On the method of solution of the differential-delay Toda equation

Physics Letters A 180 (1993) 413-418 North-Holland

PHYSICS LETTERS A

On the method of solution of the differential-delay Toda equation Javier Villarroel ~ and Mark J. Ablowitz

Program in AppliedMathematics, Universityof Colorado-Boulder, Boulder, CO 80309, USA Received 25 May 1993; accepted for publication 22 July 1993 Communicated by A.P. Fordy

The method of solution of the Toda differential-delay equation, which is a reduction of the Toda equation in 2 + 1 dimensions, is described. An important feature of the solution process is to obtain and study a novel Riemann-Hilbert problem. The latter problem requires factorization across an infinite number of strips with a suitable branching structure. Explicit soliton solutions are given.

1. Introduction During the past twenty five years numerous physically interesting nonlinear equations have been solved by the inverse scattering transform (IST). Moreover the types of nonlinear systems for which the method applies are quite varied; included are PDEs in l + 1, and 2 + 1 dimensions, discrete analogues such as differential-difference and partial difference equations, singular integro-differential equations, and interesting classes of ODEs including the well known Painlev6 equations (for a review of some o f this work see, for example, refs. [ 1-3 ] ). It might also be remarked that the four-dimensional self-dual Yang-Mills equations can also be solved by IST methods and by appropriate reductions o f the self-dual equations virtually all well-known integrable systems can be found (a review o f these results appears in ref. [2], see also refs. [4,5] ). In a recent work we have studied the ( 2 + 1 )-dimensional Toda equations [6,7] by the IST method, (O~x- ~2 0 , ) O ( x , n, t) = 2a2{exp [2 (0n+ l - 0 n ) ] - e x p [2 (0, - 0 n _ l ) ]},

( 1)

where O(x, n, t)=0n(x, t), a 2 = _+ 1, ¢2= + l, x and t are continuous variables and n is a discrete coordinate. It is straightforward to show that an admissible (and interesting) reduction is to what we shall refer to as the "differential-delay Toda equation", see (2) below. The result is that this reduction essentially follows by simply replacing the discrete variable n by the continuous variable x. The interesting differential-delay equation (2) that results is the system to be solved. This equation can also be viewed as an appropriate limit of a Volterra system [ 8 ] and therefore it can be interpreted as a model of an ecological system. Specifically, we consider here the following problem: solve

( Oxx-Ott)O(x, t) = 2 ( e x p { 2 [ 0 ( x + 1, t ) - O ( x , t ) ] } - e x p { 2 [0(x, t) - O ( x - l, t ) ] } ) ,

(2)

with O(x, 0) given and tending to a constant as Ixl--,~. Note that this is a nonlinear partial differential-delay equation. In this Letter it will be shown that the above problem is solvable, i.e. eq. (2) is integrable by means o f the inverse scattering transform method. As far as we know this is the first such l + l equation found to be integrable. As we will see shortly the inverse problem connected with (2) brings out new features not present in standard PDEs. The Toda differential-delay equation adds yet another new class of interesting nonlinear evolution equations Permanent address: Departamento de Matematicas, Universidad de Salamanca, 37008 Salamanca, Spain. Elsevier Science Publishers B.V.

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to the class of integrable equations solved by 1ST. As is c o m m o n with IST integrability in 1 + 1 dimensions, we find that a suitable R i e m a n n - H i l b e r t ( R H ) factorization problem plays a central role in the solution of this equation. In this case the R H problem requires factorization across an infinite number of strips along with a suitable Riemann surface which specifies the underlying branched structure. This RH problem is rather novel. We also note that the methods of solution connected to eqs. (1) and (2) bear no similarity to each other. Indeed in the former case one requires a Oproblem while the latter uses the aforementioned R H problem. The (nontrivial) connection between them and the reduction to the standard Toda-lattice system will be described in a forthcoming publication.

2. The Lax pair Equation (2) arises from the compatibility of the following spectral problems,

Llt= [0x +0, + i ( / n - k ) ]#(x, k ) + C ( x -

I )#(x- 1. k) e x p ( - i l . ) +C(x)#(x+ 1, k) e x p ( i / . ) = 0 ,

(3a)

and

M#= [O,+Ox+exp(il,)-exp(-il,)

]#(x, k ) + ( C # ) ( x - 1 ,

k) exp( - i / , , ) - C ( x ) / z ( x + 1, k) exp(i/,)

=0,

(3b)

where C(x) = e x p [ 0 ( x + 1 ) -O(x) ], k e C is the spectral parameter and Riemann surface of the function

I,,(k) stands for the nth branch of the

k = l - i exp (il) - i exp( - i / ) .

(4)

The inverse function has for natural domain an infinitely sheeted Riemann surface with branch cuts along the semilines {kR= nn, ( -- 1 )"k~+ ~< 0, neT}, where ~=ln [½ ( x / 5 + 1 ) ] - x / 5 . The corresponding branch points are all second order branch points. We will not dwell here on the details of the construction of the above surface: this will be presented in a forthcoming paper. It follows from the aforesaid that branches can be labeled by an integer n; our convention is that ( n - 1 ) n < R e / n < n n , and that l, has branch cuts on both kR= (n-- 1 )x and

kR=n~. For m >/0 we consider a family of eigenfunctions ]~/meach solving eqs. (3a) and (3b) with n = 1 but with different behaviors; they solve /~m(x,k)=l+ where

i

G,,(x-y)(V#,,,)(y,k),

(5)

G,, is given by H(-x)) k= --oo

+ ( ~H(X)--k=O

k=m+

Ot2k

exp[i{12k --/1 )X]

I

k=~-ooH(-x,)~2k+~exp[i{12k+'-l')x]"

Otk = 1 / [ 1 + exp (ilk) + e x p ( - - i l k )

],

(6)

and

(V#m) (x, k) = -O,#,,(x, k) + [ 1- C ( x - 1 ) ]#,, ( x - 1 ) exp( - i l , ) + [1 - C ( x ) ]#m(X+ 1 ) exp (i/~) . 414

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For m ~< - 1 we consider a different family of eigenfunctions//m which also solve eqs. (3a) and (3b) but now with n = 0. Specifically, the function v,~ solves the analog of eq. (5) with G replaced by ~ given by (~m(x,k)= (k=m+ ~ t H(x)-

k=~-ooH(--X))Ot2k+t exp[i(12k+t--lo)X]

-t- ( °~, H(X)-- ~ H(--X))OtEkeXp[i(12k--[o)X] Vov defined as Vov(X, k) = -Otv,,,(x, k)+ [1 - C ( x - 1 )] v,, ( x -

(8)

and with V/t replaced by

1 ) e x p ( - i / o ) + [1 -C(X)]Vm (X+ 1 ) exp (i/o) .

(9)

We note that both G,, and/t,~ are functions of 1~(k) although we simply write Gm(k), ltm(k). Similarly t~,, and v,,,(k) depend on k only via lo(k) (the dependence upon other branches can be proven to drop out).

3. Analytic properties We list the main properties of the above functions. (i)/~,, is well defined for mn~O.Further it is holomorphic within the above set and the limits to the boundary exist. (ii) As k approaches infinity ~m(X, k) tends to e -°tx). Similar considerations apply to urn. Thus (iii) u,, is well defined for mn
4. Inverse problem The determination of an inverse problem involves evaluating precisely the differences ,um - ~m-- 1 as we cross the lines kR=rnTt, m # 0 , 1. We have that (i) Ifn~>2 and kR=nn

lt,,(x, k=nn+e+ikl)-lt,,_~(x, k=nn-E+ikl) = F n ( k ) exp[2i(nTt--lm)x]v_~,,+l)(x, k= where the scattering data F,,(k) is given by Fn(k) = a 2 , ( k )

~ dx exp [ --2i(nn--llR)Xl

nn-E+iki) ,

V#,_I (x, k).

ka=nn v'(x~k=nn+E+ik~)-v'~-~(x~k=n7t-~+ik~)=H~(k)exp[2i(nn-~R)x]~t-'~(x~k=-nn+E+ik~) ~

(10)

( 11 )

(ii) I f n ~ < - 2 and

(12)

where H , is also scattering data given by 415

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P.(k)

H.(k)- 1+p.(-E) '

(13a)

F . ( k ) = -c~2.+, (k) f dx exp[2i(/oR --n~)x] Vov._, (x, k) ,

(13b)

p.(k) =a~ (k) ~ dx V,/t_.(k) .

(13c)

(iii) At kR = 0 one has that /to(X, kR = 0 + ) = u_l(x, kR = 0 _ )

(14)

(/to(X, kR=0+ ) - lim~o÷ /to(X, kR= E), etc. ). (iv) At kR=~ boundedness requires that both/t~ and/to admit analytic continuation to the neighbouring strips and that the relationship /tl =/to

(15)

applies. ( v ) / t , has a branch line across the semilines {kR=~, ( ~ - k [ ) ~<0} and {kR=0, (~+kj) ~<01. This stems from /t being actually a function of l~(k) and not of k only. Thus, although the functional form o f / t remains the same when we cross the above lines its argument l~ (k) does change. Note that this jump is determined by the knowledge of the function/t on one of the banks kR= 0+ along with the jump of branch l(k). To summarize. /t~ is defined on the strip {0
F.(k, t)=F.(k, 0) exp[4i sin(/~R) cosh (/~)t]

(16a)

and

H.(k, t) =H.(k, 0) exp[4i sin(/oR) cosh(lol)t ] .

(16b)

We now define ~=/t~e °(x),

u.e °(~),

nTt
n>~O,

nTt
n<~--I .

(17)

Note that as above ~Udepends upon l~(k) and not on k. Thus we have to bear in mind that kU(k) stands for ~(l~ ( k ) ) . It follows that ~ h a s an analytic structure similar to/t., u~; but its normalization is different: it tends to 1 as k approaches infinity. Conditions ( i ) - ( v i i ) above along with the normalization property define the analytic structure of the function ~ o u t of which it can be uniquely determined. Note that this function besides being sectionally analytic has also some branch cuts. This is different from other inverse problems. Thus the analytic structure of eigenfunctions associated with differential-delay equations has two novelties. Namely: they are made up of an infinite number of functions and some of them are multivalued. We now summarize the main properties of ~: (i) It is well defined on each of the strips { n ~ < k R < ( n + 1 )Tt}, Vn where it is holomorphic. The limits to the boundary do exist. (ii) For n ~ 0 , 1, - 1 it has different representations on either side of these lines. Actually

~(x,k=n~+e+ik~)-~(x,k=n~-e+ik~)=F.(k)exp[2i(n~-ljR)X]~_~+,)(x,k=-n~-~+ik~) 416

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if n>_.2 and ~U(x, k=nn+~+ik~)

- ~(x, k=nn-e+ik~) = H , ( k ) exp[2i(nn--loR)X] qu_,(x, k= -nn+~+ikl)

(19)

for n~< - 2 . (iii) It is branched across the lines {kR= _+n, ( - k ~ < 0 } (recall/tl =/to). Notice that the knowledge of ~P(l(k) ) at one o f the banks kR=n+ determines precisely ~u at the other bank (the argument o f this function changes but not its functional form). The same thing happens at k a = - n . (iv) It is continuous when going across the line k R = 0 (recall eq. ( 1 4 ) ) . (v) As k approaches infinity ~(x, k) tends to 1. The above features determine the function ~u. In principle this data allows one to recover the function ~u. Indeed assume one is given F,, n # 0, 1 and H,, n ~ 0 , - 1. We have an infinite matrix Riemann problem for/t,, n ¢ 0 and u,, n ¢ - 1 (which can be viewed as a 6-problem) and hence under favorable conditions (small norm and no poles, see the discussion about solitons below) we can expect existence and uniqueness of the solutions/tn, n # 0 and un, n ¢ - 1. Then we use eqs. (14) and (15) to obtain/to and u_ tFinally the potential is recovered from

O(x, t ) = ½In [ ~U(x, t, l o ( k = 0 + + i o o ) ) ] .

(20)

5. Solitons We now allow for solitons to appear. This can occur whenever ~u has a pole at a point which it behaves as ~U(x, k ) =

~(x) exp [il~ (k) ] - e x p [ill (ko) ]

+regular,

ko=n_ + i a around

(21)

where q~ stands for the corresponding residue. Note that ~u has a pole at the point ko but it is not singular at /~o =- n+ + i a . It can be proven that the following relationship applies, q~(x) = (2 exp[ (ll - 12)x] ~) ( k = n+ + i a ) ,

(22)

where 2 is (constant) scattering data. Let lj ( k = n+ + ia) = iyo and 12( k = n + + i a) = iy2. We introduce p = Y2- Yl and w=x/pz+4(ep/2-e -p/2) 2. Then the solitonic scattering data can be seen to evolve as follows, 2(0=2(0)

e ~°' .

(23)

In this case we have a Riemann problem which consists of determining a function whose analytic structure is the following: it has a branch cut on the semiline {kR = n, k~ >i (} and a single pole at the point ko = n_ + ia, a >t (. This information along with eq. (22) allows one to obtain the function ~u. We obtain the solitonic configuration which is given by

exp(px+ogt+Xo) .~ O(x, t) = ½In 1 + (e p - 1 ) 1 +exp(px+~ot+Xo),l"

(24)

We do not present here the explicit expression o f the N-soliton configuration; we simply mention that they arise from function ~u having N poles, kl ..... ku, all of them located on the semiline {kR=n, k~>~(}; hence ki-n_ + iai, ai >>.(. Let q~i(x) stand for the residue at k~. Then (22) yields that the q~ are obtained as solutions o f the linear system 417

Volume 180, number 6

~ i ( x ) : A i e x p [ ( Y ~ i'

PHYSICS LETTERS A

• ~(x) _ylq)

Y t " ) x ] ( 1 + j=~, exp[ _ y ¢ , , ] _ e x p [

20 September 1993

(25)

where i y ~ ) - l l ( x _ + iaj), iy m =l~ (x+ + i a j ) .

Acknowledgement T h i s work was partially s u p p o r t e d by the A i r Force Office o f Scientific Research u n d e r grant no. A F O S R 90-0039, the N a t i o n a l Science F o u n d a t i o n u n d e r g r a n t no. D M S - 9 0 2 4 5 2 8 a n d the Office o f N a v a l Research u n d e r g r a n t no. N 0 0 1 4 - 9 1 - j - 4 0 3 7 . J.V. was also partially s u p p o r t e d by grant C I C Y T A E V 9 0 - 0 3 J 2 from Spain.

References [ 1 ] J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (SIAM, Philadelphia, PA, 1981 ). [2 ] M.J. Ablowitz and P. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge Univ. Press, Cambridge, 1991 ). [3 ] S. Novikov et al., Theory of solitons. The inverse scattering method (Plenum, New York, 1984). [4] M.J. Ablowitz, S. Chakravarty and L. Takhtajan, A self dual Yang-Mills hierarchy and its reductions to integrable systems in 1+ 1 and 2+ 1 dimensions, Program in Applied Math./UCB 132, to be published in Commun. Math. Phys. [5] J. Villarroel, J. Phys. A 24 ( 1991 ) 3587. [6] J. Villarroel and M.J. Ablowitz, Phys. Lett. A 163 (1992) 293. [7] J. Villarroel and M.J. Ablowitz, Physica D 65 (1993) 48. [8] O.I. Bogoyavlenskii, Russ. Math. Surveys 46 ( 1991 ) 1.

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