On the First Integrals of KdV Equation and the Trace Formulas of Deift-Trubowitz Type

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1 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, On the Frst Integrals of KV Equaton an the Trace Formulas of Deft-Trubowtz Type MAYUMI OHMIYA Doshsha Unversty Department of Electorcal Engneerng Tatara Myakoan -3, Kyotanabe JAPAN YU YAMAMOTO Doshsha Unversty Grauate School of Engneerng Tatara Myakoan -3, Kyotanabe JAPAN Abstract: A new type of the frst ntegrals assocate wth KV equaton s constructe by applyng the trace formulas of Deft-Trubowtz type for the -mensonal Schrönger operator wth no boun states. The relatons between the well-known frst ntegrals wth the enstes expresse n terms of fferental polynomals an the present frst ntegrals are obtane. Key Wors: KV equaton, Frst ntegrals, One mensonal Schrönger operator, Trace formulas of Deft- Trubowtz type Nov. 30, 2007 Introucton In ths paper, we conser a spectral meanng of the frst ntegrals of KV equaton u t 6u u x + 3 u = 0, () x3 where u = u(x, t) s the spatally raply ecreasng real value functon. We restrct ourselves to the case such that the -mensonal Schrönger operator H t = 2 + u(x, t) x2 has no boun states,.e, σ p (H t ) =, where σ p (P ) enotes the set of screte spectrum of the operator P consere n the space L 2 (IR). The functonal I[u](t) whch correspons the functon u(x, t) to the functon of the one varable t s calle the frst ntegral or the conservaton law of the equaton (), f t I[u](t) = 0 hols for the soluton u(x, t) of KV equaton (). Moreover the functon w(x, t) s calle the local ensty of the frst ntegral I[u](t), f I[u](t) = w(x, t)x hols. In [2], Zakharov an Faeev showe that there are nfntely many frst ntegrals whch have local enstes expresse n terms of the fferental polynomals of u(x, t). On the other han, n [2], Deft an Trubowtz erve the trace formula 2 kr ± (k)f ± (x, k) 2 k c ±,j η j f ± (x, η j ) 2 = (2) 2 u(x) for the -mensonal Schrönger operator H = 2 x 2 + u(x) wth the raply ecreasng real value potental u(x), where r ± (k) are the left an rght reflecton coeffcents, f ± (x, k) are the left an rght Jost solutons, ηj 2, j =, 2,, N are the screte egenvalues of the operator H, an c ±,j are the normalzng coeffcents. We wll brefly menton these materals n the next secton. In partcular, f the operator H has no boun states, we have the trace formula kr ± (k)f ± (x, k) 2 k = u(x). (3) 2 On the other han, n [], Deft, Lun an Trubowtz erve the another trace formula k r ±(k)f ±(x, k) 2 k = u(x) (4) 2

2 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, for the raply ecreasng real value potental u(x) such that the operator H has no bouns states an satsfes the conton r ± (0) =. The man purpose of the present paper s to gve a new spectral nterpretaton of the frst ntegrals of KV equaton usng above trace formulas an, furthermore, construct a new kn of frst ntegrals. The contents of the present paper are as follows. In 2, the funamental materals are brefly explane. In 3, the trace formulas of Deft-Trubowtz type an ts generalzaton are scusse. In 4, an evoluton equaton satsfe by the Jost soluton s erve. In 5, the spectral nterpretaton of the frst ntegrals of KV equaton s gven. 2 Prelmnares In ths secton, the necessary materals are summarze. Throughout ths secton, we assume that the real value potental u(x) satsfes the ntegrablty conton ( + x 2 ) u(x) x <. Let f ± (x, k) be the Jost soluton of the egenvalue problem Hf(x, k) = f (x, k) + u(x)f(x, k) = k 2 f(x, k),.e., f ± (x, k) behave lke exp(±kx) as x ± respectvely. Let W [f, g] = fg f g be the Wronskan, then the rght (+) an left (-) reflecton coeffcents r ± (k) are efne by r ± (k) = ± W [f +(x, k), f (x, ±k)], k IR\{0}. W [f (x, k), f + (x, k)] We have r ± (k) <, k IR \ {0} n general. Moreover one of the two cases or r ± (0) = (5) r ± (k) < for all k IR. hols. It s known that the conton (5) s generc. On the other han, let η j, j N be the zeros n the upper half plane H + of the analytc functon W [f (x, k), f + (x, k)], k H +, where H + = {k k CI, Ik > 0} for the complex plane CI. Then σ p (H) = { η 2 j j N} hols, an f ± (x, η j ), j N are the square ntegrable egenfunctons assocate wth the operator H. Let c ±,j be the nomalzaton coeffcents of those egenfunctons,.e., The collectons c ±,j = f(x, η j) 2 x. (6) Σ ± = {r ± (k), η 2 j, c ±,j ; j N} are calle the scatterng ata. See [2] an [4] for etals of the scatterng theory of H. Now we conser the scatterng ata of the operator H t wth the spatally raply ecreasng potental u(x, t) whch solves KV equaton (). In ths case, the elements of the scatterng ata epen on t,.e., those are enote as r ± (k, t), c ±,j (t), an η j (t). In [3], Garner, Greene, Kruskal an Mura scovere the followng formulas; r ± (k, t) = r ± (k, 0) exp( 8k 3 t), c ±,j (t) = c ±,j (0) exp( 8η 3 j t), η j (t) = η j (0). (7) Next we explan the recurson operator Λ an the KV polynomals.the operator Λ s the formal pseu fferental operator efne by Λ = ( ) ( x 2 u (x) + u(x) x 4 3 x 3 ). (8) Put Z 0 (u) = an efne the functons Z n (u) by the recurrence relaton Z n (u) = ΛZ n (u), n IN, (9) where IN s the set of all natural numbers. Then t s known that Z n (u) are the fferental polynomals of u(x). For example, we have Z (u) = 2 u, Z 2(u) = 8 (3u2 u ). (0) We call them the KV polynomals. We refer the reaer [7, lemma 3., p.62] an [8, p.952] for more precse nformaton. Next we explan the followng Appell s lemma.

3 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, Lemma. Let y = f(x) an y = g(x) be the solutons of the 2n orer ornary fferental equaton 2 y x 2 = p(x)y, then the prouct z = f(x)g(x) solves the 3r orer ornary fferental equaton 3 z z = 4p(x) x3 x + 2p (x)z. Ths lemma s qute elementary fact an easy to prove t. See [0] an [9] for etal. By Appell s lemma an the efnton (8) of the recurson operator, we have mmeately x Λg ±(x, k) = k 2 x g ±(x, k), where g ± (x, k) = f ± (x, k) 2,.e., Λg ± (x, k) = k 2 g ± (x, k). () 3 Trace formulas of Deft-Trubowtz type By the trace formula (2) an (0), we have 2 kr ± (k)f ± (x, k) 2 k c ±,j η j f ± (x, η j ) 2 = Z (u). (2) By operatng wth the operator Λ n on the both ses of (2), then the trace formulas k 2n r ± (k)f ± (x, k) 2 k + ( ) n 2 c ±,j ηj 2n f ± (x, η j ) 2 = Z n (u) (3) mmeately follow from (9) an (). Moreover, n [6], the enttes ( ) n 2 k 2n r ± (k)f ±(x, k) 2 k c ±,j η 2n j f ±(x, η j ) 2 = Z n+ (u(x)) + u(x)z n (u(x)) 2 2 x 2 Z n(u(x)) are erve.in partcular, f the operator H has no boun states an s of the generc type,.e., the conton (5) s val, then the followng two types of the trace formulas hol; k 2n r ± (k)f ± (x, k) 2 k = Z n (u) k 2n r ± (k)f ±(x, k) 2 k = Z n+ (u(x)) u(x)z n (u(x)) + 2 x 2 Z n(u(x)) 4 The frst ntegrals wth the local enstes It s easy to see that f u(x, t) s the spatally raply ecreasng soluton of KV equaton (), then the functonal I [u](t) = u(x, t)x 2 s nepenent of t,.e., I [u] s the frst ntegral of KV equaton wth the local ensty Z (u(x, t)) = 2u(x, t). Moreover, n [5], t s shown that the functonals I n [u](t) = Z n (u(x, t))x are the frst ntegrals of KV equaton (). Hence, by the trace formulas (3), we have mmeately I n [u](t) = x + ( ) n 2 2 k 2n r ± (k, t)f ± (x, k, t) 2 k c ±,j (t)ηj 2n f ± (x, η j, t) 2 x, where r ± (k, t) an c ±,j (t) are efne by the GGKM formulas (7), an f ± (x, k, t) are the Jost solutons of the operator H t. By the efnton of the normalzaton coeffcents (6), one verfes mmeately I n [u](t) = x + ( ) n 2 k 2n r ± (k, t)f ± (x, k, t) 2 k η 2n j.

4 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, Moreover, f the operator H 0 corresponng to the ntal value u(x, 0) has no bouns states, we have the qute smple expresson I n [u](t) = x k 2n r ± (k, t)f ± (x, k, t) 2 k. (4) On the other han, f the operator H 0 s reflectonless,.e., r ± (k, 0) 0, then I n [u](t) = ( ) n 2 η 2n j (5) follows, an the rght han se of (5) s obvously nepenent of t. It s prove nepenently from these expressons that the functonal I n [u](t) oes not epen on t. In 6, usng these expressons, we gve a new proof of the fact that the functonal I n [u](t) oes not epen on t n the case that H 0 has no boun states, an construct another kn of frst ntegrals. 5 An evoluton equaton satsfe by the Jost soluton In what follows, we assume that the operator H 0 has no boun states. For the smplcty, we enote smply f(x, k, t) an r(k, t) nstea of f ± (x, k, t) an r ± (k, t), an r(k) nstea of r(k, 0). Frst we erve an evoluton equaton satsfe by the functon exp( 8k 3 t)f(x, k, t) 2. Put g = g(x, k, t) = exp( 8k 3 t)f(x, k, t) 2, (6) then, by (3), we have u(x, t) = 2 kr(k)g(x, k, t)k. Substtute ths nto KV equaton (), then 6uu x u xxx = 2 kr(k)(6ug x g xxx )k. (7) follows. By Appell s lemma mentone n 2 as Lemma, we have g xxx = 4(u k 2 )g x + 2u x g. (8) Elmnatng the term g xxx n (7) by (8), one verfes 6uu x u xxx = 2 kr(k)(2ug x 2u x g+4k 2 g x )k. Hence we have u t 6uu x + u xxx = 2 kr(k)(g t 2ug x + 2u x g 4k 2 g x )k 0. By the efnton of the Jost soluton, one obtans the asymptotc entty g t 2ug x + 2u x g 4k 2 g x C(k, t) exp(2kx), where C(k, t) s a functon of k an t. Hence we have the evoluton equaton for the functon g(x, k, t). Theorem 2. The functon g(x, k, t) solves the evoluton equaton g t 2ug x + 2u x g 4k 2 g x = 0. (9) Substtute (6) nto (9), then, by the rect calculaton, we have the evoluton equaton for the Jost soluton f = f(x, k, t). Theorem 3. The Jost solutons f = f ± (x, k, t) of the operator H t solve the evoluton equaton f t 2uf x + u x f 4k 2 f x = 0. 6 A spectral nterpretaton of the frst ntegrals of KV equaton In ths secton, we assume that the functon k n r(k), n belongs to the Schwartz space S k of k-varable functons for any n IN. For arbtrary n IN, efne the functon F n (x, t) an G n (x, t) by F n (x, t) = = G n (x, t) = k n r(k, t)f(x, k, t) 2 k k n r(k)g(x, k, t)k, k n r(k, t)f x (x, k, t) 2 k. Snce the Jost soluton f(x, k, t) behaves lke exp(kx) as x, an lke α exp(kx) + β exp( kx) as x, we have the followng lemma. Lemma 4. The functon F n (x, t) an G n (x, t) belong to the Schwartz space S x of x-varable functons for any t.

5 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, Defne the functonals J n [u](t), n by S x, one verfes J n [u](t) = = = x x F n (x, t)x k n r(k, t)f(x, k, t) 2 k k n r(k)g(x, k, t)k (20) t J n[u](t) = = 4 x (uf n)x + 4 x x k n r(k)ug x k k n r(k)ug x k. (2) It s known that the ntegraton n (20) converges. The convergence problem concerne wth the ntegraton of ths type s known to be very elcate. We refer the reaer [2] for more precse treatment concerne wth the convergence problem of the ntegraton of ths type. Now, we have the followng theorem whch s the man result of the present work. Theorem 5. Suppose that u(x, t) s the spatally raply ecraesng real value soluton of KV equaton such that the operator H 0 has no boun states. Then, the functonals J n [u](t) are nepenent of t,.e., are the frst ntegrals of KV equaton. Proof. By Theorem2, we have t J n[u](t) = = = By Lemma4, x x k n r(k)g t (x, k, t)k k n r(k) (2ug x 2u x g + 4k 2 g x )k x + 4 = x x k n r(k)(2ug x 2u x g)k x x k n+2 r(k)gk k n+2 r(k)gk. x F n+2(x, t)x = 0. follows. Moreover, snce the prouct uf n s also n Next we calculate the last term of the expresson (2). By the efnton, we have 4 x = 8 = 8 = 4 x x k n r(k)ug x k where we use the relaton Ths completes the proof. k n r(k)uff x k k n r(k)(f xx + k 2 f)f x k x (G n(x, t) + F n (x, t))x = 0, uf = f xx + k 2 f. q.e.. By (4), the relaton between the frst ntegrals J m [u] an I n [u] s state by the followng corollary. Corollary 6. For arbtrary n IN, the enttes hol. J 2n [u](t) = I n [u](t) Thus, by Theorem5, we coul construct the frst ntegrals wth the local enstes whch are not the fferental polynomals Z n (u). In ths case, we consere the problem for only the operator wthout boun states. We wll scuss a smlar problem for the operator wth the screte egenvalues n the forthcomng paper []. References: [] P. Deft, F. Lun an E. Trubowtz, Nonlnear wave equatons an constrane harmonc moton, Comm. Math. Phys. 74, 980, pp [2] P. Deft an E. Trubowtz, Inverse scatterng on the lne, Comm. Pure Appl. Math. 32, 979, pp

6 2th WSEAS Int. Conf. on APPLIED MATHEMATICS, Caro, Egypt, December 29-3, [3] C. S. Garner, J. M. Greene, M. D. Kruscal an R. M. Mura, Metho for solvng Korteweg-e Vres equaton, Phys. Rev. Letters 9, 967, pp [4] G. L. Lamb, Elements of solton theory, John Wley & Sons, New York, 980. [5] R. M. Mura, C. S. Garner an M. D. Kruskal, Korteweg-e Vres equaton an generalzatons. II. Exstence of conservaton laws an constants of moton, J. Mathematcal Phys. 9, 968, pp [6] M. Ohmya, Recurson operator an the trace formula for -mensonal Schrönger operator, J. Math. Tokushma Unv , pp [7] M. Ohmya, On the Darboux transformaton of the secon orer fferental operator of Fuchsan type on the Remann sphere, Osaka J. Math. 25, 988, pp [8] M. Ohmya, Spectrum of Darboux transformaton of fferental operator, Osaka J. Math. 36, 999, pp [9] J. Pöschel an E. Trubowtz, Inverse spectral theory, Acaemc, Orlan, 987. [0] E. T. Whttaker an G. N. Watson, A course of moern analyss, Cambrge Unv. Press, Cambrge, 927. [] Y. Yamamoto, T. Nagase an M. Ohmya, Appell s lemma an conservaton laws of KV equaton, n preparaton. [2] V. E. Zakharov an L. D. Faeev. Korteweg-e Vres equaton; A completely ntegrable Hamltonan system, Functonal Anal. an Its Appl. 5, 972, pp