IL NUOVO CIMENTO

VoT.. 78 B, N. 2

11 Dicembro 198g

Matrix Spectral Transform, Reductions and the Burgers Eq~lation ('). A. DEGASPERIS I s t i t ~ t o di F i s i c a dell'Universith . 00185 R o m a , I t a l i a I s $ i ~ t o ~u d i F i s i c a Nucleare - Sezione di l~ama

J. JP. L E O N L a b o r a ~ i r e de P h y s i q u e M a f l ~ a t i q u e ,

USTL

- 34060 M o ~ t p e l l i e r Cedez, F r a n c e

(rieevuto il 27 Gennaio 1983; manoscritto revisionato ricevuto 1'1 Agosto 1983)

Summary. - - A reduction method is introduced for the nonlinear evolution equations associated with a N • matrix spectral problem. It is shown that this technique, as applied to the 2 X 2 generalized ZakharovShabat spectral problem, yields the Burgers equation. This equation is then investigated within the spectral-transform method. PACS. 02.30. - Function theory, analysis.

1.

-

Introduction.

The spectral transform, i n t r o d u c e d as a tool to solve nonlinear evolution equations (see, for instance, (1) a n d t h e references quoted there), has been vastly investigated a n d generalized in recent times. This m e t h o d starts with a linear equation t h a t we consider here of the t y p e (a subscript variable indicates partial differentiation and bold-faced letters are matrices) (1.1)

Wx(z, k) = X ( x , k)W(x, k) ,

--~<

z<

+c~,

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (*) F. CALOOEROand A. DEGASPERZS: Spectral Trans]orm a n d Solitons, Vol. 1 (Amsterdam, 1982). 9 - II Nuovo Cimento B.

129

130

A, D E G A S P E R I S ans

J. JP. LEON

w h e r e X(x, k) is a iV • iV m a t r i x a n d k is the c o m p l e x s p e c t r a l variable. A l t h o u g h in s o m e cases it m a y be m o r e c o n v e n i e n t to s t a r t with a higher-order differential e q u a t i o n (~), it is plain t h a t a n y linear o r d i n a r y differential equation can b e set in t h e f o r m (1.1). Moreover, one could also replace (1.1) b y a difference (rather t h a n differential) linear m a t r i x equation (s). Yet, b y considering only linear equations of t y p e (1.1), one can still bring into p l a y the b o u n d a r y conditions for X(x, k) a n d its d e p e n d e n c e on t h e spectral variable k. I n fact, b o t h t h e s e features of X(x, k) are quite r e l e v a n t in t h e inverse p r o b l e m associated with (1.1). I n this r e s p e c t , ,~ faicly general investigation of the inversion t e c h n i q u e is given in (4). H e r e we limit our a t t e n t i o n to t h e simplest case, n a m e l y

(1.2)

X(x, k) = O ( x ) -

ika,

where t h e m a t r i x a is x - i n d e p e n d e n t , traceless, (1.3)

a. ~ 0 ,

Trio] = 0 ,

a n d diagonal with simple, n o n v a n i s h i n g a n d r e a l eigenvalues, (1.4)

a,~ = a, ~iJ,

a~ =/= as

for i r ~ ,

a,

= a* :/: 0

,

a n d w h e r e t h e m a t r i x Q(x) has zero diagonal elements,

(1.5)

q,,(x) = o ,

i = 1, 2, ..., i v ,

a n d a s y m p t o t i c a l l y vanishes (fast enough) as x--> :]:oo, for short

(1.6)

Q(+ ~) = 0.

I n t h e following, we shall d e n o t e b y d t h e s p e c t r a l p r o b l e m (1.1) w i t h condition (1.2) a n d p r o p e r t i e s (1.3)-(1.6). E x c e p t for t h e simplicity a n d r e a l i t y of t h e eigenvalues of t h e m a t r i x a, a s s u m e d only to simplify t h e following t r e a t m e n t , all other conditions are not restrictions, for t h e y could b e enforced b y a n a p p r o p r i a t e g a u g e t r a n s f o r m a t i o n of (1.1) (5).

(2) I. !V[. G E L ~ P A N D and L. A. DIKII: .Funct. Anal. Appl., 12, 81 (1978) (Funkts. Anal. Prilo~., 12(2), 8 (1978)). (a) •. J. ABLOWITZ:Nonlinear evolution equations--eor~tinuous and discrete, in S I A M t~ev., 19, 663 (1977); D. LEVI: The spectral trans/orm as a tool /or solving nonlinear discrete evolutior~ equations, in Lecture Notes in Physics, 98, 91 (Berlin, 1979). (4) 1). J. CAVDREY: Physica D, 6, 51 (1982); R. BEALS and R. R. C0XFMAN: Scattering and inverse scattering /or /irst order systems, Department of Mathematics, Yale University preprint (1982). (5) F. CALOOERO and A. DEGASPERIS: Spectral Trans/orm and Solitons, Vol. 2 (Amsterdam, 1984).

I~IATRIX SPECTRAL TRANSFORM, REDUCTIONS AND THE BURGERS E Q U A T I O N

131

In the following section we introduce a technique to r e d u c e the n u m b e r of i n d e p e n d e n t m a t r i x elements Q,~(x) of Q(x). T h o u g h t h e reduction conditions for the m a t r i x Q(x) t u r n out to be expressed b y nonlinear differential equations, t h e corresponding transition m a t r i x associated with ~/ is simply constrained to b e triangular. The present analysis is aimed a t investigating t h e subclass of nonlinear evolution equations, solvable b y t h e spectral transform associated with d , which is obtained b y asking t h e m a t r i x Q to satisfy t h e reduction conditions. In this paper, however, we limit this investigation to t h e 2 • case. I n fact, in sect. 3, we specialize t h e general formalism of sect. 2 to t h e 2 • 2 m a t r i x case, t h e r e b y investigating t h e reduction equations in t h e context of the generalized Zakharov-Shabat spectral problem (6.7). T h e subsequent section is t h e n d e v o t e d to the reduced nonlinear evolution equations. T h e r e it is shown how a reduction yields t h e Burgers h i e r a r c h y of equations (~,~):

(1.7a)

vt :

(1.7b)

vt-~ co(~, t)v~,

[to(D + v, t)v]~,

v ~- v(x, t) , v ~-- v(x, t) ,

where D is t h e differentiation operator (1.8)

D ~ d/dx ,

is the integro-differential operator (1.9)

= D ~- v + v~D -1 ,

D -1D ~ 1 ,

and (n(z, t) is an a r b i t r a r y entire function of t h e variable z. The Burgers equation (9) is obtained, of course, for ~o(z, t) ~ z. This finding shows t h a t the Burgers equation also has its own place in t h e general spectral-transform m e t h o d to solve nonlinear evolution equations (see also (~o)). I n this framework, we elucidate t h e connection between the Itopf-Cole transformation and the Bgcklund transformations and point out the peculiarity of the Burgers equation with respect to t h e infinite sequence of constants of t h e motion t h a t

(e) M.J. ABLOWITZ,D. J. KAUF, A. C. NEWELL and H. SEGUR: Stsd. Appl. Math., 53, 249 (1974). (7) F. CALOGERO and A. DV.GASPERIS: NUOVO Cimento B, 32, 201 (1976). (8) D. V. CHOODNOVSKY and G. V. CHOODNOVSKY: Nuovo Cimento B, 40, 339 (1977); D. LEvi, O. RAG~ISCO and M. BRUSClZI: Nuovo Cimento B, 74, 33 (1983). (9) J. M. BURGERS: The Nonlinear Di]]usion Equation (Dordrecht, 1974). (io) L. MARTINEZALONSO and G. GUERRERO: Modified hamilto,~ian systems and canonical tra~s]orraations arising ]rom the relationship between generalized Zakharov-Shabat and er~ergy.dependent SchrSdinger operators, in J. Math. Phys. (N. Y.) (in press).

1 3 9.

A. D E G A S P E R . I S

and

J. JP.

LEON

exist for t h e nonreduced class of nonlinear evolution equations (see, for instance, (m~)). As for this last point, it is shown t h a t t h e infinitely m a n y constants of t h e motion for t h e Burgers equation ,~re not i n d e p e n d e n t of one ~nother.

2 -

Reductions.

I n t h e following approach, it is convenient to introduce three orthogonal projection operators defined b y their action on a generic N • h r m a t r i x A as

(2.1)

diag(A) =

A,,"'.. 0

(2.2)

up(A) =

0 "'... i 0 ""...... 1N

low(A) =

(0 i;)~ )

,

A!~I..... 0 .....0

(2.3)

where, of course, A.j is t h e m a t r i x element of A a n d (2.1)

diag + up + low = 1 .

We consider now t h e t r a n s f o r m a t i o n of t h e linear equation (1.1) into t h e new equation

(2.5)

't"(x, k) =

X'(x, k)~'(x, k)

with (2.6)

t~'(x, k) -~ G(x, k)W(x, k),

(2.7)

X'(x, k) ---- G(x, k)X(x, k)[G(x, k)J-x + Gx(x, k)[G(x, k)] -x .

:Next we ask the gauge t r a n s f o r m a t i o n m a t r i x G(x, k) to be such t h a t t h e m a t r i x (n) l~. CALOG~.RO and A. DEGASPERIS: Commuu. Math. Phys., 63, 155 (1978).

MATRIX SPECTR.~.L TRANSFORM, R E D U C T I O N S AND THE BURGERS :EQIYATIO2q

133

X'(x, k) obtained via (2.7) be triangular, namely that

(2.8)

up(X'(x, ~)) = o .

Among the infinitely many transformation matrices t h a t bring X(x, lc) into a triangular matrix X~(x, k~) (see (2.8)) via (2.7), the matrix G(x, k) is conveniently chosen to have the following form: (2.9)

G(x, k) = "1 + g(x, k),

where the 1VX N matrix g(x, k) satisfies the equation

(2.1o)

up(g(x, k)) = g(x, k).

I t obviously implies that (2.11) low(G(x, k)) = 0 ,

diag(G(x, k)) ---- 1,

up(G(x, k)) = g(x, k).

I t is easily realized t h a t condition (2.8) on the matrix X'(x, k) and the structure (2.11) of the matrix G(x, k) do not imply auy condition on the matrix X(x, k) we start with. In fact, the 8 9 matrix elements gii(x,k) of g(x, k) can be computed by integrating the nonlinear differential equation obtained fronl (2.7) and (2.8):

(2.12)

u p ( G . G -~) + up(GXG -~) = 0,

G = G(x, k),

07", equivalently (see (2.9) and (2.10)),

(2.13)

g,+

{up[(1 + g)X(1 + g)-,]}(1 + g ) =

this is a system of 8 9 the boundary condition

0;

coupled scalar equations with, for instance,

(2.14)

G(--oo, k) = 1,

(2.15)

g(--oo, k) = 0 .

Once the matrix G(x, k) has been computed this way, the matrix X'(x, k) is explicitly given by (2.7). I t should be emphasized t h a t integrating the matrix equation (2.12) is merely an alternative way to solve the original matrix differential equation (1.1). Indeed the transformed eq. (2.5) can be explicitly solved because of the triangularity condition (2.8). Although one could deal with the general ease, we prefer to restrict our attention to the simpler ease characterized

134

A. D]!IGASPERIS a n d

z. Z~'. LEON

b y (1.2) with (1.3)-(1.6), n a m e l y ~ . ~VIoreover, we confine the explicit integr a t i o n of (2.5) with (2.8} to t h e appendix and r e p o r t below only t h e expression of t h e transition m a t r i x S(k). This m,~trix is defined by the a s y m p t o t i c behaviours (2.16a)

lim [exp

x - - ~ - - co

(2.165)

lim

[ikxr F(x, k)J ---- 1 ,

Imk ~ 0,

[exp[ikxa] F(x, k)] = S ( k ) ,

Imk = 0,

of the usual m a t r i x Jost solution F(x, k) of ~ . F o r notational convenience, we first introduce the matrices V and Q' according to the formulae

(2.t7)

X'(x, k) = V(x, k) § Q'(x, k) - - i k a ,

(2.18a)

V = diag(V) ,

(2.19a)

low(Q') : Q ' ,

and note t h a t t h e y possess t h e following properties (proved in the appendix): (2.18b)

Tr(V) = 0 ,

(2.18c)

V( •

(2.19b)

Q'( •

~ , ~) = 0 , k) = 0 .

Furthermore~ let /~ be t h e index of nilpotence of the m a t r i x Q', n a m e l y (2.19c)

[Q_'(x, k)]~-~ :/= 0 ,

[~'(x, k)J" = 0 ,

/~ < N

(note t h a t , in the generic case, t h a t is no a priori restrictions on the N • m a t r i x Q(x) we s t a r t with~ /~ ---- N). The expression of t h e transition m,~trix S(k) defined b y (2.16) t h e n reads (see the appendix) -{-co

-fCo

(2.20) --~

--r

X

--aO

--~

Imk ~ 0. I n this formula the m a t r i x G~(k) is defined b y t h e m a t r i x differential equation (2.12) t o g e t h e r with t h e a s y m p t o t i c condition (2.14) and (2.21)

lim [exp

[ikxa] G(x, k) exp [-- ikxa]] = G~(k),

Im k = 0,

MATRIX SPECTRAL TRANSFORhI, R E D U C T I O N S AND THE BURGERS E Q U A T I O N

135

t h a t readily follows from (2.7), (1.6), (2.17), (2.18c) and (2.19b). J is the m a t r i x integral operator whose action on ~ generic x - d e p e n d e n t h r X h r m a t r i x M(x) is given b y t h e formula

(2.22)

l~ote t h a t t h e u p p e r value/~ - - 2 of t h e summation i n d e x in t h e sum appearing on the r.h.s, of (2.20) takes into account the nilpotence of t h e operator ~r (2.23)

.f~-I # 0 ,

,f" --~ 0 ,

of course with t h e same index /~ as for O_'(x, k) (see (2.19c)). I n general, say for an a r b i t r a r y m a t r i x Q(x) entering in t h e spectral problem ~/, expression (2.20) of t h e transition m a t r i x is h a r d l y of any practical use. I n fact, its evaluation requires t h e integration of t h e m a t r i x equation (2.12) with (2.14) (or, equivalently (2.13), with (2.15)), and only their one can comp u t e G~(k) t h r o u g h (2.21), and V a n d Q' via the formulae

(2.24)

V(x, k) = diag(G(x, k)Q(x)[G(x, k)] -1) ,

(2.25)

Q'(x, k) = low(G(x, k) Q(x)[G(x, k)] -1) ,

which are implied b y (2.7), (1.2), (2.11), (2.17), (2.18a) and (2.19a). On t h e contrary, expression (2.20) for S(k) is of relevance in t h e context of our reduction t e c h n i q u e described below. Generally, a reduction m e t h o d is evidently aimed at reducing, b y imposing appropriate conditions, t h e n u m b e r of independent function~ which appear in the spectral problem. In fact, a reduction technique should have the two following properties: i) the corresponding reduction conditions on the spectral transform t a k e a simple form (possibly expressed b y algebraic linear equations), ii) the reduction conditions are compatible with t h e t i m e evolution (see below). Reduction techniques t h a t are different from the one discussed here h a v e been investigated in (1~,1s). Dealing here with the spectral problem ~ , the total n u m b e r oE independent functions to begin with is ~V(~V- 1) (see (1.5)). As a definition, here and in (i~) F. CALOGERO and A. DEGASP~.RIS: J. Math. Phys. (N. Y.), 22, 23 (1981). (13) A. V. M~KHAILOV: Physica D, 3, 73 (1981).

136

x . DEGASP~RIS a n d z. JP. ]LEON

the following, a reduction equation is any condition on the matrix (elements of) Q(x) such that the corresponding gauge transformation matrix G(x, k) defined by (2.12) and (2.14) be a rational function of the spectral variable k. The first implication of this definition is t h a t the transition matrix S(k) be triangular, (2.26)

up(S(k)) = 0.

Indeed this follows from (2.20) by combining (2.24) and (2.25) together with the equation (2.27)

G|

: 1,

which is implied, through (2.11) and (2.21), by the requirement that G(x, k) be rational in k. As for the spectral transform of Q(x), it is convenient to introduce the transmission matrix [3(k) and the reflected matrix r through the usual definitions (2.28)

S(k) = [1 + Pt(k)][[~(k)]-i ~

(2.29)

diag(g(k)) = [3(k) :

(2.30)

diag(,,(k)) = 0,

(2.31)

~(k) = [diag(S(k))] -1 ,

(2.32)

ct(k) : S(k) [diag(S(k))] - z -

.

Then the continuous component of the spectral transform of Q(x) is, by definition, the reflection matrix a(k). This matrix satisfies the reduction equation (2.33)

low(a(k)) = or(k),

as implied by (2.28), (2.30) and (2.26).)Ioreover, the index of nilpotence of a(k) coincides with that of Q(x) (see (2.19c)),

(2.34)

[ct(k)],-l# o ,

[ct(k)],= o ,

g<~'.

For the sake of simplicity, we omit to consider in this section the discrete part of the spectral transform. This matter will be, however, included in the subsequent discussion of the 2 X 2 Zakharov-Shabat case, for it plays a relevant role in the analysis of the reduction equation for the matrix Q(x). Indeed, a very simple instance of the relation between the reduction condition and the discrete spectrum is given by the choice (2.35)

G(x, k) : 1 ,

M A T R I X S P E C T R A L TRANSFORM~ R E D U C T I O N S

AND TIIE B U R G E R S :EQUATION

137

t h a t is the simplest matrix which satisfies (2.11) and is rational in k. According to condition (2.8), choice (2.35), as implied by (2.7) and (1.2), is consistent only if the m a t r i x Q(x) satisfies t h e reduction equation (2.36)

up(Q(x)) = 0 .

In this particular case, therefore, V(x, k) = 0 (see (2.24), (2.35), (2.36)) and, because of (2.20) a n d (2.31), the transmission matrix is (2.37)

~(k) : 1 .

This equation shows t h a t no discrete eigenvalue exists, since these should show up as poles of the trasmission matrix. Thus only the continuous component of spectral transform is present, namely

(2.38a)

dx exp [ikxo] Q(x) exp [ - ikxa] J~'- 1 ,

c~(k) -~

Im k = 0,

--co

tr

(2.38b)

JoM(x) =--fdy exp [ikya] Q.(y) exp [ - ikya] M(y) .

As for the general case, the transformation matrix G(x, k), if ration,~l in k, can have only simple poles, namely (see (2.9)) L

(2.39)

g(x, k) ---- ~ A(")(x)[ik -- $r

-1 ,

t$cl

where A('~(x) is a 2~r• t h a t of g(x, k),

(2.40)

matrix whose index of nilpotence is assumed to be

[g(x, k)].-, # o,

[g(x, k)].= o,

g
and where the number L of poles, at k = --i~(n~(x), can be arbitrarly chosen. I n fact, t h e general structure (2.39) can be inferred from eq. (2.13) or, more easily, from the alternative form of (2.13) t h a t we rewrite below (see (2.43)). To this aim, consider first t h a t (2.40) imphes

(2.41)

[c(x, k)]-,-- ~E (-)~[g(x, ~)],,

and t h a t t h e product of a n y /~ matrices whose index of nilpotence is # vanishes. Then insert (2.9) and (2.41) into eq. (2.12) and use the shorthand

138

A. DEGASPERIS a n d

J. JP. L ~ . o ~

notation

Q(-)(x)

(2.42)

:

up(Q(x)),

Q(+)(x) = lo,v(Q(x)),

to obtain t h e following m a t r i x differential equation: /.~--2

$~--2

,u--3

(2.43)

+ q'-' + . p

Q(+'Jgo' +

=0.

B y inspecting this equation, one can readly see t h a t , in fact, only simple poles in k are allowed (compare t h e first and second t e r m s with the last t e r m of t h e 1.h.s.). As a m a t t e r of fact, t h e differential equation (2.43) is compatible also with the requirement t h a t g(x, k) be a first-degree polynomial in k, say g(x, k) : So(u) ~- ikg~(u); however, this dependence on k does not allow us to satisfy t h e asymptotic condition (2.15) as far as t h e m a t r i x Q(x) vanishes as x -->--c~ (as we assume t h r o u g h o u t this paper). Because of this, in t h e following we limit our discussion only to t h e Z-pole formula (2.39). We do not investigate here t h e nonlinear m a t r i x differential equation (2.43) in the general ~V• case. We content ourselves with noticing t h a t solutions of (2.43) of t y p e (2.39) m a y exist only if t h e m a t r i x Q(x) satisfies special conditions, these being precisely the reduction equations. F o r instance, t h e trivial solution g----0, t h a t corresponds to t h e choice (2.35), exists only if Qc-)(x) _- o, and this is the simplest reduction (2.36). Less trivial reductions will be exhibited for the 2 • 2 case (see the n e x t section), each of which corresponds to a different choice of t h e n u m b e r Z of poles of G(x, k) (see (2.39)). We finally discuss in t h e present section the compatibility of the r e d u c t i o n condition with the time evolution. This m a t t e r is, of course, more c o n v e n i e n t l y dealt with b y considering t h e spectral transform of Q(x) r a t h e r t h a n Q(x) itself (and again, for the sake of simplicity, we leave out of t h e following discussion t h e discrete p a r t of the spectral transform). Following A K N S (e), let us introduce in all t h e formulae w r i t t e n above a p~rametric dependence on t h e e x t r a variable t (the time) and ask t h a t t h e m a t r i x 5ost solution of ~r evolves in t i m e according to the equation

(2.44)

Ft(x, t, k) = T(x, t, ~) F(x, t, k) - - F(x, t, k) r(k, t),

where T(x, t, k) and r(k, t) are h r • matrices. T h e r e q u i r e m e n t t h a t (2.44) be compatible with t h e spectral problem d , t h a t is with t h e equation

(2.45)

F~(x, t, k) = [Q(x, t) - - ika] F(x, t, k ) ,

MATRIX SPECTRAL TRANSFORM, REDUCTIONS AND THE BURGERS EQUATION

]~9

t h e n yields (2 A6)

Q t - T~ + [Q, T ] - - ik(a, T] = 0.

From this equation, a class of evolution equations for the matrix Q(x, t) is obtained by assuming tha~ T(x, t, k) be polynomial in k. However, our interest here is confined only to the corresponding class of equations governing the evolution of the spectral transform ct(k, t) of Q_(x,t). Such evolution equations can be derived by noticing first t h a t (2.46), together with the polynomial k-dependence of T and the asymptotic vanishing behaviour of Q as x --~ • implies t h a t

(2.47)

T(--c~, t, k) = T ( ~ o % t, k).

Then combining (2.44) with the asymptotic behaviour (2.:16a) yields

(2.48)

T(--c~, t, k) = r(k, t),

(2.49)

diag(r(k, t)) = r(k, t).

Finally, the evolution equation for the transmission matrix S(k, t) follows from (2.16b), (2.44), (2.47)-(2.49) a n d reads (2.50)

St(k, t) = Jr(k, t), S(k, t)],

Imk : 0.

Since the m a t r i x r(k, t) is diagonal (see (2.49)), this last evolution equation obviously implies that, if S(k, t) is triangular at the initial time to, t h e n it stays t r i a n g u h r during the whole evolution. This guarantees t h a t the spectral transform a(k, t) (see (2.32)) satisfies the reduction equation (2.33) at any time t,

(2.51)

low(~(k, t)) -- ~(k, t),

if it does so at t = to. This conclusion can be immediately read out of the evolution equation

(2.52)

a,(k, t) = Jr(k, t), a(k, t)],

hnk = 0,

t h a t follows from (2.50), (2.28) and (2.49). For future reference, we also report here the evolution equation for t h e transmission matrix:

(2.53)

{~,(k, t) = 0 .

140

A. D ~ A S P E ~ S

and

~.

JP. L E O N

3. - The 2 X2 Zakharov-Shabat spectral problem. The reduction technique introduced in the previous section is specialized here to the 2 • matrix case, namely to the spectral problem (1.1) with (1.2) and

(3.1a)

Q(x)= (

qC-)(oX))

(3.')

o =(: --:)'

0

qc§

&ca

(3.1c)

jdx Iq'•

<

(see (1.2)-(1.6)), this being the generalized Zakharov.Shabat spectral problem (5-7). I n this case it is readily seen t h a t , because of (2.10), the m a t r i x g(x, k) has the form (3.2)

g(x,k)=

h(x,k)(:

:),

where the function h(x, k), as implied by (2.43), satisfies the Riccati equation (3.3)

h~ ~- 2ikh ~ q(-)(x) -- q(+)(x) h ~ ~ 0 ,

h ~ h(x, k).

I t is also easily verified t h a t the matrix (2.9) with (3.2) transforms the Zakharov-Shabat equation (1.1) (wiLh (1.2) and (3.1)) into eq. (2.5) with (2.17) and (3.4)

(3.5)

V(x,

k) ~ qC+~(x)h(x, k ) ~ ,

O_'(x,k)=q(+'(x)(~ :).

Consider now the condition t h a t the matrix (3.2) be rational in k (see (2.39)). In this case this condition reads L

(3.6)

h(x, k) :

~ aC'O(x)[ik -- ~('O(x)]-1 , n=l

since, as ill the general case, only simple poles are comp,~tible with eq. (3.3) (see, however, the r e m a r k m a d e after eq. (2.43)). Transformation (2.6) is, therefore, characterized by the 2Z functions

MATRIX SPECTRAL TRAMSFOR~I, REDUCTIONS

141

AND THE BURGERS EQUATION

aC")(x), ~(')(x), n ~-- 1, 2, . . . , L , and these, together with qC+)(x) and q H ( x ) , have

to satisfy the following differential equations that obtain by inserting (3.6) in (3.3) : (3.7a) ~")(x) = qC+~(x)al,,)(x) ,

n = 1, 2, . . . , Z ,

r.

(3.7b) a(_")(x) : 2q(+)(x) ~ aC")(x)a(,,)(x)[~(,O(x) - - ~(,,,)(x)]- 1 - 2a(")(x) ~c,O(x) , m~n

n ----

1, 2, . . . ,

L,

I,

(3.7c) 2 ~ a("~(x) ~- q ( ) ( x ) = 0 . These are 2L ~ 1 differential equations for 2L ~- 2 functions and, therefore, they lead, upon elimination of the functions ~r and a("~(x), to a (generally complicate) relationship between q(+)(x) and q(-)(x). This relation is the reduction equation and, in the following, we will refer to its solution as to the/~-pole reduction, just to recall that it originates from a transformation matrix (2.9) with Z poles (see (3.2) ~nd (3.6)) in the k-plane. Let us discuss now the 1-pole reduction (the zero-pole reduction corresponds to the trivial case qC-)(x) ----- 0). In this case, eqs. (3.7) with /~----1 can be easily solved and found to yield the reduction equation

(3.s)

q(+~(x) q(-)(x) - - [ln qC-)(x)],,.

This can be satisfied by introducing the function ~(x) through the equations (3.9a)

q(-~(x) ~- exp [~(x)]

(3.9b)

q(+)(x) ~- q~:(x) exp [-- ~(x)].

,

In terms of ~(x), the solution of (3.7) then reads

(3.10a)

aCl)(x) ~ a ( x ) ---- - - 89 e x p [(p(x)] ,

(3.10b)

~c,)(x) _= ~ ( x ) =

- - 89~ , ( x ) ,

so that function (3.6) takes the expression (3.11)

h ( x , k) ~-- - - [~x(x) -~ 2 i k ] -1 exp [~(x)].

Let us emphasize that this solution of the Riccati equation (3.3) does yield the solution of the generalized Zakh,~rov-Shabat equation, with qC+)(x) and qC-)(x) given by (3.9), for any choice of the function ~(x); the explicit

142

A. D:EGASPERIS a n d

z . aP. L~O~r

expression of the general solution tY(x, k) can be easily recovered from the formulae derived in the appendix. However, from now on we focus our attention on those functions ~v(x) such that (3.1c) holds with qr177 given by expressions (3.9). This condition guarantees that the corresponding matrix Q(x), given by (3.1a) with (3.9), possesses the spectral transform (for the definition of the spectral transform of Q(x) see below and (~,~'~)). In addition to the requirement that ~(x) be appropriately smooth for finite x and real (for the sake of simplicity), we ask ~(x) to have the following asymptotic behaviour: (3.12)

q~(x) --> T 2P (~)x + c(• @ ~(:L)(x) ,

x ---> 4- c~ ,

where pc• > 0,

(3.13) c(• are constants and (3.14a)

V w ( 4- ~ ) = 0 ,

(3.14b) (3.14o)

**t

exp [4-

-->0,

X

exp[4-

-->0 ,

~. --.->- 4 - (~:) .

"--> 4 - ~

~

Condition (3.13), together with the expression of the transmission coefficients (see appendix) (3.15)

fl~•

= [k 4- ip(=V)J/[k ~ i p W ] ,

shows that the Zakharov-Shabat spectral problem with (3.9), (3.12)-(3.34) possesses two discrete eigenvalues, namely at k = ipC+) and at k------ip(-). Indeed, in the theory of the spectral transform it is proved that the poles of fl~+~(k) (fiN(k)) in the upper (lower) half-plane are in one-to-one correspondence with the discrete eigenvalues of the spectral problem. For the sake of completeness, we report below the explicit expression of the spectral transform of Q(x), with (3.1a), (3.9), (3.12)-(3.14). The spectral transform of Q(x) is generally defined as the set of quantities (3.16)

sEQ] =

=

2,..., N},

l ~ > 0 and ~_(2) are their where k~~) are the discrete eigenvalues with • I ~--~'(~) associated parameters defined by the equations (3.17a)

,-*+~lim[exp[--ik~'x]~)(x)]='~)(~)'

(3.17b)

,--+~lim[exp[ik:-)x]qJ"-'(x)J=Y:-'(1O)'

143

MATRIX SPECTRAL TRANSFORM, REDUCTIONS AND THE BURGERS EQUATION

(3.17c)

n = 1, 2~ ..., 37 ,

Q:~'---- [y~l] ~ ,

fdx

(3.17d)

(01

0)%, (x) ----I,

n ----i, 2, ..., 37,

--r

where q~*~(x) is the normalized vector solution of the Zakharov-Shabat spectral problem with k = k,r177. I n the present case, 37 = 1, k r =- • ip r i.e. (see (3.12)) (3.18)

} ( ~ -_ i~,(:F

(3.19a)

^r -- -- 2[p r -f- pl-)] "~exp [-- c(-)]

(3.19b)

@y'--~ 89exp [c(+)],

~),

-t-r ' ~ 0 z ( X ) 2~" ]} exp

[-- 2 i k x ]

,

...i . j

(3.20b)

ell(k) = 0 ,

where c(• a n d qr are related to g(x) by (3.12) a n d (3.9b). I n particular, it should be noticed t h a t the class of matrices Q(x) (3.1a) t h a t satisfy t h e 1-pole reduction equation (3.8) contains the well-known onesoliton expression (3.21)

qCi~(x) = -~ 2p exp [ ~: 2 i s ( x - - A ) ] / c o s h [ 2 p ( x - - ~)],

where p, s, $ and ~ are arbitrary parameters; in fact, this case corresponds to the particularly simple spectral transform (3.22a)

ac•

---- 0 ,

(3.22b)

k c*)

s • ip

(3.22c)

e[~ - -

Tip

exp[2i(T s 2 -

/f=l, ip~)] .

We have, therefore, completely characterized t h e class of matrices Q(x) (see (3.1a)) t h a t satisfy the 1-pole reduction and t h a t are spectral-transformable. I n fact, in x-space t h e y are the solution of the reduction equation (3.8) and are t h e n generally given by formulae (3.9) with (3.12)-(3.14); in k-space, on t h e other hand, t h e y are characterized by the property of having only two points in the discrete spectrum (one eigenvalue in the upper hMf-plane and one in the lower half-plane) and a triangular reflection matrix (see (A.14)), i.e. a ( - ~ ( k ) = O. The connecting formulae are given by

144

x. D~GASPERIS

and

J. JP. L ~ O ~

(3.18)-(3.20). Moreover, it is easy to verify t h a t t h e functions qC+)(x) and qH(x) t h a t are obtained via t h e inverse m e t h o d of Gel'fand-Levitan-Marchenko (see, e.g., (7)) from the spectral t r a n s f o r m (3.16) with all(k) = 0 and N = 1 a u t o m a t i c a l l y satisfy t h e r e d u c t i o n equation (3.8). Similar conclusions can be r e a c h e d also in t h e case of t h e Z-pole reduction. I n this case, however, eqs. (3.7) are more complicated, and we limit our treatm e n t to t h e mere display of t h e expressions t h a t characterize t h e 2-pole reduction; t h e y read (3.23a)

q~+)(x)q(-)(x) = [ln qH(x) ~ v ( x ) ] ~ ,

(3.23b)

v~(x) = [q(-I(x)]-~ exp Iv(x)].

H e r e v(x) is an a r b i t r a r y function, with the only condition t h a t the functions qC• obtained from (3.23) (3.24a)

qC-)(x)

= [~,,(x)]-~ exp E~(x)],

(3.24b)

qr

= v.dx)[2~(x)

-- In

v,,(x)],,

exp [-- v(x)]

satisfy condition (3.1c). Concerning the properties t h a t characterize t h e Z-pole reduction in k-space, we note t h a t assumption (A.18) on t h e position of t h e poles ~(')(x) as x -~ -4-~, together with t h e a s y m p t o t i c vanishing of t h e residues a("~(x), i.e. aC,~(:J=c~)= 0 (see (3.6), (3.2), (2.9), (2.14), (2.21)" and (2.27)), imply, as explicitly shown b y t h e differential equation (3.7b) as x -+ -boo, t h a t

~,~-(+)"-/-0 ,

(3.25)

n = 1, 2, ..., Z ,

where t h e numbers /,, ~'(• are defined b y (A.18). T h e implication of condition (3.25) is t h a t the spectral t r a n s f o r m of a m a t r i x Q(x) satisfying t h e Z-pole reduction equation must possess the Z discrete eigenvalues (3.26)

k c• n

-- --

-4- "~p,~ "-r

,

n =

1, 2,

"~

'

L

*

Indeed, t h e explicit expression (A.17) of t h e transmission coefficients shows t h a t fl~§ (flc-)(k)) has Z poles in t h e upper (lower) half-plane, t h e r e b y proving t h a t the corresponding Z a k h a r o v - S h a b a t spectral problem possesses 2L discrete eigenvalues. The conclusion is t h e n t h a t t h e L-pole reduction equation in k-space has t h e simple f o r m (see (3.16)) (3.27)

~tc-)(k) ---- 0 ,

N = L.

As a final remark, we n o t e t h a t , of course, t h e symmetrical c,~se, a(+)(k) = 0, N = Z, would obtain b y starting from the v e r y beginning with a m a t r i x

145

MATRIX SPECTRAL TRANSFORM, R E D U C T I O N S AND THE BURGERS E Q U A T I O N

G ( x , k) having t h e alternative triangularity property (see (2.9) and (3.2))

(0 :)

G ( x , k) = 1 + h(x, k) 1

"

4. - The Burgers class o f evolution equations.

I n sect. 2 we have shown t h a t the L-pole reduction of the N • spectral problem (1.1) with (1.2) is compatible with the isospectral flow described by the evolution equation (2.46). This means that, if t h e m a t r i x Qo(x) satisfies the L-pole reduction equation, t h e n t h e matrix Q ( x , t) t h a t obtains b y integrating the evolution equation (2.46), with the initial condition Q ( x , to) ~ Q_o(x), satisfies the same L-pole reduction equation at all times. Here we consider the (class of) evolution equations (2.46) in the simplest case of 2 • 2 matrices. The corresponding spectral problem is, of course, t h a t associated with the Zakharov-Shabat equation (1.1) with (1.2) and (3.1); moreover, it is more convenient to rewrite the evolution equation (2.46) in the following form (6-7): (4.1)

Q, : ay(L, t) Q ,

Q ~ Q ( x , t) ,

where a is the constant matrix (3.1b), y(z, t) is an arbitrary polynomiM in z, i.e. M

r(z, t) = Z r . , ( t ) ( -

(4.2)

2iz).,,

o < ~ < ~,

and L is the integro-differential operator whose action on a generic off-diagonal matrix M ( x ) is specified by the equation r

111

To the evolution equation (4.1) in x-space there corresponds in k-space t h e following evolution equation for the spectral transform of Q(x, t) (see (3.16), (A.14) and (2.52)) : (4.4a)

%-(+)= T y(k, t)a (• ,

a ~:) ---- a(+)(k, t) ,

(4.4b)

dk~=-)/dt --=- 0 ,

n = 1, 2, ..., N ,

(4.4c)

d~)(t)/dt

ev

I0 - ] l

,

~, :

1, 2, ..., N .

hen evident t h a t the reduc~ic, n condition (3.27) is compatible with tu:i,~'~ equa~i,.n (4.1) for tile reduced matrix Q_(x, t).

I t i,,

the

= 9 ,,rk /~ ~(• , t) ~ ( t )

Nux, vo Cimento

B.

146

A. DEGASPERIS and J. JP. LEON

[ Let us focus our attention first on the 1-pole reduction ~the 0-pole reduction q(-)(x~ t) = 0 merely reduces (4.1) to the class of linear evolution equations (4.5)

q~+~ =

--

~' ( --~~ G ,

t ) q(+),

q(+)= q(+}(x,t)).

I n this case q(+)(x, t) and q(-)(x, t) satisfy the reduction equation (3.8) and this implies that the matrix evolution equation (4.1) can be rewritten as the scalar evolution equation (4.6)

(lnq(-)), ~ y(s

q(-) = qH(x~ t) ,

t).1 ,

where .Sf is the first-order linear differential operator defined by the formula

(4.7)

.2~](x) = (2i)-l(f,(x) -- ](x)[ln qH(x, t)]~},

where l(x) is an arbitrary (differentiable) function. I n fact~ this result follows from the identity

(4.8)

D

1/q C-~ q(+)/] --~

D s

f~.l,

that, of course, holds only if (3.8) is satisfied; h e r e / ) is operator (1.8). In fact, (4.6) is a straight consequence of (4.2), (4.8) and of the formula 0

(4.9)

q(-)~

.D 2

(1D 1 ) O , ( x , t)(1/qr_ ) q~+,]= (1D

D3) [lnq(-'(x, t)],,

that obtains by taking into account the reduction equation (3.8). The evolution equation (4.6) can be given a nester form by introducing the function (4.10)

v(x, t) ~-- --[ln qr

t)],

and replacing the polynomial (or entire function) ~(z, t) that characterizes eq. (4.6) with the expression

(4.11)

~(z, t) ~-- YoCt) -- 2iza~(2iz, t)

that defines the polynomial (or entire function) w(z, t); in fact, one finally obtains in this way precisely the Burgers class of evolution equations (1.7). Indeed, for w(z, t ) : z~ (1.7) is the well-known Burgers equation (8) (4.12)

vt ~ vx~ ~ 2vv~ 9

v ~-- v(x~ t) .

MATRIX SPECTRAL TRANSFORM, R E D U C T I O N S AND THE BURGERS E Q U A T I O N

147

This equation, being of applicative relevance as a model for diffusion and shock formation (9), has been largely investigated. An i m p o r t a n t tool in this respect is, of course, the following transformation t h a t linearizes t h e Burgers equation, or rather the whole class (4.13)

vt = D w ( D + v, t) v ,

v =~ v(x, t) ;

this transformation, t h a t has been introduced indipendently by ttoPF and COLE (~4) and reads

(4.1~)

O~(x, t) = v(x, t) O(x, t ) ,

introduces (up to a x-independent factor) the new d e p e n d e n t variable O(x, t) t h a t satisfies the linear evolution equation

(4.15)

0 , = Dto(D, t)O ~- a(t)O ,

0 ~ O(x, t)

(a(t) being an arbitrary x-independent function) whenever v(x, t) in (4.14) is a solution of (4.13). We note t h a t , while this transformation is local, the spectral-transform approach clearly relies on nonlocal properties (particularly on the asymptotic behaviour as x --> • c~) of the functions q(+~(x, t) and q H ( x , $) t h a t enter into t h e Zakharov-Shabat spectral problem; therefore, in order to apply the spectral-transform approach to investigate eq. (4.13), one has to specify the class of functions v(x, t) t h a t yield a 2 • 2 matrix Q(x, t) (see (3.1a)) t h a t possesses the spectral transform (3.16). This is done b y noticing t h a t , according to (4.10) and (3.9a), one has

(4.16)

v(x, t) = - - qJ~(x, t)

and that, as shown in the previous section, the asymptotic behaviour specified by (3.12)-(3.14) implies t h a t (4.17a)

v(x, t) --> 4- 2p~ :~ -4- uc:~)(x, t ) ,

(4.17b)

ur

t) exp [ :j: pC+)x] --> 0 ,

x --> :j: c~ ,

(4.17c)

-a - r ~% t) exp [ -i- p~:~)x] -> 0 ,

x --~ : k o 3 ,

x --> :j: oo ,

with p(• > 0 (see (3.13)). We should point out in this respect t h a t conditions (4.17) on the class of solutions of the Burgers equation (or, more generally, of eq. (4.13)), t h a t can be analysed via t h e spectral transform, are strictly (14) j. D. COL~: Q. Appl. Math., 9, 225 (1950); E. ttOPF: Commun..Pure Appl. Math., 3, 201 (1950).

1~8

A. D E G A S P E R I S

and

j.

jP.

LEON

a consequence of the 1-pole reduction equation (3.8) (see also (3.9) and (4.16)). ~ e also note that~ within this class of solutions v(x~ t)~ the function O(x~t), introduced through the Hopf-Cole transformation (4.14), diverges exponentially at both ends x ~ ~ : ~ ; indeed~ it is readily seen from (4.14) and (4.10) t h a t O(x~ t)~--[q(-)(x, t)] -~ (modulo a x-independent factor). F r o m t h e previous section we also conclude t h a t the matrix Q(x~ t)~ corresponding to a solution v(x~ t) of eq. (4.13) t h a t satisfies the asymptotic conditions (4.17)~ always possesses one soliton (see (3.18)) in addition to a continuous spectrum component (see (3.20)). I n particular~ the one-soliton formula (3.21) yields the familiar kink solution (4.18)

v(x, t) = p(-)-- pC+)_~ (pC-) ~_ pr

t gh((pr

~- pC+))[x -- ~(t)]} ,

where the kink position is given by the expression t

(4.19)

~(t) = ~o + (p(-) + p(+))-~fdv[pC§

2p(+), ~) § pHw(2p(-),

v)] ,

to

~o(z~ t) being~ of course~ t h e function t h a t characterizes the particular evolution equation of class (4.13). I n addition to possessing the nice property of being linearized by the Cole-Hopf transformation (4.14)~ t h e Burgers equation (as well as the whole hierarchy (4.13)) has been found to have other interesting features. ] n fact~ it has been shown t h a t t h e Burgers equation has infinitely m a n y symmetries (15,1e); this does not imply~ however~ the existence of infinitely m a n y independent constants of the motion. On the other hand~ it is well known (5.7) t h a t t h e nonlinear evolution equation (4.1) has infinitely m a n y conserved quantities t h a t are in involution; their explicit expression (1~) is -{-~ (4.20)

C . : (n ~-l)-ltrlafdxO_(x,t)(2iL).+l[xQ(x~t)]},

n=0,1,2,...,

where L is the integro-differential operator defined by (4.3). I t is now plain t h a t formula (4.20)~ if Q(xy t) on the r.h.s, of (4.20) satisfies the l-pole reduction~ i.e. if Q(x~ t) is given by (3.1a) with (3.9) and (4.16)~ yields an infinite sequence of conserved quantities for the Burgers class of evolution equations (4.13). However~ t h e effect of the l-pole reduction on the matrix Q(x~ t) is to m a k e the conserved functionals (4.20) (with the exceplion of two) be (15) p. j. OLVER: J. Math. Phys. (N. Y.), 18, 1212 (1977). (le) A. S. I~OKAS: J. Math. -Phys. (N. Y.), 21, 1318 (1980).

M.~.TRIX SPECTRAL TRANSFORM, R E D U C T I O N S

AND THE B U R G E R S E Q U A T I O N

1~9

all linearly dependent on each other. This is easily shown within the spectraltransform approach by writing the expression of the conserved quantities C. in terms of the spectral transform (3.16) of Q(x, t); in fact, the relevant formula reads (4.21)

C . = (2i)"+a/(n + 1) -1 i (

(2ig)-11

[k:+''+' - - ( ~ e k ~.r ) .+1 --k~,-)-+' + ( R e k ~ ) ~+~]-

d]~ k n h l [ X - - 0~r

t)0~{-)(k, t ) ] / ,

" = 0, 1, 2,

J

--r

The reduction condition (3.8), th,~t in k-space merely states t h a t only one soliton is present, N ~ 1, and t h a t the reflection coefficient ac-)(k: t) vanishes, a H ( k , t ) ~ O, readily implies that the conserved quantities (4.21) take the simple expression (4.22)

C. --~ 2"+~(n ~- 1)-1[( -

pC+)),,+1_ p~-).+l] ,

n = O, 1, 2, .,-

where we have set (see (3.18) and (3.12)) (4.23)

k"-1e L )- -- • ip ct)

~

Thus the first two constants of the motion

(4.24a)

Co -~ - - f d x v ~ ( x , t) : - - 2(p c+)~- pr

,

--r

(1.24b)

Cl = --fax v(x, t) v.(x, t) = 2(p,+,'-- p,,2),

and only these, are independent of each other, while all the others can be, of course, expressed in terms of C0 and C1. There exists, however, a third independent constant of the motion t h a t is not included in the infinite sequence (4.20). For the sake of simplicity, we display this additional conserved quantity only for the Burgers equation (4.12)~ its expression being +r

(4.25)

o = j d x Ix -- 2tv(x, t)]

t);

the time independence of integral (4.25) implies that the centre of mass -I-r

(4.26)

+co

x(t)=fdxxvxCx, --or;

-1 , --

~o

15O

A. DEGASPERIS a n d

J. JP. LEON

associated with t h e solution v(x, t) of t h e Burgers equation (4.12), moves with c o n s t a n t speed, namely (4.27)

x(t) = v(t--to) § Xo,

(4.28)

V = 2C1/Co = - - 2(p (+)- p(-)) ,

(4.29)

Xo :

I n addition already p o i n t e d e q u a t i o n (4.12). (4.13) f r o m t h e (4.30)

(2Clto-- c)/Co.

to the existence of infinitely m a n y symmetries, it has been out (17,18) t h a t B~,cklund transformations exist for t h e Burgers I n fact, t h e y can be easily derived for the class of equations simple B~cklund t r a n s f o r m a t i o n

O'(x, t) = B(D)O(x, t ) ,

t h r o u g h t h e ttopf-Cole e q u a t i o n (4.14) t h a t relates O(x,t) to v(x,t). I n d e e d (4.30), w h e r e B(z) is a polynomial with t i m e - i n d e p e n d e n t coefficients, maps a solution O(x, t ) , o f t h e linear equation (4.15) into a n o t h e r solution O'(x, t) of the same equation (see (1.8)). I f t h e n v'(x, t) corresponds to O'(x, t) via t h e ttopf-Cole transformation, i e

(4.31)

O" (x, t) = v'(x, t)o'(x, t) ,

it is easily seen t h a t v'(x, t) is a new solution of t h e Burgers equation (4.21), and is r e l a t e d to v(x, t) b y t h e Bi~cklund t r a n s f o r m a t i o n (4.32)

v'(x, t) = [B(D + v ) v ] [ B ( D q- v).1] -x ,

v =- v(x, t ) .

This obtains from the o p e r a t o r i d e n t i t y (4.33)

B ( D ) I ( x ) : ](x)B[D Jr ],(x)/](x)],

where B(z) is an entire function and ](x) is an a r b i t r a r y infinitely differentiable function. Thus, for instance, for (4.34)

B(z) ~- a ~- bz ,

t h e Bii,cklund t r a n s f o r m a t i o n (4.32) reads (4.35)

v'(x, t) ~ {av(x, t) ~- b[v~(x, t) ~- v2(x, t)]} [a + by(x, t ) ] - ' .

(17) A. S. FOKAS and B. FUCHSSTEINER: Theor. Math. Appl., 5, 423 (1981). (is) j . W]~Iss: Tire Painlevd property ]or partial di]]erential equations, II: Bdcklund trarts]orraation, Lax pairs and the Schwarzian derivative, Preprint LJI-R-82-202, June 1982.

MATRIX SPECTRAL TRANSFORM~ R E D U C T I O N S AND T H E BURGERS E Q U A T I O N

151

On the other hand, it has been shown (7) t h a t B~cklund transformations exist for t h e class of nonlinear evolution equations (4.1), a simple instance of such transformation~ being given by the formula (4.36)

Q'~-- Qz ~- 2isa(Q'-- Q) ~- (w'-- w + 2p)(Q' -f- Q) = 0 ;

here s and p are two (generally complex) parameters,

(4.37)

w(x, t) =--fdy qC+)(y,t) qr

t),

$$

and w'(x, t) is defined in the same way in terms of qr t) (see (3.1a)). Transformation (4.36) has been shown to map a solution Q(x, t) of the evolution equation (4.1) into another solution Q_'(x, t) of the same evolution equation, provided t h e integration (x-independent) constants t h a t enter in Q'(x, t) by solving (4.36) for a given Q(x, t) have an appropriate time dependence (note t h a t the variable t appears in (4.36) only parametrically). As for the connection between t h e B~cklund transformations (such as (4.36)) provided by the spectral-transform approach, t h e reduction technique discussed here and the B~cklund transformations (4.32) for the Burgers class of evolution equations (4.13), we limit t h e present discussion to few remarks by pointing out only few relevant results t h a t can be easily verified by direct computation. We firstly note t h a t the B~cklund transformation (4.36) takes a matrix Q(x, t) t h a t satisfies the 0-pole reduction, i.e. qc-)(x, t ) ~ O, into a matrix Q'(x, t) t h a t satisfies instead t h e 1-pole reduction (3.12), i.e.

q~+)'(x, t)qC-~'(x, t) -~ [lnqH'(x, t)]~x 9 This is consistent with the characterization of the spectral transform t h a t satisfies the L-pole reduction equation, namely a(-)(k, t) --~ 0, N ---- L (see (3.16)), as shown in sect. 3. Indeed, it is known t h a t the B~icklund transformation (4.36) adds two discrete eigenvalues, namely k r : s 3= ip (with 4- I m k c• > 0), to the discrete component of the spectral transform (3.16) of Q(x. t). Therefore, since the ~pectral transform of a matrix Q.(x, t) satisfying t h e 0-pole reduction equation has no discrete component (i.e. lq = 0 in (3.16)), while a m a t r i x O.'(x, t) satisfying the 1-pole reduction has two discrete eigenvalues (i.e. N ---- 1 in (3.16)), one should expect t h a t t h e B~cklund transformation (4.36) maps a matrix Q(x, t) t h a t satisfies t h e 0-pole (or, more generally, L-pole) reduction equation into a matrix O.'(x, t) t h a t satisfies the 1-pole (or, more generally, (L ~ 1)-pole) reduction equation. Concerning the time evolution, the B~cklund transformation (4.36) has, therefore, the property of transforming a solution (i.e. qI+)(x, t)) of the linear

152

A. D E G A S P E R I S

and

J.

Je.

LEON

evolution equation (4.5), if Q(x, t) satisfies the 0-pole reduction, into a solution v'(x, t) of the Burgers equation (4.13), where v'(x, t) is related to Q'(x, t) b y formula (4.10) (with primed quantities). The connection of this transformation with the known results on the Burgers equation is established b y the fact that the B~cklund transformation (4.36) (of course, with q(-)(x, t) -~ O) is, in fact, the product of the ttopf-Cole transformation (4.14) (with ri(x, t)----~ q(+)(x, t)) times a Bgcklund transformation (4.35) for the Burgers equation. We close this section by emphasizing that our attention has been focused here only on the 1-pole reduction; it seems, however, interesting to extend a similar analysis to the 2-pole reduction, whose explicit expression is shown by eqs. (3.24), or, more generally, to the L-pole reduction, whose characterization has been provided here only for the spectral transform (i.e. not in x-space for Q(z)) by conditions (3.27).

J J P L wishes to acknowledge the hospitality of the Departement of Physics in Rome and AD thanks Prof. P. C. SABATIF_~for his friendly hospitality. This work has been partially supported by the ~ATO Research Grant 57o. 057.81.

APPENDIX

Consider first the matrix X'(x, k), obtained via transformation (2.7)from the matrix X(x, k); then (2.17), together with the properties (2.18) and (2.19), follows directly from conditions (1.3)-(1.6) and from eqs. (2.11) satisfied b y the matrix G(x, k). In fact, since

(An)

X(•

k) = -- ika

is diagonal and since (A.2)

diag(Gx(x, k)[G(x, k)]- 0 = 0,

as implied b y (2.9) and (2.10), there follows from (2.7) that (2.18c) is satisfied. The validity of (2.19b) follows from the limit of (2.7), as x --> ~= co, and from the condition (see (2.19a)) (A.3)

low(G=(x, k)[G(x, k)] -1) : low(G(x, k)a[G(x, k)] -1) = 0,

that is implied by (2.11). :Now we provide the general solution of the triangular system of ODEs (2.5) with (2.17). We first note that 2

--co

MATRIX

SPECTRAL

TRANSFORM,

REDUCTIONS

AND

THE

BURGERS

EQUATION

153

is the nonsingular matrix solution of (2.5), satisfying the equation (A.5)

det[F'(x, k)] ---- 1

and the boundary condition (A.6)

lim [exp

x--~--co

[ikxa] F'(x,

k)] ---- 1 ,

if the matrix ~(x, k) satisfies the integral equation

,I)(x, k) = 1 + J,I)(x, k),

(A.7)

where J is the integral operator defined by (2.22). One should then notice that the integral operator J is a nilpotent operator because Q'(x, k) satisfies (2.19c). This property is specified by (2.23), where/x is the index of nilpotence defined by (2.19c). This implies t h a t the :Neumann series expansion of the solution ~(x, k) of the integral equation (A.7) reduces to the sum

(A.S)

r

k) = ~: Y - - 1 , n~D

that provides, therefore, together with (A.4), the explicit expression of F(x, k). Of course, the general (vector) solution of (2.5) finally reads (A.9)

T ' ( x , k) = F'(x, k)c ,.

where c is an arbitrary constant (x-independent) vector. I t is clear now that the knowledge of the transformation matrix G(x, k) implies t h a t also the Jost solution F(x, k) of eq. (1.1) with (1.2) is known via (2.6). Indeed boundary condition (2.14) implies t h a t F(x, k) = [G(x, k)]-'F'(x, k)

(A.lo)

satisfies precisely the asymptotic behaviour (2.16a); finally, this expression, together with definition (2.16b), and (2.21), (A.4), (A.8) and (2.22), yields expression (2.20) of the transition matrix S(k). We close this appendix by specializing the previous formulae to the 2 • case treated in sect. 3, in the particular case in which the matrix G(x, k) is rational in k, as shown by (2.9), (3.2) and (3.6). In this case, of cottrse, p = 2 and, therefore, (A.8) reads x

(A.11)

~(x, k) = 1 ~-

y exp [-- 2iky] [I {[k ~- i~'n)(- c~)]/[k -~ i~,,'(y)]} 2.

1S4

A.

DEGASPERIS and

J.

JP.

LEON

in fact, this expression obtains by taking into account definition (2.22), (3.5) and the expression

V(x, k)~ --{~ln[ik--~(")(x)]}a

(A.12)

t h a t is implied by (3.4), (3.6) and (3.7a). Moreover, in this case one has

[G(x,k)]-~~ l--h(x,k)(~

(A.13)

~),

and this, combined with (A.4), (A.12) and (A.11), yields the expression of the J o s t solution F(x, k). F r o m this expression a n d definition (2.16b) one can easily compute the transition matrix S(k). ~Ve merely report below the expression of the reflection coefficients a(• a n d transmission coefficients fl(• defined, through (2.32) and (2.31), by

(0

(A.14)

aCk) =

(A.15)

~(k) ~- (fl(O(k)

ac+)(k)

fl:-)~k))"

We obtain ~-a~

(A.16a) a(+)(k) =

x q(+'(x) 1-[ ([k ~-

ips

+ i~("(x)]} 2 exp [-- 2ikx],

n--i

(A.16b)

0d-)(k) ---- 0,

(A.17)

tic•

= [I

[(k H- ip~-))/(k -- ip~+~)]~l,

where we have set (A.18)

9

~(n)(•

---- • p~)

,

n : 1,

2,

...,

JL

~

RIASSUNTO

Si introduce un metodo di riduzione per le equazioni di evoluzione non lineari associate ad un problema spettrale matriciale iv)
MATRIX SPECTRAL TRANSFORM, REDUCTIONS AND THE BURGERS ~.QUATION

155

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