THE

HARRY

DYM

PLANE:

INVERSE

EXACT

SOLUTIONS

EQUATION SPECTRAL

Boris G. Konopelchenko

1.

ON

THE

COMPLEX

TRANSFORM

and Jyh-Hao

AND

Lee

INTRODUCTION The Harry Dym equation zzz

is one of the most exotic soliton equations. It was discovered in the unpublished work by H. Dym (see [1]) and rediscovered in more general form in [2] within the classical string problem. The (1 + 1)-dimensional Harry Dym (HDR) equation (z G R) possesses many properties typical for soliton equations (see, e.g., [3, 4] and references therein). The inverse spectral transform (IST) method has been applied to the HDR equation in [5]. Equation (1) also has a (2 + 1)-dimensional integrable generalisation [6]. On the other hand, the HD equation has successfully resisted all attempts to construct its solutions in explicit form. The cusp solitons of the HDR equation constructed in [5] are given by in~lplieit formulae and can be analysed only numerically. The reciprocal link between the HDR and KdV equations also provides us with the implicit solutions since it includes the simultaneous change of both dependent and independent variables (see, e.g., [3, 4]). The interest in the HD equation has been bolstered essentially by the recent papers [7-11]. They demonstrated that the HD equation on the complex plane (z E C) (so will refer to such an equation as the HDc equation) is relevant to such physical problems as the Hele-Shaw problem, the Saffman-Taylor problem, and the chiral dynamics of closed curves in the plane. Here we study the HDc equation (1) on the complex plane (z E C) by the inverse spectral transform (IST) method. The initial value problem is solved by the c%method for the solutions q which decay sufficiently fast as [z I .-+ ~ . It is shown that the HDc eigenfunction introduced in a standard manner has an essential singularity at the origin of the plane of the spectral parameter. The renormalisation of this eigenfunction allows us to eliminate this essential singularity but the implicity of the corresponding formulae arises as a penalty. We show also that the HDc equation can be embedded into the framework of the c%dressing method in a manner which is free of an essential singularity. However the corresponding c%dat.a depends explicitly on the solution of the HDc equation. As a result, the c%dressing provides us only with an implicit exact solution of the HDc equation. A wide class of such solutions corresponding to the delta-function type (9-data is presented. The analysis both of the initial value problem and the c%dressing demonstrate that the duality between the essential singularity of the eigenfunction and the implicity of the inverse problem (or (%) data is the characteristic feature of the H D equation. 2,

INITIAL

VALUE

PROBLEM

To analyze the initial value problem for the HDe equation we start with its well-known [1-4] representation as the compatibility condition for the linear system

(2)

~bzz + A2(1 + q)r = 0,

r -

2{2(1 +

3_

- ((1 +

1

- 2iA) = o,

(3)

Budker Institute of Nuclear Physics, Novosibirsk-90, 6300090 Russia. Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, R.O.C. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 337-344, May, 1994. Original article submitted June 12, 1993. 0040-5779/94/9902-0629512.50 (~) 1994 Plenum Publishing Corporation

G29

whereA e C , z = z + iy ~ C, ~ = o-~, ~b, = ao~ z. In the present paper, we will look for the complex-valued solutions q of the HDe equation smooth and bounded on the plane Z and such that q -+ 0 as ]z] --+ oo. Such solutions evidently should depend on both z and 2, where the bar denotes the complex conjugation. Since the HDr equation (1) and the linear system (2), (3) contain only 0,, the 2-dependence ofq(z, 5, t) and r is determined by the condition of boundedness of q and r on the plane z. In the analysis of the initial value problem we also assume that q decays sufficiently fast as ]z] --+ oo. Following the c~ method approach (see [12-13] and reviews [14-17]), we consider the solutions r of Eq. (2) of the form r 2; A, A) = #(z, 2; A, A), where # is a function bounded in z and # --+ 1 as ]z] --+ oo. Such a solution of EQ. (2) is bounded for all z. The function # obeys the equation #,z + 2iAtt, + A~q# = 0. We specify the function # by requirement that # also solves the following integral equation:

#(z, 2; A, it) = 1 + A 2 I r e dz'

A

d2G(z - z, 2 - 5'; A, A)q(z', 2')#(z', 2'; A, A),

(4)

where G(z - z'; 5 - 2'; A, A) is the Green function of the operator 0z2 + 2iAO~ bounded on the entire complex plane z. Simple calculations give (5) G(~, ~; A, i) = 4 - 1~ ( e - ~ i ( x ~ + ~ ) i- ) " Since -gh0G = 57( 1 9 -~e - ~ i ( ~ + ~ ) , the solutions of Eq. (4) are analytic nowhere in the complex plane of the spectral parameter A. Following the 0-method, one gets

@(z, ~; ~, ~) Oh

= ~(~, 2; -~, -~)T(:~, ~, t)~ - ~ ( ~ + ~ ) ,

(6)

where

T(A,A,t)= 2~i 1 ~l f f c dz A d ~ ( ~ + ~ ' ) g ( z , ~, t)~(z, ~; ~, ~).

(7)

Equatio n (6) has been derived in collaboration with R. Beals. Thus, we have a rather usual quasilocal 0-problem. To use the 3-problem (6) for generating the inverse problem equation via the generalised Cauchy formula we have to analyse the behavior of # at infinity, A --+ oo. One has

#(z,~; A,]O -- ei~(~'~'O~ ((l § q)-88 + O ( ~ ) )

(8)

as k -+ cr where

1 //e ~ = a ; ~ ( ~ / i u q - 1) = g~-/~/

dz' ~ , _Ae'(gl+q(z,,~,,t) ~

1).

(9)

N o t e t h a t ~ is a bounded complex function on z and ~. Thus, the function p h a s an essential singularity at A --+ oo. Hence, the 3-problem (9) is not an appropriate one for the generation of the inverse problem equation. To overcome this difficulty, we introduce a new function I*~ via

~(z, 2; ~, 5,) = ~(z, 2; ~, i)~(~,~,0~.

(10)

The function #~ has no essential singularity at infinity and can be treated as the renormalised function #. For finite A the function #~ obeys the 0-equation

0X

= #r(-~' -i)T(~' ~' t)e-2~(z~+~+~)"

(11)

To eliminate the ambiguity of the r.h.s, of (11) at infinity A --+ oo we impose on the data T the constraint T(A, A, t) : 0 (VA, IAI > a), where a is some (large) constant. For such data the 0-problem (11) is welldefined and free of an essential singularity on the entire complex plane including infinity. 630

Such a renormalised/)-problem (11) together with the relation #~ (A -4 oo) = (1 + q)- 88 provides us with the inverse problem equation p~(A)=(l+q)- 88

1 /idA'

AdA' , A')e-2i(~'z+f'z+E~')T(A' ~ t). ~ 5 - 7 #~(-A' , ,

(12)

One can choose #~(A = 0) = 1. Then evaluating (13) at A = 0, one gets the following reconstruction formula:

q(z, Y., t) = - 1 + { 1 - ~il / ~

dA AAdA#~(-A'-A)e-2i(~z+Xz+e~)T(A' A, t) 1-4~ .

(13)

The integral equation (12) and the formula (13) solve, in principle, the inverse problem for the spectral problem (2). Due to the term c(z, 5, t) this solution is an implicit one. Then, using Eq. (3), one easily obtains T(A, A,t) = T0(A, A)e -si~%, where To is an arbitrary function. These formulas allow us to solve, in principle, the initial value problem for the HDc equation by the standard I S T scheme, q(z, 5, 0) .4 T(A, A, 0) .4 T(A, A, t) .4 q(z, 2, t). However, in virtue of the term r 5, t) in the r.h.s, of (12), (13), such a procedure is more implicit compare to the KdV equation. This is the penalty for eliminating the essential singularity. 3.

(9-DRESSING

Now we will apply the 0-dressing method to derive and solve the HDr equation. This very powerful and effective method has been proposed in [18] and then developed in [19-22]. It is based on the so-called nonlocal c%problem or its particular versions. We start with the scalar quasilocal c%problem

O)C(A, O1

def = (Rx)(A, i) = n(A,

(14)

It is assumed that the function X is normalised canonically X .4 1 as A .4 ~ and x .4 Xo + AX1, +A2X2 + 9" " as A .40. The z, ~, t-dependence in (14) is introduced via the equations [Dz, [~] = [D,, R] = [Dt,/~] = 0, where D~ = i i i i ~fz , D~ = Oz - i - -zfz, and f(z, ~, t) is some function. These operators De, D~, Dt Dt = at -- 4i - yft, obviously commute to each other for any function f. Here we introduce the inverse powers of the spectral parameter A in contrast to the previous section. Such an introduction of the spectral parameter is more convenient for the cg-dressing. One has _

_

.

z__

~

L

8i

In order to eliminate an essential singularity in R, we choose Ro as Ro(A, A) = 0(c~ - IAI).~0(A, ~), where Ro is an 1, ,~>0 arbitrary function, a is some (small) positive constant, and 0(~) = 0, ~ < 0. According to the general c%dressing approach [16-22] one should construct the operators of the form

L=

E

U~o,~l,~2,~a(z,2, t)A2'~~ lr~?2r~3-~--t

(16)

which obey the two conditions [0-~, L]X = 0 and (Lx)(A) .4 0 as A .4 oc, where X is a solution of the 0-problem (14). Such operators Li provide us with the linear problems LiX = 0, i = 1 , . . . , a, which give rise to the corresponding nonlinear integrable equation [16-22]. We start with the quantity D~X. It has the second-order pole at the origin. One can compensate it by the term -~zU2x . The requirement of absence of the second singularity in the quantity D2zX + -~U2x gives U = 1 + f~. Then the condition of vanishing of the residue in the first-order pole implies 1 + fz = )@2. As a result, the first desired linear problem is of the form 1

2

Dz2X+~TU X = 0 ,

where

U=l+fz.

(17)

Here we emphasize that the automatic absence of the term DzX is a consequence of our choice of the inverse powers of A in the operators D , , D~, Dt (30). Note that one cannot construct a similar linear problem using the operator D~. 631

To construct the second linear problem we consider the quantity DtX. It has the third-order pole at A = 0. It is easy to show that the third-order singularity is absent in the quantity 4 DtX - - - ~ D ~ x .

(18)

To compensate the second-order singularity in the quantity (18) we add the term V-~X. The proper V is given by V = - 2 ( i ~ -12 ) ~ = 2(U -~)~. Further demanding the absence of the first-order singularity in the quantity

O,x

-

4

-P-5

D

,x

= Xt -

+

i -fAx

4

'

-

+

(19)

one gets iftxo + 4U-1xI~ - 2(U-1),X1 = 0. Eliminating X1 and X0 from the last equation, one obtains

2(1)

(20)

l + . f , ) 8 9 zz

f~ -- (1 + f . ) a

Finally, the quantity (19) obeys the condition L x -+ 0 as A --+ ec. Hence, the second linear problem is

D t X - ,-7r

+

A-U

U 2(-771)z,x --

0.

(21)

A-

Equations (17) and (21) form a desired linear system. It is compatible by construction and implies the HDc equation (1) for 1 + q = U 2 . It is not difficult to show that in virtue of the relation 1 + q = (1 + f.)2

(22)

the HDc equation is the differential consequence of Eq. (20). Similar to other cases, the transition to the function r defined by r = g(A)e~(~-+~)+i~ converts the linear system 1 (17), (21) into the system (2), (3) with the substitution A ~ y. It is not difficult to show that the usual 0-dressing with the problem (14) and the operators Dz, D~, Dt with f = const leads to the mKdV equation on the complex plane. The only way to arrive at a linear problem of the form (17) with variable U is to admit the essential singularity of X at A = 0. Thus the 0-dressing method clearly demonstrates the duality between the essential singularity of the eigenfunction and the implicity of the 0-data for the HDc equation. Note also that Eq. (20) and the HDc equation (1) do not contain the derivatives with respect to f. The dependence of their solutions on 5 is fixed by the dependence of the 0-data R on f. 4.

EXACT

SOLUTIONS

In order to construct exact solutions of the HDc equation we need the solutions of the c~-problem (14) or the corresponding integral equation = 1 +

1

f/C

dpA . _

?

X ( - # , - # ) R o ( # , p ) e,

_2i(_z ]__~_[_]__~_ 8i

~, r~ u"

(23)

in closed form. Such solutions are associated, as usual, with the delta-function type 0-data R0. Indeed, choosing R0 as N R0 = 9" ~,~=1 ~ =M0 T,~ 6(~) (A - A,~a), where T,~ and An~ are some complex constants, 6(~)(A) is the ~-th derivative delta-function, and N, M are arbitrary positive integers, one gets the expression for X(A, A) with the (~-th order poles at the points A,~ (n = 1 , . . . , N, ~ = 1 , . . . , M). This expression contain the quantities X (~) ~ X(-A, To obtain the system of equation for X(~), one should consider Eq. (23) with A --+ -A, then differentiate it c~ times with respect to A and, finally, evaluate at the points A,~/~ (/3 = 1 , . . . , M). The result is the closed system of N M linear algebraic equations for X(~) (n = 1 , . . . , N; ~ = 1 , . . . , M). Solving this system, one gets X(A, A) and consequently, one calculates exact solutions of the H D c equation by the formula 1 + q = U 2 = X~ 4. These solutions are implicit since the exponent in (23) contains the function f, which obeys Eq. (20), and hence, the H D c equation (1). 632

One can construct these implicit solutions in a slightly different manner. Solving the algebraic system for X(~) mentioned above, then evaluating Eq. (23) at A = 0, one obtains a closed implicit formula for the function f(z, 2, t). It provides us with the implicit solutions of Eq. (20). Further, the use of the relation 1 + q = (1 + f~)2

(24)

gives rise to the solutions of the HDc equation (1). Here we will consider the simplest solutions of the HDc equation. For the c%data tg0 of the form N

Ro = ~r E T~5(A- A=), one gets the following equation for f: det 2 C

1 + fz - det2(1 + C)'

(25)

where , ?l, m : 1, C,~,~ - .k~ - + Am - - v n r n - - c T~ ' , N. The relation (25) is, in fact, the nonlinear ordinary differential equation for the function f(z, 2, t). It provides us with the implicit solutions of Eq. (20). Finally, using (24), one gets the implicit solutions of the HDc equation (1). So the problem of constructing exact explicit solutions of the HDc equation is reduced to the problem of coustructing exact explicit solutions ofo.d.e. (25). A similar situation also takes place for more general 0-data. In the simplest case N = 1, Eq. (25) is reduced to

f, = - s i n -2

where ~ - )h and 2ir

+-+-+-# # = log(T~-[(). Integrating Eq. (26), one gets

tg

+-+-+

+r

=

+r

,

+ T r m + a ( 2 , t),

(26)

(27)

where m = 0, 4-1, +2 and a(~, t) = ~ + + 7 The concrete choice of m and a(~, t) fixes the solution of Eq. (26). We choose m = 0 and a = 1. The corresponding equation

tg( z_ + 2 + f(z, 2, t) #

-#

- -# +

4t

~-5 + r

f(z, 2, t) #

(28)

defines the function f completely but implicitly. The solution of Eq. (28) is obviously periodic in the real variable [ = ~ + ~, effectively one-dimensional and bounded. We emphasize that the solution of the HDc equation given by (28) is completely different from the simplest solution of the HDn equation found in [5]. The reason is that it contains the combination ~ + ~. z The same is true for all solutions given by (25). They cannot be reproduced to the solutions of the HDn equation found in [5] by any choice of the parameter Ak. It is easy to see that the solution of (25) at N = 2 is two-dimensional and periodic in the variables ~2 ~ except that A1/A2 is a real and non-integer constant and describes the scattering of t w o waves. Other simple solutions of the HDc equation correspond to N

R0 = 7r E

T ~ 5 ' ( A - An).

(29)

In this case the function X has the second-order poles. The algebraic system of equations for X(-An) and X~(-A~) can be readily obtained from (23). Here we will consider the simplest case N = 1. One finally gets the following equation for f: (1 + fz)- 89 = 1 -

4 + 8i#(w'(#) - ~-/'z)+ 4#2(e 2i(~(t*)+"z+~~ - 1) A

(30)

12~ where A is given by A = 2 - 4 # 2 + 2 i ( w ' ( , ) - ~-/-~2)+ 4--~e-21(~(u)+~+~'~ and w ' ( p ) = - - ~ - .-r. Formula (30) describes the function completely but again implicitly. Comparing (30) and (26), we see that the solution (30) of the type (29) is essentially different from the previous one. 633

5.

ON THE SAFFMAN-TAYLORPROBLEM

As was shown in [7, 8], the HDc equation (1) is closely connected with the Saffman-Taylor problem of the motion of the interface of two fluids (one with high viscosity and one with low viscosity). Namely, if q solves the HDc equation (1), then the interface function ~'(z, t) is given by the formula ~ = 1 + q [7]. Thus the exact solutions of the HDc equation considered in the previous section provide us with the exact implicit expression for the interface function via 2, t) =

i+ i //c dz'

A_ d2'

(i+

t))

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