darryl d. holm and zhijun qiao- r-matrix for a geodesic flow associated with a new integrable peakon...

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R-MATRIX FOR A G EODESIC FLOW ASSOCIATED WITHA NEW INTEGRABLE PEAKON EQUATION

Darryl D. Holm, T-7Zhijun Qiao, T-7

Proceedings of Nonlinear Evolution Equations DynamicsSystems (NEEDS) 2001Cambridge, UKJuly 24-31 ,200 1

r Los AlamosNATIONAL LABORATORYLos Alamos National Laboratory, an affirmative actiodequal opportunity employer, is operated by the Universityof California for the U.S.Department of Energy under contract W-7405-ENG-36. By acceptance of this article, the publisher recognizes hat the U.S. Governmentretains a nonexclusive, royalty-free icense o publish or reproduce he published form of this contribution, or to allow others to do so. for US.Government purposes. Los Alamos National Laboratory requests that the publisher dentify his article as work performed under theauspices of the US . Departmentof Energy. Los Alamos National Laboratory strongly supports academic reedom and a researchers right topublish; as an institution,however, the Laboratory does not endorse the viewpoint of a publ icationor guarantee ts technical correctness.

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R-matrix for a geodesic flow associatedwith a new integrable peakon equation

Darryl D. Holm' and Zhijun Qiao',2'Theoretical Division and CNLS, MS B-284

Los Alamos National LaboratoryLos Alamos, NM 87545, USA

21nstituteof Mathematics, Fudan UniversityShanghai 200433, PR China

Email: [email protected]

November 27, 2001

AbstractWe use the r-matrix formulation to show the integrability of geodesicflow on an N-dimensional space with coordinates q k , with k = 1,...,N ,equipped with the co-metric g i j = e-141--431(2 - -Iq1-q3l). This flow isgenerated by a symmetry of the integrable partial differential equation(pde) mt +um, + 3mux = 0, m = u - a2uZx a s a constant), whichwas recently proven to be completely integrable and possess peakonsolutions by Degasperis, Holm and Hone. The isospectral eigenvalueproblem associated with this integrable pde is used to find a new Laxrepresentation for its N-peakon solution dynamics. By employing thisLax matrix we obtain the r-matrix for the integrable geodesic flow.

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Keywordsr-matrix structure.

Peakon equation, Lax representation, Hamiltonian, Lax matrix,

A M S Subject: 35Q53; 58F07; 35Q35PACS: O3.4 0.G ~; 3.40Kfi 47.10.+g

1 IntroductionThe b = 3 peakon equation and its isospectral problemWe begin with the case b = 3 of the b-weighted peakon equation. This is theevolutionary equation defined on the real line as,

lim m=O, (1.1)in which the subscripts denote partial derivatives with respect to the indepen-dent variables x and t . For any values of the dimensionless constant b andconstant lengthscale Q , this equation admits exact N-peakon solutions

2m t + u m , + b m u , = O , m = u - Q u,,, 1 4 - + ~

N

i= lin which the 2N time-dependent functions p i ( t ) and q i ( t ) , i = 1 , 2 , . . ,N,satisfy a system of ordinary differential equations whose character depends onthe value of the bifurcation parameter b. The case b = 2 is the dispersionlesslimit of the integrable Camassa-Holm (CH) equation that was discovered forshallow water waves in [2]. As shown in [3], the CH equation with dispersion is'one full order more accurate in asymptotic approximation beyond Korteweg-de Vries (KdV) for shallow water waves, yet it still preserves KdV's solitonproperties such as complete integrability via the inverse scattering transform(IST) method.

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An equation equivalent to the case b = 3 of the peakon equation (1.1)was singled out for special attention among a family of related equations, byDegasperis and Procesi in [l]. The peakon equation (1.1) was shown to becompletely integrable for the case b = 3 by Degasperis, Holm and Hone [4],who found the followingLax pair consisting of a third order eigenvalue problemand a second-order evolutionary equation for the eigenfunction,

1Q2= - + + - A m + ,1 2

Q2 X 3 xClt = - -qxx - + (21, + -)+.Compatibility $J~ = q t zXxmplies Eq. (1 .1 ) with b = 3 provided dX/& = 0. . .Thus, Eq. (1.1)with b = 3 is integrable by the inverse spectral transform forthe isospectral eigenvalue problem (1 .3 ) .

Equation (1 .1 ) with b = 3,(1.5)mt +umx + 3 m u X= 0 , m = u 0 u,, ,

was shown to be integrable by the inverse spectral transform and to possessan infinite sequence of conservation laws in Degasperis et al. [4]. The first fewof these are, in the notation of [4],

H-1 = L J u 3 d x , Ho = J m d x , (1.6)6H I = + 5 v : + 4 v 2 ) d x ,We shall pay

(1.7)the quantity Y is defined as

2 -1 - 2 -1v := ( 4 - ax) u= (4- 8,) (1- a y

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Lax matrix for N-peakon dynamicsSubstituting the N-peakon solution,

into the isospectral eigenvalue problem (1.3) yields [4]

Setting + ( q i ( t ) , )= +i(t) hen gives the following matrix eigenvalue problem,N2

1 -

j= 1(1.10)

where

Let L denote the N x N matrix &. In Ref. [4], the authors used the twoconserved quantities trL and trL2 to solve the 2-peakon subdynamics of thethe N-peakon dynamics q k , pk, with k = 1, ..,N , cu = 1, satisfying

Amongst other results, the authors in [4] iscovered the two-peakon collisionrules for N = 2 and gave explicit formulas for its phase shifts as functions ofthe asymptotic speeds of the tw o peakons.

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2 A geodesic flow associated b = 3 peakonsThe quantity used for determining the two-peakon N = 2 collision laws in [4],

is also the quadratic conservation law H1 in (1.6) for the b = 3 peakon equation(1.5), when H1 is evaluated on the N-peakon solution (1.8) with Q = 1.

The canonical Hamiltonian dynamics generated by HI is geodesic motion onan N-dimensional space with co-metric gij = e-1qa-q31(2- e-lqa-qjl). As we shallshow by finding its r-matrix structure in the remainder of the present paper,the geodesic motion canonically generated by the conservation law H1 = TrL2in (2.1) provides a new 2N-dimensional integrable system,

These geodesic HI-dynamics for Pk,qk, are not the same as the N-peakondynamics in (1.12). Rather, we are studying the restriction to the peakonsector of the Hl-flow in the hierarchy of integrable equations associated withthe isospectral problem for equation (1.5).

R-matrix results for the geodesic HI-dynamicsTo find the r-matrix structure for these Hl-dynamics for pk, q k , we now intro-duce an alternative Lax matrix for the peakon dynamics of Eqn. (1.5),

N

i , j= l

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where

The Lax matrix (2.4) also satisfies,. N

(2.7)which is the Hamiltonian for the canonical dynamics in Eqs. (2.2) and (2.3).In Eq. (2.6), we have

and the function A ( z ) Satisfies the following relations,

a aa Y

( z + - ) A ( z ) A ( - z ) = 0. (2.13)We shall work in the canonical matrix basis Ei j ,

( E i j ) k l = J i k J j l , i , j , k, = 1,...,N .To find the r-matrix structure for the HI-dynamics in Eqs. (2.2) and (2.3), weconsider the so-called fundamental Poisson bracket [6]

{ L l ,L2 ) = [r12,L l ] - T a l ,L21, (2.14)6


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where

k , l= lN

i r j t sJN

(L11 L2} = ( L i j i Lkl}Ezj @ Ekl.a , j , k , l = l

Here { L i j ,L k l } is the standard Poisson bracket of two functions, 1 is the N x Nunit matrix, and the quantities are to be determined. In Eq. (2 .14) , e , . ]denotes the usual commutator of matrices.

After a lengthy calculation for both sides of Eq. (2 .14) , we obtain thefollowing key equalities (whose detailed verification is given in the Appendix) :

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[ rs t ,!,Ij1 = 0, for different s, t , , .where r j l = x k , m r k m , j l E k m , r l l = x k , m r k m , l l E k m l are two N x N matriceswhose entries are to be determined, L is the Lax matrix, and [ e , L ] k l stands forthe k-th row and the l-th colum element of [-, ] .

In matrix notation, all the above equalities can be rewritten as[ r j l ,L] = B j , j f . 1 , (2.15)[ T u , L ] = B 1 l , (2.16)

where B j , B are the following two N x N matrices:

By solving Eqs. (2.15) and (2.16), we have the following r-matrix structure:

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Perhaps not unexpectedy, this non-contant r-matrix for the geodesic H I-dynamics differs from the constant r-matrix associated with the CH equation( b= 2) discovered by Ragnisco and Bruschi in [5].

Concluding remarksIn this paper, we found the r-matrix formulation' for the integrable geodesicmotion generated canonically by the quadratic quantity H1 in (2.1). Thisquantity arises by restriction t o the peakon sector of a quadratic conservationlaw in the hierarchy of integrable equations associated with the isospectralproblem for equation (1.5). The quadratic quantity HI is also conserved forthe 2-peakon dynamics of the 1+1 integrable partial differential equation (1.5)that was singled out in [l]and was proven to be completely integrable bythe isospectral transform in [4]. We also introduced a new Lax matrix L forthe N-peakon flows of the integrable equation (1.5) that facilitated the r-matrix calculations and for which H1 = itrL2. In later work, we shall discussadditional flows in the hierarchy of integrable equations associated with theisospectral problem for equation (1.5) and study their relationships to classicalfinite-dimensional integrable systems [7].

AcknowledgmentsD. D. Holm is grateful to A. Degasperis and A. N. W. Hone for their help andcollaboration in many discussions that parallel the work reported here. Z. Qiaowould like to express his sincere thanks to J. M. Hyman and L. G. Margolinfor their warm invitations and friendly academic discussions. This work wassupported by the US. epartment of Energy under contracts W-7405-ENG-36and the Applied Mathematical Sciences Program KC-07-01-01; and also theSpecial Grant of National Excellent Doctorial Dissertation of China.

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References[l]A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and

Perturbation Theory, edited by A. Degasperis and G. Gaeta, World Sci-entific (1999) pp.23-37.

[2] R. Camassa and D. D. Holm, An integrable shallow water equation withpeaked solitons, Phys. Rev. Lett. 71 (1993) 1661-1664.

[3] H. Dullin, G. Gottwald and D. D. Holm, An integrable shallow waterequation with linear and nonlinear dispersion. Phys. Rev. Lett. 87 (2001)1945-1940.

[4]A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equationwith peakon solutions. In preparation (2001).

[5] 0. Ragnisco and M. Bruschi, Peakons, r-matrix and Toda lattice, PhysicaA 228 (1996) 150-159.

.[6] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theoryof Solitons (Springer-Verlag, Berlin, 1987).

[7] D. D. Holm and Z. Qiao, On the relationship between the b=3 peakonequation and finite-dimensional integrable systems. In preparation (2001).

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AppendixThe following computations are needed in verifying Eq. (2.14).First, we calculate the left hand side of Eq. (2.14).

where the superscript I means Eq. (2.12) with the argument.Thus, we obtain the following formula,

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The first term of Eq. (2.19) is:

Therefore) we have

(2.19)

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