Multipeakons and a theorem of Stieltjes

Transcription

1 Multipeakons and a theorem of Stieltjes arxiv:solv-int/99030v Mar 999 R. Beals Yale University New Haven, CT D. H. Sattinger University of Minnesota Minneapolis J. Szmigielski University of Saskatchewan Saskatoon, Saskatchewan, Canada November 0, 207 Inverse Problems, 5, 999, Letters L-L4 Abstract A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts. AMS (MOS) Subject Classifications: 35Q5, 35Q53. Key words: Multipeakons, Stieltjes, continued fractions. Camassa and Holm [2],[3] introduced the strongly nonlinear equation ( 4 D2 )u t = 3 2 (u2 ) x 8 (u2 x) x 4 (uu xx) x, () as a possible model for dispersive waves in shallow water and showed that it could formally beintegrated using thespectral problem (D 2 +k 2 m )ψ = 0, Research supported by NSF grant DMD Research supported by NSF Grant DMS Research supported by the Natural Sciences and Engineering Research Council of Canada

2 where m = 2( 4 D2 )u. We showed in a previous article that a Liouville transformation ([], equation (5.8)) maps this spectral problem to the string problem v (y)+k 2 g(y)v(y) = 0, < y < ; v(±,z) = 0, (2) where m(x) = g(y)sech 4 (x), and y = tanh x. The Liouville transformation is independent of m, and m may assume both positive and negative values. The scattering data are invariant under the transformation, and consist of a discrete set of eigenvalues kj 2 and associated coupling constants c j. The evolution of the data under the flow () is k j = 0 and ċ j = 2c j /kj; 2 []. The n-peakon solution has the form u(x,t) = n p j (t)exp( 2 x x j (t) ), (3) j= where the p j, x j evolve according to a completely integrable Hamiltonian system [3]. A number of properties of the two-peakon and n-peakon solutions were obtained from an analysis of the dynamical system in [3]. In this note we use a continued fraction expansion and a theorem of Stieltjes to give explicit algebraic formulas for the n-peakon solutions. The multi-peakon solution arises when g(y) dy is a sum of delta functions: g(y)dy = n g j δ(y y j )dy, < y < < y n <. (4) j= Equation (2) is then interpreted in the following sense: v = 0, y j < y < y j ; v (y j +) v (y j ) = zg j v(y j ), (5) for 0 j n. We let y 0 = and y n+ =, and we have taken z = k 2 as the spectral parameter. We denote the lengths of the subintervals by l j = y j+ y j, for j = 0,..., n. Let ϕ and ψ be the solutions of (5) that satisfy the boundary conditions ϕ(,z) = 0, ϕ (,z) =, ψ(,z) = 0, and ψ (,z) =. The eigenvalues {λ j } of (5) are the zeros of ϕ(,z), which are simple. The coupling constants {c j } are characterized by the identities ϕ(y,λ j ) = c j ψ(y,λ j ). Differentiating with respect to y and setting y =, we find that c j = ϕ (,λ j ). 2

3 To construct ϕ we define q j = ϕ(y j,z), and p j = ϕ (y j,z). Thus p j is the value of the derivative ϕ in the interval (y j,y j ). The p j and q j can be obtained recursively from (5) by q = l 0, p =, and q j q j = l j p j, p j p j = zg j q j. (6) It follows inductively that ϕ(,z) is a polynomial of degree n. We assume for most of this note that the n point masses g j are positive, so that the eigenvalues are negative: λ n < λ n <...λ < 0. By classical oscillation theory, the c j alternate signs, with c > 0. The scattering data is encoded in the Weyl function w(z) = ϕ (,z)/ϕ(,z). Since ϕ(y,0) = +y, ϕ(,0) = 2; and therefore ϕ(,z) = 2 n j= ( z/λ j). Combining these results, we obtain w(z) z = n 2z + a j, a j = z λ j 2 c j j= ( λ j /λ k ). (7) k j The residues a j of w(z)/z at z = λ j are positive, since both the coupling constants c j and the successive products alternate signs. Following Krein [4], we may write the Weyl function as a continued fraction l n + zg n + l n (8) This follows by induction on (6). We have p =, q = l 0, q 2 = q + l p 2, p 2 = p +zg l 0 ; hence p 2 q 2 = p 2 l 0 +p 2 l = = l +, zg + l 0 3

4 Assuming (8) for n masses, and adding an additional mass at y n (y n,), we have p n+ = p n +zg n q n, q n+ = q n +l n p n+ ; and p n+ q n+ = p n+ q n +l n p n+ = = l n + zg n + p n q n A classical result of Stieltjes [5] recovers the coefficients of the continued fraction (8) from the Laurent expansion of w(z)/z at infinity, obtained from (7) by expanding each (z λ j ) : w(z) z = k=0 ( ) k A k z k+, A k = { + n 2 j= a j k = 0; n j= ( )k λ k j a j, k. (9) Since the eigenvalues are negative andthe a j are positive, each A k is positive. Stieltjes showed that such a Laurent series can be uniquely developed in a continued fraction, b 2k = ( 0 k )2, b 2k+ = ( k )2. k b z + k 0 k 0 k+ b 2 + b 3 z +... Here 0 0 = = 0 and 0 k, k, k, are the k k minors of the Hankel matrix A 0 A A 2... A A 2 A 3... H = A 2 A 3 A whose upper left hand entries are, respectively, A 0 and A. Comparing this continued fraction with that for the Weyl function, we obtain l j = ( n j) 2, g 0 j = ( 0 n j+) 2. (0) n j 0 n j+ n j+ n j 4

5 Because the a j are positive, the 0 k are positive, k n +, and the k are positive, k n. With n eigenvalues and coupling coefficients, 0 n+2 vanishes and the continued fraction terminates. The time dependence of l j and g j under () is determined explicitly from the evolution of the coupling coefficients, (7), and (9). The multi-peakon solution is given by u(x,t) = 2 exp( 2 x x )dm(x,t), where dm = g(y)(dy/dx)dx is the pull-back of g dy under the transformation y = tanhx. Hence n dm(x,t) = g j (t)sech 2 (x j (t))δ(x x j (t))dx. j= The positions x j (t) are obtained recursively from x j = 2 log +y j y j, y j = k<j l k. Since l k = 2 we obtain x j = 2 log Λ j Λ + j, Λ j = k<j l k, Λ + j = k j l k. () The n-peakon solution is thus given by (3) with p j = g 2 jλ j Λ+ j. These formulas imply that the asymptotic positions of the peaks are given by ( λj ) log, t ; (2) λ k x n j+ t/λ j + 2 log2a j(0)+ k<j x j t/λ j + 2 log2a j(0)+ k>j k>j log ( λ j λ k ) t. (3) From these formulas it can be determined that the phase shift in the peakon with speed /λ j as t ± is ( log λ ) j ( λj ) log. (4) λ k λ k k<j 5

6 Moreover, the asymptotic height of this same peakon is /λ j. From this we may conclude that the n-peakon solution is asymptotically a sum of single peakons moving with speeds /λ j. The two-peakon solution has the form (3) with p (t) = λ2 a +λ 2 2 a 2 λ λ 2 (λ a +λ 2 a 2 ), p 2(t) = a +a 2 λ a +λ 2 a 2 ; (5) x (t) = 2 log 2(λ λ 2 ) 2 a a 2 λ 2 a +λ 2 2a 2, x 2 (t) = 2 log2(a +a 2 ), (6) where a j (t) = a j (0)e 2t/λ j. The explicit form for the relative positions and momenta x 2 x and p 2 p was given in [3]. We have concentrated here on the multi-peakon case, for which the masses g j all have the same sign and the solutions do not develop singularities. The procedure is purely algebraic, however, so the formulas hold also for mixed positive and negative masses: the peakon-antipeakon case. In this case singularities can occur. Acknowledgement The authors thank H P McKean for some helpful discussions and comments, particularly with regard to the issue of the positivity of m. References [] Beals, R., D. H. Sattinger, & J. Szmigielski. Acoustic scattering and the extended Korteweg-deVries hierarchy. Advances in Mathematics, 40:90 206, 998. [2] Camassa, R. and D. D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 7:66 664, 993. [3] Camassa, R., D. D. Holm, & J. M. Hyman. A new integrable shallow water equation. Advances in Applied Mechanics, 3: 33, 994. [4] Krein, M. G. On inverse problems for an inhomogeneous string. Dokl. Akad. Nauk. SSSR, 82: ,

7 [5] Stieltjes, T. J. Sur la réduction en fraction continue d une série procédent suivant les puissances descendantes d une variable. In Oeuvres Complètes de Thomas Jan Stieltjes, volume I, pages P. Noordhoff, Groningen,