov [~1 and Fokas and Ablowitz [5] can be expressed in terms of the new transform b ±(k, 1) [6], iR ±(k, 1) =b~(k, 1)O(k—l) +E~(1, k)O(l—k) —.

I

±

+

j dzb—(k, z)b~—(1, z)O(k—z)

(8)

.

The following characterization equations for b ±(k, 1) have also been obtained [6],

25 November 1991

Consequently, the evolution of the PB between b has an exponential form too.

±

2. The evaluation of the variational derivatives of ± the spectral data b (k, 1) with respect to u( r) is performed in the usual way [3,7}. “Differentiating” (5) with respect to u(r) yields, the following expression, —

6F~(k, r)

=

G ±(r, r’, k)F ± (k r’)

(13)

.

u(r) Here G±(r,r’, k) is the Green function of eq. (5). These functions are analytically continued in the upper (G + ) and lower (G ) half k-planes, respectively. The self-adjointnessof eq. (5) on the real axis Im k= 0 implies a symmetry relation between the Green functions ofeq. (5) and the adjoint equation,

b~(k,1)+b~(k,l)

=±i(J —

k

—

J

G~(r, r’, k)=~7t(r’, r, k) )dzb±(k,z)~(1,z),

(9)

=

.

(14)

Duetoeq. (14)anysolutionoftheadjointequation, may be written in the following form,

P

b~(k,l)+b—(l, k)

(Im k=0)

~,

F~(k,r)=exp(ik2y—ikx)

_iJ dzb~(k,z)5(1, z).

(10)

+

J

d2r’ G(r’, r, k)u(r’) exp(ik2y’—ikx’). (15)

k

The time dependence of the solutions of the auxiliary linear problem is determined by the second auxiliary linear equation:

Formulae (13)—(15) allow us to calculate the vanational derivatives öb’ (k, 1) /~u( r) from (7). The final answer is given in terms of the functions F ±(k

iF, + 4F~+ 6uF~+ 3u~-F—3 w~F=0.

r) and F ±(k, r): ~b~-(k 1) P~(kr)F~(1r) = öu(r) 2it

(11)

The two linear differential operators (5) and (11) are the so-called Lax pair of KPI [3]. This paper presents the description of the KPI Hamiltonian structure in terms of the new spectral data b~(k, 1). The PB are calculated and the infinite set of integrals of motion is obtained. The evaluation of the PB is carried out in the long-time asymptotic. This method of calculation follows from the Jacobi identity. In fact, the time dependence of the PB between two functions on a phase space is determined only by the time dependence of these functions. It is easy to prove by direct manipulations, using (1), (5) and (11), that the new spectral transform depends on time exponentially: bt(k, / t)=bT~(k,1) exp[4it(k3—13)} 368

.

(12)

~b~(k, 1) 6u (r)

=

—

(16)

,

F~(k,r)F~(1,r) 2ir

(17)

The substitution of the last two equations into the PB definition (2) yield the usual [3] expression in terms of the solutions F —, F {b~(k1), b~(q,p)} -.

=

J

J.~

~2,

[(P ±(k)F ±(1)O~P±(q)F ±(p)

47t —

~ ±(q)F ±(p) 8

~P±(k)F ±(1)]

.

(18)

The above mentioned sequence of the Jacobi identity allows us to use the asymptotics of the functions

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F~and P~for are of the form

PHYSICS LETTERS A

ItI—~oo.This long-time asymptotics j~(k)=

F±(k)=[1+CltI~~~2+O(t_!)]exp(ik2y_ikx)

J

dpR(p,k)exp[itw(p)]

—=

(19)

(this will be proved below). The further calculations

25 November 1991

+

are very easy and we obtain

J

dp O((s—p) (s’—p) )R(p, k)f 1 (k).

(25)

—=

{b ±(k, 1), b ±(q, p) }

In the long-timelimit the integration in the first term on the r.h.s. can be carried out with the help of the

=~i[sgn(k—/)+sgn(p—q)]ô(l—q)ô(k—p) (20) {b~(k,1),b~(q,p)} _—~i[sgn(1—k)+sgn(q-—p)]ô(l—p)ô(k—q). (21) The long-time asymptotic of the auxiliary problem solution can be obtained from the integral equation of the inverse problem,

f(k)—l-I- JdPd~f_Cv)R(~~~) 2iti 2—q2)+4it(p3—q3)] q—k+iO xexp[ix(p—q)+iy(p —

.

(22)

The functions f ±(k) =F ±(k) exp(ik2y—ikx) are analytical and bounded in the lower and upper half k-plane, respectively. The above equation arises after projecting the boundary condition (6), rewritten in terms off ±(k), on the lower half k-plane. It is possible to introduce “slowly varying” variables v~,v~: xzv~t+O(l), y=v~t+O(l).It is convenient to use the parametrization Vx= l2ss’, v~,=—6(s+s’). By using the well-known formula of the theory of distributions [8], lim exp[ixf(k)J(k+iO)’ Ix-.oo

=—2iriO(—sgn(x)f’(k))ô(k)exp[ixf(k)], (23) we perform the integration over q, and make the substitution f(k)=1+O((s—k)(s’—r))f1(k)exp[—itw(k)] f=sgn(t), 3—6k2(s+s’) + l2kss’. (24) w(k) =4k As a result ofthese steps an equation of the following type appears,

stationary phase method. The main term of the asymptotic solution of eq. (24) can be written in the form fl(k)={hg!(k)exp[itw(s)] 2 +kg2(k) exp[itw(s’)]}ItI~” h=exp[~iit sgn(s—s’)] (26) The function g 1 (k) satisfies the following integral equation of Volterra type, .

g1(k)=R(s,k) +

J

dpO(~(s—p)(s’—p))g,(p)R(p,k). (27)

The equation for g 2(k) is obtained from (27) by changing s to s’ and vice versa. If the kernel R (I, k) is bounded everywhere in the (k, 1)-plane, it can easily be checked that the Neumann series of the above Volterra equation are convergent. The solutions of eq. (27), g!2(k), determine the asymptotic form of f — (k) which we obtain by projecting the expression (24) on the lower half k-plane, using (26). The last step is for smoothening of the step-function. The main term of the long-time asymptotic of the form (19) is evaluated now with the help of the stationary phase method. In conclusion we note that this asymptotic is uniform with respect to the spatial variables x, y [9]. 3. Now we have to obtain the so-called “trace identity” in order to introduce action—angle type variables in terms of the spectral transform b±(k, 1). Let us consider the characterization equation (9) with k=1. to the definition (7),and the lower functions b ±(k,According k) are analytical in the upper half k-plane, respectively, and tend to zero as I ki —+cc. The projection (9) with k= Ion the appropriate half 369

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PHYSICS LETTERS A

half plane k> I or k
k-plane leads us to the following equation, b~(k k)—

+

— —

J

~

2it

b~(p,q)b-~(p,q) p—k+iO

25 November 1991

(28)

spectral us to introduce data: b(k,I)=b~(k,l) canonical variables orb(k,/)=b(k,1) in terms of the k>l(ork
with

As is mentioned above (12), direct evaluation shows that 8,b ±(k, k) = 0. Thus, b ±(k, k) is the set of conserved quantities enumerated by the continuous complex parameter k. In order to find the well-known integrals of motion, let us consider the following equation for the functions f~(k)=F~(k) Xexp(ik2y—ikx), ~ (29)

case in the further calculations. So, PB of standard type appear:

The coefficients of the expansion in inverse powers of k,

{b(k,

{b(k, 1), b(q, p)}=O, {b(k, 1), E(q,p)}=ö(l—p)ô(k—q)/2i.

Of special interest is the PB between b(k, k) and the canonical variables introduced above. Using relations (28) and (35) we obtain k), b(/, m)} i / b(m,l)

= l+~ =! (2ik~’ satisfy the system of recurrence equations

f(k)

f(IL...

(30)

—8~’u

(31)

Substituting (30) in (7) we obtain the spectral data expansion of the form ~ I~(k)”.

—

0,

b~(k,/)=—E~(1,k).

‘!

0,

12 =

— ~V’

(37)

Taking into account the above parameterization and reductions we rewrite the integrals I,, in the following form, —

The first five terms of expansion (32) contain the well-known integrals of motion the momentum components and the Hamiltonian: d 2r U

2m_k_i0+/_k_i0)~ (36) Thesecond expansion of (see b(k, k) in powers starts with the order (33)). Thus, of the1/k expansion of

(32)

n=O

$

b(1,m)\

the r.h.s ofeq. (36) in inverse powers ofkgives the reduction relation

n=l,2,3

=

(35)

]~J~-~-~i~ 4

~

~ ~

(38)

As follows from eq. (36), {b(k, k), b(I, m)}=0. So, all the integrals of motion I~are in involution with respect to the PB (2).

1 3=—P,,, I4=—~H. (33) The higher order terms I~(n> 4) have no simple interpretation, The trace identities can be obtained by comparing with (32) the result of the expansion in inverse powers of k computed from the r.h.s. of eq. (28): =+ —

JJ

dpdq sgn(p—q)b~(p, q)b~(p, q)q” 2it

.

(34) It is convenient to constrain the parameterization of 2, the the spectral transform b ±(k, 1) by the half of P 370

4. The complete description of KPI phase space includes the infinite set of constraints first discovered by Manakov. During his investigations Manakov showed that the conservation of the Cauchy daturn class in time is provided by this set of constraints [101.Recently, Grinevich and Novikov related the KPII constraints with the decrease rate ofthe inverse scattering solutions at spatial infinity [81. These constraints can be derived from the KPI equation (1) directly, without the usage of the inverse scattering transform gets from (1) thatmethod. In the limit lxi —fcc one

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w(oo,y)= Jdxu(x,.v)=0.

PHYSICS LETTERS A

(39)

25 November 1991

(43)

n=!

satisfies the system of recurrence equations Differentiating eq. (39) by time and manipulation in an appropriate way with (1) gives the second constraint =

f

(n+!)

(ik~)~ [if

w~)

(n-! ~+i(8x8;

‘8~)f (~

x

Jdxfdsu(s,y)=0.

(40)

This recurrence procedure allows us in principle to obtain all constraints of the above mentioned set. The next two are of the form

J J

f~)= —8;’u(ik~)~, f (2) = (ik~)—2 ( ‘~2 + i~ — ~

dx (u2~o;!wYY) =0,

dx (2w~+8; ‘3~u2—8;28~w)=0.

(41)

_8~(uf~)]. (44) Substituting (43) into definition (7) and expanding the exponent in powers ofk~we derive the respective expansion of the spectral data b ±(p, 1). There is a hypothesis that in the KPI case the preserving of the Cauchy datum class of fast decreasing functions is provided by the condition b(p, l)=o((p—l)”) as p—/--~O.The first four orders vanish as is equivalent to eqs. (39)—(41). The soliton sector of phase space is not considered in this Letter. Solitons will be incorporated in a forthcoming paper which will be published elsewhere.

-

We note that the linear KPI (LKPI) possesses an analogical set of constraints. The application of the procedure described above to the LKPI gives us the infinite discrete set of constraints of the form J~xo~a~w=o.

Acknowledgement The author wishes to express his gratitude to Dr. G.K. Shimkina for improvement of the language of this paper.

(42) References

The solution of the LKPI equation (which can be obtained by the Fourier transform method) belongs to the Cauchy datum class of the fast decreasing functions if the Fourier transform of the initial conditions a(k~,icY) satisfies the following condition, a ( k~, k~)= o ( k~)as k~—*0. We can express this condition in terms of the a(k~,kY) expansion in powers of k~.So, eqs. (42) are equivalent to the vanishing of the above mentioned expansion terms. Let us note that in the linear limit the functions b(p, 1) transfer to a(k~,IcY) where k~=p—Iand kY=p2—/2. It is evident from the above speculations that nonlinear conditions can be derived from the expansion of b(p, I) in powers of k~.As follows from (29) the expansionf~(p) in powers of k~(p=~(k~/k~—k~)),

[1] B.B. Kadomtsev and VI. Petviashvili, Soy. Phys. DokI. 15 (1970) 539. [2] C.S. Gardner, J. Math. Phys. 12 (1972)1548. [3] V.E. Zakharov et al., Theory ofsolitons (Plenum, New York, 1984). [4] S.V. Manakov, Physica D 3 (1981) 420. [5] AS. Focas and M.J. Ablowitz, Stud. Appi. Math. 69 (1983) 211. [6] M. Boiti, J.J.-P. Leon and F. Pempinelli, Phys. Lett. A 141 (1989) 96. [7] V.D. Lipovsky, Funkt. Anal. AppI. 20 (1986) 79. [8] V.S. Vladimirov, The equations of mathematical physics [9] [10] [11]

(Nauka, Moscow, 1975) [in Russian]. M.V. Fedoriuk, The steepest descent method (Nauka, Moscow, 1977) [in Russian]. S.V. Manakov, private communication (1981). P.G. Grinevich and S.P. Novikov, Funkt. Anal. AppI. 22 (1988) 23.

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