L]~TT~ERE AL ~UOVO CIM]ENTO

VOL. 23, N. 9

28 0 t t o b r e 1978

Nonlinear Evolution Equations and Ordinary Differential Equations of Painlev~ Type. M. 5 . ABLOWITZ (*) a n d

A.

RAMANI (**)

Program in Applied Mathematics, Princeton University - Princeton, N . J . 08540 H. S~GU~

Aeronautical Research Associates o] Princeton, Inc. 50 Washington Road, P.O. Box 2229, Princeton, N . J . 08540 (rieevuto il 24 Luglio 1978)

The purpose of this paper is to establish firmly the deep and important connection between nonlinear partial differential equations that are solved by inverse scattering transforms and nonlinear ordinary differential equations without movable critical points. In 1888 KOVALESKAYAwas awarded the Borden prize for her major contribution to the theory of the motion of a rigid body about a fixed point. Her main idea was to carry out the apparently nonphysical calculation of determining t h e choices of parameters for which the equations of motion admitted no movable critical points (i.e., the locations of any branch points or essential singularities did not depend on the constants of integration). In all such cases she then solved the equations explicitly. In all other cases, the solution is still unknown (1). As noted in a previous paper (2), ordinary differential equations (0DE's) without movable critical points (hereafter referred to as being of P-type) also arise in connection with nonlinear evolution equations solvable by inverse scattering transforms (3). As an example, it was demonstrated there that the solution of a certain linear integral equation also solves Painlevd's second equation (PH; hereafter we denote the six transcendendants of Painlevd as P~ .... , Pw)- In this paper we further exploit this connection to reveal much more of the underlying structure of these problems. I) We prove t h a t all solutions of certain linear integral equations necessarily possess the P-property (i.e., no movable critical points).

(*) (**) (1) (2) (3)

P e r m a n e n t a d d r e s s : C l a r k s o n College, P o t s d a m , N. u 13676. P e r m a n e n t a d d r e s s : L P T t t E , U n i v e r s i t 6 P a r i s Sud, 91405 Orsay, F r a n c e . V. V. GOLUBEV: Lectures on Integral of E q u a t i o n o! M o t i o n ... (Moscow, 1953). M. J. fl-BLOWITZ a n d It. SEGUR: P h y s . Rev. Left., 38, 103 (1977). M. J. fl-BLO~VITZ, D. J. ]~.~UP, A. C. NEWELL a n d H. SEGUR: Stud. A p p l . 3lath., 53, 249 (1974).

333

~

M.J.

ABLOWITZ, A. RAMANI

and

I t . S~GVR

II) W e give a direct method, which, in any given case, establishes t h a t the solution of the linear integral equation necessarily satisfies a nonlinear ODE obtained b y a similarity reduction of an appropriate nonlinear evolution equation. This result provides global information about the corresponding ODE. I I I ) W e describe a singular-point analysis, similar to those of Kovalevskaya (1) and Painlev~ (4) b y which one m a y test whether a given ODE satisfies necessary conditions in order for it to be of the P-type. IV) The final objective of this paper is to state precisely what we think the connection is between nonlinear evolution equations solvable b y inverse scattering transforms and ODE's of P-type. W e now discuss I). Consider co

(1)

K(x, y;

= F(x + y) +

jK(x, s,

s, Y) d s ,

x

where x is any point in the finite complex plane, the integral is taken along a smooth contour C from x to ~-co which deviates only a finite distance from the real axis, and y on C. Iu the cases we shall consider here N(x, s, y) is related t o / ~ ; e.g., (A)

N = F ( s ~- y ) , co

(B)

2r = f.F(x + z)_F(z + y) dz , x

(c)

N -- if '(s + z)F(z + y) dz.

I n (B) we allow either i) -P(z) : / V ( z ) or ii) F(z) : (/~(z*))* (where (*) denotes complex conjugate) and in (C) we allow ii). In any case/V is analytic in each of its arguments, provided F(z) is analytic. W i t h these choices, (1) becomes the linear integral equation used to solve certain nonlinear evolution equations of physical significance (6,5). F u r t h e r generalizations are obtained b y replacing the scalar functions in (1) b y matrices (6,7). Here, we use (1) to solve 0 D E ' s . The result in (3) was a particular example, using (B) with / ~ ( x ) : = Ai(x/2), where Ai(x) is the A i r y function. The powerful results of F r e d h o l m theory apply to linear integral equations with compact kernels. The proof of Fredholm's first theorem (6)can be modified to apply co

provided: I1V(x, s, y)l <~M(x, y) < oo for all s, and ~M(x, y) dy = T(x) < co. x

(4) E. L . INCE." Ordinary Di/]erential Equations (New Y o r k , N. Y., 1947). (~) D. J . K•uP a n d A. C. ~IEWELL." J o u r n . M a t h . P h y s . , 19, 798 (1978). (6) V. E. ZAK~AROV a n d A. B. SHs F u n k t . A n a l . P u l o z h . , 8, 43 (1974) (Funct. A n a t . A p p l . , 8, 226 (1974)). (~) H . CORNILLE: J o u r n . M a t h . P h y s . , 18, 1855 (1977). (B) W . POGORZELSKI" Integral E q u a t i o n s and T h e i r A p p l i c a t i o n s , Vol. 1 ( L o n d o n , 1966).

NONLINEAR

EVOLUTIOI~

EQUATIONS

ETC.

33~

Under these conditions, eq. (1) has a unique solution which can be expressed as

K(x, y; 4)

= F(x § y) d- 4f.F(x d- s),Y(x, s, y ; 4) d s , x

(2) ~ ( x , s, y; 4)

C(x, s, y; 4) -

D(x; 4)

C and D are nontrivial entire functions of 2 which are given as absolutely convergent series. In i m p o r t a n t subclasses where F(x) is analytic and r a p i d l y decreasing (e.g., _F(x) -~ Ai(x/2)) C and D are analytic in each of their arguments. Hence the only singularities of K come from the zeros of D, and can only be poles. Therefore, independently of the contour C, K must have t h e / ) - p r o p e r t y , and (2) proves the operator in (1) to be invertible except at zeros of D(x, 4). W i t h certain modifications, this argument can be extended to cases in which E(x) has (fixed) singularities. Next, consider II). The essence of the method is to apply two operators ~1, L~ to (1), such t h a t Z~_F = 0. The action of these operators on (1) yields equations f o r K . To illustrate, consider case (B), ~----iv. Rewrite the integral equation as co

(3)

K(x, y; 4) = F(x § y) § 4fK(x, x d- v; 4)F(2x § v + u ) F ( x + u § y) dv du = 0 0

and define r~

(4a)

.Ks(x, x d- u; 4) = r E ( x , x + v; 4)F(2x + v d- u) dv o

so t h a t co

(4b)

.K(x, y; 4) = F(x + y) + 4fKz(x, x + u; 4)F(x § u + y) d u . o

A p p l y the operator L i = (~x--~y) to (3), and find r

(5)

(~i--~2) K (x, y; 4) : 4f[(~l ?- ~2)Ke(x, x JF u; 4)]F(x d- u Jr y) du , o

where ~ , ~ are derivatives with respect to the first and second arguments of the co function. But applying (~i § ~2) to (4), and assuming the operator I - - ~ S / ~ F ~ I - - 4 ~ to be invertible (this was demonstrated b y (2)), gives o (6a)

(Old- O2)K2(x, Y; 4)

=

- - 2 K ( x , x; 4 ) K ( x , y ; 4).

Using (6a) in (5) yields (6b)

( ~ - - ~ ) K ( x , y; ~) = --2,~K(x, x; 4)K~(x, y; 4).

33~

M.J.

A B L O W I T Z , A. RAMANI

and H . S ~ G U R

To obtain an ODE, we next apply another operator L 2 on (3) and group terms such t h a t ( I - - 2 5 ) L ~ K = t~, for some /~. By using L e F = 0, and (6), we can find R = (I--4~)~K. In this w a y we obtain directly the equation: ~2K = / ~ K . For example, using the operator (3) ~ = (~x = ~y)~-- (x § y)/2 on (3) yields (7)

.5~K(x, y, 4) = - - 24K(x, x; 4)2K(x, y; 4)

after a trivial integration and noting F = ( I - - ~ ) K . On y = x, this is the equation for PH. This approach is analogous to that in (e,7), where the authors relate K to P D E s . We have used this method to establish that each of the similarity solutions, decaying as x--~ ~ , associated with sine-Gordon (related to Pro), nonlinear SchrSdinger, and modified KdV (P~I), are the K(x, x; 4) of (1B); similarity solutions of KdV (related to PII) and some of its higher order analogues are associated with the K of (1A) ; similarity solutions of the massive Thirring model (~), and the derivative nonlinear SchrSdinger equations (5)are related to the K of (1C). In the latter P D E we find that its similarity solution is related to P i v , i.e., for i ~ + YJx~~ i(~P2~*)~ = 0, with ~ = t-~f(x/t89 and ] = ~ exp [i0], then ~ satisfies P~v. I t should be noted t h a t : i) the direct approach outlined above applies even to potentials which have (( bad ~ properties--e.g., poles, and weak decay as x--~-- or ii) Those similarity solutions which are generated by co

F(x) ~ A~(x) =

(2=)-lfexp

[ikx ~- ik2~+1/(2n ~- 1)] dk

(i.e., related to nonlinear evolution equations whose linear dispersion relation satisfies w(k) ~ Ck ~n+l) have very rapid decay as x--~ co. The Fredholm theory discussed earlier applies, iii) There are some solutions which decay weakly at infinity. We cannot use F r e d h o l m theory to prove the P - p r o p e r t y ; nevertheless for some range of 2, x the operator ( I - - 4 ~ ) is invertible, iv) In case (A), a consequence of (2) is K(x, x; 2) = 4 ~ In D(x, 4). The importance of this form was first observed by Painlev4 (4). In the P D E s solved by (A), this is the dependent-variable transformation used by HmOTA (10) to find N-soliton formulae. F o r x real, it is possible to obtain global information about some of these solutions. Fo r example, consider case (B) with F(x)=Am(x), given above, and ~ ( x ) : F ( x ) . (I-)J) is invertible for any real x if 4 is real and ~ ~ 1. The proof depends simply on noting that, for any real x, ~ is a positive operator with an /~2-norm bounded above by 1. When 2 >1, there is at least one location, x, on the real axis, where K(x, x; 4) has a pole. As 4~1, the location of this pole tends to - - c~. In particular, the critical branch of PH is achieved when 4 = 1. Details will be published elsewhere. I I I ) Consider an ODE of order n, whose solution has dominant algebraic behaviour near some (movable) singular point, z0; y ( z ) ~ a(Z--Zo)-q ~- ~(z). If q is not an integer, this represents a (movable) algebraic branch point, hence the equation is not of P-type. If q is an integer, then for every possible ~, develop the expansion for ~(z; ~). The series for ~ contains arbitrary constants which are added in at powers (z--z0)~; r satisfies an n-th order algebraic equation. The roots with Re ( r ) < - - q are not of interest.

(9) A . V. /~IIKHXILOV: J E T P Lett., 23, 320 (1976). (10) It. HIROTX: Phys. Rev. Left., 27, 1192 (1971).

NONLIN]~AR E V O L U T I O N

E Q U A T I O N S ETC.

337

A n y root w i t h Re (r) > - - q, r not an integer, also represents a m o v a b l e b r a n c h point. The interesting cases h a v e all roots w i t h R e (r) > - - q as real integers. If r = integer, r > - q, t h e n t h e solution at the first such integer is generally of t h e form ( z - z0)~. 9(fl + y In (z--zo)), fl arbitrary. If ? =fi 0 t h e e q u a t i o n is not of t h e P - t y p e . If t h e equation at the order r e q u i r e d is identically satisfied, t h e n ~ = 0 . W e refer to this as a resonance criterion. I n this case we proceed to check whether all resonance criteria (for r = integer, r > - - q ) are met. If so, these are the lowest t e r m s in a L a u r e n t series9 (Usually t h e r e will be n - - 1 c o n s t a n t s to be added; the o t h e r c o n s t a n t is %.) These formal m e t h o d s can be m a d e rigorous using Painlevd's m e t h o d s (4). As an example, consider (8)

W" = z ~ W + 2W 3,

where w = 1/~ + ~(~), ~ = z - - z o and Q(~) satisfies

The left-hand side o~ (9) has a h o m o g e n e o u s solution ~3, and hence we expect a resonance at order ~. Indeed, assuming ~ a 0+al~+a~ 2 + a a ~ a + - . . we find a 0 = 0 , a l = - - z ~ / 6 , a s = --~r~z~-x/4 and at order 3, 0 " a 3 = 8 9 ~-~. If m @ 0 , 1 then In ~ terms are p r o d u c e d and t h e e q u a t i o n is n o t of t h e P - t y p e . W h e n m = 0, 1 we h a v e satisfied the necessary conditions, and indeed Painlevd has p r o v e n (4) the equations h a v e no m o v a b l e critical points. IV) Our final o b j e c t i v e is to state precisely w h a t we t h i n k t h a t is t h e connection is between nonlinear e v o l u t i o n equations solvable b y inverse scattering transforms and ODEs of t h e P-type. F o r a working definition, we say t h a t an e v o l u t i o n e q u a t i o n t h a t is in the I S T class is the solution of a linear i n t e g r a l equation of t h e f o r m (1), or a n y derivative of t h e solution, also solves the e v o l u t i o n equation. (The r e a d e r is w a r n e d t h a t the sine- Gordon e q u a t i o n does not satisfy this definition, a l t h o u g h an integral transformationof it does.) Using this definition, we m a k e t h e following conjecture. : Consider an e v o l u t i o n equation in t h e I S T class. Then e v e r y O D E obtained by an e x a c t r e d u c t i o n of the evolution e q u a t i o n is of the P - t y p e . E x a m p l e s of t h e s e reductions are t h e O D E ' s t h a t gives r e a d y solutions, travelling w a v e solutions and scaling-type similarity solutions. If correct, this c o n j e c t u r e provides an a priori t e s t of w h e t h e r or not a g i v e n e v o l u t i o n e q u a t i o n can be solved exactly by IST. Besides those O D E s directly related to Px'P~v or to elliptic functions, we h a v e tested b y the local analysis above O D E s r e l a t e d to the nonlinear SchrSdinger equation, some of the (( h i g h e r order ~) K d V and modified K d V equations, Boussinesq, and massive Thirring. T h e y all satisfy t h e necessary conditions for t h e O D E to be of t h e P - t y p e . (We caution t h a t s o m e t i m e s a t r a n s f o r m a t i o n is necessary, as for sine-Gordon.) If an e v o l u t i o n e q u a t i o n reduces to an O D E t h a t is not of the P - t y p e , t h e n we do not expect it to be solved b y IST. Using t h e singularity analysis described above, we h a v e e x a m i n e d t h e similarity solutions of some evolution equations for which numerical evidence shows t h a t s o l i t a i y w a v e s do n o t interac~ as solitons: (( cubic K d V ~; ut + 6u3u~ + u ~ = 0 (t~), (( complex modified K d V ~>ut + (lu)~u)~ + u ~ = 0 02), <~double (11) 9[9 KRUSKAL." p r i v a t e c o m m u n i c a t i o n . (~) C. K A R N E r , A. SEN a n d F. CHU: p r i v a t e c o m m u n i c a t i o n .

~

~I. J. ABLOWlTZ, A. RAMANI

and

i{. sv, o u R

sine-Gordon ))ux~ : sin u ~- fl sin 2u (i3) (after the t r a n s f o r m a t i o n w = exp [iu] is used on t h e O D E ) . I n e v e r y case, t h e O D E is not of the P - t y p e . F i n a l l y , t h e two-dimensional cubic nonlinear SchrSdinger e q u a t i o n in radial co-ordinates, i ~ - ~ ~ , r + ( 1 / r ) ~ r l • ]~p]2~ : 0 has a similarity solution of t h e form %o= exp [ilt]/~(r). The s i n g u l a r i t y analysis shows t h a t R(r) does n o t h a v e the P - p r o p e r t y . Therefore, we conjecture t h a t c a n n o t be found b y IST.

W e are grateful for v e r y helpful discussions w i t h M. D. KRUSKAL. This w o r k was s u p p o r t e d b y N F S , G r a n t ~ o . MCS 75-07568A03, and b y t h e U.S. A r m y Research Office.

(13) M. J. ABLOWITZ,M. D. E:RI~SKALand J. F. :LADIK: submitted S.I.A.~I. Appl. Math.