Rational KP and mKP-solutions in Wronskian form

Vol. 30 (1991)

REPORTS ON MATHEMATICAL

No. 2

PHYSICS

RATIONAL KP AND mKP-SOLUTIONS

IN WRONSKIAN

FORM

F. GESZTESY Department

of Mathematics,

University

of Missouri.

Columbia,

MO 65211, USA

and W. SCHWEIGER Institute

for Theoretical

Physics,

University

(Received December

of Graz, A-8010 Graz. Austria 10, 1990)

Rational solutions of the Kadomtsev-Petviashvili (KP) and modified Kadomtsev-Pctviashvili (mKP) equations in Wronskian form are constructed for the first time. The relationship between the Wronskian technique and Hirota’s bilinear formalism is discussed. It is shown that each pair of KP solutions giving rise to a mKP solution is connected via the KP-Bticklund transformation.

1. Introduction

Quite some efforts have been devoted to the study of meromorphic solutions of soliton equations in the literature. This is partly due to the fact that one can find explicit representations in terms of N x N determinants (with entries given by elementary functions) and/or Riemann theta functions, resp. Hirota T-functions in analogy to N-soliton solutions for certain classes of meromorphic solutions (such as rational ones), and also due to the intriguing connections between the motion of the corresponding poles w.r.t. time and completely integrable N-particle systems as discovered in [4]. Corresponding references in the context of the (modified) KU-eq. are e.g. [l, 2, 3, 4, 5, 6, 9, 12, 13, 33, 37, 39, 441 and in the (modified) KP-context e.g. [2, 8, 26, 27, 29, 30, 31, 40, 43, 441. A different representation of solutions of soliton equations in terms of Wronskian determinants originated with Satsuma’s paper [42] in 1979. He considered the (m)KdVsolitons in terms of Wronskians. This has been generalized by Freeman and Nimmo to the case of KP-solitons [14, 151 and rational KdV-solutions [35]. In the course of their work also the connection with Hirota’s bilinear formalism and T-functions has been established. More recently, the full power of this approach has been uncovered and its place in Sato’s theory of soliton equations has been described e.g. in [20, 381. These developments make it desirable to find the Wronskian form of other KP-solutions than the solitons as well. Surprisingly this has not yet been achieved to the best of our present knowledge. Although the rational KP-solutions have formally been found in 1977 by performing a certain “long-wave” limit [2, 291, no Wronskian representations of these solutions seem to exist in the literature. The purpose of this paper is to fill this gap w51

206

F.GESZTESY and W. SCHWEIGER

and derive the rational solutions for both, the KP and the mKP-equation in Wronskian form. In Section 2, after reviewing the standard determinant form of N-soliton solutions for the KP and mKP-eq., we rewrite them in Wronskian form following [14, 151 and perform the “long-wave” limit of [2, 291. We then prove in Theorem 2.5 and 2.9 that the limiting expressions are rational functions in (t, 2, y) and indeed satisfy the KP and mKP-eq. This answers a question posed by Nimmo and Freeman 1351as to whether there exist other nonlinear equations than the KdV-equation that allow for rational solutions in Wronskian form. To the best of our knowledge, these Wronskian representations for rational (m)KP-solutions are novel. In fact, for the mKP-eq., we could not even find rational solutions (except for the simplest one (2.83)) in the literature. At the end of Section 2 we briefly describe the special cases of rational (m)KdV-solutions. In Section 3 we connect the Wronskian formalism used in Section 2 with Hirota’s formalism and Bticklund transformations for the KP-eq. As a consequence of Theorem 2.9 we obtain in Theorem 3.1 Wronskian solutions of the bilinear KP-BBcklund transformation. 2. Rational

(m)KP-solutions

in Wronskian

in constructing rational solutions V(t, 2, y) and (KP) equation

In what follows we are interested $(t, 2, y) of the Kadomtsev-Petviashvili KP(V)

:= V, - 6VV, + V,,,

:= +t - 642& + L,

+ 3(d,*V,,)

(mKP,)

and the modified Kadomtsev-Petviashvili mKP,(4)

form

+ 3(8,-‘&,)

= 0,

(2.1)

equations + 6~&(d&)

= 0,

E = fl,

(2.2)

respectively, where we have introduced the abbreviation (%?f)(&

5, Y) :=

j

ds’.f(t,

2’7 Y)

(2.3)

03 for appropriate f’s According to a method developed in [2, 291 we will recover first rational solutions by performing a certain “long-wave” limit of corresponding N-soliton solutions and then give a direct proof that they indeed satisfy the KP and mKP,-equations, respectively. We start by briefly reviewing N-soliton solutions of the KP and mKP,equations as obtained with the help of the dressing method [45] (see also [41]) and inversion of a generalized Miura transformation [16], respectively. THEOREM

2.1. Let (t, Z, y) E R” and define c,,,(t,

lcE c,

Y) := c, ed4p2,

GIEC\(O),

- d)y

+ d) - 6k2(p, + qn)lt)> Qn6 c \ ((9, l
+ [4(d

2%EC,

pj # pl for .i # 1,

c,,,(& y) :=

2

c,,,(t, Y) > n

E=

(2.4) fl, (2.5)

RATIONAL SOLUTIONS IN WRONSKIAN FORM

A,+(& z, Y>:=

&$,

2, Y) :=

[c,,,(G Y)(P, + q,)-le-(P~+q~n)zl~m=l, o

207

N EN ,

N=O

[~~,&+ Y)(P,

0,

,

EEN

qm)-1e-(P~+qm)81~m=1,

(2.6) (2.7)

N=O

with fi = N if k # p,, 1 < n 5 N and K = N - 1 otherwise (without loss of generality we have assumed in this case k = pi). Then V,,t(t,~,y)

:= lc* - 282 ln{det[l + A,,,(t,z,y)]},

N E No, E = fl,

(2.8)

is an N-soliton solution of the KP-equation KP(Vv,,)

(t,~, y) := -k + 4(N+W,c

a, In

6=&l,

= 0,

(2.9)

det[l + i~,,tt, z, Y)I det,l + nN (t z y), ,E

3

,

N,

fi

E NO,

6 =

fh

(2.10)

1

is an (N + N)-soliton solution of the mKP,-equation mKP&CN++)

= 0,

E = ztl.

(2.11)

Originally, the N-soliton solution (2.8) of the KP-equation was derived in [45]. A direct proof that (2.8) fulfills the KP-equation can be found in [34]. The soliton solutions (2.10) of the mKP,-equations were first derived in [21] using the machinery of T-functions and vertex operators. A simple derivation of (2.10) based on a Miura-type transformation that connects solutions of (2.1) and (2.2) appeared in [16]. A direct proof that (2.10) fulfills the m.KP,-equations will be an immediate consequence of the techniques to be developed subsequently (see Remark 2.13). Remark 2.2: A criterion for the solutions (2.8) and (2.10) to be nonsingular (and hence Cm(R3)) is given in [36] and extended in [16].

For our purposes, it appears to be most convenient to rewrite the soliton solutions (2.8) and (2.10) in Wronskian form. For the KdV and mKdV-eqs. this device goes back to Satsuma [42]. LEMMA

2.3. Define

cx, ..-

(-I)“-lb+&)

GL

N P +4 flfi, j=l .

n=l,...,

N, N>2,

(2.12)

3#n N=l 1,

and a,(i, z, y) := (-l)n+l

exp{-q,z

- &y

+ [4q: - 61c*g,]t}+

+cu, exp{p,s - &y

- [4p”, - 6k*p,]t},

n = 1,. . . , N.

(2.13)

208

F. GESZTESY

and W. SCHWEIGER

Then + A,v]} = d, In W(at , . . , aA~) - c p,, 1=1

8,ln{det[l where W(al,

. . . , a,v) denotes the Wronskian

of the functions

(2.14)

al : . . , aN w.r.t. x.

Proof: We follow Freeman and Nimmo [14, 1.51, who were the first to give soliton solutions of the KP-equation in Wronskian form. With the definitions

(2.15)

(2.16)

1

N 1 ~ [ Pn + 4m n,m=l ’

A:= DI(& 2, Y) := [b,,

q-p,x

+ c(pi - qi)y

(2.17)

+ [4(~? + 4:) ~ 6k2(p,

+ ~n>l~)l~,,~l,

(2.18)

and 02(x)

:=

[&,

eq+w)l~m=l

!

(2.19)

2, ~J)AD~(x).

(2.20)

we may write (2.6) as A~,t(t, Defining

additional

2, Y) =

P1P2&(tr

matrices v := [pk-y,=,

)

(_l)“‘“‘&q;P’

1 ?

and

(2.21) ‘1;

M/’ := it is easily shown by exploiting

PlAP2

=

[ the Vandermonde

(-v-’ anI

fi

(PJ + 4m)

J=, (pj - Pn) J#IL

Hence

I

= v-‘W.

(2.23)

TL,Wt=l

A = Pl-‘V+WP,-’ and therefore

(2.22)

n,m=l structure of V that

(2.24)

(cf. eq. (2.20)) 1 + AAv,r = 1 + PlP2D1Pl-1V~1WP2-‘D2 = P2DlV-‘(VD,’

+ WDz)P;J-l.

(2.25)

RATIONAL SOLUTIONS IN WRONSKIAN FORM

Note, that the inverse matrices P,-‘, V-l, i # j. Using ~,ln{P+’

P;’

209

exist as long as pi # pj, i,j

= l,...,

N,

(2.26)

f(x)} = Q + 8, ln f(z)

and det(AB) = det AdetB

(2.27)

in eq. (2.25) proves the assertion of the lemma. n EXAMPLE

2.4. N = 1

al(t,x, y) = fiexp

- [2(Pii - 4;) - 3@(Pl

x cash

- q1)lt

x

- V(P? + d) - 3k2(p1 + ql)]t + i 01

=

(PI +

In al

,

Ql)/Cl.

V,,,(r, 2, Y) = k2 - 282 ln[al(t, 2, Y)I = ,@ _

tpl + d

cosh-2

2 - [2(p: + 4:) - 3k2(p1 + 41)lt + + Inal E =

I

kl.

,

(2.29)

Assuming lc = pl we obtain (cf. (2.10) and (2.14)) &,,(t, 2, Y> = - & ln[al(t, 2, ~11 Pl + 41 5 _ E

2 + [p: - 2q: + 3pfql]t + i In (~1 , t = zkl.

(2.30)

In order to study rational solutions as a certain limit of soliton solutions it is useful to introduce &l := Pn + qn,

n=

l,...,N,

(2.31)

vn := Pn - qnr

n=

l,...,N.

(2.32)

and By means of these new parameters a, In W(ai, . . . , a~) = 8, In IV@,. . . , bN),

(2.33)

210

F. GESZTESY and W. SCHWEIGER

with b,,(t,z,y)=exp{~~}[(-l)“+‘exp{-~~+t~y+~[r;~+3v~-6kZ]t}+

+a, exp

&I -x 1 2

- Eyy

Taking now the limit K~ 4 0, = 1 ,..., N, keeping V, constant, CX~= (-l)“(l

- ?[~.a

+ 3~; ~ 61c2]t

,

>I

n = l>. . , N.

n = 1,. . , N, in such a way that n = 1,. . , N, and

+ &~n)

Q/K~

(2.34)

= O(l),

i,j

n = 1,. . . , N,
(2.,35)

7 . . , OAJ) n K, + 0(/P+‘), n=l

(2.36)

+ O(K;),

we obtain lV(bl, . . , bN) = (-1) ~W(Ol,. with O,(t,

2, y) := exp {7x}

E = *1,

k E c,

[x - ev,y - 3(vi

- 21c2)t + En],

<,EC,

1/n E C,

n=l,...,

(2.14), (2.33), and (2.36) yields an expression in Wronskian form.

Combining (2.Q the KP-equation

(2.37)

N. for rational

solutions

of

THEOREM 2.5. Let vv,E(t,x,y)

= k2 - 28: lnW(&,

. . . ,ON),

(2.38)

6 = &l.

Then V,v,, is a rational function of (t, x, y) and

Proof

Due

to the cancellation

(2.39)

KP(Kv,,)

= 0.

of the

exponential

Fx by applying 1 > -28: In(.) to W(OI,. . ,O,) (cf. eq. (2.26)) V V,t is seen to be rational in (t,x, y). For the proof of (2.39) we introduce some helpful abbreviations. The Wronskian PV in (2.38) is an N x N determinant of a matrix, say with columns G(O), . . ,6(1”p1), W(O1,. . . ) 0,)

= det(8(‘),

terms

exp

. , scNpl)),

(2.40)

with

oyt:

T 2,

y) =

7

n=O>...,N-1.

(2.41)

>

In a more compact

notation

originally

introduced

(N=l)

:= W(O1,.

in [14, 151 we shall abbreviate

In general, NTK will re.present N ~ K + 1 consecutive we introduce the shorthand . . . 6(Lv-K). Furthermore, (N=l,h)

:= W(@ ,...,

(2.42)

. , ON).

@,-I,f&,@,+

,,...

(ordered)

>@,v)

columns

G(O), . .

RATIONAL

SOLUTIONS

= det(@),

IN WRONSKIAN

. . . , &j(2-2), j-j, @,

if the i-th column in the Wronskian (N:l)

211

FORM

. . , $N-‘))

(2.43)

is replaced by some column fi.

For the

special case fi = 6(K) and i = N we will use (~?2,

K) := (N?l,

e(“))

= det($O), . . . , ($(N-2), @cK)).

(2.44)

Now, substituting (2.38) into (2.1) and integrating the resulting equation with respect to CCunder the boundary condition Vzv,, + k2 as 1x1+ 03 yields

KqKv,E) =

+ 3WY,)W wzt + wzzzz - WzW, - 4W,,,W,

- 6rC2(Wx,W - IV;)-

+ 3wjz - 3Wy’.

(2.45)

Without loss of generality we assume at this point that lc’ = 0. (Solutions for I; # 0 are simply regained via a Galilei transformation (t, x) --) (t, II:+ 6rC2t).) For lt? = 0 the functions 0, of eq. (2.37) read O,(t,z,?/)=exp{~z}[z-~~~~-3y~t+S~],

n=l,...,N,

(2.46)

and satisfy the partial differential equations (2.47) and O,,t = -4.[8,;,,,

- (?)‘@_I

n = 1)“‘) N.

)

(2.48)

With the notation introduced above, the z-derivatives of the Wronskian W take the form w, = (C2, w,, r/t;;,, = (NT4, W 2222 = (NT5,

= (C3,N

N),

(2.49)

- 1,N) + (NT2,N

N - 2, N - 1, N) + 2(N=3,

(2.50)

N - 1, N + 1) + (N=2,

N - 3, N - 2, N - 1, N) + 3(iC4,

+3(Nr3,

+ 1))

N - 1, N + 2) + 2@3,

N + 2))

(2.51)

N - 2, N - 1, N + 1)+

N, IV + 1) + (NT2,

N + 3).

(2.52)

As a consequence of relation (2.47) the y-derivatives become 2+1

N-l w,

=

E

(C-3,

N - 1, N) - (C2,

Iv + 1) + C(Cl, i=o

(2

W~,=(~~5,N-3,N-2,N-1,N)+2(~~3,NIN+1)-(N=3,N-l,N+2)-(N=4,N-2,N-l,N+l)+(N=2,N+3)i+l

N-l _

2

C i=N-2

($I1,

(3

i+l @i+29

+ N~~~,C~, i=O

(2

@i)j+

O(i))

,

(2.53)

212

F. GESZTESY

and W. SCHWEIGER

i+1

N-2 -zC(~~~,(~~(i)_~(N+l)),

(2.54)

i=O

with vL’ denoting a column (2.55)

vfl = (vt&, . . . , vj,&?j,#-. Similarly, equation (2.48) implies for the t-derivatives of W IVt = -4{(N=4,N,N-2,N-l)+(N~3,N+l,N-l)+(N~2,N+2)i+l

N-l -

c

1, (2

(c-

(2.56)

O(i))} ,

i=o and Wzt = 4{-(Nr5,N

- 3,N - 2, N - 1, N) + (Nr3,N,

N + 1) - (NT2,

N + 3)+

N

Inserting the expressions (2.49)-(2.54) and (2.56), (2.57) into the right-hand side of (2.45) and gathering terms appropriately (taking into account that k2 = 0) results in (Wz, + ~zzzz + 3W,,)W

- w,w,

- 4w,,,w,

= 12{(N=3,

N, N + l)(Nrl)

- (N=3,

+ (N=3,

N - 1, N)(Nr2,

N + l)}+

N - 1, N + l)(N=2,

N)+

i+l

N-l -

+ 3w& - 3w,2

c(N=l,

(z’@)(N=2,

-

N)

i=O

6

(Nr1,

(3

@cN))(N=l)

+

c

s=o

zy.vm2 N-l

i+l

N-l

N-l _

%+I -

+ ~(N=l,(;)20(“))(N=3,N-l,N) i=o

(N=l,

(g&),

N-l

6 ‘N’)(N=l)+

RATIONAL

Due to the Plucker (K-3,

relation

N, N + l)(_C-1)

SOLUTIONS

IN WRONSKIAN

=

- (fC3,

N - 1, N + l)(C2,

the first curly bracket

cancel.

& [ det(e(&$O),. , ,($+-2),

& [det{(e(z)=

“0 (0), . . ! ,(&o(“.-21,

Iv)+

N - 1, IV)(N=2, The terms within

(2.59)

N + 1) = 0 the second

bracket

may

e(&$~~r))x x

=

213

[17]

+(N=3, the terms within be rewritten as

FORM

det(e-($@(o!, . . :e-(&(,y-l)

e($@(“))T

x(e-(&(0)

1 .

.

.

)I z =o

x (2.60)

1

Since the product of the N x N matrices within the curly brackets is independent whole expression vanishes. The same trick applies to the remaining brackets This proves the theorem.

of .z the in (2.58). n

Remark 2.6: Formally, the proof and Nimmo [14, 151 which says that

of Theorem

V(t, 2, y) = -28,

2.5 generalizes

In W(ar , . . , a.&~)

the result

of Freeman

(2.61)

214

F. GESZTESY and W. SCHWEIGER

if a,, n = 1,. . . , IV fulfill the partial differential

is a solution of the KP-equation (cf. (2.47) (2.48))

equations

an.y = +a,,,,,

n = l>...!N,

(2.62)

an-t = -4a~n.xss,

n = l,...,N.

(2.63)

and

Observing, I;? = 0)

however,

that for the specific choice (without

+*y

a, = exp eqs. (2.62)

- $t

we again assume

n = l,...,N,

(2.64)

1

{

(2.63) are satisfied

o,,

loss of generality

and that (cf. eq. (2.26))

&lnIY(at

,...,

aN) = d,lnW(Ot

,...,

(2.65)

O,),

the validity of Theorem 2.5 follows from [14, 1.51. Nevertheless, it is instructive to consider our alternative proof of Theorem 2.5, since it immediately applies as it is also to the KdV case (cf. Remark 2.14) as well as to other reductions of the KP-equation such as the Boussinesq equation. Remark 2.7: The rational particular choice \

solutions

of the IiP-eq.

quoted

in [43] correspond

(2.66)

n = l,...,N. EXAMPLE

to the

2.8. N = 1

v,,,(t, 2,

y) =

k* - 282In @ICC2:

= k* + N = 2, En! n = 1,2, according

Y)

2 (z - ~yy - 3(vf - 2k*)t +
,

E= Itl.

(2.67)

to (2.66)

V21~(t,2, y) = k* - 28; In W[@(t,

z, y),

@2(t,

2, y)]

= k2 + 2[z - cvlY - 3(vf - 2k2)t]* + [cc - EVZY~ 3(v; - 2k2)t12 ~ [8/(~1 - ~2)~1 [L-c- EVlY - 3(vT - 2k*)t][x

- w2y - 3(v; - 2k2)t] + [4/(v1 -

v2)*]

(2.68)

e=ztl.

Next, we consider rational solutions of the mKP,-equations. According to Theorem 2.1 we must distinguish between the cases k = p,~ and Ic # p,, n = 1, . , N. For k = pN we introduce

= (-lY(Pn

+ qn) cn

Av p, + qn = a,, I-I ~ : ,=, PI -P,

n=

l,...,N-1.

(2.69)

>

215

RATIONAL SOLUTIONS IN WRONSKIAN FORM

ForIc#pp,,n=l,...,Nwedefine cl,

t-l)“-‘(Pn

=

4n)

+ :,

N pj + qn _ rI-Pj -Pn J=,

k + qnan f-p,

J#,L

=

1+ +

ck!n

2

+ W.2,)

I

lc # ?,

:

n =

2

l,...

(2.70)

,N.

Taking now the limit K, + 0, n = 1, . . . , N, in eq. (2.10) with cy, chosen according to eq. (2.35) motivates THEOREM

2.9. If Ic = F,

let

$2~~_1,~(t,x, y) = &ln[W(O1,

fl .

(2.71)

6 = *l.

(2.72)

E=

. . , @j~_~)/lfV(O~, . . . , OAT)],

N,let

IfI;#$,n=l,...,

/W(&,

. ,O,)

,

I Then c&N-~.~ and c#Q.~,~ are rationalfinctions mKP,(4ZNpI,E) Proof: In the case Ic = 7

of (t, x, y) satisfying m,KPt(&v,E)

= O!

(2.73)

= 0.

the function C$has the general form (2.74)

C#I = 8, In 5 , where the functions f and f satisfy eq. (2.45) with Ic = y.

Inserting (2.74) into (2.2)

and using (2.45) gives

LLfz - Lf -

xfzz +&f

()

- E&f+ F

2sf

=0

(2.75)

as a necessary and sufficient condition for 4 (as defined in (2.74)) to fulfill the mKP,equations. In particular, for fl = W(@I, . . . , ON-I) and f = W(O1, . . , ON) insertion of the expressions (2.49)-(2.53) into the left-hand side of (2.75) gives 2{(N=3,

N)(N=l)

- (N=3,

+ (N=2)+-3, N-2 _

c(j+?2, 2=0

N - l)(N=2, N - l,N)}

N)+ ~ ( F)2

(N=2)(N=l)-

2+1 ($@(‘))(N?-1)

+ (N+&N’?~,

($2@cti, i=o

216

F. GESZTESY

NY3

=

0

0

N- ~ 3

and W. SCHWEIGER

(N - 2)

(N ~ 1)

N

?

(A-2)

(N-1)

N

r

(2.76)

,

where F = (0. . . 0. 1)r is an N-dimensional column. This proves that (2.71) is a solution of the mh-PC-equations. In the case k: # +, n = 1,. . N, the function 4 takes on the general form @I =

where

,? and f satisfy eq. (2.45). 2L.L

- Lf

as a necessary and sufficient equations. Defining &(L equation

--k + i),, In f

Inserting

.\’

- box) + c(f,f

for 4 (as defined

2, y) = o,,(t: 5, ;I/) + he?.‘.

(2.78) with f = l,I-(@l, . . ,6,)

2{[(A~I,$@))

(2.77) into (2.2) and using (2.45) yields

~ ff.r.T + 2KEJ condition

(2.77)

f’

= (NY

- &)

in (2.77))

= 0

(2.78)

to fulfill the mKP,-

n. = l,...,N, l), and f = IIT(Oi:

(2.79) . .0,1;) yields

.\’ _ (ni_rI,~(~~-+“),(~~I)1v

s

~[(N~l~k~(T--ij)~(N~1,~(‘L.))](~~21M)-(~~~l)(~~~3,~~-

In order to eliminate the terms containing z-derivative. Furthermore, we utilized [email protected] _ #~+I) ’ J .I

($G(‘),

= (k-F)@;“‘.

I,N)}+

we used

j = l,....

again

the trick with the

N,

(2.81)

RATIONAL

SOLUTIONS

IN WRONSKIAN

FORM

217

and

(2.82) Thus &N,~ is a solution

of the mKP,-equations.

Remark 2.10: A direct proof that the “long-wave” limit procedure of [2, 291 indeed yields rational solutions of the KP-equation has first been given by Matsuno [30]. His result is phrased in terms of determinants in analogy to (2.8) and not in terms of Wronskians (such as (2.14)). Freeman and Nimmo [14, 151 (based on the work of Satsuma [42] in the (m)KdV-case) were the first to write down the KP-soliton solutions in Wronskian form (see also [17, 201). To the best of our present knowledge, our Wronskian representation of rational solutions for the KP- and mKP,-equations are new. In fact, for the mhTP,-equations we could not even find rational solutions (except for 41.~) in the literature. EXAMPLE

2.11.

=

N

= 1

-k+

z-~~,~-3(~:-2k~)t+<,

+ ~ k-7

[ - [X - fV]tJ - 3(vf ~ 2/i+

+ [,I_‘,

1

-1 1 f = fl

k#$.

We would like to emphasize that the explicit form of 0 did neither of Theorem 2.5 nor in the proof of Theorem 2.9. In fact, we have LEMMA

enter

.

(2.84)

in the proof

2.12. In order to verify that Vv,, = k’ - 23; In T/l’(at:

. . , a,)

(2.85)

is a solution of the KP-equation, it @ices (apart from integrability conditions on V, and V,, w.r.t. x near CC) that a,,, II = 1. . . , N fulfil the partial differential equations an.y = +[a,,,,,

- fijl.nanlr

n = l,....N.

F = fl

(2.86)

n = l....,N,

(2.87)

and a ,l,t = -4[an.Xzs

~ ~~.nanl + 6kz[a,,,,

- &an]?

n = 1,. , N complex constants. Similarly, the conditions (2.86) and with 31,,~, LG.,, &, (2.87) on aTL,Y and a,,t (apart from integrability conditions on c&, &v w.r.t. x near E) guarantee that &~_l.~

= d, ln[PV(aj,.

: uj~~_,)/W7(a,,

, a~)]

(2.88)

218

F. GESZTESY and W. SCHWEIGER

is a solution of the mKP,-equation. 42N,t

=

-k

In order that

+ 8,

ln[W(i-ii,

. . , ZAhr)/W(al,.

(2.89)

. . , ulAF)]

solves the mKP,-equation, one has to ensure, in addition to (2.86) and (2.87) for an,y and a+ that a,,, and Z,>t furfill (2.86) and (2.87) and that

_(jjq

(~~/A&

N) - (Nrl)(N=3,

3”9)(N72,

N - 1, N) = 0.

In the special case Ic = 0, Pl,n = p2,n = 0, n = 1,. . . , N, Lemma observed in [14, 151. We shall further discuss Lemma 2.12 in Section 3.

2.12 has been

Remark 2.13: According to Lemma 2.3 the soliton solutions of Theorem rewritten in Wronskian form with a, as defined in (2.13) and (cf. (2.70))

2.1 can be

exp{ -qnz - eq,2,y + [4qi - 4k2q,]t} +

Z,(t. Z, y) = (-l)n+l +k+qrl IC_P~, n Since in this particular

(2.90)

exp{p,z

- V~Y - [4~: - 41c2~,lt],

n = l,...,N.

(2.91)

case

an,y + ca,,,,,

= G,,

+ G,,,

= 0,

n = l,...,N,

(2.92)

and a,,t + 4an,zzz = Gt the validity quence of

of Theorem

+ 4%,,,,

= 0,

2.1 follows from Lemma Ci,

n = l,...,N,

(2.93)

2.12 and the fact that (2.90) is a conse-

- a n,Z = (k+q,)a,,

n = l,...,

N.

(2.94)

Remark 2.14: In the special case where a,, Z,, n = 1,. . . , N are independent of y, (2.85), (2.88), and (2.89) become Wronskian solutions of the KdV and mKdV-eq., provided a,, and Zi, are chosen in such a way that (2.87), (2.90) and an.xx - rlnan = &l,.ZZ - %&C = 0, are satisfied.

Using

and (2.54) (with

(2.95), a simple

($)*

replaced

and 2.9 apply to Wronskian

by 7,)

solutions

Remark 2.15: In particular,

calculation

n = l,...,N,

(2.95)

shows that the right-hand

become

of the KdV

zero.

Thus the proofs

and mKdV-eq.

sides of (2.53) of Theorem

2.5

as well.

(2.85), (2.88) and (2.89) with

[(2np1)/31,2n-l-3p(_4t)p

%(Gq =

c p=.

(2n - 1 - 3P)!P! ’

n = l,. ‘. ‘N’

(2.96)

RATIONAL

SOLUTlONS

IN WRONSKIAN

219

FORM

and w+w31

iin

=

.+-‘-“P(_4qP

cp=.

(2n - 1 - 3p)!p! +

[@-v/31 +f

z2+-“p(_4t)~

cp=.

(2n - 2_

3P)IP!’

(2.97)

?z = l,...,N,

(where [(2n - 1)/3] and [(2n - 2)/3] d enote the integer parts of these fractions) are seen to be rational solutions of the KdV and mKdV-eq. since a,,t + 4an,zss = &,t + 4G,zzz

n = l,...,N,

= 0,

(2.98)

and (2.90) holds as a consequence of IcE, -a,,

n = l,...,N.

= Ica, - a,_i,

(2.99)

For the proof of this statement along the lines of Theorems 2.5 and 2.9 one can again use that the right-hand side of (2.53) and (2.54) (with v, = 0) vanishes due to n = l,...,N,

an,xx - a,-1 = an,zz - a,_1 = 0,

(2.100)

and the general result that for any matrix B with matrix elements bi,, i, j = 1, . . . , N, >v

det k=l ITic

(

b,l ...

... .

bl.k?x

b,vl

. . .

b,vk,zs

1::

il.)

=

[bk:iz

gdet

i::

bykir)

.

(2.101)

. . .

The rational Wronskian solution (2.85) (with a, as defined in (2.96)) for the KdV-eq. was first derived in 13.51.The &C&‘-solutions (2.88) and (2.89) in Wronskian form (with a, and a, as defined in (2.96) and (2.97) to the best of our knowledge are new. Rational solutions of the mKdV-eq. equivalent to (2.88) could also be derived from results in [3]. 3. Blickhmd transformations

and Hirota’s r-functions

In this section we further illustrate the content of Lemma 2.12 and make the connection between the Wronskian formalism of Satsuma and Freeman and Nimmo employed in Section 2, Hirota’s formalism and KP-Backlund transformations. We briefly recall Hirota’s approach. Defining the bilinear differential operators (F(D,)G(Dz)HP,)f :=

g)(t> [J’(4

-

2, Y>

&)G(&

-

&l>ff(a,

-

+).f(t,

x, y>g(t’>

I’,

Y’)]

t=t’

X=X’

(3.1)

Y=Y’

for appropriate

F, G, H

and f, g, and introducing Hirota’s T-function by the transforma-

tion V(t, z,

Y> = -28;

In[r(x,

Y, t>l

(3.2)

220 we obtain

F. GESZTESY

the bilinear

form of the KP-equation

KP(V) Equation

(3.3) rewritten rrt,

and W. SCHWEIGER

= a,{+[D,(D,

[18, 191 + 02) + 3D$r}

= 0.

(3.3)

in terms of a,, d,, 3, reads

- rr,-rt + rrzT,s.,-.r~ 4r,r,,,

+ 3r;,

+ 3rr,,

- 3$

= 0,

(3.4)

where we have assumed that V --f 0 as 1x1 + K. Next, we recall the Backlund transformation for the KP-eq. in bilinear form [18]. Let T be a solution of (3.3). Then integration of (D;+eD,)r.7=0.

t=*tl

(3.5)

and (D,+D;-3ED,D,)r.7=0> yields another solution of &., ar,, 3, read

E=ztl

7 (resp. v) of (3.3).

Equations

x(t. 2, Y) := W(~,

(3.6)

(3.5) and (3.6) rewritten

in terms

Y, t)l~(~ Y, Ql ,

(3.9)

the system of partial differential equations (3.5) and (3.6) is seen to be just the bilinear form of the (first) modified Kadomtsev-Petviashvili eq. ([21, 22, 23, 251) mKP,(4) = 0 or, equivalently, m@,(x)

:= xtx - 6x&m

As a consequence of eq. (3.5) (see [7, 16, 21, 23, 241)

V, v and x are connected v = xp + Xrz v = 2; - j&

via the Miura-type

satisfies

transformation

txy ,

- E&

E = 51.

An alternative formulation of the KP-Backlund transformation (3.11) is given in refs. [7] and [32]. Rephrasing now the first part of Lemma 2.11 in the present 7.V

(3.10)

+ xzsss + 3xyy + 6~xz.z~~ = 0.

E := lW(Ui? . . . , U-V), t= ztl, T(J,, := 1;

(3.11) based on eqs. (3.10) and context,

we obtain

that

NEN. (3.12)

(3.3) if (3.13) (3.14)

RATIONAL SOLUTIONS IN WRONSKIAN FORM

221

But then also ?N,e defined by ?N,E := W(u’,, . . . ) U’N), a;(& 5, Y) := aj,z(t, 2, Y), satisfies (3.3). By means of these r-functions

=

0,

(3.15)

l
the second part of Lemma 2.12 yields

THEOREM 3.1. Define TN,~ and ?N,~ as in &KP(VN,,)

NEN,

E= &l,

?c,, := 1;

(3.12)-(3.15) such that &qcv,,)

=

(3.16)

0,

where VN,,(& z, Y): = -282 ln[7N,4Z7 Y, t)l

, (3.17)

NENo.

v~,,(t, 2,~): = -282 ln[G,e(Z, Y,t)],

Then both the pairs (TN-l,+, TN,c) and (TN,,, ?N+) are Wronskian solutions of the KPBiicklund transformation (3.5) and (3.6) or, equivalently, the functions XN,c(t,

2, Y) :=

ln[~N-I,6(?

i% t)/m,dZ,

!h t)l?

N

E N,

(3.18)

iv E

NO,

(3.19)

and X^N,&

2,

Y> := &N,&

!I, t)/FN,&

Y, t)l,

are solutions of the (first) modified Kadomtsev-Petviashvili eq. (3.10), i.e., mb,(XN,c)

= 0,

~‘b&N,r)

= 0,

E=

*1.

(3.20)

Proof: Assume k = 0, then (2.88) directly corresponds to (3.18). The correspondence between (2.89) and (3.19) (for Ic = 0) is established by identifying uj with u; and ?;3 with uj, respectively. In particular, for this choice of uj and ?;j eq. (2.90) is easily proved to hold. Thus, the assertion of the Theorem is an immediate consequence of Lemma 2.12 and the equivalence of the mKP-eqs. (3.20) and the KP-Backlund transformation (3.5) (3.6). n Remark 3.2: Theorem 3.1 in the special case ,Di,j = Pz,~ = 0 in (3.13) and (3.14) is well known and due to Freeman and Nimmo [15] (see also [17, 201). It is just our formal generalization to ,C?i,j# 0, pz,j # 0 (cf. Remark 2.6) which yields new Wronskian solutions of the bilinear KP-Backlund transformation (3.5) (3.6) and allows to cast the rational KP and mKP-solutions into the Wronskian form (2.38) and (2.71), (2.72) (see (2.47) and (2.48)). The considerations in this section (and hence the ones in Section 2) now generalize in a natural way to the n-th order KP-hierarchy [lo, 11, 21, 28, 381. We shall dwell on that in a future publication.

222

F. GESZTESY

and W. SCHWEIGER

REFERENCES [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

[33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]

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