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Coherent structures for the Ishimori equation
Physica D 48 (1991) 367-395 North-Holland
Coherent structures for the Ishimori equation I. Localized solitons with stationary boundaries V.G. Dubrovsky Institute of Nuclear Physics, Novosibirsk-90, 630090, USSR
and B.G. Konopelchenko 1 Department of Mathematics, Yale University, Yale Station, New Haven, CT 06520, USA
Received 2 July 1990 Accepted 2 September 1990 Communicated by A.C. Newell
The initial-boundary value problem for the Ishimori-II equation is studied. The time evolution of the inverse problem data in the case of nontrivial boundaries is found. A general formula for exact solutions of the Ishimori-lI equation with nontrivial boundaries is derived. Localized soliton solutions of the Ishimori-lI equation with time-independent boundaries are studied in detail. It is shown that there exist essentially three different types (ss, sb, bb) of localized solitons for the Ishimori-ll equation. Some explicit examples of such solutions are presented.
I. Introduction Nonlinear evolution equations in 1 + 1, 2 + 1 and multi-dimensions integrable by the inverse spectral transform (IST) m e t h o d form a wide class of differential equations with a n u m b e r of remarkable properties (see e.g. refs. [1-3]). A main feature of such equations in 1 + 1 dimensions is the existence of soliton solutions which are localized in one dimension. Solutions of the (2 + 1)-dimensional integrable equations which are localized only in one dimension (plane solitons) were constructed years ago [4, 5]. T h e n the lumps (rational nonsingular solutions) were also discovered [5]. Exact solutions exponentially localized on the plane but decreasing at t ~ + ~ have b e e n constructed in ref. [6]. A n d only recently the travelling solutions of the soliton type of the (2 + 1)-dimensional equations exponentially localized in all directions on the plane have been found. For the first time this has b e e n d o n e in ref. [7] by the use of the B~icklund transformations. Spectral theory of such localized solitons for the D a v e y - S t e w a r t s o n I (DS-I) equation and their connection with the initial-boundary value problem for the DS-I equation have been studied by different m e t h o d s in a series of papers [8-13]. T h e localized solitons (or dromions) of the DS-I equation possess properties which are different from the properties of one-dimensional solitons. In IPermanent address: Institute of Nuclear Physics, Novosibirsk-90, 630090, USSR. 016%2789/91/$03.50 © 1991 -Elsevier Science Publishers B.V. (North-Holland)
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particular, they can be driven by a change of the boundaries [11-13]. All this demonstrates the richness of the coherent structure associated with the (2 + D-dimensional integrable equations. The present paper is the first one from a series of papers devoted to the study of the coherent structures for the Ishimori equation
s , ( x , y, ,) + s x (sxx +
+
= o,
@xx -- Og2q)yy q" 20~2S" (S~. x Sy) = 0,
(1.1)
where S = (S~, S 2, S 3) is the three-dimensional unit vector (S 2 = 1), q~ is the scalar field and a 2 = ± 1. Eq. (1.1) is the (2 + 1)-dimensional integrable generalization of the Heisenberg ferromagnet equation (isotropic Landau-Lifshitz equation) S, = S × S**. It has been introduced by Ishimori in ref. [14]. An important feature of eq. (1.1) is the existence of classes of topologically nontrivial and nonequivalent solutions which are classified by the topological charge [14]
Q=
f dxdyS'(&×S,,).
(1.2)
This topological invariant is connected with the mapping S (2) ~ S (2) defined by the unit vector
S(x, y, t) with the boundary value S~ = (0, 0, - 1). The Ishimori equation (1.1) is of great interest also since it is the first example of an integrable nonlinear spin-one field model on a plane. The applicability of the IST method to the Ishimori equation (1.1) is based on its equivalence to the commutativity condition [Ll, L 2] = 0 of the operators [14]:
L l = a ~ v + PO,, L 2 = i O, + 2PO 2 + (Ix + aPyP - ieg3Pq~x + i % ) Ox,
(1.3)
where P = S" o- and o- = (o-l, % , 0"3) are Pauli matrices and 0x = a/0 x, Oy-= O/Oy, ~t - a/Ot. The standard initial value problem for the Ishimori-I ( a = i) and Ishimori-II ( a = 1) equations with vanishing boundary values has been solved in refs. [15-17] with the use of the ~-method ( a = i) and the nonlocal R i e m a n n - H i l b e r t problem method ( a = 1) respectively. The exact rational-exponential solutions ( a = i) and solutions with functional parameters (o~ = 1) have also been constructed in refs. [15, 16]. Our aim is to solve the initial-boundary value problem for the Ishimori-II equation with nontrivial boundaries. In the characteristic coordinates ~: = ½(y + x), r I = ½(y - x ) in virtue of the equalities q~e = f_n d r / ' S "(Se x S , , ) + 2u2(~:, t ) ,
¢ , = f~_ d ~ ' S ' ( S ~ , x S , ) +
2ut(r/,t),
(1.4)
eq. (1.1) at a = 1 is equivalent to the equation
l(f, drI,S.(SE×S,,)+2Uz(,t))S - l ( f~_~ d~' S "( S¢' × Sn)
2u,( ~, t ) )S, = O,
(1.5)
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
369
where u~(~7,t) and U2(~,t) are arbitrary scalar functions. The initial-boundary value problem is to solve eq. (1.5) with given initial value S(s¢, ~7,0) and boundary values u1(~7,t) and u2( ¢, t). We will use the method proposed by Fokas and Santini for the DS-I equation [10-13]. Firstly we present the solution of the forward and inverse spectral problems for the auxiliary linear problem L~O = 0 slightly different from that given in ref. [16]. Consideration of the nontrivial boundaries requires a modification of the operator L2. The obtained equation L2¢, = 0 gives rise to the following linear equation for the Fourier transform S(~¢,~7,t) of the inverse problem data St - ½i(S,¢ + S,,7) + u 2 ( £ , t ) S ~ - u , ( ~ l , t ) S n = O ,
(1.6)
which coincides with the linearized equation (1.5) for S+= S 1 + iS 2. For real S, separation of variables reduces the problem of solving eq. (1.6) to solving the linear equations 2iXt(sc, t) + Xg¢ + 2 i u 2 ( e , t ) X ¢ =
O,
2iYt(r/, t) + Ynn - 2iu,(~7, t ) Y n = 0.
(1.7)
Using the exact solutions of eqs. (1.7) and the solution of the factorized inverse problem for the auxiliary linear problem L~@ = 0, we construct the exact solutions of the Ishimori-II equation (1.5). They are representable in the form S(~:, r/, t) = - tr(o'g~r3g-I ), ~(~:, ~7, t) = 2iln(det g) + 2 ~'1u2(¢', t) + 2 ~ l u l ( ~ 7 ' , t),
(1.8)
where
_ i_ ( (x,(1- pap*b)-'pap*X*) - (Y.,p*(1- b,op*)-'x.) } (X,(1-pap'b)-'pY)
-(Y*,p*(1-bpap*)-'bpY)
(1.9)
where (X, Y) :-- EiXiY~, X i and Yi are the solutions of eqs. (1.7), p# are arbitrary constants and a ik =
f
d rl ' Yk* ( rl ' , t ) O, , Yi ( ~O' , t ) ,
bik = - j Coods~ ' Xk ( ¢', t) a¢, X,* ( ¢', t ) .
(1.10)
Charge Q for these solutions is also given by the compact formula +o~
i
ff
OeO.ln(det g) d e d n .
(1.11)
In the present paper we consider in detail the case of the stationary" boundaries ul(~0) and u2(~:), In this case the linear problems (1.7) are gauge equivalent to the 2 × 2 matrix Zakharov-Shabat spectral problem with the reduction which gives rise to the modified KdV equation. The inverse problem for that spectral problem has been solved in ref. [18] and the corresponding discrete spectrum includes the solitons (s) and the breathers (b). As a result, we have essentially three different types of exact localized
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V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
solutions of the Ishimori-II equation: Sss, Ssb and Sbb, which correspond to the choice of the functions X and Y as the soliton or breather eigenfunctions. The simplest examples of such solutions are presented explicitly. We discuss also more complicated exact solutions of the Ishimori-II equation. The paper is organized as follows. In section 2 the solution of the inverse spectral problem for the Ishimori-II equation is presented. The compact formulae in the case of the degenerated inverse problem data are derived in section 3. The time dependence of the Ishimori-II eigenfunction is found in section 4. It is shown that the Fourier transforms of the inverse problem data obey the linear equations (1.6). It is demonstrated also that the linearized equation (1.5) coincides with (1.6). The compact formulae for the exact solutions of the Ishimori-II equation via the eigenfunctions of the linear problems (1.7) are derived in section 5. Some explicit formulae for the exact localized solutions of the Ishimori-II equation are presented in section 6.
2. The inverse spectral transform formalism for the Ishimori-II equation
I - x ) , TI = ~(y - x ) the first auxiliary linear problem Ll~b = 0 In the characteristic coordinates ~: = ~(y for the Ishimori-II (Ish-II) equation is of the form (a, 0
0) O~ ~b+ ½Q(a¢ - On)~b= O"
(2.1)
We assume that S ~ S= = (0, 0, - 1) at ~, "q --, oo and Q := P + ~r3. The spectral parameter A is introduced by the transition to the function X defined as [16]
X("rl'A)=~b(~'rl)( e-i'/:~O e'~/~0).
(2.2)
The function X obeys the equation 0
(2.3)
Oa X - (i/2A)[o'3,X] + ½Q(O~- on + i/A)X = 0.
The operator (~ formally inverse to the operator l L ° = ( an0 0~0) - ( i / 2 A ) [ ° ' 3 ' ' ] acts as follows [16]:
:= ( where (~b,, ~1= 62'
4~,2) (~22
'
O '(e
T/t))
'/7))
612( ,
(2.4)
V.G. Dubrovsky and R G. Konopefchenko / Coherent structures for the Ishimori equation, I
371
is a 2 x 2 matrix. The kernel G(~ - ~', r / - r/') of the operator G is the Green function for the problem (2.3). Since the integrals 0~ l and 0~"1 can be chosen as 0~- ~ = f ~ = d ~ ' and ~ - l = fn+=d~7', the Green function G is defined nonuniquely. This freedom allows us to construct the bounded analytic Green functions. Here we will use the following Green functions:
f' d,'<,(e,r/')
f'
+ d r / ' e i(n-n')/a ~t)t2( b~, ')~')
-=
(G+(.,
a ) , / , ) ( ~ , n ) :=
f't de' e-i"-e')/a ~bz,(e',r/)
f*- ~ d'¢
d,7'<,(e,r/')
f" oo (o-(.,a)6)(¢,r/)
:=
)
'/'22(~5',r/)
f"_=dr/' e i("-'')/x q~2(¢, r/') ]
f* d~-'e-i(¢-¢')/'~cbz,(.~',r/) +oo
ff de'*2=(e',r/)
,1
(2.5)
It is easy to see that the Green function G+(A) is analytic and bounded at the upper half-plane Im A > 0 while the Green function G-(A) is analytic and bounded at the lower half-plane Im A < 0. Note that the Green functions (2.5) differ from the Green functions used in ref. [16] by the signs of the lower limits of integration in qS~ and q~22 respectively. With the use of the Green functions G + and G - we define the solutions X + and X- of eq. (2.3) via the integral equations X±(~,rt,a)
a~Ln¢O'- a' +
= 1 - {G-+-( • , , a ~
i/a)X±(',
(2.6)
a)}(¢,'7),
where 0' := 0/0¢', a' := a/0r/'. As far as the Green functions G +, G - , the solutions X +, X- of the integral equations (2.6) are analytic and bounded in the upper and lower half-planes, Im a > 0 and Im a < 0, respectively. Then since G + - G-4= 0 at Im a = 0 one has also X + - X - ~ 0 at Im a = 0. Thus one can define the function X = X +,
forlmA>0,
X = X -,
forlmA<0,
which is analytic and bounded at the whole complex plane and has a jump across the real axis. So we arrive at the standard Riemann-Hilbert problem. As in reL [16] we assume that the homogeneous integral equations (2.6) have no nontrivial solutions. To specify the Riemann-Hilbert problem one must express, according to the standard procedure, the jump X + - X - at Im A = 0 via X-. To do this we firstly obtain the integral equation for X + - X-. Using (2.6), one gets
(x+-x-)(~,r/,a)=r(~,~,*)-(&
1 , a )~o(
0'
-
~' +i/~)(x+-x-))(e,r/)
(2.7)
where
v(~,r/,a) =
o
f_+[ d. e"~'-~"J~~(0(e, , ' ) ( ~ - ~o,+ i/,)x +),2t
-f+?de'e-'
+i/*)X-(~Y',n)):,
o]
(2.8)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
372 and
f_n d ~ ' e i ( n - n ' ) / x ~ , 2 ( ~ , ~ ' )
f" dn'~,,(¢,w')
(6(.,,~)4,)(¢,n) =
(2.9)
f~ d#' e-i~e-e'v* ~2,(~,, a )
fe d~'~22(
~)
Note that the diagonal elements of the matrix F in (2.8) are equal to zero in contrast to the similar matrix considered in ref. [16]. This is due to our choice of the Green functions (2.5). We will look for the expression for X + - X - via X- in the form (x+-x-)(~,rt,h)
dl
(2.10)
-]-fx(i)Z,(~,~7)f(l,a)~;'(~,~),
= -
where "~x(~:' r/) := ( ei'~/aO e -in/aO )
(2.11)
and f(l, A) is the 2 × 2 matrix which must be found. Substituting (2.10) into the r.h.s, of (2.7), we obtain
f
dl X+ - x - = F -
-~75-
A = Q ( o ' - ~'+
i/a)x-(l)
f n-~o dr/' ei(n_n,)/a 2A12 1
~All
dr/'
f ~ dsc ' e -i(~-~')/* -~ 2-A21
(2.12)
f e_ dK' ~l A
22
with ~tf(l,k)
~A
1•
On the other hand from (2.6) one has "~ dl 'iTx-(l)2f,f(l,a)Z;
1
f_
=X+-X -
_ 7~-2,f ( l , a ) ~ ;
_f~
J_~ l 2
f
~
!
f -~'q d'q' e i~"
dr/' ~BI1
f+~ d~:' e-i(e-e')/t
-~')#
~BI2
1 ~1Z t f ( l , h ) y ~ ; l
½B21 l
with
B = Q ( O ' - ~'+ i / l ) x - ( l ) .
(2.13)
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373
Transforming (2.12) with the use of the identity
((0'- ~"+ i/A)x-(l) ~,f(l, A) ~ " ) ( s c, r/) = ((0'- ~'+ i/1)x-(l))(Gn) 2dt(~, ~7)f(1, A) ,~-'(~:, n),
(2.14)
performing the matrix multiplication in (2.13) and subtracting the obtained equations, one gets di
f[[
-ff~,f(I, A) ~ ; '
.,r,;'( ~, n)
-f_ d~'e~e'/*½(e(o'-~'+i/~)x-)~ _~
g
0
_~de'eie'/'½(a(a'-g'+i/l)x-)2~ 0 tf2~
f12 ) ~,a-1(~, r/). f22
(2.15)
Multiplying (2.15) by X, from the right and acting on (2.15) by
f*~d~e -i'/p
1 2~r
0
)
from the left, we obtain
f,1(P,A)
f12(P,A) )
f21( P, A) f22( P, A) (
0 = -VS(p,A )
T~(p, A) )
0
(+~dl[ 0 -J..~ --ff(T~(p,1)
0 fll(l'A) 0)[
f12(l, A) t
y=(l, a) ] '
(2.16)
where +
1
T,d p , a) = ~
f f ded, e -ie/'-''/~ ½(Q(O-~+i/a)x+)12, +~
1
r2q(p, a)= ~ f f dedve ''/'+'e/" ½(Q(O-~+ i/A)x-)=,.
(2.17)
--oo
In matrix form eq. (2.16) looks like f(p,A)+
f+= at _ -~T-(p,I) f(1, A)= T+(p,A)- T-(p,A),
(2.18)
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V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the lshimori equation, I
where (2.19)
T+ = (00
Thus, we have proved that the jump X + - X - indeed is given by (2.10) with the matrix f(l, A) which obeys eq. (2.18) (see also ref. [16]). Note that f is easily expressed via T~z and T~]
f(p,A) =
0
T~2( p, A)
- r~(p,a)
- f +_~ d l ~T - " 2lip , l) r ~ 2 ( l , a ) Joe
(2.20)
So, we have arrived at the standard regular nonlocal Riemann-Hilbert problem. Its solution is given by the rather standard linear singular integral equation [16] +~xa
x-(a)=l+
1
dp dk X - ( P ) X p f ( p , k ) ~ '
-41ff T k2
k-XTN
(2.21)
Eq. (2.21) is the inverse problem for the linear problem (2.9). The functions T~2(A,ix) and T~(A,/.t) are the inverse problem data. The reconstruction formula for the potential P(~,rt, t) is obtained by substitution of the Taylor expansion X( ~:, T~, a) = XO( ~, 77) -t- a)(l( ~, ~) q- •2X2 ( ~, ~) q-...
(2.22)
near a = 0 into eq. (2.3). Consideration of the terms of the order A-1 gives the formula [16]
P( ~, rl, t) = S( ~, rt, t) " rr = _g~sg-l,
(2.23)
where
g ( ~ , n , t ) =X(sC, r/,t,Z = 0 ) +oc
=l +~
1
ff
dl dk
f2 k3 Z ( e , rl, l) Y2)(e, rl) f ( l , k) X c ' ( s~, , ) .
(2.24)
One can also obtain the compact formulae for the topological charge (1.2) and the scalar field ~p. Note firstly that
! S.(&xS,)=4~
i tr(PP, Py). -g-4w
(2.25)
Using expression (2.23) for P, one obtains tr(PPxP,,) = 2tr(~r3((g- l),,gx - (g-')xgy))"
(2.26)
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375
On the other hand substituting (2.22) into (2.3), one gets in zero order in A
c r 3 g - l g x - g - l g y = li[ g-ix1, cr3] •
(2.27)
Differentiating (2.27) with respect to x and y, one obtains o 3 ( g - 1)y g x + O ~ 3 g - ' g x y - ( g - l g y ) y = ½iBy[g-lXl,tr3],
(g-lgx)x--O'3(g-l)xgy--or3g-lgxy=
½io'3 0x [ g - 1X1, o'3] .
The summation of these two formulae gives
Or3((g-l)yg x --(g-l)xgy ) = (g-lgy)y- (g-lgx)x+ li(oy q- or30x) [ g-IXl, or3] • Substituting (2.28) into (2.26), taking into account well-known formula tr(gxg- 1) = Ox In det g, we obtain
that tr((Cqy-t-O'30x)[g-lxDor3])
(2.28) :
0 and using the
tr(PPxG) = 2 t r ( ( g - l g y ) ~ - ( g - l g x ) x ) = 2(0yz - ~2) Indet g = 2 0,0n lndet g.
(2.29)
So the topological charge Q is +oo
Q
i ~(~r
ff
d~:dr/0e0n lndet g.
(2.30)
Eq. (2.29) gives rise also to the following reconstruction formula for the scalar field ~p: q~(~, r/, t) = 2iln det g + 2 0~-1u2(~ ', t) + 2 0~lUl(rl ', t).
(2.31)
Eqs. (2.21), (2.23), (2.24) and (2.31) completely solve the inverse problem for the linear equation (2.3). For the real S the matrix P is self-Hermitian and the relations (2.23), (2.24) imply that g(G r/, t) is the unitary matrix gg* = 1. So Idet g[ = 1 and therefore the scalar field ~o is a real one for the real u~ and u 2. Note that similar expressions for the topological charge and the scalar field can be derived for the Ishimori-I equation (a = i):
Q=
1
ff
dx dY (Ox2 + 02) lndet g
(2.32)
--oo
and ~p(x, y) = 21ndet g. Eqs. (2.31), (2.33) are in complete agreement with the formula ~ = 2 i a l n d e t g different method in ref. [19].
(2.33) -1, derived by a
V.G. D&m&y
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3. ~~en~~t~
and B.G. KorwpeMzenko / Coherenf structures for the ishimori equation, I
spectral data
To study the initi~I-bounda~ value problem for the fshimori equation and to construct the exact solutions we need, similar to the DS-I equation El&-123, the solution of the inverse problem equations with the degenerated inverse problem data, i.e,
For this purpose it is convenient to rewrite the matrix equations @lOI as the system of equations for the columns of x. One has
(3.21 and
!y the inverse prublem equation f2.21) looks like
(3.4)
(3.5)
f3.61 substitution
of (3.1) into (3.4) and (3.5) gives res~ctjvely
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
377
Let us introduce the quantities 1 Fj:= ~
f+®dl {xiz(l)) '-oo ~ -Tj(l) e-in//IX~z(l ) '
(3.9)
\ 1
•
6;:= --~-L+~-~2Sy(l)e'W'{ X~(') [ /XI,(0 ]"
(3.10)
Eqs. (3.7), (3.8) imply now the following system of equations for Fj. and Gy: 1 N_ dA Sj(A) 2-~-= f-~+°~ d._~lzsj(l)eie/, , - Ek=l F f f o- +®"~'-
Gj=(1)
N+
1--~f +°~-~z Tj( l ) e-i./, + E Gk L k=l
o
e i¢/a ( T k ( A )
e-i~/a) +,
+~o _~_ oo
Tj(A)e-i'7/X(Sk(A)ein/a)-.
(3.11)
If one denotes 1 ~i~/,, ~(e) ,= 2-~-f +~_=vS;(t) d; 1
•J(,):= 2-~-~f eT-~i ,~(/' '), _ ajk := f_+? - ~ Tj(A) e-in/x (Sk(A) ein/x) -, ~ik:= -- f_+~-~-z Sj(A) eiWX(f'k(h)e-iWx) +,
(3.12)
then the system (3.11) is represented in the compact form N+
N
k=l
k=l
Ft'- E aikak=(O)"l'i(~), aj- ~ ~jkFk=(~)O'j.(,).
(3.13)
This system is easily solvable and the solution is of the form
N
(N+)
Fj= E (I-A)fp t Y"k=la"kcrk , p=l
"rp
N+
aj = E
. = 1
(1-B)~p I
( N_ Ek =
o'p
l~pk~'k
) (3.14)
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
378
where A and B are N_ x N_ and N÷ × N+ matrices of the form N+
N_
As~ = E OLik[~kj,
Blm = ~ 3U, al,,,.
k=l
(3.15)
k=l
Knowledge of Fj and G~ allows us to find the matrix X (a). Namely, (3.7) and (3.8) imply
X~I(A)
=
XI2(}t)
=
-- 2 ~
0
X22(a)
E F/.(~.(a)e -is~/A) , j=l E a i ( s i ( 1 ~ ) ein/a) -.
_1_~
1
(3.16)
i=1
Finally, one finds S(~:, ~7, t) by eq. (1.8) via a'-(sc, r/, a = 0). Thus, the inverse problem for the Ishimori-II equation with the degenerated inverse data, as usually, is solved explicitly. The corresponding potential P(sc, rl, t) depends on the 2(N++ N ) arbitrary functions of one variable. Note that in the degenerated case the Fourier transforms of the inverse problem data +oo
dl dA T~(l,A)Ae_in/t_i~/a
=
-i f f Tr V dl
(3.17)
--Qo
have the factorized form too: N_
S(~¢, rt) = 2at Y'~ r~(rl) ~i(s¢), i=1 N+
7~(~:, r/) = 2-rr E ~rs(~:) ~(~7),
(3.18)
i=1
where
'1"i('r/)
ri(~) =
and o-i(¢) are given by the formulas (3.12) and
~
i
f2 i
f
~ -A~/(A) e-''/A' dA -~-ASj(A) e iÈ/A.
(3.19)
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379
4. Time evolution of the inverse problem data
In the case of the nontrivial boundary conditions ~°n 6 ~ - 2 2u1(~7'/)'
~06n--~-~z~_ ~ 2u2(sc, t)
(4.1)
the second operator L z, associated with the Ishimori-II equation, must be modified in order that the linear problems Ll~0 = 0,
/~2~0= 0
(4.2)
will be compatible and equivalent to the Ishimori equation. The operator L 1 remains unchanged /,1 = L r The modified operator L z can be constructed analogously to the DS-I equation case [10, 11]. Namely, we will look for this operator in the form L 2 = L 2 + f _+~ -dl ffy(k,l),
(4.3)
where L 2 is given by (1.3) and the kernel y ( k , l) must be found. So the time evolution of the eigenfunction qJ is defined by the equation L2qJ + f _+~ ~dlT 7 t " k , l") ~ b ( I ) = L 2 ~ O + O "
(4.4)
Since the Ishimori equation is equivalent to the condition [L 1, L 2] = 0 and LlqJ = 0, then eq. (4.4) implies LlO = 0,
(4.5)
i.e. O and ~Oare the solutions of the same differential equation. The relation between O and 0 can be found by comparing the integral equations satisfied by these functions. Let us choose ~0 as ~b(,, r/, A) = X_ ( , , .q, A) ( e i o x
0 ) e-in/,~ , :
(4.6)
where X obeys the integral equation (2.6). This function ~Oobeys the integral equation ~O(~'r/'A)=( ei6/A0 e -in/xO )_{~-(.,A)½Q(a,_~,+i/A)~b(.)}(~¢,~7),
(4.7)
where =
(4.8)
The first column of qJ obeys the equation (i//11 ) (e i x) qJ21 = O - { ( ~ - ( " A ) ½ Q ( a ' - a'+i/A)O}(~,r/).
(4.9)
380
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the lshimori equation, I
(°-)
The quantity O obeys the same equation (4.5) as ~b, hence, the first column of 0 obeys the same integral equation as \ ,~, . The only difference is the inhomogeneous term, i.e.
(1911 L'ql'" ) ~, , O 2 1 ) = ( O 2 1 . ] - { G - ( ' , A ) I Q ( "dr- +
(4.10)
i/A)O}(sc,~7).
The quantities ~ii~ and 02i ~ are found from eq. (4.4). Taking into account (1.3) and (1.4) and considering the limit r / ~ -0% one finds i 011~= - 2t---5 e ie/A +
i
-ff~u2(~,t)e ie/A, 021~ = 0.
(4.11)
Comparing now the integral equations (4.9) and (4.11), one gets o.(a))
= _ __
i
)
2A2 162,(A )
+ 2A J_~ 7gY{A- - T
62,(I)
'
(4.12)
where --~1 ( + ° ° d ~ e i e P u 2 ( ~ , t ) .
Y(P) = 2~r .#-oo
(4.13)
Now let us consider equation for the second column
0221
t X~2 e-inlA"
It is of the form
(5t :( o @221
e-irllX
, J2~t
(4.14)
The second column of 0 obeys eq. (4.10) with another inhomogeneous term O. Its form can be found by the consideration of eq. (4.4) in the limit se ~ - oo. One gets
0~2~ = O,
022. = ~
i
e -i~/* + -~-£u~(n, t) e - ~ / ~ .
(4.15)
Comparison of the integral equations for the second columns of 0 and O, as a result, gives
012 i,b22]
-oo ~ Y t "~'- 1 ]{~22(1) '
(4.16)
where
¢/(p) = ~
_ drlui(rl,t)e -inp. 'sT
(4.17)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
381
So, we have found all components of the matrix O. Thus, the time evolution of the quantities [~//ll)=(XH)ei'/;t ~/(/~') = t ~'21
and
¢;(A)=/x~2]
IX~2] e-in/a
X21
is defined by the following equations:
i __i /.+~o d l t *t lip(0f-O,1)2~s+JiR(8 e O,1)0 -~-7*+2Aj_= ~-~)qs(l) =
_
_
~t=½iP(a~-On)2~+¼iR(8~
an)q;+
~ , t
1
(4.18)
and _
i
^
"
dl^[1 _ -iTy[-~-l)~(l),
(4.19)
where
R:=Pt-Pn +(Pe+Pn)P+i(1-P) fn_®d~7'S'(SexSn,)+i(1-P) fe_ood~'S.(Se, xSn).
(4.20)
Similar to the DS-I equation, the nontriviality of the boundaries essentially changes the evolution of the eigenfunction ~ in time. Eqs. (4.18) and (4.19) allow us to find the time evolution of the inverse problem data. Consideration of eq. (4.18) at ~:~ -oo gives rise to the following equation: 1" i i__ [+® dl i l o't(A,rl,t ) =~lO'nn+ul(n,t)o~n---~o'+ 2A2J_o° -/TyrO--
1) tr(l, ~t) T r/
(4.21)
for the quantity ~+~ g dl Z_21(l,A,t)e -in~t= ~r(x,n,t):=J_®
lim ~b21.
(4.22)
Analogously for the quantity
lim 1~12,
--
(4.23)
"r/---~- - o o
one obtains from (4.19)
"rt(~,A, t )
1. = -- ~17"~:~-- U 2 ( ~ , t)'r~: +
i
i /-+~°dl [1
1]
-~-~'r + 2A J-o~ lTYt A- - T ] z(l'sC)"
(4.24)
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V.G. Dubrovskyand B.G. Konopelchenko /Coherent structuresfor the Ishimoriequation, 1
Further, for the full Fourier transforms of the inverse problem data
dl dA AT_, l A)e_in/~_ie/x
TS
r
,
= _ i f +-== -~-A°'(A, dA r/, t) e-ie/a
(4.25)
and
dt + ~(~',v,t) := -iff ~7,~T,2(/,,~)
ei~/l+irl/a
=-if+_f ~2a,c(a,Gt)ei'/a,
(4.26)
one gets the simpler evolution equations. They are S~t(~,'q, t) - ½i(S~¢ + S~,7) + u2(~,t) S ¢ - u l ( ~ 7 , t ) Sn = 0
(4.27)
7~t(G ~7, t) + ½i(T~e+ T , ~ n ) + u 2 ( G t ) T ~ - u , ( ~ , t ) T n = O .
(4.28)
and
Thus, the time evolution of the full Fourier transforms of the inverse problem data for the Ishimori-II equation is given by the simple linear differential equations which contain the boundaries u~(rI, t) and u2(G t) as the potentials. Eqs. (4.27) and (4.28) play a fundamental role for all further constructions. Note that they contain the first-order derivatives over ~ and ~7 in contrast to the DS-I equation case. Eqs. (4.27) and (4.28) are closely connected with the Ishimori-II equation in the weak fields limit. Indeed, in the weak limit 8S 1 := $1 ~ 1, ~S 2 := $2 << 1 and ~S 3 = 0 (since S~- ~S = 0). Neglecting the nonlinear terms, one obtains from (1.5) the system of two equations: ~ S + t - ~ 1 i (S+e~+ S+nn)+uz(~,t)S+¢
ul(r/,t)S+.=O,
,~_, + ½i(S_e~ + S-nn) + u2(~' t) S_e - Ul(r/, t) S-n = 0,
(4.29)
where S~:= 51 +_ iS 2. So, eqs. (4.27), (4.28) which define the time evolution of the inverse problem data coincide with the linearized Ishimori-II equation.
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
383
The exact interrelation between the inverse problem data and S+ and S_ follows from the definitions (2.17) and (4.25), (4.26). In the weak fields limit the formulae (2.17) become d-oo
ff
1
W,+~(p,*)--~
d £ d r l e -ie/p-in/a ½S_(0¢- 0n + i/A)X~2,
--oo
1
--
1f f d~d~e~'/~+~e/*-~S+(ae-O~ + i/,
)xiS.
(4.30)
Then from the integral equations (2.6) it follows that in the weak limit e i(v-n')/* ½S_(0¢ - On, + ilA)x~2,
X~2= - f ; d , '
x~-2--1- f ~ =da '~~S+(Oa,-
0n + i/,~)X~,
XH = 1 - ½fn dr/'S_(0¢ - 0n, + i/3.)Xf, , X2] = - f+~= dsC, e-i(e-e,)/, 5S + l"
(0¢,
--
0n + i/A)XS.
(4.31)
So X~2-=0,
X~2---1,
X~----1,
(4.32)
X~----0.
Substituting (4.32) into (4.30), one gets -}-~
1
r (p,a) ---
ff
d~drle_in/a_i¢/p ~ ix S _- ,
+~
T~(p,A)
---
~
1
ff
d~d.qein/p+i¢/a
~ -i~ S - +.
(4.33)
Combination of (4.33) and (4.25), (4.26) gives rise to the following interrelation between S, 7~ and S+, S_: S(sC, r/,t) ='rrS+({:, rt,/),
T(sC,r/,t) = ~S_(sc,'r/,t).
(4.34)
The coincidence of the equations which define the time evolution of the Fourier transform of the inverse problem data and linearized soliton equation takes place also for the DS-I equation [10-12]. This is a general phenomenon. Formulae (4.33) and (4.34) allow us to describe the constraints on the inverse problem data which correspond to the real field S(~:, 77, t).
384
V.G. Dubrovsky and B,G. Konopelchenko / Coherent structures for the Ishimori equation, I
Indeed, for real S one has S÷= S*. Then relations (4.33) and (4.34) imply
ATS(l, A) = -t(T,2(A, l))
(4.35)
and S ( ~ , r / , t ) = T*(~:,r/,t),
S(~,r/, t) = 7~*(~,r/, t).
(4.36)
Iterating formulae (2.17), we conclude that the constraints (4.35), (4.36) take place also for the nonweak real S.
5. Exact formulae for the general coherent structures of the Ishimori-lI equation The linear equations (4.27), (4.28) are the key equations in the whole method of the study of the coherent structures for the Ishimori-II equation. In what follows we will consider the case of the real functions S(¢, 7, t). In virtue of (4.36), (4.35), in this case it is sufficient to study the single equation iSt +½(S¢¢ + L ~ ) + iu2(~, t ) S . ~ - i u l ( r / , t) L = 0 .
(5.1)
Our aim here is to derive the general formulae for the exact solutions of the Ishimori-II equation via the solutions of eq. (5.1). Solutions of eq. (5.1) can be constructed by the method of the separation of variables g(~, ~7, t) = X(~:, t) V(~7, t)
(5.2)
similar to the DS-I equation [10-12]. In our case the functions X and Y obey the equations 1
iX~ + ~Xee + iuz(g , t) X e = 0,
(5.3)
iYt+~V,, 1
(5.4)
_
iu,(~7, t ) Y,
=
0.
The combination of the exact formulae (3.14)-(3.16) and the factorized solutions of eq. (5.1) will allow us to construct the localized solitons of the Ishimori-II equation. The general factorized solution of (5.1) is of the form S(~, r/, t) = 2 T g E p i j X i ( ~ , t) Yj(T], t )
(5.5)
/j
where Po' are constants and Xi(¢, t), Yi(r/, t) are the solutions of eqs. (5.3) and (5.4), respectively. The equivalent form of S is
g(~, r/, t) = 2"rr• r/(r/, t) ~i(~, t)
(5.6)
i
with
zi(~,t) = ~.,PijYj(rl,t), J
~'i(~,t) =Xi(~,t).
(5.7)
V.G. Dubrovsky and B.G. Konopelehenko / Coherent structures for the Ishimori equation, I
385
For such a S(s¢, rl, t) one has -.~- oo
d~d-q
ff
2~x ~-'X"(~'t) Yj('q't)PiJ eif/a+in/, i,j
= ~.,T~(l,t) ~.(,~,t),
(5.8)
i
where
Ti(l,t) = ~Pij j
f+
~
~d~7 e'n/tYj(r/,t), -
--¢m
7~/(h, t) = ~- _
~
e if/x X/(~, t).
(5.9)
Using (4.35), one also gets
+~o ~dn Yj*(rl, t) e -'~/~ T~2(I, A, t) = EP* iA gf+o~ ~o ~dE Xi.(~,t)e_ie/l Loo ij
= •Si(l,t) L(A,t),
(5.10)
i
where
Si(l,t)=S+_~d2~xi*(<,t)e-i
=
~Pkjf+o~ ~dr/ j
Ai YJ*(~' t)
e -in/'~.
--oo
(5.11)
The formulae (5.8)-(5.11) give the concrete expressions for the degenerated inverse problem data. Using their formulae, one can obtain the concrete expressions for the other needed quantities from section 3. Taking into account that (7~k(A' t) e-ie/~)-+
i
2x/~ A f~¢~d~'X~(~''t)ei(g'-g)/A
(5.12)
and (S~(A' t) ei'~/a) "+-
~
i
"q
•
~jP~Jf±ood'q'e'(n-~"/~Ys (rl''t)'
(5.13)
V.G. Dubrousky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
386
one gets ~ri(~,t) =Xi*(~,t),
rj(rl,t)
= ~_~PjkYk(~7,t), k
= (papt)jk, O~jk = E pjlPkmalm * l,m
= bi, = - -f¢
(¢, (A ' t)e -i~/*)
Xk(~', t) a¢,Xi*(~', t), ---~
1
)t~O ~
(Sk(a,t)ei~/~) - ~
'
~1
Y'~Yj*(~7 , t)(pt)jk.
(5.14)
J
Using these expressions and eqs. (3.14), we obtain
Fj =
E(1
-
t -'(Ep(papt)kpX*(~'t)
pap b)j k
(5.15)
F~tPktYt(rl, t)
,-,( X~(~,t) ) Gj= E(lk - b p a p )jk ~,,mbktPlmrm(rl,t) •
(5.16)
Further, eqs. (3.16) give
1 - (X,(1 -paptb)-'paptX*)
_(X,(l_paptb)-,py)
g=X-(A=0)=
(Y., p,(1- b,,a,,*)-'X*) l+(Y*,pt(1-bpapt) 'bpY)
(5.17)
where we denote ( X , Y ) = Y'.iXiY,. At last, rewriting the reconstruction formulae (2.23), (2.31) in the components X(A = 0) we get
2x2:(o) x:,(o) s+ = - x,,(o)
S =
x2:(o)
- x,2(o)
x:,(o)
'
2X,,(0) X,2(0) x , , ( o ) x22(o) - x , 2 ( o ) x2,(o) '
Xl,(O) x~=(o) + x,~(o) x2,(o) s3 = - x , , ( o ) x~2(o) - x,~(o) x~,(o) "
(5.18)
and q~(sc, r/, t) = 2i In[x,,(0) X22(0) - X,2(0) X2,(0)]
+ 2f e d~'u2(#',t) + 2f" dw'u,(n',t)
+ const.
(5.19)
t
387
Eqs. (5.18), (5.19), (5.17) are the main result of the present paper. They allow us to construct explicit exact solutions of the Ishimori-II equation with nontrivial boundaries using the exact solutions of the linear equations (5.3), (5.4). Noting that the summations in (5.15), (5.16), (5.17) may be also continuous, these formulae correspond to any set of the exact solutions of eqs. (5.3), (5.4). At this point the situation is similar to the DS-I equation [10-12]. An essential difference between the Ishimori-II and DS-I equations is that in our case the linear equations (5.3), (5.4) contain the first-order derivatives over ~ and 7/ and the pure imaginary potentials iul(r/, t) and iu2(~, t). For the DS-I equation the corresponding linear equation is the one-dimensional nonstationary Schr6dinger equation [10-12]. This last equation is associated with the IST integration of the Kadomtsev-Petviashvili (KP) equation [1-4, 10-12]. In our case the linear problem (5.3) (or (5.4)) is associated with the IST integrability of the modified KP equation. Indeed, the modified KP equation [20] u t + U x x x + ~ 3U 2 ux g 3 O x l u y y + 3 U x ~.l u v = O
(5.20)
is equivalent to the commutativity condition [L~, L 2] = 0 of the operators [21] L 1 = iOy + ~ + iu%<, L 2 = 0, + 40;~ + 6iuO 2 + [ + 3iu x + 3(3~- tuy) - 3u2] Ox + c.
(5.21)
This fact is not so surprising if one takes into account the formal gauge equivalence between Ishimori and DS equations mentioned in refs. [15, 16] and the gauge equivalence between KP and mKP equations studied in ref. [20]. Indeed, it is easy to see that eqs. (5.3), (5.4) convert into the nonstationary Schr6dinger equations 2i)(, + Xf< + (u 2 + iu2¢ + 2 ~;lu2,(~:', t ) ) X = O, 2iYt + Ynn
+(u2-iuin-20;iul,(~l',t))Y=O, (5.22)
under the gauge transformations
)(=Xexp(iS~d<'u2(<',t)), Y=Yexp(-iS~d~l'ul(~7',t)).
(5.23)
We see that the solution of the general initial-boundary value problem for the Ishimori-II equation is connected with the solution of the initial value problem for the modified IZd~ equation. This last problem will be considered in a separate paper.
6. Time-independent boundaries In this paper we restrict ourselves to the case of the time-independent boundaries u~ =u~(r/), u 2 = u2(st). In this case we will be able to use the previously known results for the special Zakharov-Shabat spectral problem. For the stationary boundaries ul(r/) and u2(£) eq. (5.1) admits the full separation of variables: S(~, 7, t) = T ( t ) X ( ~ ) Y(~q).
(6.1)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
388
The functions T, X and Y obey the following linear equations:
T,(t) + ½i(A 2 + A'2)T= 0, X~g + 2iUz(s~) X~ + h2X(~:)
= 0,
(6.2)
Y,n - 2iul(rl)Y n + a'2y(rl) = 0
(6.3)
where A and A' are parameters. The one-dimensional spectral problems (6.2), (6.3) can be investigated by different methods. Here we will use their equivalence to the specialized Zakharov-Shabat spectral problem. Firstly we note that the spectral problems (6.2) and (6.3) are equivalent to the problems ~sesc nL [A2 4- u2(~) -iu2~(sc)] .,Y = 0
(6.4)
1 2 + [A'2+uZ(r/) + iul,7(,r/)]12=O
(6.5)
and
via the gauge transformations
So, the problems (6.2) and (6.3) are gauge equivalent to the spectral problems of the Schr6dinger type
q ~ + [A 2 + u2(p) 4- iup(p)]q~+-(p,A) = 0
(6.7)
with the very special potential
V(p) =
- [u2(o) +
iuo(P)],
(6.8)
where u(p) is a real function. Then, it is not difficult to see that the spectral problem (6.7) can be rewritten in the following matrix form 01p + iu01 = h02,
0 2 p - iu02 = - h 0 1 .
(6.9)
Indeed, the elimination of 0[12 or 01 from (6.9) gives rise to the scalar equations
Olpp+(A2+u2+iUp)tOl=O,
02pp-k(A2+u2--i.#)02=O.
(6.10)
So, one can identify q~+(P) = 01(P),
¢ - ( P ) = 02(P)-
(6.11)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
389
At last, transiting to the variables 01 "-
$1 + i$2 2
'
02 :=
if/1 --
2i
i$2
'
(6.12)
we arrive at the specialized Zakharov-Shabat spectral problem
.)
Ol
It follows from all these formulae that 2(~:) = - 0 2 ( ~ ) - i01(~) ,
17(~) = 0 , ( ~ ) + iO2(n),
(6.14)
where 01 and 02 are the solutions of the spectral problem (6.13). So we are able to use the exact results associated with the problem (6.13) for the construction of the exact solutions of the problems (6.2), (6.3). Of course, for the calculation of the function X(~:) we must use u2(~) as the potential u in (6.13) and for calculating the function Y(rI) one must use ul(~7) as u in (6.13). The spectral problem (6.13) has been studied in detail in the papers [18, 22] in connection with the IST integration of the modified KdV equation. It has been shown in refs. [18, 22] that the spectrum of the problem (6.13) contains a continuous part defined on the real axis Im A = 0 and a discrete part which consists of the points located symmetrically with respect to the imaginary axis Re A = 0. The points laying on the imaginary axis correspond to the soliton potentials u and each pair of the points _+a + i/3 correspond to the breathers. Since the problem (6.13) is not self-adjoint then the discrete spectrum of (6.13) contains also multiple points which are associated with the multiple poles of the reflection coefficients [23]. The eigenfunctions 0 n and the potentials u which correspond to the discrete spectrum are found as usually [1-4] explicitly in the closed form. The general soliton-breather eigenfunction which corresponds to N 1 solitons and N 2 breathers is of the form [18, 22] On(P):=O(P'An)=~"(I+Mm
)nm
t
1 e
+
0
k
where the (N~ + 2 N 2) × ( N l + 2 N 2) matrix M looks like
ei(X, +Am)pCm Mnm
an + Am
,
(6.16)
where n , m = 1 , 2 , . . . , N 1 + 2 N 2 and An = + a n + i/3 n. The corresponding general transparent N 1soliton + Nz-breather potential u is given by the formula
u(p)
2cl ~p Imdet(1 + i M ) dp I m l n d e t ( 1 + i M ) = 2 arctg R e d e t ( 1 + i M ) '
d2~ Indet(1 + M 2 ) . u2(p) = dp---
(6.17)
V.G. Dubrousky and B.G. Konopelchenko /Coherent
390
structures for the Ishimori equation, I
If all A,, = ifi, (Im p, = 0) then eq. (6.17) gives the general N,-soliton potential. At all A, = t_cx, + i/I, (Im cy, = Im J3,z= 0) we have the genera1 N,-breather potential. Correspondingly eq. (6.15) gives the N,-soliton ~igenfunction and the &-breather eigenfunction. Eqs. (6.15)--(6.17) alIow us to construct the exact solutions of the problems (6.21, (6.3). For the potentials ~~(8) and u,(n) given by eq. (6.17) the eigenfunctions X and Y are of the form
where the functions 6, and 6, are given by the formula (6.15). Note that the function XCN,&<) corresponds to the N,-soliton + N,-breather boundary function u,(l) while Y$,, rj,,(?~) corresponds to the fii-soliton + ~~-breather bounda~ function u,(n). Now using eqs. (5,14)-(5.19), (6.1%(6.181, we obtain the exact explicit solutions of the Ishimori-II equation with the general stationary transparent boundaries u2(<) and U-&V)_ These solutions are defined by the arbitrary N,-soliton + N,-breather boundary function u,(5), N,-soliton + N,-breather boundary function u,(q) and arbitrary constants plk. The matrices a and b involved in eq. (5.17) can be represented in a more compact form. Indeed, taking into account (6.181, one gets
(6.19) and
(6.20) So, one has
adrt) = A,c,,t~), b;, = ATcik(Or
(6.21)
where cik(p) :=I’
--r
dp’~(h,,p’));*(Aj,p’).
(6.22)
V.G. Dubrovskyand B.G. Konopelchenko/Coherentstructuresfor the Ishimoriequation, 1
391
Note once more that Cmt(rl) is calculated via the functions (Y(Ai,-O) , X(Ak,~7))= (i//i,[]/2) which correspond to the potential u = ul(r t) in (6.9) while cik(~) is associated with the potential u = u2(~:). The soliton (s) and breather (b) are quite different transparent potentials. As a result, we have four different types of the exact solutions of the Ishimori-II equation:
Sss(N,M)(~,rl,t),
Ssb(N,M)(~,TI,t),
Sbs(N,M)(~,'q,t),
Sbb(N,M)(~,TI,I),
which correspond to the choices of X and Y as the soliton's or breather's eigenfunctions. The solutions Ssb and Sbs are, obviously, related by the interchange sc ~, ~/. So, we have three essentially different types of the exact solutions:
Sss(N.M), Ssb(N,M), Sbb(N,M)
(6.23)
where the integers N and M correspond to the N-soliton (breather) boundary u2(~:) and M-soliton (breather) boundary ul(r/). All these solutions can be calculated explicitly. Here we present the simplest examples of such exact solutions. Using the formulae given in the appendix, one gets the simplest (5.5) localized soliton with N = M = 1: S(s,sXl,l)l =
2e-V¢-vn
--~ e-2Vn
>(
)
c 2/z e-2~'g c°s[ ½(/x2+v2)t] +
( 4--~ cd
e-z~'g-2~" + l -
( cd e-2~-2vn 4- - - ~ - - - 1 ) 2 ( ~ v e - 2 V ~ 7 +
+ ~
c
tZV)sin[½(~2+v2)t -c'd-
e - 2t~tj
]
)2
S(s,s)(l, 1)2 = 2 e - / ~ . v-o
-~-1-
4--~e-Z"e-a~/
) cos[½(.2+vZ)t]
×
cd
/zv - 1 e_2~,_2~ n + -c-d-
c
+
e -2~,
+
e -2~n + ~c e -2~,¢
)2
2/* e-2~'¢) sin[ ½(/z2 +
v2)t]
2 e - 2v'q - 2t.t,f S(s,s)(1,1) 3 = - 1 +
cd
tx v - 1 e - 2 ~ - z " n + 'c-d
+
e_2~,n + ~
e -z~'~
/.L e _ 2vrt 2 ~ e - 2 ~ ¢ + 2-C q~(s's×l'l)=--2arctg [( c ) 2 ][( d ) 2 ] 1 j[£/.' e-4~'e + 1 )-flu e-4Vn + I + ~ e -4~:-4vn - - -
cd
+ 4 arctg( 2-~- e-2~'~) + 4 arctg(~-~v e-2~n) + const. ,
(6.24)
where/x > 0, v > 0 and c, d are real constants. It is easy to see that this solution is of the breather type in finite t.
392
V,G. Dubrovstcy and B, G, Konopelchenko / Coherent structures for the Ishimori equation, 1
The simplest (s, b)-type localized soliton for the Ishimori-II equation is ( N = M = 1):
P = P~
, papt=alt,
P~z = 1 0
p Y = p I ~ Y j = Y~, 2(1 + b * a n ) X * Y ~ *
S(s,b×l,l)+ = S(s,bX~,~)~ + iS(s,bX1,D2 =
l1 - a l i b i 2
2[XlzlYll 2 S(s,bXI,1)3 = - 1 +
I1 - a l i b i z '
q%,t, xl, l) = 2i ln(det g ) + 2 a~-lu2(s~') + 2 0~"1u1(rl'),
(6.25)
where
1 -a~b* "
detg=
l-anb
atl =
i ( a + ifl)e ~+2'~m 4/3[eh(Z/3r t + g,) + ( i f l / a ) s i n ( 2 a r / + q~ - 6)] '
b= -
t~/d ( d / 2 / . t ) e -2~e + i '
X=X(A
i e - tt~+ itx2t/2 = i/z,~:,t) = - - ( d / 2 / z ) e -2u~" -- i ' (e-ic~-/3n-i~
+ia + i ei~'7+t~n +q') e-i(a2-t32)t/2+ctt~t
YI = Y(2ti = oe + i/3, r/, t) ........ 2[clh(2~r/+ ~ ) + i ( / 3 / a ) s i n ( Z a r / + ~o " 6)]
e_e, = t - a n b c l - a n b a 2 ~
+~82
a+i/3=¢-~flZeia,
'
c=l-aubcl-a~lbei%
'
u,(~7) = - 2 ~-~ arctg( fl sin(2a~7 + ~ - ~ )
)
u 2( ~ ) = 2 ad-d-ia r c t g [ ( d / 2 / z ) e - 2 ~ # ] . %
At last, the simplest N = M = 1 localized soliton of the (b, b) type is of the form
P = ~ P21
P22 ]
'
papt =
+
=
2(1 + a22b~])X('Y ~
I1 S(b,b×l,l)3 = -- 1 +
-a=b1112
21XlI21Y212 I1 -- a2zb1112'
~o(b,b×l,~) = 2 i l n ( d e t g ) + 2 O~-lu2(# ') + 2 O~ut(~7'),
(6.26)
F.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
393
where 1 - a22b11 det g = 1 - az2bll" i ( a + i/3) e q'l-2aclt a22 = -- 4/3[ch(2/3rl + ~1) + i ( f l / a ) s i n ( 2 a r / + ~1 - 3)] ' i( - a + i/3) e o2+2aclt bll -- 4/3[ch(2/3~: + 4'2) + i ( / 3 / a ) sin(2a¢ + ~o2 - 3)] ' X 1 = X ( A l = a + ifl,~:,t) (e ia¢+/36+~2 h- i e -ia6-/3¢-i~'2+i'~) e -i(a2-¢2)t/2+a#t 2[ch(2/3~: + ~2) - i ( f l / a ) sin(2o~: + ~02 - 3)] I"2 = Y(A'2 = - a + i/3, n, t) (ei,~n -/3n + i t ¢ I -i~ d'- i e -ian +/3n+~) 2[ch(2flr/+ ~tl) + e-4,~ =
Ickla 218~fl
2
e
-i(a2-~2)t/2-a~t
i(fl/a) s i n ( 2 a r / +
(k=l,2);
~o1 -- (~)]
a+ifl=V/a-~fl2ei~;
Ck=lcklei~k
(k=l,2),
/3 sin(2ar/+ ~1 -- ~) ),
u'(r/) = - 2d-~ arctg( a
~
~-~-7)-
1 Ua(~¢) = - 2
(/3 s i n ( 2 a , + ~%- ~) ) arctg ~- ch(2flrl + ~2) "
Notice that the poles of the reflection coefficient which determine the breathers ul(r/) and u2(~) are chosen in the (b,b) solution of eq. (6.26) to be the same: a 1 = h'~ = a + i/3, h 2 = h'2 = - a + i/3. All these solutions are of the breather type, all of them decrease exponentially at ~:2 + r/2 ~ oo at all directions, but their forms are quite different. The solution $(b,s) is obtained from Sts.b) by the interchange ~: ,~, ~7These examples demonstrate the richness of the types of the coherent structures for the Ishimori-II equation. In more detail the properties of the localized solitons for the Ishimori-II equation, the case of time-dependent boundaries and scattering of the localized solitons will be considered in the second paper.
Acknowledgements We are thankful to A. Degasperis and P. Santini for the useful discussions. One of the authors (B.G.K.) is very grateful to the Department of Mathematics of the Yale University and, especially, to R. Beals for the warm hospitality.
394
V.G. D u b r o v s k y a n d B . G . K o n o p e l c h e n k o / C o h e r e n t
s t r u c t u r e s f o r the l s h i m o r i equation, 1
Appendix H e r e we present for completeness some intermediate formulae which are n e e d e d for the calculation of the exact solutions of the Ishimori-II equation. T h e simplest soliton potential for the spectral p r o b l e m (6.13) corresponds to N~ = 1, N 2 = 0, A = i/* (/* > 0) and cl = b(i/*)/a'(i/*) -- ic (Ira c = 0) [3, 18, 22]. It is of the form
u(p) = 2
_~p
(c arctg ~
) 2/* sgn c e -2*'p = - ch(Z/*p + P0)
(1.1)
where e p,, = 2 / , / I c ] . T h e c o r r e s p o n d i n g eigenfunction ,~ is e - it.tp + it~2t / 2
f,(p) = -i
(A.2)
( c / 2 / * ) e -2t*p + i"
T h e quantities a and b are ~, 1 a = ~ ( d / 2 v ) e -2~n + i '
b-
/1. 1 c ( c / 2 / * ) e -2~*~ + i"
(A.3)
T h e functions Xl(s~) and Yl(rl) are
- i e -~+it~2t/2 Xl(s~) = ( c / 2 / * ) e - 2 ~ - i '
e -~'~j+ivzt/2 Yl(r/) = ( d / 2 u ) e -2~n - i
(A.4)
Using eqs. (A.2)-(A.4) and choosing p = 1, one obtains from eqs. (5.17)-(5.18) the solution (6.24). T h e simplest b r e a t h e r potential corresponds to N 1 = 0, N 2 = 1, A1 = a + i/3, A2 = - a + i/3, a,/3 > 0, c I = b ( A O / a ' ( A I) = ic, c 2 = b ( A 2 ) / a ' ( A 2) = ic* [3, 18, 22]. It looks like
/,/(p)
-2 d
17
where tg 6 = f l / a ,
arctg/3 s i n ( 2 a p + q~ - 6) ach(2/3p +t0) '
e* = 2 f l ~
(A.5)
+ / 3 2 / a l c ] , c = Icle i*. T h e c o r r e s p o n d i n g eigenfunction l?l(p) is of the
form i e i°~p+flp+4' +
e-iap-flp
-iq~+i8
)?'(P) = 2ch(Z/3p + to) - 2 i ( / 3 / a ) s i n ( Z a p - 6 + q~) "
(A.6)
T h e function Yl(P) looks like i e ictp + flp + ~t .OF e - iap - flp - iq~+ i~
Y1(P) = 2 c h ( 2 / 3 p + to) + 2 i ( / 3 / a ) s i n ( 2 a p - 6 + q~) "
(A.7)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
395
We present here also one general formula. Using eq. (6.15) for 1~1, 1~q2 and eq. (6.17) for u, it is not difficult to show that the functions X~ and Ym can be represented in the following compact form:
~_,(1-iM)~/exp(iAp~-iA2nt/2)exp(-ifgd~,u2(~,))
Xn(~,') =X(An,sc, t ) = _
p
= - - e -i'~{'/2 Y'.
P
(1 - iM)~7
Ym(~7,t)=Y(Am,rl,t ) = i ~ ( 1
e ixp~: (det(1
+ iM))*
det(1 + i M )
'
(; d'O'Ul('O') )
+lM)mpexp(iAprl -iAM/2)exp i
p
= ie_iA~,/2~( 1 + p
(A.8)
iM)2,~ei~p n det(1 + iM) (det(1 + i M ) ) *
"
(1.9)
These formulae essentially simplify all calculations.
References [1] V.E. Zakharov, S.V. Manakov and S.P. Novikov. Theory of Solitons. The Inverse Problem Method (Nauka, Moscow, 1980); Consultant Bureau (1984). [2] M.J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform (SIAM, Philadelphia, 1981). [3] A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985). [4] Y. Satsuma, J. Phys. Soc. Japan 40 (1976) 286. [5] L.A. Bordag, A.R. Its, V.B. Matveev, S.V. Manakov and V.E. Zakharov, Phys. Lett. A 63 (1979) 205. [6] H. Cornille, J. Math. Phys. 20 (1979) 144. [7] M. Boiti, J.J.-P. Leon, L. Martina and F. Pempinelli, Phys. Lett. A 132 (1988) 432. [8] M. Boiti, J.J.-P. Leon and F. Pempinelli, Phys. Lett. A 141 (1989) 101. [9] M. Boiti, J.J.-P. Leon and F. Pempinelli, preprints Montpellier, PM/88-40 (1988); PM/88-44 (1988); PM/89-17 (1989); PM/90-04 (1990). [10] A.S. Fokas and P.M. Santini, Phys. Rev. Lett. 63 (1989) 1329. [11] A.S. Fokas and P.M. Santini, Dromions and a boundary value problem for the Davey-Stewartson 1 equation, Physica D 44 (1990) 99. [12] P.M. Santini and A.S. Fokas, The initial-boundary value problem for the Davey-Stewartson equation . . . . preprint 684, Universita di Roma (1989). [13] P.M. Santini, Physica D 41 (1990) 26. [14] Y. Ishimori, Progr. Theor. Phys. 72 (1984) 33. [15] B.G. Konopelchenko and B.T. Matkarimov, Phys. Lett. A 135 (1989) 183. [16] B.G. Konopelchenko and B.T. Matkarimov, J. Math. Phys. (1990), to appear; preprint Saclay SPhT/88/229 (1988). [17] R. Beals and R.R. Coifman, Inverse Problems 5 (1989) 81. [18] M. Wadati, J. Phys. Soc. Japan 32 (1972) 168. [19] B.G. Konopelchenko and B.T. Matkarimov, Stud. Appl. Math. 82 (1990) 319; preprint INP 89-123 (1989). [20] B.G. Konopeichenko, Phys. Lett. A 92 (1982) 323. [21] B.G. Konopelchenko and V.G. Dubrovsky, Phys. Lett. A 102 (1984) 15. [22] M. Wadati, J. Phys. Soc. Japan 34 (1973) 1289. [23] M. Wadati and K. Ohkuma, J. Phys. Soc. Japan 51 (1982) 2029.
Coherent structures for the Ishimori equation I. Localized solitons with stationary boundaries V.G. Dubrovsky Institute of Nuclear Physics, Novosibirsk-90, 630090, USSR
and B.G. Konopelchenko 1 Department of Mathematics, Yale University, Yale Station, New Haven, CT 06520, USA
Received 2 July 1990 Accepted 2 September 1990 Communicated by A.C. Newell
The initial-boundary value problem for the Ishimori-II equation is studied. The time evolution of the inverse problem data in the case of nontrivial boundaries is found. A general formula for exact solutions of the Ishimori-lI equation with nontrivial boundaries is derived. Localized soliton solutions of the Ishimori-lI equation with time-independent boundaries are studied in detail. It is shown that there exist essentially three different types (ss, sb, bb) of localized solitons for the Ishimori-ll equation. Some explicit examples of such solutions are presented.
I. Introduction Nonlinear evolution equations in 1 + 1, 2 + 1 and multi-dimensions integrable by the inverse spectral transform (IST) m e t h o d form a wide class of differential equations with a n u m b e r of remarkable properties (see e.g. refs. [1-3]). A main feature of such equations in 1 + 1 dimensions is the existence of soliton solutions which are localized in one dimension. Solutions of the (2 + 1)-dimensional integrable equations which are localized only in one dimension (plane solitons) were constructed years ago [4, 5]. T h e n the lumps (rational nonsingular solutions) were also discovered [5]. Exact solutions exponentially localized on the plane but decreasing at t ~ + ~ have b e e n constructed in ref. [6]. A n d only recently the travelling solutions of the soliton type of the (2 + 1)-dimensional equations exponentially localized in all directions on the plane have been found. For the first time this has b e e n d o n e in ref. [7] by the use of the B~icklund transformations. Spectral theory of such localized solitons for the D a v e y - S t e w a r t s o n I (DS-I) equation and their connection with the initial-boundary value problem for the DS-I equation have been studied by different m e t h o d s in a series of papers [8-13]. T h e localized solitons (or dromions) of the DS-I equation possess properties which are different from the properties of one-dimensional solitons. In IPermanent address: Institute of Nuclear Physics, Novosibirsk-90, 630090, USSR. 016%2789/91/$03.50 © 1991 -Elsevier Science Publishers B.V. (North-Holland)
368
V.G. Dubrousky and B.G. Konopelchenko / Coherent structures for the Isltimori equation, I
particular, they can be driven by a change of the boundaries [11-13]. All this demonstrates the richness of the coherent structure associated with the (2 + D-dimensional integrable equations. The present paper is the first one from a series of papers devoted to the study of the coherent structures for the Ishimori equation
s , ( x , y, ,) + s x (sxx +
+
= o,
@xx -- Og2q)yy q" 20~2S" (S~. x Sy) = 0,
(1.1)
where S = (S~, S 2, S 3) is the three-dimensional unit vector (S 2 = 1), q~ is the scalar field and a 2 = ± 1. Eq. (1.1) is the (2 + 1)-dimensional integrable generalization of the Heisenberg ferromagnet equation (isotropic Landau-Lifshitz equation) S, = S × S**. It has been introduced by Ishimori in ref. [14]. An important feature of eq. (1.1) is the existence of classes of topologically nontrivial and nonequivalent solutions which are classified by the topological charge [14]
Q=
f dxdyS'(&×S,,).
(1.2)
This topological invariant is connected with the mapping S (2) ~ S (2) defined by the unit vector
S(x, y, t) with the boundary value S~ = (0, 0, - 1). The Ishimori equation (1.1) is of great interest also since it is the first example of an integrable nonlinear spin-one field model on a plane. The applicability of the IST method to the Ishimori equation (1.1) is based on its equivalence to the commutativity condition [Ll, L 2] = 0 of the operators [14]:
L l = a ~ v + PO,, L 2 = i O, + 2PO 2 + (Ix + aPyP - ieg3Pq~x + i % ) Ox,
(1.3)
where P = S" o- and o- = (o-l, % , 0"3) are Pauli matrices and 0x = a/0 x, Oy-= O/Oy, ~t - a/Ot. The standard initial value problem for the Ishimori-I ( a = i) and Ishimori-II ( a = 1) equations with vanishing boundary values has been solved in refs. [15-17] with the use of the ~-method ( a = i) and the nonlocal R i e m a n n - H i l b e r t problem method ( a = 1) respectively. The exact rational-exponential solutions ( a = i) and solutions with functional parameters (o~ = 1) have also been constructed in refs. [15, 16]. Our aim is to solve the initial-boundary value problem for the Ishimori-II equation with nontrivial boundaries. In the characteristic coordinates ~: = ½(y + x), r I = ½(y - x ) in virtue of the equalities q~e = f_n d r / ' S "(Se x S , , ) + 2u2(~:, t ) ,
¢ , = f~_ d ~ ' S ' ( S ~ , x S , ) +
2ut(r/,t),
(1.4)
eq. (1.1) at a = 1 is equivalent to the equation
l(f, drI,S.(SE×S,,)+2Uz(,t))S - l ( f~_~ d~' S "( S¢' × Sn)
2u,( ~, t ) )S, = O,
(1.5)
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
369
where u~(~7,t) and U2(~,t) are arbitrary scalar functions. The initial-boundary value problem is to solve eq. (1.5) with given initial value S(s¢, ~7,0) and boundary values u1(~7,t) and u2( ¢, t). We will use the method proposed by Fokas and Santini for the DS-I equation [10-13]. Firstly we present the solution of the forward and inverse spectral problems for the auxiliary linear problem L~O = 0 slightly different from that given in ref. [16]. Consideration of the nontrivial boundaries requires a modification of the operator L2. The obtained equation L2¢, = 0 gives rise to the following linear equation for the Fourier transform S(~¢,~7,t) of the inverse problem data St - ½i(S,¢ + S,,7) + u 2 ( £ , t ) S ~ - u , ( ~ l , t ) S n = O ,
(1.6)
which coincides with the linearized equation (1.5) for S+= S 1 + iS 2. For real S, separation of variables reduces the problem of solving eq. (1.6) to solving the linear equations 2iXt(sc, t) + Xg¢ + 2 i u 2 ( e , t ) X ¢ =
O,
2iYt(r/, t) + Ynn - 2iu,(~7, t ) Y n = 0.
(1.7)
Using the exact solutions of eqs. (1.7) and the solution of the factorized inverse problem for the auxiliary linear problem L~@ = 0, we construct the exact solutions of the Ishimori-II equation (1.5). They are representable in the form S(~:, r/, t) = - tr(o'g~r3g-I ), ~(~:, ~7, t) = 2iln(det g) + 2 ~'1u2(¢', t) + 2 ~ l u l ( ~ 7 ' , t),
(1.8)
where
_ i_ ( (x,(1- pap*b)-'pap*X*) - (Y.,p*(1- b,op*)-'x.) } (X,(1-pap'b)-'pY)
-(Y*,p*(1-bpap*)-'bpY)
(1.9)
where (X, Y) :-- EiXiY~, X i and Yi are the solutions of eqs. (1.7), p# are arbitrary constants and a ik =
f
d rl ' Yk* ( rl ' , t ) O, , Yi ( ~O' , t ) ,
bik = - j Coods~ ' Xk ( ¢', t) a¢, X,* ( ¢', t ) .
(1.10)
Charge Q for these solutions is also given by the compact formula +o~
i
ff
OeO.ln(det g) d e d n .
(1.11)
In the present paper we consider in detail the case of the stationary" boundaries ul(~0) and u2(~:), In this case the linear problems (1.7) are gauge equivalent to the 2 × 2 matrix Zakharov-Shabat spectral problem with the reduction which gives rise to the modified KdV equation. The inverse problem for that spectral problem has been solved in ref. [18] and the corresponding discrete spectrum includes the solitons (s) and the breathers (b). As a result, we have essentially three different types of exact localized
370
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
solutions of the Ishimori-II equation: Sss, Ssb and Sbb, which correspond to the choice of the functions X and Y as the soliton or breather eigenfunctions. The simplest examples of such solutions are presented explicitly. We discuss also more complicated exact solutions of the Ishimori-II equation. The paper is organized as follows. In section 2 the solution of the inverse spectral problem for the Ishimori-II equation is presented. The compact formulae in the case of the degenerated inverse problem data are derived in section 3. The time dependence of the Ishimori-II eigenfunction is found in section 4. It is shown that the Fourier transforms of the inverse problem data obey the linear equations (1.6). It is demonstrated also that the linearized equation (1.5) coincides with (1.6). The compact formulae for the exact solutions of the Ishimori-II equation via the eigenfunctions of the linear problems (1.7) are derived in section 5. Some explicit formulae for the exact localized solutions of the Ishimori-II equation are presented in section 6.
2. The inverse spectral transform formalism for the Ishimori-II equation
I - x ) , TI = ~(y - x ) the first auxiliary linear problem Ll~b = 0 In the characteristic coordinates ~: = ~(y for the Ishimori-II (Ish-II) equation is of the form (a, 0
0) O~ ~b+ ½Q(a¢ - On)~b= O"
(2.1)
We assume that S ~ S= = (0, 0, - 1) at ~, "q --, oo and Q := P + ~r3. The spectral parameter A is introduced by the transition to the function X defined as [16]
X("rl'A)=~b(~'rl)( e-i'/:~O e'~/~0).
(2.2)
The function X obeys the equation 0
(2.3)
Oa X - (i/2A)[o'3,X] + ½Q(O~- on + i/A)X = 0.
The operator (~ formally inverse to the operator l L ° = ( an0 0~0) - ( i / 2 A ) [ ° ' 3 ' ' ] acts as follows [16]:
:= ( where (~b,, ~1= 62'
4~,2) (~22
'
O '(e
T/t))
'/7))
612( ,
(2.4)
V.G. Dubrovsky and R G. Konopefchenko / Coherent structures for the Ishimori equation, I
371
is a 2 x 2 matrix. The kernel G(~ - ~', r / - r/') of the operator G is the Green function for the problem (2.3). Since the integrals 0~ l and 0~"1 can be chosen as 0~- ~ = f ~ = d ~ ' and ~ - l = fn+=d~7', the Green function G is defined nonuniquely. This freedom allows us to construct the bounded analytic Green functions. Here we will use the following Green functions:
f' d,'<,(e,r/')
f'
+ d r / ' e i(n-n')/a ~t)t2( b~, ')~')
-=
(G+(.,
a ) , / , ) ( ~ , n ) :=
f't de' e-i"-e')/a ~bz,(e',r/)
f*- ~ d'¢
d,7'<,(e,r/')
f" oo (o-(.,a)6)(¢,r/)
:=
)
'/'22(~5',r/)
f"_=dr/' e i("-'')/x q~2(¢, r/') ]
f* d~-'e-i(¢-¢')/'~cbz,(.~',r/) +oo
ff de'*2=(e',r/)
,1
(2.5)
It is easy to see that the Green function G+(A) is analytic and bounded at the upper half-plane Im A > 0 while the Green function G-(A) is analytic and bounded at the lower half-plane Im A < 0. Note that the Green functions (2.5) differ from the Green functions used in ref. [16] by the signs of the lower limits of integration in qS~ and q~22 respectively. With the use of the Green functions G + and G - we define the solutions X + and X- of eq. (2.3) via the integral equations X±(~,rt,a)
a~Ln¢O'- a' +
= 1 - {G-+-( • , , a ~
i/a)X±(',
(2.6)
a)}(¢,'7),
where 0' := 0/0¢', a' := a/0r/'. As far as the Green functions G +, G - , the solutions X +, X- of the integral equations (2.6) are analytic and bounded in the upper and lower half-planes, Im a > 0 and Im a < 0, respectively. Then since G + - G-4= 0 at Im a = 0 one has also X + - X - ~ 0 at Im a = 0. Thus one can define the function X = X +,
forlmA>0,
X = X -,
forlmA<0,
which is analytic and bounded at the whole complex plane and has a jump across the real axis. So we arrive at the standard Riemann-Hilbert problem. As in reL [16] we assume that the homogeneous integral equations (2.6) have no nontrivial solutions. To specify the Riemann-Hilbert problem one must express, according to the standard procedure, the jump X + - X - at Im A = 0 via X-. To do this we firstly obtain the integral equation for X + - X-. Using (2.6), one gets
(x+-x-)(~,r/,a)=r(~,~,*)-(&
1 , a )~o(
0'
-
~' +i/~)(x+-x-))(e,r/)
(2.7)
where
v(~,r/,a) =
o
f_+[ d. e"~'-~"J~~(0(e, , ' ) ( ~ - ~o,+ i/,)x +),2t
-f+?de'e-'
+i/*)X-(~Y',n)):,
o]
(2.8)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
372 and
f_n d ~ ' e i ( n - n ' ) / x ~ , 2 ( ~ , ~ ' )
f" dn'~,,(¢,w')
(6(.,,~)4,)(¢,n) =
(2.9)
f~ d#' e-i~e-e'v* ~2,(~,, a )
fe d~'~22(
~)
Note that the diagonal elements of the matrix F in (2.8) are equal to zero in contrast to the similar matrix considered in ref. [16]. This is due to our choice of the Green functions (2.5). We will look for the expression for X + - X - via X- in the form (x+-x-)(~,rt,h)
dl
(2.10)
-]-fx(i)Z,(~,~7)f(l,a)~;'(~,~),
= -
where "~x(~:' r/) := ( ei'~/aO e -in/aO )
(2.11)
and f(l, A) is the 2 × 2 matrix which must be found. Substituting (2.10) into the r.h.s, of (2.7), we obtain
f
dl X+ - x - = F -
-~75-
A = Q ( o ' - ~'+
i/a)x-(l)
f n-~o dr/' ei(n_n,)/a 2A12 1
~All
dr/'
f ~ dsc ' e -i(~-~')/* -~ 2-A21
(2.12)
f e_ dK' ~l A
22
with ~tf(l,k)
~A
1•
On the other hand from (2.6) one has "~ dl 'iTx-(l)2f,f(l,a)Z;
1
f_
=X+-X -
_ 7~-2,f ( l , a ) ~ ;
_f~
J_~ l 2
f
~
!
f -~'q d'q' e i~"
dr/' ~BI1
f+~ d~:' e-i(e-e')/t
-~')#
~BI2
1 ~1Z t f ( l , h ) y ~ ; l
½B21 l
with
B = Q ( O ' - ~'+ i / l ) x - ( l ) .
(2.13)
V.G.DubrovskyandR G. Konopelchenko/ Coherentstructuresfor theIshimoriequation,I
373
Transforming (2.12) with the use of the identity
((0'- ~"+ i/A)x-(l) ~,f(l, A) ~ " ) ( s c, r/) = ((0'- ~'+ i/1)x-(l))(Gn) 2dt(~, ~7)f(1, A) ,~-'(~:, n),
(2.14)
performing the matrix multiplication in (2.13) and subtracting the obtained equations, one gets di
f[[
-ff~,f(I, A) ~ ; '
.,r,;'( ~, n)
-f_ d~'e~e'/*½(e(o'-~'+i/~)x-)~ _~
g
0
_~de'eie'/'½(a(a'-g'+i/l)x-)2~ 0 tf2~
f12 ) ~,a-1(~, r/). f22
(2.15)
Multiplying (2.15) by X, from the right and acting on (2.15) by
f*~d~e -i'/p
1 2~r
0
)
from the left, we obtain
f,1(P,A)
f12(P,A) )
f21( P, A) f22( P, A) (
0 = -VS(p,A )
T~(p, A) )
0
(+~dl[ 0 -J..~ --ff(T~(p,1)
0 fll(l'A) 0)[
f12(l, A) t
y=(l, a) ] '
(2.16)
where +
1
T,d p , a) = ~
f f ded, e -ie/'-''/~ ½(Q(O-~+i/a)x+)12, +~
1
r2q(p, a)= ~ f f dedve ''/'+'e/" ½(Q(O-~+ i/A)x-)=,.
(2.17)
--oo
In matrix form eq. (2.16) looks like f(p,A)+
f+= at _ -~T-(p,I) f(1, A)= T+(p,A)- T-(p,A),
(2.18)
374
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the lshimori equation, I
where (2.19)
T+ = (00
Thus, we have proved that the jump X + - X - indeed is given by (2.10) with the matrix f(l, A) which obeys eq. (2.18) (see also ref. [16]). Note that f is easily expressed via T~z and T~]
f(p,A) =
0
T~2( p, A)
- r~(p,a)
- f +_~ d l ~T - " 2lip , l) r ~ 2 ( l , a ) Joe
(2.20)
So, we have arrived at the standard regular nonlocal Riemann-Hilbert problem. Its solution is given by the rather standard linear singular integral equation [16] +~xa
x-(a)=l+
1
dp dk X - ( P ) X p f ( p , k ) ~ '
-41ff T k2
k-XTN
(2.21)
Eq. (2.21) is the inverse problem for the linear problem (2.9). The functions T~2(A,ix) and T~(A,/.t) are the inverse problem data. The reconstruction formula for the potential P(~,rt, t) is obtained by substitution of the Taylor expansion X( ~:, T~, a) = XO( ~, 77) -t- a)(l( ~, ~) q- •2X2 ( ~, ~) q-...
(2.22)
near a = 0 into eq. (2.3). Consideration of the terms of the order A-1 gives the formula [16]
P( ~, rl, t) = S( ~, rt, t) " rr = _g~sg-l,
(2.23)
where
g ( ~ , n , t ) =X(sC, r/,t,Z = 0 ) +oc
=l +~
1
ff
dl dk
f2 k3 Z ( e , rl, l) Y2)(e, rl) f ( l , k) X c ' ( s~, , ) .
(2.24)
One can also obtain the compact formulae for the topological charge (1.2) and the scalar field ~p. Note firstly that
! S.(&xS,)=4~
i tr(PP, Py). -g-4w
(2.25)
Using expression (2.23) for P, one obtains tr(PPxP,,) = 2tr(~r3((g- l),,gx - (g-')xgy))"
(2.26)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structuresfor the Ishimori equation, I
375
On the other hand substituting (2.22) into (2.3), one gets in zero order in A
c r 3 g - l g x - g - l g y = li[ g-ix1, cr3] •
(2.27)
Differentiating (2.27) with respect to x and y, one obtains o 3 ( g - 1)y g x + O ~ 3 g - ' g x y - ( g - l g y ) y = ½iBy[g-lXl,tr3],
(g-lgx)x--O'3(g-l)xgy--or3g-lgxy=
½io'3 0x [ g - 1X1, o'3] .
The summation of these two formulae gives
Or3((g-l)yg x --(g-l)xgy ) = (g-lgy)y- (g-lgx)x+ li(oy q- or30x) [ g-IXl, or3] • Substituting (2.28) into (2.26), taking into account well-known formula tr(gxg- 1) = Ox In det g, we obtain
that tr((Cqy-t-O'30x)[g-lxDor3])
(2.28) :
0 and using the
tr(PPxG) = 2 t r ( ( g - l g y ) ~ - ( g - l g x ) x ) = 2(0yz - ~2) Indet g = 2 0,0n lndet g.
(2.29)
So the topological charge Q is +oo
Q
i ~(~r
ff
d~:dr/0e0n lndet g.
(2.30)
Eq. (2.29) gives rise also to the following reconstruction formula for the scalar field ~p: q~(~, r/, t) = 2iln det g + 2 0~-1u2(~ ', t) + 2 0~lUl(rl ', t).
(2.31)
Eqs. (2.21), (2.23), (2.24) and (2.31) completely solve the inverse problem for the linear equation (2.3). For the real S the matrix P is self-Hermitian and the relations (2.23), (2.24) imply that g(G r/, t) is the unitary matrix gg* = 1. So Idet g[ = 1 and therefore the scalar field ~o is a real one for the real u~ and u 2. Note that similar expressions for the topological charge and the scalar field can be derived for the Ishimori-I equation (a = i):
Q=
1
ff
dx dY (Ox2 + 02) lndet g
(2.32)
--oo
and ~p(x, y) = 21ndet g. Eqs. (2.31), (2.33) are in complete agreement with the formula ~ = 2 i a l n d e t g different method in ref. [19].
(2.33) -1, derived by a
V.G. D&m&y
376
3. ~~en~~t~
and B.G. KorwpeMzenko / Coherenf structures for the ishimori equation, I
spectral data
To study the initi~I-bounda~ value problem for the fshimori equation and to construct the exact solutions we need, similar to the DS-I equation El&-123, the solution of the inverse problem equations with the degenerated inverse problem data, i.e,
For this purpose it is convenient to rewrite the matrix equations @lOI as the system of equations for the columns of x. One has
(3.21 and
!y the inverse prublem equation f2.21) looks like
(3.4)
(3.5)
f3.61 substitution
of (3.1) into (3.4) and (3.5) gives res~ctjvely
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
377
Let us introduce the quantities 1 Fj:= ~
f+®dl {xiz(l)) '-oo ~ -Tj(l) e-in//IX~z(l ) '
(3.9)
\ 1
•
6;:= --~-L+~-~2Sy(l)e'W'{ X~(') [ /XI,(0 ]"
(3.10)
Eqs. (3.7), (3.8) imply now the following system of equations for Fj. and Gy: 1 N_ dA Sj(A) 2-~-= f-~+°~ d._~lzsj(l)eie/, , - Ek=l F f f o- +®"~'-
Gj=(1)
N+
1--~f +°~-~z Tj( l ) e-i./, + E Gk L k=l
o
e i¢/a ( T k ( A )
e-i~/a) +,
+~o _~_ oo
Tj(A)e-i'7/X(Sk(A)ein/a)-.
(3.11)
If one denotes 1 ~i~/,, ~(e) ,= 2-~-f +~_=vS;(t) d; 1
•J(,):= 2-~-~f eT-~i ,~(/' '), _ ajk := f_+? - ~ Tj(A) e-in/x (Sk(A) ein/x) -, ~ik:= -- f_+~-~-z Sj(A) eiWX(f'k(h)e-iWx) +,
(3.12)
then the system (3.11) is represented in the compact form N+
N
k=l
k=l
Ft'- E aikak=(O)"l'i(~), aj- ~ ~jkFk=(~)O'j.(,).
(3.13)
This system is easily solvable and the solution is of the form
N
(N+)
Fj= E (I-A)fp t Y"k=la"kcrk , p=l
"rp
N+
aj = E
. = 1
(1-B)~p I
( N_ Ek =
o'p
l~pk~'k
) (3.14)
V.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
378
where A and B are N_ x N_ and N÷ × N+ matrices of the form N+
N_
As~ = E OLik[~kj,
Blm = ~ 3U, al,,,.
k=l
(3.15)
k=l
Knowledge of Fj and G~ allows us to find the matrix X (a). Namely, (3.7) and (3.8) imply
X~I(A)
=
XI2(}t)
=
-- 2 ~
0
X22(a)
E F/.(~.(a)e -is~/A) , j=l E a i ( s i ( 1 ~ ) ein/a) -.
_1_~
1
(3.16)
i=1
Finally, one finds S(~:, ~7, t) by eq. (1.8) via a'-(sc, r/, a = 0). Thus, the inverse problem for the Ishimori-II equation with the degenerated inverse data, as usually, is solved explicitly. The corresponding potential P(sc, rl, t) depends on the 2(N++ N ) arbitrary functions of one variable. Note that in the degenerated case the Fourier transforms of the inverse problem data +oo
dl dA T~(l,A)Ae_in/t_i~/a
=
-i f f Tr V dl
(3.17)
--Qo
have the factorized form too: N_
S(~¢, rt) = 2at Y'~ r~(rl) ~i(s¢), i=1 N+
7~(~:, r/) = 2-rr E ~rs(~:) ~(~7),
(3.18)
i=1
where
'1"i('r/)
ri(~) =
and o-i(¢) are given by the formulas (3.12) and
~
i
f2 i
f
~ -A~/(A) e-''/A' dA -~-ASj(A) e iÈ/A.
(3.19)
V.G. Dubrovsky and B.G. Konopelehenko / Coherent structuresfor the lshimori equation, I
379
4. Time evolution of the inverse problem data
In the case of the nontrivial boundary conditions ~°n 6 ~ - 2 2u1(~7'/)'
~06n--~-~z~_ ~ 2u2(sc, t)
(4.1)
the second operator L z, associated with the Ishimori-II equation, must be modified in order that the linear problems Ll~0 = 0,
/~2~0= 0
(4.2)
will be compatible and equivalent to the Ishimori equation. The operator L 1 remains unchanged /,1 = L r The modified operator L z can be constructed analogously to the DS-I equation case [10, 11]. Namely, we will look for this operator in the form L 2 = L 2 + f _+~ -dl ffy(k,l),
(4.3)
where L 2 is given by (1.3) and the kernel y ( k , l) must be found. So the time evolution of the eigenfunction qJ is defined by the equation L2qJ + f _+~ ~dlT 7 t " k , l") ~ b ( I ) = L 2 ~ O + O "
(4.4)
Since the Ishimori equation is equivalent to the condition [L 1, L 2] = 0 and LlqJ = 0, then eq. (4.4) implies LlO = 0,
(4.5)
i.e. O and ~Oare the solutions of the same differential equation. The relation between O and 0 can be found by comparing the integral equations satisfied by these functions. Let us choose ~0 as ~b(,, r/, A) = X_ ( , , .q, A) ( e i o x
0 ) e-in/,~ , :
(4.6)
where X obeys the integral equation (2.6). This function ~Oobeys the integral equation ~O(~'r/'A)=( ei6/A0 e -in/xO )_{~-(.,A)½Q(a,_~,+i/A)~b(.)}(~¢,~7),
(4.7)
where =
(4.8)
The first column of qJ obeys the equation (i//11 ) (e i x) qJ21 = O - { ( ~ - ( " A ) ½ Q ( a ' - a'+i/A)O}(~,r/).
(4.9)
380
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the lshimori equation, I
(°-)
The quantity O obeys the same equation (4.5) as ~b, hence, the first column of 0 obeys the same integral equation as \ ,~, . The only difference is the inhomogeneous term, i.e.
(1911 L'ql'" ) ~, , O 2 1 ) = ( O 2 1 . ] - { G - ( ' , A ) I Q ( "dr- +
(4.10)
i/A)O}(sc,~7).
The quantities ~ii~ and 02i ~ are found from eq. (4.4). Taking into account (1.3) and (1.4) and considering the limit r / ~ -0% one finds i 011~= - 2t---5 e ie/A +
i
-ff~u2(~,t)e ie/A, 021~ = 0.
(4.11)
Comparing now the integral equations (4.9) and (4.11), one gets o.(a))
= _ __
i
)
2A2 162,(A )
+ 2A J_~ 7gY{A- - T
62,(I)
'
(4.12)
where --~1 ( + ° ° d ~ e i e P u 2 ( ~ , t ) .
Y(P) = 2~r .#-oo
(4.13)
Now let us consider equation for the second column
0221
t X~2 e-inlA"
It is of the form
(5t :( o @221
e-irllX
, J2~t
(4.14)
The second column of 0 obeys eq. (4.10) with another inhomogeneous term O. Its form can be found by the consideration of eq. (4.4) in the limit se ~ - oo. One gets
0~2~ = O,
022. = ~
i
e -i~/* + -~-£u~(n, t) e - ~ / ~ .
(4.15)
Comparison of the integral equations for the second columns of 0 and O, as a result, gives
012 i,b22]
-oo ~ Y t "~'- 1 ]{~22(1) '
(4.16)
where
¢/(p) = ~
_ drlui(rl,t)e -inp. 'sT
(4.17)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
381
So, we have found all components of the matrix O. Thus, the time evolution of the quantities [~//ll)=(XH)ei'/;t ~/(/~') = t ~'21
and
¢;(A)=/x~2]
IX~2] e-in/a
X21
is defined by the following equations:
i __i /.+~o d l t *t lip(0f-O,1)2~s+JiR(8 e O,1)0 -~-7*+2Aj_= ~-~)qs(l) =
_
_
~t=½iP(a~-On)2~+¼iR(8~
an)q;+
~ , t
1
(4.18)
and _
i
^
"
dl^[1 _ -iTy[-~-l)~(l),
(4.19)
where
R:=Pt-Pn +(Pe+Pn)P+i(1-P) fn_®d~7'S'(SexSn,)+i(1-P) fe_ood~'S.(Se, xSn).
(4.20)
Similar to the DS-I equation, the nontriviality of the boundaries essentially changes the evolution of the eigenfunction ~ in time. Eqs. (4.18) and (4.19) allow us to find the time evolution of the inverse problem data. Consideration of eq. (4.18) at ~:~ -oo gives rise to the following equation: 1" i i__ [+® dl i l o't(A,rl,t ) =~lO'nn+ul(n,t)o~n---~o'+ 2A2J_o° -/TyrO--
1) tr(l, ~t) T r/
(4.21)
for the quantity ~+~ g dl Z_21(l,A,t)e -in~t= ~r(x,n,t):=J_®
lim ~b21.
(4.22)
Analogously for the quantity
lim 1~12,
--
(4.23)
"r/---~- - o o
one obtains from (4.19)
"rt(~,A, t )
1. = -- ~17"~:~-- U 2 ( ~ , t)'r~: +
i
i /-+~°dl [1
1]
-~-~'r + 2A J-o~ lTYt A- - T ] z(l'sC)"
(4.24)
382
V.G. Dubrovskyand B.G. Konopelchenko /Coherent structuresfor the Ishimoriequation, 1
Further, for the full Fourier transforms of the inverse problem data
dl dA AT_, l A)e_in/~_ie/x
TS
r
,
= _ i f +-== -~-A°'(A, dA r/, t) e-ie/a
(4.25)
and
dt + ~(~',v,t) := -iff ~7,~T,2(/,,~)
ei~/l+irl/a
=-if+_f ~2a,c(a,Gt)ei'/a,
(4.26)
one gets the simpler evolution equations. They are S~t(~,'q, t) - ½i(S~¢ + S~,7) + u2(~,t) S ¢ - u l ( ~ 7 , t ) Sn = 0
(4.27)
7~t(G ~7, t) + ½i(T~e+ T , ~ n ) + u 2 ( G t ) T ~ - u , ( ~ , t ) T n = O .
(4.28)
and
Thus, the time evolution of the full Fourier transforms of the inverse problem data for the Ishimori-II equation is given by the simple linear differential equations which contain the boundaries u~(rI, t) and u2(G t) as the potentials. Eqs. (4.27) and (4.28) play a fundamental role for all further constructions. Note that they contain the first-order derivatives over ~ and ~7 in contrast to the DS-I equation case. Eqs. (4.27) and (4.28) are closely connected with the Ishimori-II equation in the weak fields limit. Indeed, in the weak limit 8S 1 := $1 ~ 1, ~S 2 := $2 << 1 and ~S 3 = 0 (since S~- ~S = 0). Neglecting the nonlinear terms, one obtains from (1.5) the system of two equations: ~ S + t - ~ 1 i (S+e~+ S+nn)+uz(~,t)S+¢
ul(r/,t)S+.=O,
,~_, + ½i(S_e~ + S-nn) + u2(~' t) S_e - Ul(r/, t) S-n = 0,
(4.29)
where S~:= 51 +_ iS 2. So, eqs. (4.27), (4.28) which define the time evolution of the inverse problem data coincide with the linearized Ishimori-II equation.
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
383
The exact interrelation between the inverse problem data and S+ and S_ follows from the definitions (2.17) and (4.25), (4.26). In the weak fields limit the formulae (2.17) become d-oo
ff
1
W,+~(p,*)--~
d £ d r l e -ie/p-in/a ½S_(0¢- 0n + i/A)X~2,
--oo
1
--
1f f d~d~e~'/~+~e/*-~S+(ae-O~ + i/,
)xiS.
(4.30)
Then from the integral equations (2.6) it follows that in the weak limit e i(v-n')/* ½S_(0¢ - On, + ilA)x~2,
X~2= - f ; d , '
x~-2--1- f ~ =da '~~S+(Oa,-
0n + i/,~)X~,
XH = 1 - ½fn dr/'S_(0¢ - 0n, + i/3.)Xf, , X2] = - f+~= dsC, e-i(e-e,)/, 5S + l"
(0¢,
--
0n + i/A)XS.
(4.31)
So X~2-=0,
X~2---1,
X~----1,
(4.32)
X~----0.
Substituting (4.32) into (4.30), one gets -}-~
1
r (p,a) ---
ff
d~drle_in/a_i¢/p ~ ix S _- ,
+~
T~(p,A)
---
~
1
ff
d~d.qein/p+i¢/a
~ -i~ S - +.
(4.33)
Combination of (4.33) and (4.25), (4.26) gives rise to the following interrelation between S, 7~ and S+, S_: S(sC, r/,t) ='rrS+({:, rt,/),
T(sC,r/,t) = ~S_(sc,'r/,t).
(4.34)
The coincidence of the equations which define the time evolution of the Fourier transform of the inverse problem data and linearized soliton equation takes place also for the DS-I equation [10-12]. This is a general phenomenon. Formulae (4.33) and (4.34) allow us to describe the constraints on the inverse problem data which correspond to the real field S(~:, 77, t).
384
V.G. Dubrovsky and B,G. Konopelchenko / Coherent structures for the Ishimori equation, I
Indeed, for real S one has S÷= S*. Then relations (4.33) and (4.34) imply
ATS(l, A) = -t(T,2(A, l))
(4.35)
and S ( ~ , r / , t ) = T*(~:,r/,t),
S(~,r/, t) = 7~*(~,r/, t).
(4.36)
Iterating formulae (2.17), we conclude that the constraints (4.35), (4.36) take place also for the nonweak real S.
5. Exact formulae for the general coherent structures of the Ishimori-lI equation The linear equations (4.27), (4.28) are the key equations in the whole method of the study of the coherent structures for the Ishimori-II equation. In what follows we will consider the case of the real functions S(¢, 7, t). In virtue of (4.36), (4.35), in this case it is sufficient to study the single equation iSt +½(S¢¢ + L ~ ) + iu2(~, t ) S . ~ - i u l ( r / , t) L = 0 .
(5.1)
Our aim here is to derive the general formulae for the exact solutions of the Ishimori-II equation via the solutions of eq. (5.1). Solutions of eq. (5.1) can be constructed by the method of the separation of variables g(~, ~7, t) = X(~:, t) V(~7, t)
(5.2)
similar to the DS-I equation [10-12]. In our case the functions X and Y obey the equations 1
iX~ + ~Xee + iuz(g , t) X e = 0,
(5.3)
iYt+~V,, 1
(5.4)
_
iu,(~7, t ) Y,
=
0.
The combination of the exact formulae (3.14)-(3.16) and the factorized solutions of eq. (5.1) will allow us to construct the localized solitons of the Ishimori-II equation. The general factorized solution of (5.1) is of the form S(~, r/, t) = 2 T g E p i j X i ( ~ , t) Yj(T], t )
(5.5)
/j
where Po' are constants and Xi(¢, t), Yi(r/, t) are the solutions of eqs. (5.3) and (5.4), respectively. The equivalent form of S is
g(~, r/, t) = 2"rr• r/(r/, t) ~i(~, t)
(5.6)
i
with
zi(~,t) = ~.,PijYj(rl,t), J
~'i(~,t) =Xi(~,t).
(5.7)
V.G. Dubrovsky and B.G. Konopelehenko / Coherent structures for the Ishimori equation, I
385
For such a S(s¢, rl, t) one has -.~- oo
d~d-q
ff
2~x ~-'X"(~'t) Yj('q't)PiJ eif/a+in/, i,j
= ~.,T~(l,t) ~.(,~,t),
(5.8)
i
where
Ti(l,t) = ~Pij j
f+
~
~d~7 e'n/tYj(r/,t), -
--¢m
7~/(h, t) = ~- _
~
e if/x X/(~, t).
(5.9)
Using (4.35), one also gets
+~o ~dn Yj*(rl, t) e -'~/~ T~2(I, A, t) = EP* iA gf+o~ ~o ~dE Xi.(~,t)e_ie/l Loo ij
= •Si(l,t) L(A,t),
(5.10)
i
where
Si(l,t)=S+_~d2~xi*(<,t)e-i
=
~Pkjf+o~ ~dr/ j
Ai YJ*(~' t)
e -in/'~.
--oo
(5.11)
The formulae (5.8)-(5.11) give the concrete expressions for the degenerated inverse problem data. Using their formulae, one can obtain the concrete expressions for the other needed quantities from section 3. Taking into account that (7~k(A' t) e-ie/~)-+
i
2x/~ A f~¢~d~'X~(~''t)ei(g'-g)/A
(5.12)
and (S~(A' t) ei'~/a) "+-
~
i
"q
•
~jP~Jf±ood'q'e'(n-~"/~Ys (rl''t)'
(5.13)
V.G. Dubrousky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
386
one gets ~ri(~,t) =Xi*(~,t),
rj(rl,t)
= ~_~PjkYk(~7,t), k
= (papt)jk, O~jk = E pjlPkmalm * l,m
= bi, = - -f¢
(¢, (A ' t)e -i~/*)
Xk(~', t) a¢,Xi*(~', t), ---~
1
)t~O ~
(Sk(a,t)ei~/~) - ~
'
~1
Y'~Yj*(~7 , t)(pt)jk.
(5.14)
J
Using these expressions and eqs. (3.14), we obtain
Fj =
E(1
-
t -'(Ep(papt)kpX*(~'t)
pap b)j k
(5.15)
F~tPktYt(rl, t)
,-,( X~(~,t) ) Gj= E(lk - b p a p )jk ~,,mbktPlmrm(rl,t) •
(5.16)
Further, eqs. (3.16) give
1 - (X,(1 -paptb)-'paptX*)
_(X,(l_paptb)-,py)
g=X-(A=0)=
(Y., p,(1- b,,a,,*)-'X*) l+(Y*,pt(1-bpapt) 'bpY)
(5.17)
where we denote ( X , Y ) = Y'.iXiY,. At last, rewriting the reconstruction formulae (2.23), (2.31) in the components X(A = 0) we get
2x2:(o) x:,(o) s+ = - x,,(o)
S =
x2:(o)
- x,2(o)
x:,(o)
'
2X,,(0) X,2(0) x , , ( o ) x22(o) - x , 2 ( o ) x2,(o) '
Xl,(O) x~=(o) + x,~(o) x2,(o) s3 = - x , , ( o ) x~2(o) - x,~(o) x~,(o) "
(5.18)
and q~(sc, r/, t) = 2i In[x,,(0) X22(0) - X,2(0) X2,(0)]
+ 2f e d~'u2(#',t) + 2f" dw'u,(n',t)
+ const.
(5.19)
t
387
Eqs. (5.18), (5.19), (5.17) are the main result of the present paper. They allow us to construct explicit exact solutions of the Ishimori-II equation with nontrivial boundaries using the exact solutions of the linear equations (5.3), (5.4). Noting that the summations in (5.15), (5.16), (5.17) may be also continuous, these formulae correspond to any set of the exact solutions of eqs. (5.3), (5.4). At this point the situation is similar to the DS-I equation [10-12]. An essential difference between the Ishimori-II and DS-I equations is that in our case the linear equations (5.3), (5.4) contain the first-order derivatives over ~ and 7/ and the pure imaginary potentials iul(r/, t) and iu2(~, t). For the DS-I equation the corresponding linear equation is the one-dimensional nonstationary Schr6dinger equation [10-12]. This last equation is associated with the IST integration of the Kadomtsev-Petviashvili (KP) equation [1-4, 10-12]. In our case the linear problem (5.3) (or (5.4)) is associated with the IST integrability of the modified KP equation. Indeed, the modified KP equation [20] u t + U x x x + ~ 3U 2 ux g 3 O x l u y y + 3 U x ~.l u v = O
(5.20)
is equivalent to the commutativity condition [L~, L 2] = 0 of the operators [21] L 1 = iOy + ~ + iu%<, L 2 = 0, + 40;~ + 6iuO 2 + [ + 3iu x + 3(3~- tuy) - 3u2] Ox + c.
(5.21)
This fact is not so surprising if one takes into account the formal gauge equivalence between Ishimori and DS equations mentioned in refs. [15, 16] and the gauge equivalence between KP and mKP equations studied in ref. [20]. Indeed, it is easy to see that eqs. (5.3), (5.4) convert into the nonstationary Schr6dinger equations 2i)(, + Xf< + (u 2 + iu2¢ + 2 ~;lu2,(~:', t ) ) X = O, 2iYt + Ynn
+(u2-iuin-20;iul,(~l',t))Y=O, (5.22)
under the gauge transformations
)(=Xexp(iS~d<'u2(<',t)), Y=Yexp(-iS~d~l'ul(~7',t)).
(5.23)
We see that the solution of the general initial-boundary value problem for the Ishimori-II equation is connected with the solution of the initial value problem for the modified IZd~ equation. This last problem will be considered in a separate paper.
6. Time-independent boundaries In this paper we restrict ourselves to the case of the time-independent boundaries u~ =u~(r/), u 2 = u2(st). In this case we will be able to use the previously known results for the special Zakharov-Shabat spectral problem. For the stationary boundaries ul(r/) and u2(£) eq. (5.1) admits the full separation of variables: S(~, 7, t) = T ( t ) X ( ~ ) Y(~q).
(6.1)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, 1
388
The functions T, X and Y obey the following linear equations:
T,(t) + ½i(A 2 + A'2)T= 0, X~g + 2iUz(s~) X~ + h2X(~:)
= 0,
(6.2)
Y,n - 2iul(rl)Y n + a'2y(rl) = 0
(6.3)
where A and A' are parameters. The one-dimensional spectral problems (6.2), (6.3) can be investigated by different methods. Here we will use their equivalence to the specialized Zakharov-Shabat spectral problem. Firstly we note that the spectral problems (6.2) and (6.3) are equivalent to the problems ~sesc nL [A2 4- u2(~) -iu2~(sc)] .,Y = 0
(6.4)
1 2 + [A'2+uZ(r/) + iul,7(,r/)]12=O
(6.5)
and
via the gauge transformations
So, the problems (6.2) and (6.3) are gauge equivalent to the spectral problems of the Schr6dinger type
q ~ + [A 2 + u2(p) 4- iup(p)]q~+-(p,A) = 0
(6.7)
with the very special potential
V(p) =
- [u2(o) +
iuo(P)],
(6.8)
where u(p) is a real function. Then, it is not difficult to see that the spectral problem (6.7) can be rewritten in the following matrix form 01p + iu01 = h02,
0 2 p - iu02 = - h 0 1 .
(6.9)
Indeed, the elimination of 0[12 or 01 from (6.9) gives rise to the scalar equations
Olpp+(A2+u2+iUp)tOl=O,
02pp-k(A2+u2--i.#)02=O.
(6.10)
So, one can identify q~+(P) = 01(P),
¢ - ( P ) = 02(P)-
(6.11)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
389
At last, transiting to the variables 01 "-
$1 + i$2 2
'
02 :=
if/1 --
2i
i$2
'
(6.12)
we arrive at the specialized Zakharov-Shabat spectral problem
.)
Ol
It follows from all these formulae that 2(~:) = - 0 2 ( ~ ) - i01(~) ,
17(~) = 0 , ( ~ ) + iO2(n),
(6.14)
where 01 and 02 are the solutions of the spectral problem (6.13). So we are able to use the exact results associated with the problem (6.13) for the construction of the exact solutions of the problems (6.2), (6.3). Of course, for the calculation of the function X(~:) we must use u2(~) as the potential u in (6.13) and for calculating the function Y(rI) one must use ul(~7) as u in (6.13). The spectral problem (6.13) has been studied in detail in the papers [18, 22] in connection with the IST integration of the modified KdV equation. It has been shown in refs. [18, 22] that the spectrum of the problem (6.13) contains a continuous part defined on the real axis Im A = 0 and a discrete part which consists of the points located symmetrically with respect to the imaginary axis Re A = 0. The points laying on the imaginary axis correspond to the soliton potentials u and each pair of the points _+a + i/3 correspond to the breathers. Since the problem (6.13) is not self-adjoint then the discrete spectrum of (6.13) contains also multiple points which are associated with the multiple poles of the reflection coefficients [23]. The eigenfunctions 0 n and the potentials u which correspond to the discrete spectrum are found as usually [1-4] explicitly in the closed form. The general soliton-breather eigenfunction which corresponds to N 1 solitons and N 2 breathers is of the form [18, 22] On(P):=O(P'An)=~"(I+Mm
)nm
t
1 e
+
0
k
where the (N~ + 2 N 2) × ( N l + 2 N 2) matrix M looks like
ei(X, +Am)pCm Mnm
an + Am
,
(6.16)
where n , m = 1 , 2 , . . . , N 1 + 2 N 2 and An = + a n + i/3 n. The corresponding general transparent N 1soliton + Nz-breather potential u is given by the formula
u(p)
2cl ~p Imdet(1 + i M ) dp I m l n d e t ( 1 + i M ) = 2 arctg R e d e t ( 1 + i M ) '
d2~ Indet(1 + M 2 ) . u2(p) = dp---
(6.17)
V.G. Dubrousky and B.G. Konopelchenko /Coherent
390
structures for the Ishimori equation, I
If all A,, = ifi, (Im p, = 0) then eq. (6.17) gives the general N,-soliton potential. At all A, = t_cx, + i/I, (Im cy, = Im J3,z= 0) we have the genera1 N,-breather potential. Correspondingly eq. (6.15) gives the N,-soliton ~igenfunction and the &-breather eigenfunction. Eqs. (6.15)--(6.17) alIow us to construct the exact solutions of the problems (6.21, (6.3). For the potentials ~~(8) and u,(n) given by eq. (6.17) the eigenfunctions X and Y are of the form
where the functions 6, and 6, are given by the formula (6.15). Note that the function XCN,&<) corresponds to the N,-soliton + N,-breather boundary function u,(l) while Y$,, rj,,(?~) corresponds to the fii-soliton + ~~-breather bounda~ function u,(n). Now using eqs. (5,14)-(5.19), (6.1%(6.181, we obtain the exact explicit solutions of the Ishimori-II equation with the general stationary transparent boundaries u2(<) and U-&V)_ These solutions are defined by the arbitrary N,-soliton + N,-breather boundary function u,(5), N,-soliton + N,-breather boundary function u,(q) and arbitrary constants plk. The matrices a and b involved in eq. (5.17) can be represented in a more compact form. Indeed, taking into account (6.181, one gets
(6.19) and
(6.20) So, one has
adrt) = A,c,,t~), b;, = ATcik(Or
(6.21)
where cik(p) :=I’
--r
dp’~(h,,p’));*(Aj,p’).
(6.22)
V.G. Dubrovskyand B.G. Konopelchenko/Coherentstructuresfor the Ishimoriequation, 1
391
Note once more that Cmt(rl) is calculated via the functions (Y(Ai,-O) , X(Ak,~7))= (i//i,[]/2) which correspond to the potential u = ul(r t) in (6.9) while cik(~) is associated with the potential u = u2(~:). The soliton (s) and breather (b) are quite different transparent potentials. As a result, we have four different types of the exact solutions of the Ishimori-II equation:
Sss(N,M)(~,rl,t),
Ssb(N,M)(~,TI,t),
Sbs(N,M)(~,'q,t),
Sbb(N,M)(~,TI,I),
which correspond to the choices of X and Y as the soliton's or breather's eigenfunctions. The solutions Ssb and Sbs are, obviously, related by the interchange sc ~, ~/. So, we have three essentially different types of the exact solutions:
Sss(N.M), Ssb(N,M), Sbb(N,M)
(6.23)
where the integers N and M correspond to the N-soliton (breather) boundary u2(~:) and M-soliton (breather) boundary ul(r/). All these solutions can be calculated explicitly. Here we present the simplest examples of such exact solutions. Using the formulae given in the appendix, one gets the simplest (5.5) localized soliton with N = M = 1: S(s,sXl,l)l =
2e-V¢-vn
--~ e-2Vn
>(
)
c 2/z e-2~'g c°s[ ½(/x2+v2)t] +
( 4--~ cd
e-z~'g-2~" + l -
( cd e-2~-2vn 4- - - ~ - - - 1 ) 2 ( ~ v e - 2 V ~ 7 +
+ ~
c
tZV)sin[½(~2+v2)t -c'd-
e - 2t~tj
]
)2
S(s,s)(l, 1)2 = 2 e - / ~ . v-o
-~-1-
4--~e-Z"e-a~/
) cos[½(.2+vZ)t]
×
cd
/zv - 1 e_2~,_2~ n + -c-d-
c
+
e -2~,
+
e -2~n + ~c e -2~,¢
)2
2/* e-2~'¢) sin[ ½(/z2 +
v2)t]
2 e - 2v'q - 2t.t,f S(s,s)(1,1) 3 = - 1 +
cd
tx v - 1 e - 2 ~ - z " n + 'c-d
+
e_2~,n + ~
e -z~'~
/.L e _ 2vrt 2 ~ e - 2 ~ ¢ + 2-C q~(s's×l'l)=--2arctg [( c ) 2 ][( d ) 2 ] 1 j[£/.' e-4~'e + 1 )-flu e-4Vn + I + ~ e -4~:-4vn - - -
cd
+ 4 arctg( 2-~- e-2~'~) + 4 arctg(~-~v e-2~n) + const. ,
(6.24)
where/x > 0, v > 0 and c, d are real constants. It is easy to see that this solution is of the breather type in finite t.
392
V,G. Dubrovstcy and B, G, Konopelchenko / Coherent structures for the Ishimori equation, 1
The simplest (s, b)-type localized soliton for the Ishimori-II equation is ( N = M = 1):
P = P~
, papt=alt,
P~z = 1 0
p Y = p I ~ Y j = Y~, 2(1 + b * a n ) X * Y ~ *
S(s,b×l,l)+ = S(s,bX~,~)~ + iS(s,bX1,D2 =
l1 - a l i b i 2
2[XlzlYll 2 S(s,bXI,1)3 = - 1 +
I1 - a l i b i z '
q%,t, xl, l) = 2i ln(det g ) + 2 a~-lu2(s~') + 2 0~"1u1(rl'),
(6.25)
where
1 -a~b* "
detg=
l-anb
atl =
i ( a + ifl)e ~+2'~m 4/3[eh(Z/3r t + g,) + ( i f l / a ) s i n ( 2 a r / + q~ - 6)] '
b= -
t~/d ( d / 2 / . t ) e -2~e + i '
X=X(A
i e - tt~+ itx2t/2 = i/z,~:,t) = - - ( d / 2 / z ) e -2u~" -- i ' (e-ic~-/3n-i~
+ia + i ei~'7+t~n +q') e-i(a2-t32)t/2+ctt~t
YI = Y(2ti = oe + i/3, r/, t) ........ 2[clh(2~r/+ ~ ) + i ( / 3 / a ) s i n ( Z a r / + ~o " 6)]
e_e, = t - a n b c l - a n b a 2 ~
+~82
a+i/3=¢-~flZeia,
'
c=l-aubcl-a~lbei%
'
u,(~7) = - 2 ~-~ arctg( fl sin(2a~7 + ~ - ~ )
)
u 2( ~ ) = 2 ad-d-ia r c t g [ ( d / 2 / z ) e - 2 ~ # ] . %
At last, the simplest N = M = 1 localized soliton of the (b, b) type is of the form
P = ~ P21
P22 ]
'
papt =
+
=
2(1 + a22b~])X('Y ~
I1 S(b,b×l,l)3 = -- 1 +
-a=b1112
21XlI21Y212 I1 -- a2zb1112'
~o(b,b×l,~) = 2 i l n ( d e t g ) + 2 O~-lu2(# ') + 2 O~ut(~7'),
(6.26)
F.G. Dubrovsky and B.G. Konopelchenko /Coherent structures for the Ishimori equation, I
393
where 1 - a22b11 det g = 1 - az2bll" i ( a + i/3) e q'l-2aclt a22 = -- 4/3[ch(2/3rl + ~1) + i ( f l / a ) s i n ( 2 a r / + ~1 - 3)] ' i( - a + i/3) e o2+2aclt bll -- 4/3[ch(2/3~: + 4'2) + i ( / 3 / a ) sin(2a¢ + ~o2 - 3)] ' X 1 = X ( A l = a + ifl,~:,t) (e ia¢+/36+~2 h- i e -ia6-/3¢-i~'2+i'~) e -i(a2-¢2)t/2+a#t 2[ch(2/3~: + ~2) - i ( f l / a ) sin(2o~: + ~02 - 3)] I"2 = Y(A'2 = - a + i/3, n, t) (ei,~n -/3n + i t ¢ I -i~ d'- i e -ian +/3n+~) 2[ch(2flr/+ ~tl) + e-4,~ =
Ickla 218~fl
2
e
-i(a2-~2)t/2-a~t
i(fl/a) s i n ( 2 a r / +
(k=l,2);
~o1 -- (~)]
a+ifl=V/a-~fl2ei~;
Ck=lcklei~k
(k=l,2),
/3 sin(2ar/+ ~1 -- ~) ),
u'(r/) = - 2d-~ arctg( a
~
~-~-7)-
1 Ua(~¢) = - 2
(/3 s i n ( 2 a , + ~%- ~) ) arctg ~- ch(2flrl + ~2) "
Notice that the poles of the reflection coefficient which determine the breathers ul(r/) and u2(~) are chosen in the (b,b) solution of eq. (6.26) to be the same: a 1 = h'~ = a + i/3, h 2 = h'2 = - a + i/3. All these solutions are of the breather type, all of them decrease exponentially at ~:2 + r/2 ~ oo at all directions, but their forms are quite different. The solution $(b,s) is obtained from Sts.b) by the interchange ~: ,~, ~7These examples demonstrate the richness of the types of the coherent structures for the Ishimori-II equation. In more detail the properties of the localized solitons for the Ishimori-II equation, the case of time-dependent boundaries and scattering of the localized solitons will be considered in the second paper.
Acknowledgements We are thankful to A. Degasperis and P. Santini for the useful discussions. One of the authors (B.G.K.) is very grateful to the Department of Mathematics of the Yale University and, especially, to R. Beals for the warm hospitality.
394
V.G. D u b r o v s k y a n d B . G . K o n o p e l c h e n k o / C o h e r e n t
s t r u c t u r e s f o r the l s h i m o r i equation, 1
Appendix H e r e we present for completeness some intermediate formulae which are n e e d e d for the calculation of the exact solutions of the Ishimori-II equation. T h e simplest soliton potential for the spectral p r o b l e m (6.13) corresponds to N~ = 1, N 2 = 0, A = i/* (/* > 0) and cl = b(i/*)/a'(i/*) -- ic (Ira c = 0) [3, 18, 22]. It is of the form
u(p) = 2
_~p
(c arctg ~
) 2/* sgn c e -2*'p = - ch(Z/*p + P0)
(1.1)
where e p,, = 2 / , / I c ] . T h e c o r r e s p o n d i n g eigenfunction ,~ is e - it.tp + it~2t / 2
f,(p) = -i
(A.2)
( c / 2 / * ) e -2t*p + i"
T h e quantities a and b are ~, 1 a = ~ ( d / 2 v ) e -2~n + i '
b-
/1. 1 c ( c / 2 / * ) e -2~*~ + i"
(A.3)
T h e functions Xl(s~) and Yl(rl) are
- i e -~+it~2t/2 Xl(s~) = ( c / 2 / * ) e - 2 ~ - i '
e -~'~j+ivzt/2 Yl(r/) = ( d / 2 u ) e -2~n - i
(A.4)
Using eqs. (A.2)-(A.4) and choosing p = 1, one obtains from eqs. (5.17)-(5.18) the solution (6.24). T h e simplest b r e a t h e r potential corresponds to N 1 = 0, N 2 = 1, A1 = a + i/3, A2 = - a + i/3, a,/3 > 0, c I = b ( A O / a ' ( A I) = ic, c 2 = b ( A 2 ) / a ' ( A 2) = ic* [3, 18, 22]. It looks like
/,/(p)
-2 d
17
where tg 6 = f l / a ,
arctg/3 s i n ( 2 a p + q~ - 6) ach(2/3p +t0) '
e* = 2 f l ~
(A.5)
+ / 3 2 / a l c ] , c = Icle i*. T h e c o r r e s p o n d i n g eigenfunction l?l(p) is of the
form i e i°~p+flp+4' +
e-iap-flp
-iq~+i8
)?'(P) = 2ch(Z/3p + to) - 2 i ( / 3 / a ) s i n ( Z a p - 6 + q~) "
(A.6)
T h e function Yl(P) looks like i e ictp + flp + ~t .OF e - iap - flp - iq~+ i~
Y1(P) = 2 c h ( 2 / 3 p + to) + 2 i ( / 3 / a ) s i n ( 2 a p - 6 + q~) "
(A.7)
V.G. Dubrovsky and B.G. Konopelchenko / Coherent structures for the Ishimori equation, I
395
We present here also one general formula. Using eq. (6.15) for 1~1, 1~q2 and eq. (6.17) for u, it is not difficult to show that the functions X~ and Ym can be represented in the following compact form:
~_,(1-iM)~/exp(iAp~-iA2nt/2)exp(-ifgd~,u2(~,))
Xn(~,') =X(An,sc, t ) = _
p
= - - e -i'~{'/2 Y'.
P
(1 - iM)~7
Ym(~7,t)=Y(Am,rl,t ) = i ~ ( 1
e ixp~: (det(1
+ iM))*
det(1 + i M )
'
(; d'O'Ul('O') )
+lM)mpexp(iAprl -iAM/2)exp i
p
= ie_iA~,/2~( 1 + p
(A.8)
iM)2,~ei~p n det(1 + iM) (det(1 + i M ) ) *
"
(1.9)
These formulae essentially simplify all calculations.
References [1] V.E. Zakharov, S.V. Manakov and S.P. Novikov. Theory of Solitons. The Inverse Problem Method (Nauka, Moscow, 1980); Consultant Bureau (1984). [2] M.J. Ablowitz and H. Segur, Solitons and Inverse Scattering Transform (SIAM, Philadelphia, 1981). [3] A.C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, 1985). [4] Y. Satsuma, J. Phys. Soc. Japan 40 (1976) 286. [5] L.A. Bordag, A.R. Its, V.B. Matveev, S.V. Manakov and V.E. Zakharov, Phys. Lett. A 63 (1979) 205. [6] H. Cornille, J. Math. Phys. 20 (1979) 144. [7] M. Boiti, J.J.-P. Leon, L. Martina and F. Pempinelli, Phys. Lett. A 132 (1988) 432. [8] M. Boiti, J.J.-P. Leon and F. Pempinelli, Phys. Lett. A 141 (1989) 101. [9] M. Boiti, J.J.-P. Leon and F. Pempinelli, preprints Montpellier, PM/88-40 (1988); PM/88-44 (1988); PM/89-17 (1989); PM/90-04 (1990). [10] A.S. Fokas and P.M. Santini, Phys. Rev. Lett. 63 (1989) 1329. [11] A.S. Fokas and P.M. Santini, Dromions and a boundary value problem for the Davey-Stewartson 1 equation, Physica D 44 (1990) 99. [12] P.M. Santini and A.S. Fokas, The initial-boundary value problem for the Davey-Stewartson equation . . . . preprint 684, Universita di Roma (1989). [13] P.M. Santini, Physica D 41 (1990) 26. [14] Y. Ishimori, Progr. Theor. Phys. 72 (1984) 33. [15] B.G. Konopelchenko and B.T. Matkarimov, Phys. Lett. A 135 (1989) 183. [16] B.G. Konopelchenko and B.T. Matkarimov, J. Math. Phys. (1990), to appear; preprint Saclay SPhT/88/229 (1988). [17] R. Beals and R.R. Coifman, Inverse Problems 5 (1989) 81. [18] M. Wadati, J. Phys. Soc. Japan 32 (1972) 168. [19] B.G. Konopelchenko and B.T. Matkarimov, Stud. Appl. Math. 82 (1990) 319; preprint INP 89-123 (1989). [20] B.G. Konopeichenko, Phys. Lett. A 92 (1982) 323. [21] B.G. Konopelchenko and V.G. Dubrovsky, Phys. Lett. A 102 (1984) 15. [22] M. Wadati, J. Phys. Soc. Japan 34 (1973) 1289. [23] M. Wadati and K. Ohkuma, J. Phys. Soc. Japan 51 (1982) 2029.