An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

An integrable semi-dis cr etization of the Camas sa-Holm equation and its determinan t solution Y asuhir o Ohta 1 , Ken- ichi Maruno 2 ‡ and Bao-F eng F eng 2 1 Departmen t of Mathematics, K obe Univ ersity , Rokko, K obe 657-850 1, Japan 2 Departmen t of Mathematics, The University of T e xas-Pan American, Edinb urg, TX 7 8541 Abstract. An integrab le semi-discretization of the Camassa-Holm equatio n is presented. The keys of its co nstruction are bilinea r fo rms and deter minant struc ture of solution s of the CH equation . Deter minant for mulas of N -soliton solution s of the con tinuous and semi-discrete Camassa-Holm equations are p resented. Based on determina nt formulas, we can g enerate multi-soliton, multi-cuspo n and multi-soliton- cuspon solutions. Numerical compu tations using the integrab le sem i-discrete Camassa-Holm equ ation are p erforme d. I t is shown that the integrable semi-discrete Camassa-Holm e quation giv es very accur ate nu merical results even in the c ases of cuspon-cu spon and solito n-cuspo n interactions. The n umerical computa tion for an initial v alue condition , which is not an exact solution, is also presented. 3 November 2 018 P A CS numbers: 02.30 .Ik, 05.45 .Yv T o be submitted to : J. Phys. A: Math. Gen. 1. Introduction The Camassa-Holm (CH) equation w T + 2 κ 2 w X − w T X X + 3 ww X = 2 w X w X X + ww X X X . (1.1) has attracted considerable interest si nce i t has been deriv ed as a model equati on for shall ow- water wave s [1]. Here, w = w ( X , T ) , κ is a positive parameter and the subscripts T and X appended to w ( X , T ) denote partial differentiation. Originally , this equation has been found in a mathem atical search of recursion opera tors connected with the integrable p artial differential equations [2]. The CH equation has been shown to be completely integrable. In the case of κ = 0, the CH equation admit s peakon solutions which are represented by piecewise analytic functions [3]. Schiff obtained single- and two-soliton solutions in a parametric form by using the B ¨ acklund transform ation [4]. An approach based on the in verse scattering t ransform method (IST) provides an explicit form of the in v erse mapping in terms o f Wronsk ian [5, 6, 7, 8]. The N -soliton solutio n was also con structed by using the Hi rota b ilinear m ethod [9, 10, 11, 12] (See also [13]). The ke y point of com putations of the N -solit on soluti on by ‡ e-m ail: kmaru no@utpa.ed u An inte grable semi-discr etization of the Camassa-Holm equation 2 the Hirot a bilin ear method in thos e papers is the relationship between t he CH equation and the AKNS shall ow water wa ve equati on [14]. When κ 6 = 0, cusped solit ary wav e solution s, as well as analytic soliton solutions, were f ound in [15, 16, 17]. In [18, 19], the interaction of cusped soliton (cuspon) was studied in detail. It is e xtremely dif ficult to perform numerical com putations of the C H equation due to t he singularities of cuspon and peakon so lutions. So far , se veral numerical computati ons of the CH equati on were presented [20, 21, 22, 23, 24, 25, 26]. Howe ver , none of thes e nu merical methods give s satisfac tory results for soliton-cuspon and cuspon-cuspon interactions. Integrable discretizations o f soli ton equations ha ve recei ved considerable attenti on recently [27, 28]. Ablowitz and Ladik proposed how to construct integrable di screte analogues of soliton equations based on Lax pairs [29, 30]. Hi rota proposed another method to construct integrable discrete analogues of s oliton equations based on bilinear equations [31, 32, 33]. Applications of integrable d iscretizations of s oliton equations were considered in various fields [34, 35, 36]. The purpose in t he present p aper is to present an integrable semi-discretization of the CH equation throug h a bi linear approach and to perform num erical computati ons by using integrable scheme. W e show that the integrable scheme gives very accurate numerical results e ven for the soliton-solition and cuspon-cuspon interac tions. The outline of this p aper is as foll ows. In § 2, we gi ve bil inear form s and a determi nant formula of the N -so liton soluti on of equatio n (1.1). As far as we are concerned, bil inear forms directly related to th e CH equation have not been known yet. These bilin ear forms can help us to understand mathematical structure of the CH equation more thorough ly . Based on these bi linear equation s, we obtain the det erminant form ula for the CH equation by using determinant technique [37, 38, 39, 40]. In § 3, we giv e an integrable semi-discrete Camassa- Holm equation and its determinant formula of the N -sol iton solutio n. In § 4, we present resul ts of numerical computations by using the proposed integrable semi-discrete CH equation. 2. Bilinear equations and determinant formulas of the Camassa-Holm eq uation In this section, we give bilinear equat ions and the N -soliton soluti on of th e CH equation . All of the existing works in constructing N -soliton sol utions of the CH equation using H irota bi linear method take advantage of t he relationship between t he CH equation and the AKNS shallow water wa ve equation [9, 10, 11, 12, 14]. So far , the b ilinear forms which are directly related to the CH equation remain unknown. In t he present paper , we giv e bil inear equations obtained from the CH equati on directly and derive the determinant soluti on by using the determinant technique. Our formulation in this section i s crucial for the deriv ation of integrable semi- discretization of the CH equation. An inte grable semi-discr eti zation of the C amassa-Holm equation 3 Lemma 2.1. Bi linear equations                       1 2 D t D x − 1  f · f = − gh , 1 2 D t ( D y − 2 cD x ) f · f = − D x g · h , 1 2 D s D x f · f = D t g · h ,  1 2 D s ( D y − 2 cD x ) − 2  f · f = ( D t D x − 2 ) g · h , (2.1) have a determinant solution f = τ 0 , g = τ 1 , h = τ − 1 , τ n =          ψ ( n ) 1 ψ ( n + 1 ) 1 · · · ψ ( n + N − 1 ) 1 ψ ( n ) 2 ψ ( n + 1 ) 2 · · · ψ ( n + N − 1 ) 2 . . . . . . . . . ψ ( n ) N ψ ( n + 1 ) N · · · ψ ( n + N − 1 ) N          , wher e ψ ( n ) i = a i , 1 ( p i − c ) n e ξ i + a i , 2 ( q i − c ) n e η i , ξ i = p i x + p 2 i y + 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = q i x + q 2 i y + 1 q i − c t + 1 ( q i − c ) 2 s + η i 0 . Pr oof. Consi der the following Casorati determinant solutio n, τ n =           ψ ( n ) 1 ψ ( n + 1 ) 1 · · · ψ ( n + N − 1 ) 1 ψ ( n ) 2 ψ ( n + 1 ) 2 · · · ψ ( n + N − 1 ) 2 . . . . . . . . . ψ ( n ) N ψ ( n + 1 ) N · · · ψ ( n + N − 1 ) N           , (2.2) where ψ ( n ) i ’ s are arbitrary functions of four cont inuous independent variables, x , y , t and s , which satisfy linear dispersion relations, ∂ x ψ ( n ) i = ψ ( n + 1 ) i + c ψ ( n ) i , (2.3) ∂ y ψ ( n ) i = ∂ 2 x ψ ( n ) i , = ψ ( n + 2 ) i + 2 c ψ ( n + 1 ) i + c 2 ψ ( n ) i , (2.4) ∂ t ψ ( n ) i = ψ ( n − 1 ) i , (2.5) ∂ s ψ ( n ) i = ψ ( n − 2 ) i . (2.6) Thus we can choose ψ ( n ) i as follows: ψ ( n ) i = a i , 1 ( p i − c ) n e ξ i + a i , 2 ( q i − c ) n e η i , An inte grable semi-discr eti zation of the C amassa-Holm equation 4 ξ i = p i x + p 2 i y + 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = q i x + q 2 i y + 1 q i − c t + 1 ( q i − c ) 2 s + η i 0 . For simplicity , we introduce a con venient notation, | n 1 , n 2 , · · · , n N | =          ψ ( n 1 ) 1 ψ ( n 2 ) 1 · · · ψ ( n N ) 1 ψ ( n 1 ) 2 ψ ( n 2 ) 2 · · · ψ ( n N ) 2 . . . . . . . . . ψ ( n 1 ) N ψ ( n 2 ) N · · · ψ ( n N ) N          . (2.7) In this notation, the solution for the above bilinear forms, τ n , is re written as τ n = | n , n + 1 , · · · , n + N − 1 | . (2.8) W e sh ow that the above τ n actually sati sfies the bilinear equations (2.1) by u sing the Laplace expansion technique [39, 40]. The differ ential formulas for τ are giv en by ( ∂ x − N c ) τ n = | n , n + 1 , · · · , n + N − 2 , n + N | , (2.9) ( ∂ y − 2 c ∂ x + N c 2 ) τ n = | n , n + 1 , · · · , n + N − 2 , n + N + 1 | − | n , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | , (2.10) ∂ t τ n = | n − 1 , n + 1 , · · · , n + N − 1 | , (2.11) ∂ s τ n = | n − 2 , n + 1 , · · · , n + N − 1 | − | n − 1 , n , n + 2 , · · · , n + N − 1 | , (2.12) ( ∂ t ( ∂ x − N c ) − 1 ) τ n = | n − 1 , n + 1 , · · · , n + N − 2 , n + N | , (2.13) ∂ t ( ∂ y − 2 c ∂ x + N c 2 ) τ n = | n − 1 , n + 1 , · · · , n + N − 2 , n + N + 1 | − | n − 1 , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | , (2.14) ∂ s ( ∂ x − N c ) τ n = | n − 2 , n + 1 , · · · , n + N − 2 , n + N | − | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N | , (2.15) ( ∂ s ( ∂ y − 2 c ∂ x + N c 2 ) − 2 ) τ n = | n − 2 , n + 1 , · · · , n + N − 2 , n + N + 1 | − | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N + 1 | − | n − 2 , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | + | n − 1 , n , n + 2 , · · · , n + N − 3 , n + N − 1 , n + N | , (2.16) which are proved by using the linear dispersion relations (2.3)-(2.6). (See Appendix) The first equation of eqs. (2.1) Let us introduce an identity for 2 N × 2 N determinant,       n − 1 n + 1 · · · n + N − 2 n + N n Ø n + N − 1 n − 1 Ø n + N n n + 1 · · · n + N − 2 n + N − 1       = 0 . An inte grable semi-discr eti zation of the C amassa-Holm equation 5 Applying the Laplace expansion to the left-hand side, we obt ain the algebraic bilinear identity for determinants, | n − 1 , n + 1 , · · · , n + N − 2 , n + N | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | −| n , n + 1 , · · · , n + N − 2 , n + N | × | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n + 1 , · · · , n + N − 2 , n + N − 1 , n + N | × | n − 1 , n , n + 1 , · · · , n + N − 2 | = 0 , (2.17) which is re written by using (2.8), (2.9), (2.11) and (2.13), into t he differential bilinear equation, ( ∂ t ( ∂ x − N c ) − 1 ) τ n × τ n − ( ∂ x − N c ) τ n × ∂ t τ n + τ n + 1 τ n − 1 = 0 , i.e., ( ∂ t ∂ x τ n − τ n ) τ n − ∂ t τ n ∂ x τ n + τ n + 1 τ n − 1 = 0 . Setting n = 0, f = τ 0 , g = τ 1 , h = τ − 1 , the above bilinear equatio n leads to the first equati on of (2.1). The second equation of eqs. (2.1) Let us introduce two identities for 2 N × 2 N determinant s,       n − 1 n + 1 · · · n + N − 3 n + N − 1 n + N n Ø n − 1 Ø n + N n n + 1 · · · n + N − 1       = 0 ,       n − 1 n + 1 · · · n + N − 2 n + N + 1 Ø n + N − 1 n − 1 Ø n + N + 1 n n + 1 · · · n + N − 2 n + N − 1       = 0 . Applying the Laplace expansion to the left-hand si de, we obtain the algebraic bilinear identities for determinants, | n − 1 , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | − | n , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | × | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n − 1 , n , n + 1 , · · · , n + N − 3 , n + N − 1 | × | n + 1 , n + 2 , · · · , n + N − 1 , n + N | = 0 , (2.18) | n − 1 , n + 1 , · · · , n + N − 2 , n + N + 1 | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | − | n , n + 1 , · · · , n + N − 2 , n + N + 1 | × | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n + 1 , · · · , n + N − 1 , n + N + 1 | × | n − 1 , n , n + 1 , · · · , n + N − 3 , n + N − 2 | = 0 . (2.19) T aking t he d iff erence of these two bil inear identi ties, it is rewritten by u sing (2. 8)-(2.11) and (2.14) into the diffe rential bilinear equation, ∂ t ( ∂ y − 2 c ∂ x + N c 2 ) τ n × τ n − ( ∂ y − 2 c ∂ x + N c 2 ) τ n × ∂ t τ n + ( ∂ x − N c ) τ n + 1 × τ n − 1 − τ n + 1 ( ∂ x − N c ) τ n − 1 = 0 , An inte grable semi-discr eti zation of the C amassa-Holm equation 6 i.e., ( ∂ t ∂ y τ n ) τ n − ∂ y τ n ∂ t τ n − 2 c (( ∂ t ∂ x τ n ) τ n − ∂ x τ n ∂ t τ n ) + ( ∂ x τ n + 1 ) τ n − 1 − τ n + 1 ( ∂ x τ n − 1 ) = 0 , which is nothing but the second equati on of (2.1). The third equation of eqs. (2.1) Let us introduce two identities for 2 N × 2 N determinant s,       n − 2 n + 1 · · · n + N − 2 n + N n Ø n + N − 1 Ø n + N n n + 1 · · · n + N − 2 n + N − 1       = 0 ,       n − 1 n n + 2 · · · n + N − 2 n + N n + 1 Ø n + N − 1 Ø n + N n + 1 n n + 2 · · · n + N − 2 n + N − 1       = 0 . Applying the Laplace expansion to the left-hand si de, we obtain the algebraic bilinear identities for determinants, | n − 2 , n + 1 , n + 2 , · · · , n + N − 2 , n + N | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | − | n − 2 , n + 1 , n + 2 , · · · , n + N − 2 , n + N − 1 | × | n , n + 1 , n + 2 , · · · , n + N − 2 , n + N | + | n − 2 , n , n + 1 , n + 2 , · · · , n + N − 2 | × | n + 1 , n + 2 , · · · , n + N − 2 , n + N − 1 , n + N | = 0 , (2.20) | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | − | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N − 1 | × | n , n + 1 , n + 2 , · · · , n + N − 2 , n + N | + | n − 1 , n , n + 1 , n + 2 , · · · , n + N − 2 | × | n , n + 2 , · · · , n + N − 2 , n + N − 1 , n + N | = 0 . (2.21) T aking the dif ference of these two bilinear identit ies, it is re writ ten by using (2.12) and (2.15) into the differential bili near equation, ∂ s ( ∂ x − N c ) τ n × τ n − ∂ s τ n × ( ∂ x − N c ) τ n + ( ∂ t τ n − 1 ) τ n + 1 − τ n − 1 ( ∂ t τ n + 1 ) = 0 , i.e., ( ∂ s ∂ x τ n ) τ n − ∂ s τ n ∂ x τ n − ( ∂ t τ n + 1 ) τ n − 1 + τ n + 1 ( ∂ t τ n − 1 ) = 0 . Setting n = 0, f = τ 0 , g = τ 1 , h = τ − 1 , the abov e bilinear equation leads to the third equation in (2.1). The f ourth equation of eqs. (2.1) Let us introduce four identities for 2 N × 2 N determinants,       n − 2 n + 1 · · · n + N − 2 n + N + 1 n Ø n − 2 Ø n + N + 1 n n + 1 · · · n + N − 2 n + N − 1       = 0 , An inte grable semi-discr eti zation of the C amassa-Holm equation 7       n − 2 n + 1 · · · n + N − 3 n + N − 1 n + N n Ø n − 2 Ø n + N n n + 1 · · · n + N − 1       = 0 ,       n − 1 n n + 2 · · · n + N − 2 n + N + 1 n + 1 Ø n − 1 Ø n + N + 1 n + 1 n n + 2 · · · n + N − 1       = 0 ,       n − 1 n n + 2 · · · n + N − 3 n + N − 1 n + N n + 1 Ø n − 1 Ø n + N n + 1 n n + 2 · · · n + N − 1       = 0 . Applying the Laplace expansion to the left-hand si de, we obtain the algebraic bilinear identities for determinants, | n − 2 , n + 1 , · · · , n + N − 2 , n + N + 1 | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | − | n , n + 1 , · · · , n + N − 2 , n + N + 1 | × | n − 2 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n − 2 , n , n + 1 , · · · , n + N − 2 | × | n + 1 , · · · , n + N − 2 , n + N − 1 , n + N + 1 | = 0 , (2.22) | n − 2 , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | × | n , n + 1 , · · · , n + N − 1 | − | n , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | × | n − 2 , n + 1 , · · · , n + N − 1 | + | n − 2 , n , n + 1 , · · · , n + N − 3 , n + N − 1 | × | n + 1 , · · · , n + N − 1 , n + N | = 0 , (2.23) | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N + 1 | × | n , n + 1 , · · · , n + N − 1 | − | n , n + 1 , · · · , n + N − 2 , n + N + 1 | × | n − 1 , n , n + 2 , · · · , n + N − 1 | + | n − 1 , n , n + 1 , · · · , n + N − 2 | × | n , n + 2 , · · · , n + N − 1 , n + N + 1 | = 0 , (2.24) | n − 1 , n , n + 2 , · · · , n + N − 3 , n + N − 1 , n + N | × | n , n + 1 , · · · , n + N − 1 | − | n , n + 1 , · · · , n + N − 3 , n + N − 1 , n + N | × | n − 1 , n , n + 2 , · · · , n + N − 1 | + | n − 1 , n , n + 1 , · · · , n + N − 3 , n + N − 1 | × | n , n + 2 , · · · , n + N − 1 , n + N | = 0 . (2.25) T aking an appropriate linear combination of these four bilinear identities, i t is re written by using (2.16) into the differ ential bilinear equation, ( ∂ s ( ∂ y − 2 c ∂ x + N c 2 ) − 2 ) τ n × τ n − ( ∂ y − 2 c ∂ x + N c 2 ) τ n × ∂ s τ n + ∂ t τ n − 1 × ( ∂ x − N c ) τ n + 1 − ( ∂ t ( ∂ x − N c ) − 1 ) τ n − 1 × τ n + 1 − τ n − 1 ( ∂ t ( ∂ x − N c ) − 1 ) τ n + 1 + ( ∂ x − N c ) τ n − 1 × ∂ t τ n + 1 = 0 , i.e., ( ∂ s ∂ y τ n ) τ n − ∂ y τ n ∂ s τ n − 2 c (( ∂ s ∂ x τ n ) τ n − ∂ x τ n ∂ s τ n ) − 2 τ n τ n − ( ∂ t ∂ x τ n + 1 ) τ n − 1 + ∂ x τ n + 1 ∂ t τ n − 1 + ∂ t τ n + 1 ∂ x τ n − 1 − τ n + 1 ∂ t ∂ x τ n − 1 + 2 τ n + 1 τ n − 1 = 0 , which leads to the fourth equation in (2.1). An inte grable semi-discr eti zation of the C amassa-Holm equation 8 Theor em 2.2. Bilinear e quations          −  1 2 D t D x − 1  f · f = gh , 2 c f f = ( D x + 2 c ) g · h , − 2 f f = ( D t D x + 2 cD t − 2 ) g · h , (2.26) have a determinant solution f = τ 0 , g = τ 1 , h = τ − 1 , τ n =          ψ ( n ) 1 ψ ( n + 1 ) 1 · · · ψ ( n + N − 1 ) 1 ψ ( n ) 2 ψ ( n + 1 ) 2 · · · ψ ( n + N − 1 ) 2 . . . . . . . . . ψ ( n ) N ψ ( n + 1 ) N · · · ψ ( n + N − 1 ) N          , wher e ψ ( n ) i = a i , 1 ( p i − c ) n e ξ i + a i , 2 ( − p i − c ) n e η i , ξ i = p i x + 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = − p i x − 1 p i + c t + 1 ( p i + c ) 2 s + η i 0 . Pr oof. In t he pre vious Lemma, apply the 2-reduction condition q i = − p i . This condition giv es a constraint D y = 0 , into bilinear equations. Thus we have                   1 2 D t D x − 1  f · f = − gh , cD t D x f · f = D x g · h , 1 2 D s D x f · f = D t g · h , ( − cD s D x − 2 ) f · f = ( D t D x − 2 ) g · h . (2.27) After simple manipulatio ns, we have          −  1 2 D t D x − 1  f · f = gh , 2 c f f = ( D x + 2 c ) g · h , − 2 f f = ( D t D x + 2 cD t − 2 ) g · h . (2.28) Let us consider a determinant solutio n. When we apply the 2-reduction condition q i = − p i , An inte grable semi-discr eti zation of the C amassa-Holm equation 9 on the determinant solution in the pre vious Lemma, we will hav e τ n =          ψ ( n ) 1 ψ ( n + 1 ) 1 · · · ψ ( n + N − 1 ) 1 ψ ( n ) 2 ψ ( n + 1 ) 2 · · · ψ ( n + N − 1 ) 2 . . . . . . . . . ψ ( n ) N ψ ( n + 1 ) N · · · ψ ( n + N − 1 ) N          , where ψ ( n ) i = a i , 1 ( p i − c ) n e ξ i + a i , 2 ( − p i − c ) n e η i , ξ i = p i x + 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = − p i x − 1 p i + c t + 1 ( p i + c ) 2 s + η i 0 . Thus the theorem was prov ed. Theor em 2.3. The C H equation ( ∂ T + w ∂ X )( w X X − w ) + 2 w X  w X X − w − κ 2  = 0 , i.e., w T + 2 κ 2 w X − w T X X + 3 ww X = 2 w X w X X + ww X X X , wher e κ 2 = 1 / c, is decomposed into bilinear equations          −  1 2 D t D x − 1  f · f = gh , 2 c f f = ( D x + 2 c ) g · h , − 2 f f = ( D t D x + 2 cD t − 2 ) g · h , (2.29) thr oug h the hodogr aph transfor mation ( X = 2 cx + log g h , T = t , and the dependent variable transformation w =  log g h  t . Pr oof. Consi der the dependent va riable transformation u = g f , v = h f . From bilinear equations (2.29), we obtain        − (( log f ) x t − 1 ) = u v , 2 c = ( D x + 2 c ) u · v , − 2 = ( D t D x + ( 2 l og f ) x t + 2 cD t − 2 ) u · v . (2.30) An inte grable semi-discr eti zation of the C amassa-Holm equation 10 Eliminati ng ( log f ) x t from the third equation using the first equation in (2.30), we obtain ( 2 c = ( D x + 2 c ) u · v , − 2 = ( D t D x + 2 cD t − 2 uv ) u · v . (2.31) Thus we ha ve    2 c =  log u v  x + 2 c  uv , − 2 =  ( log uv ) x t +  log u v  x  log u v  t + 2 c  log u v  t − 2 uv  uv . (2.32) Eliminati ng  log u v  x + 2 c from the second equation using the first equation in (2.32), we hav e      2 c =  log u v  x + 2 c  uv , − 2 =  ( log uv ) x t + 2 c uv  log u v  t − 2 uv  uv . Introducing new dependent variables φ = u v = g h , ρ = uv = gh f 2 , we ha ve    2 c ρ = ( log φ ) x + 2 c , − 2 = ρ ( l og ρ ) x t + 2 c ( log φ ) t − 2 ρ 2 . (2.33) Let w = ( log φ ) t =  log g h  t . From the first equation in (2.33), we obtain − ( log ρ ) t = ( log φ ) x t 2 c + ( log φ ) x = ρ 2 c ( log φ ) x t = ρ 2 c w x . Thus we ha ve      − ( log ρ ) t = ρ 2 c w x , ρ 2 c ( log ρ ) x t + w + 1 c = 1 c ρ 2 . (2.34) Replacing ( log ρ ) t in the second equation by − ρ 2 c w x using the first equation, we obtain      − ( log ρ ) t = ρ 2 c w x , − ρ 2 c  ρ 2 c w x  x + w + 1 c = 1 c ρ 2 . Consider the hodograph transformation ( X = 2 cx + log φ , T = t . An inte grable semi-discr eti zation of the C amassa-Holm equation 11 Then we hav e ∂ X ∂ x = 2 c + ( log φ ) x = 2 c +  log g h  x = 2 c f 2 gh = 2 c ρ , ∂ X ∂ t = ( log φ ) t =  log g h  t = w ,    ∂ x = 2 c ρ ∂ X , ∂ t = ∂ T + w ∂ X . Using these results, we obtain    − ( ∂ T + w ∂ X ) l og ρ = w X , − w X X + w + 1 c = 1 c ρ 2 . (2.35) Eliminati ng ρ from the first equation using the second equation in eqs.(2.35), we obtain ( ∂ T + w ∂ X ) l og  − w X X + w + 1 c  = − 2 w X . Thus we finally obtain the CH equation ( ∂ T + w ∂ X )( w X X − w ) + 2 w X  w X X − w − 1 c  = 0 . The theorem was approv ed. Furthermore we hav e t he follo wing theorem. Corollary 2.4. The CH equation has a determinant form of N -sol iton solutions. Pr oof. From T heorem 2.2 and 2.3, the proof is obvious. 3. A semi-discrete Camassa-Holm equation Lemma 3.1. Bi linear equations                         1 − ac a D t − 1  f ( k + 1 , l ) · f ( k , l ) = − g ( k + 1 , l ) h ( k , l ) ,  1 − bc b D t − 1  f ( k , l + 1 ) · f ( k , l ) = − g ( k , l + 1 ) h ( k , l ) ,  1 − ac a D s − D t  f ( k + 1 , l ) · f ( k , l ) = D t g ( k + 1 , l ) · h ( k , l ) ,  1 − bc b D s − D t  f ( k , l + 1 ) · f ( k , l ) = D t g ( k , l + 1 ) · h ( k , l ) , (3.1) have a determinant solution f ( k , l ) = τ 0 ( k , l ) , g ( k , l ) = τ 1 ( k , l ) , h ( k , l ) = τ − 1 ( k , l ) , An inte grable semi-discr eti zation of the C amassa-Holm equation 12 τ n ( k , l ) =          ψ ( n ) 1 ( k , l ) ψ ( n + 1 ) 1 ( k , l ) · · · ψ ( n + N − 1 ) 1 ( k , l ) ψ ( n ) 2 ( k , l ) ψ ( n + 1 ) 2 ( k , l ) · · · ψ ( n + N − 1 ) 2 ( k , l ) . . . . . . . . . ψ ( n ) N ( k , l ) ψ ( n + 1 ) N ( k , l ) · · · ψ ( n + N − 1 ) N ( k , l )          , wher e ψ ( n ) i ( k , l ) = a i , 1 ( p i − c ) n ( 1 − a p i ) − k ( 1 − b p i ) − l e ξ i + a i , 2 ( q i − c ) n ( 1 − aq i ) − k ( 1 − bq i ) − l e η i , ξ i = 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = 1 q i − c t + 1 ( q i − c ) 2 s + η i 0 . Pr oof. Consi der the following Casorati determinant solutio n, τ n ( k , l ) =          ψ ( n ) 1 ( k , l ) ψ ( n + 1 ) 1 ( k , l ) · · · ψ ( n + N − 1 ) 1 ( k , l ) ψ ( n ) 2 ( k , l ) ψ ( n + 1 ) 2 ( k , l ) · · · ψ ( n + N − 1 ) 2 ( k , l ) . . . . . . . . . ψ ( n ) N ( k , l ) ψ ( n + 1 ) N ( k , l ) · · · ψ ( n + N − 1 ) N ( k , l )          , where ψ ( n ) i ’ s a re a rbitrary functions of tw o c ontinuous independent va riables, t and s , and two discrete independent variables, k and l , which satisfy the linear disp ersion relations, ∆ k ψ ( n ) i = ψ ( n + 1 ) i + c ψ ( n ) i , (3.2) ∆ l ψ ( n ) i = ψ ( n + 1 ) i + c ψ ( n ) i , ( 3.3) ∂ t ψ ( n ) i = ψ ( n − 1 ) i , (3.4) ∂ s ψ ( n ) i = ψ ( n − 2 ) i , (3.5) where ∆ k and ∆ l are defined as ∆ k ψ ( k , l ) = ψ ( k , l ) − ψ ( k − 1 , l ) a and ∆ l ψ ( k , l ) = ψ ( k , l ) − ψ ( k , l − 1 ) b , respectiv ely . Thus we can choose ψ ( n ) i as follows: ψ ( n ) i ( k , l ) = ( p i − c ) n ( 1 − a p i ) − k ( 1 − b p i ) − l e ξ i + ( q i − c ) n ( 1 − aq i ) − k ( 1 − bq i ) − l e η i , ξ i = 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = 1 q i − c t + 1 ( q i − c ) 2 s + η i 0 . W e use the following notation for simpl icity: | n 1 k 1 , l 1 , n 2 k 2 , l 2 , · · · , n N k N , l N | =          ψ ( n 1 ) 1 ( k 1 , l 1 ) ψ ( n 2 ) 1 ( k 2 , l 2 ) · · · ψ ( n N ) 1 ( k N , l N ) ψ ( n 1 ) 2 ( k 1 , l 1 ) ψ ( n 2 ) 2 ( k 2 , l 2 ) · · · ψ ( n N ) 2 ( k N , l N ) . . . . . . . . . ψ ( n 1 ) N ( k 1 , l 1 ) ψ ( n 2 ) N ( k 2 , l 2 ) · · · ψ ( n N ) N ( k N , l N )          . An inte grable semi-discr eti zation of the C amassa-Holm equation 13 In this notation, τ n ( k , l ) is re writt en as τ n ( k , l ) = | n k , l , n + 1 k , l , · · · , n + N − 1 k , l | , or suppressing the index k and l , τ n ( k , l ) = | n , n + 1 , · · · , n + N − 1 | . The first equation of eqs. (3.1) By using t he Casoratian technique developed in [37] and [38], it is po ssible to derive th e following differential and dif ference formulas for the τ function, τ n − 1 ( k , l ) = | n − 1 , n , · · · , n + N − 2 | , (3.6) τ n ( k + 1 , l ) = 1 ( 1 − ac ) N − 2 | n k + 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | , = 1 ( 1 − ac ) N − 1 | n , · · · , n + N − 2 , n + N − 1 k + 1 | , (3.7) τ n + 1 ( k + 1 , l ) = 1 ( 1 − ac ) N − 1 | n + 1 , · · · , n + N − 1 , n + N k + 1 | , = 1 a ( 1 − ac ) N − 2 | n + 1 , · · · , n + N − 1 , n + N − 1 k + 1 | , (3.8) ∂ t τ n = | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | , (3.9) 1 − ac a τ n ( k + 1 , l ) = 1 a ( 1 − ac ) N − 2 | n , · · · , n + N − 2 , n + N − 1 k + 1 | , (3.10) 1 − ac a ∂ t τ n ( k + 1 , l ) − τ n ( k + 1 , l ) = 1 − ac a ( 1 − ac ) N − 2 | n − 1 k + 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | − | n k + 1 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | = 1 a ( 1 − ac ) N − 2 | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | + 1 ( 1 − ac ) N − 2 | n k + 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | − 1 ( 1 − ac ) N − 2 | n k + 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | = 1 a ( 1 − ac ) N − 2 | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | , (3.11) Let us introduce an identity for 2 N × 2 N determinant,       n + 1 k · · · n + N − 2 k n + N − 1 k n + N − 1 k + 1 n − 1 k n k Ø Ø n + N − 1 k n + N − 1 k + 1 n − 1 k n k n + 1 k · · · n + N − 2 k       = 0 . Applying the Laplace expansion to the left-hand side, we obt ain the algebraic bilinear identity for determinants, | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | × | n , n + 1 , · · · , n + N − 2 , n + N − 1 | An inte grable semi-discr eti zation of the C amassa-Holm equation 14 −| n , n + 1 , · · · , n + N − 2 , n + N − 1 k + 1 | × | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n + 1 , · · · , n + N − 2 , n + N − 1 , n + N − 1 k + 1 | × | n − 1 , n , n + 1 , · · · , n + N − 2 | = 0 , (3.12) which is re written into the dif ferential bilinear equation,  1 − ac a ∂ t τ n ( k + 1 , l ) − τ n ( k + 1 , l )  τ n ( k , l ) − 1 − ac a τ n ( k + 1 , l ) ∂ t τ n ( k , l ) + τ n + 1 ( k + 1 , l ) τ n − 1 ( k , l ) = 0 . Setting n = 0, f = τ 0 , g = τ 1 , h = τ − 1 , the above bilinear equatio n leads to the first equati on in (3.1). The second equation of eqs. (3.1) The proof is similar to the proof of the first equation. The third equation of eqs. (3.1) W e use the following differe ntial and differenc e formulas for the τ function τ n − 1 ( k , l ) = | n − 1 , n , · · · , n + N − 2 | , (3.13) τ n ( k + 1 , l ) = | n k + 1 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | , (3.14) τ n + 1 ( k + 1 , l ) = | n + 1 k + 1 , · · · , n + N − 1 k + 1 , n + N k + 1 | , = − 1 a | n + 1 k + 1 , · · · , n + N − 1 k + 1 , n + N − 1 | , (3.15) ∂ t τ n ( k , l ) = | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | , (3.16) 1 − ac a τ n ( k + 1 , l ) = 1 a | n , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | = 1 a | n k + 1 , n + 1 , n + 2 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | , (3.17) ∂ t τ n ( k + 1 , l ) = | n − 1 k + 1 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | , (3.18) ∂ t τ n + 1 ( k + 1 , l ) = | n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 , n + N k + 1 | (3.19) = − 1 a | n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 , n + N − 1 | , (3.20) ∂ t τ n − 1 ( k , l ) = | n − 2 , n , · · · , n + N − 3 , n + N − 2 | , (3.21) ∂ s τ n ( k , l ) = | n − 2 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n , n − 1 , n + 2 , · · · , n + N − 2 , n + N − 1 | , (3.22) 1 − ac a ∂ s τ n ( k + 1 , l ) = 1 − ac a | n − 2 k + 1 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | + 1 − ac a | n k + 1 , n − 1 k + 1 , n + 2 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | , = | n − 1 k + 1 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | + 1 a | n − 2 , n + 1 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | − 1 a | n − 1 , n k + 1 , n + 2 k + 1 , · · · , n + N − 2 k + 1 , n + N − 1 k + 1 | , (3.23) An inte grable semi-discr eti zation of the C amassa-Holm equation 15 Let us introduce following identiti es for 2 N × 2 N determinant ,      n − 1 k n k + 1 n + 2 k + 1 · · · n + N − 1 k + 1 n k n + 1 k Ø n + N − 1 k n − 1 k Ø n k n + 1 k n + 2 k · · · n + N − 2 k n + N − 1 k      = 0 , and       n − 2 k n + 1 k + 1 · · · n + N − 1 k + 1 n k Ø n + N − 1 k n − 2 k Ø n k n + 1 k · · · n + N − 2 k n + N − 1 k       = 0 . Applying the Laplace expansion to the left-hand si de, we obtain the algebraic bilinear identities for determinants, we obtain the algebraic bilinear identities for determinants, | n − 1 , n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 | × | n , n + 1 , n + 2 , · · · , n + N − 2 , n + N − 1 | −| n , n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 | × | n − 1 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n + 1 , n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 | × | n − 1 , n , n + 2 , · · · , n + N − 2 , n + N − 1 | −| n k + 1 , n + 2 k + 1 , · · · , n + N − 1 k + 1 , n + N − 1 | × | n − 1 , n , n + 1 , n + 2 , · · · , n + N − 2 | = 0 , (3.24) and | n − 2 , n + 1 k + 1 , · · · , n + N − 1 k + 1 | × | n , n + 1 , n + 2 , · · · , n + N − 2 , n + N − 1 | −| n , n + 1 k + 1 , · · · , n + N − 1 k + 1 | × | n − 2 , n + 1 , · · · , n + N − 2 , n + N − 1 | + | n + 1 k + 1 , · · · , n + N − 1 k + 1 , n + N − 1 | × | n − 2 , n , n + 1 , · · · , n + N − 2 | = 0 , (3.25) By dividing the differenc e of above two equations by a , we arriv e at a bilinear equation 1 − ac a ∂ s τ n ( k + 1 , l ) τ n ( k , l ) − 1 − ac a τ n ( k + 1 , l ) ∂ s τ n ( k , l ) − ∂ t τ n ( k + 1 , l ) τ n ( k , l ) + τ n ( k + 1 , l ) ∂ t τ n ( k , l ) − ∂ t τ n + 1 ( k + 1 , l ) τ n − 1 ( k , l ) + τ n + 1 ( k + 1 , l ) ∂ t τ n − 1 ( k , l ) = 0 . Setting n = 0, f = τ 0 , g = τ 1 , h = τ − 1 , the abov e bilinear equation leads to the third equation in (3.1). The f ourth equation of eqs. (3.1) The proof is similar to the proof of the third equation. Theor em 3.2. Bilinear e quations                         1 − ac a D t − 1  f k + 1 · f k = − g k + 1 h k ,  1 + ac a D t − 1  f k + 1 · f k = − g k h k + 1 ,  1 − ac a D s − D t  f k + 1 · f k = D t g k + 1 · h k ,  1 + ac a D s + D t  f k + 1 · f k = D t g k · h k + 1 , (3.26) An inte grable semi-discr eti zation of the C amassa-Holm equation 16 have a determinant solution f ( k , l ) = τ 0 ( k , 0 ) , g ( k , l ) = τ 1 ( k , 0 ) , h ( k , l ) = τ − 1 ( k , 0 ) , τ n ( k , l ) =          ψ ( n ) 1 ( k , l ) ψ ( n + 1 ) 1 ( k , l ) · · · ψ ( n + N − 1 ) 1 ( k , l ) ψ ( n ) 2 ( k , l ) ψ ( n + 1 ) 2 ( k , l ) · · · ψ ( n + N − 1 ) 2 ( k , l ) . . . . . . . . . ψ ( n ) N ( k , l ) ψ ( n + 1 ) N ( k , l ) · · · ψ ( n + N − 1 ) N ( k , l )          wher e ψ ( n ) i ( k , l ) = a i , 1 ( p i − c ) n ( 1 − a p i ) − k ( 1 + a p i ) − l e ξ i + a i , 2 ( − p i − c ) n ( 1 + a p i ) − k ( 1 − a p i ) − l e η i , ξ i = 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = − 1 p i + c t + 1 ( p i + c ) 2 s + η i 0 . Pr oof. App lying the 2-reduction condition q i = − p i , b = − a , the τ function satisfies τ n ( k + 1 , l + 1 ) = 1 N ∏ i = 1 ( 1 − a p i )( 1 + a p i ) τ n ( k , l ) . Let f k = f ( k , 0 ) , g k = g ( k , 0 ) , h k = h ( k , 0 ) , Then                         1 − ac a D t − 1  f k + 1 · f k = − g k + 1 h k ,  1 + ac a D t − 1  f k + 1 · f k = − g k h k + 1 ,  1 − ac a D s − D t  f k + 1 · f k = D t g k + 1 · h k ,  1 + ac a D s + D t  f k + 1 · f k = D t g k · h k + 1 . Let us consider a determinant solutio n. When we apply the 2-reduction condition q i = − p i , b = − a , on the determinant solution in the pre vious Lemma, we will hav e τ n ( k , l ) =          ψ ( n ) 1 ( k , l ) ψ ( n + 1 ) 1 ( k , l ) · · · ψ ( n + N − 1 ) 1 ( k , l ) ψ ( n ) 2 ( k , l ) ψ ( n + 1 ) 2 ( k , l ) · · · ψ ( n + N − 1 ) 2 ( k , l ) . . . . . . . . . ψ ( n ) N ( k , l ) ψ ( n + 1 ) N ( k , l ) · · · ψ ( n + N − 1 ) N ( k , l )          An inte grable semi-discr eti zation of the C amassa-Holm equation 17 where ψ ( n ) i ( k , l ) = a i , 1 ( p i − c ) n ( 1 − a p i ) − k ( 1 + a p i ) − l e ξ i + a i , 2 ( − p i − c ) n ( 1 + a p i ) − k ( 1 − a p i ) − l e η i , ξ i = 1 p i − c t + 1 ( p i − c ) 2 s + ξ i 0 , η i = − 1 p i + c t + 1 ( p i + c ) 2 s + η i 0 . Thus the theorem was prov ed. W e propose a semi-discrete analogue of the CH equation          ∆ 2 w k = 1 δ k M δ k M w k + κ 2 δ k κ 4 δ 2 k − 4 a 2 κ 4 − a 2 ! , ∂ t δ k =  1 − δ 2 k 4  δ k ∆ w k , where a dif ference operator ∆ and an a verage operator M are defined as ∆ F k = F k + 1 − F k δ k , M F k = F k + 1 + F k 2 . Theor em 3.3. The semi-discr ete CH equation          ∆ 2 w k = 1 δ k M δ k M w k + κ 2 δ k κ 4 δ 2 k − 4 a 2 κ 4 − a 2 ! , ∂ t δ k =  1 − δ 2 k 4  δ k ∆ w k , (3.27) is decomposed into bilinear equations                         1 − ac a D t − 1  f k + 1 · f k = − g k + 1 h k ,  1 + ac a D t − 1  f k + 1 · f k = − g k h k + 1 ,  1 − ac a D s − D t  f k + 1 · f k = D t g k + 1 · h k ,  1 + ac a D s + D t  f k + 1 · f k = D t g k · h k + 1 , (3.28) thr oug h the transformation δ k = 4 a ( κ 2 + a ) g k + 1 h k f k + 1 f k + ( κ 2 − a ) g k h k + 1 f k + 1 f k , and w k =  log g k h k  t . wher e κ 2 = 1 / c . An inte grable semi-discr eti zation of the C amassa-Holm equation 18 Pr oof. Let us start fr om bilinear equations (3.28). By simpl e manipulatio n of eqs. ( 3.28), we obtain          − 2  1 a D t − 1  f k + 1 · f k = g k + 1 h k + g k h k + 1 , 2 ac f k + 1 f k = ( 1 + a c ) g k + 1 h k − ( 1 − ac ) g k h k + 1 , − 2 a f k + 1 f k = (( 1 + ac ) D t − a ) g k + 1 · h k − (( 1 − ac ) D t + a ) g k · h k + 1 . Let u k = g k f k , v k = h k f k . Then we hav e                      − 2  1 a  log f k + 1 f k  t − 1  = u k + 1 v k + u k v k + 1 , 2 ac = ( 1 + ac ) u k + 1 v k − ( 1 − ac ) u k v k + 1 , − 2 a =  ( 1 + ac )  log u k + 1 v k  t +  log f k + 1 f k  t  − a  u k + 1 v k −  ( 1 − ac )  log u k v k + 1  t −  log f k + 1 f k  t  + a  u k v k + 1 . (3.29) Substitutin g the first equation into the thi rd equation in eqs.(3.29), we obtain                2 ac = ( 1 + ac ) u k + 1 v k − ( 1 − ac ) u k v k + 1 , − 2 a = ( 1 + ac ) u k + 1 v k  log u k + 1 v k  t − ( 1 − ac ) u k v k + 1  log u k v k + 1  t − a 2 ( u k + 1 v k + u k v k + 1 )(( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 ) + a 2 c ( u k + 1 v k − u k v k + 1 ) . Simplifying the above equations, we hav e              2 ac = ( 1 + ac ) u k + 1 v k − ( 1 − ac ) u k v k + 1 , − 2 a = 1 2 (( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 )  log u k + 1 v k + 1 u k v k  t + ac  log u k + 1 u k v k + 1 v k  t − 2 au k + 1 v k + 1 u k v k , (3.30) Let φ k = u k v k = g k h k , ρ k = u k v k = g k h k f 2 k . From the first equation in (3.30), we obtain 2 ac u k v k + 1 = ( 1 + a c ) φ k + 1 φ k − ( 1 − ac ) , 2 ac u k + 1 v k = ( 1 + a c ) − ( 1 − ac ) φ k φ k + 1 . Multiply ing these two equations, we obtain ( 2 ac ) 2 ρ k + 1 ρ k =  ( 1 + ac ) φ k + 1 φ k − ( 1 − ac )   ( 1 + ac ) − ( 1 − ac ) φ k φ k + 1  . An inte grable semi-discr eti zation of the C amassa-Holm equation 19 From the second equation of (3.30), we obtain − 2 a = 1 2 (( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 )  log ρ k + 1 ρ k  t + ac ( l og φ k + 1 φ k ) t − 2 a ρ k + 1 ρ k . Thus we ha ve ( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 4 ac  log ρ k + 1 ρ k  t + 1 2 ( log φ k + 1 φ k ) t + 1 c = 1 c ρ k + 1 ρ k . Let us define a lattice parameter δ k = 4 ac ( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 . Then δ k = 2 ( 1 + ac ) u k + 1 v k − ( 1 − ac ) u k v k + 1 ( 1 + ac ) u k + 1 v k + ( 1 − ac ) u k v k + 1 = 2 ( 1 + ac ) g k + 1 h k − ( 1 − ac ) g k h k + 1 ( 1 + ac ) g k + 1 h k + ( 1 − ac ) g k h k + 1 = 2 ( 1 + ac ) φ k + 1 − ( 1 − ac ) φ k ( 1 + ac ) φ k + 1 + ( 1 − ac ) φ k = 2 1 + ac 1 − ac φ k + 1 φ k − 1 1 + ac 1 − ac φ k + 1 φ k + 1 . Thus we ha ve φ k + 1 φ k = 1 − ac 1 + ac 1 + δ k 2 1 − δ k 2 , where a latt ice parameter δ k is a functio n d epending on ( k , t ) . The lattice parameter corresponds to ∂ X ∂ x = 2 c ρ = 2 c + ( log φ ) x in the continuous case. At time t , X = X 0 + K − 1 ∑ k = 0 δ k is a x -coordinate of the k -th lattice point. Thus we have the following system:                        ( 2 ac ) 2 ρ k + 1 ρ k =  ( 1 + ac ) φ k + 1 φ k − ( 1 − ac )   ( 1 + ac ) − ( 1 − ac ) φ k φ k + 1  , 1 δ k  log ρ k + 1 ρ k  t + 1 2 ( log φ k + 1 φ k ) t + 1 c = 1 c ρ k + 1 ρ k , δ k = 2 1 + ac 1 − ac φ k + 1 φ k − 1 1 + ac 1 − ac φ k + 1 φ k + 1 . (3.31) Let w k = ( log φ k ) t =  log g k h k  t . From the first equation in (3.31), we obtain − ( log ρ k + 1 ρ k ) t = 1 + ac 1 − ac  φ k + 1 φ k  t 1 + ac 1 − ac φ k + 1 φ k − 1 + −  φ k φ k + 1  t 1 + ac 1 − ac − φ k φ k + 1 An inte grable semi-discr eti zation of the C amassa-Holm equation 20 =     1 + ac 1 − ac φ k + 1 φ k 1 + ac 1 − ac φ k + 1 φ k − 1 + φ k φ k + 1 1 + ac 1 − ac − φ k φ k + 1      log φ k + 1 φ k  t = 1 + ac 1 − ac φ k + 1 φ k + 1 1 + ac 1 − ac φ k + 1 φ k − 1  log φ k + 1 φ k  t = 2 δ k ( w k + 1 − w k ) . From the third equation in (3.31), we obtain 2 δ k + 1 = 2 1 + ac 1 − ac φ k + 1 φ k 1 + ac 1 − ac φ k + 1 φ k − 1 , 2 δ k − 1 = 2 1 + ac 1 − ac φ k + 1 φ k − 1 . Multiply ing these two equations, we obtain 4 δ 2 k − 1 = 4  1 + ac 1 − ac φ k + 1 φ k − 1   1 − 1 − ac 1 + ac φ k φ k + 1  = 4 ( 1 − ac )( 1 + ac ) ( 2 ac ) 2 ρ k + 1 ρ k =  1 a 2 c 2 − 1  ρ k + 1 ρ k . Thus we ha ve a system                − ( log ρ k + 1 ρ k ) t = 2 w k + 1 − w k δ k , 1 δ k  log ρ k + 1 ρ k  t + w k + 1 + w k 2 + 1 c = 1 c ρ k + 1 ρ k , 4 δ 2 k − 1 =  1 a 2 c 2 − 1  ρ k + 1 ρ k . Let r k = log ρ k = log g k h k f 2 k . Then we hav e                    − ∂ t ( r k + 1 + r k ) = 2 w k + 1 − w k δ k , 1 δ k ∂ t ( r k + 1 − r k ) + w k + 1 + w k 2 + 1 c 1 − 4 a 2 c 2 δ 2 k 1 − a 2 c 2 = 0 , 4 δ 2 k − 1 =  1 a 2 c 2 − 1  e r k + 1 + r k . Let r ′ k = ∂ t r k . An inte grable semi-discr eti zation of the C amassa-Holm equation 21 Then                    − ( r ′ k + 1 + r ′ k ) = 2 w k + 1 − w k δ k , 1 δ k ( r ′ k + 1 − r ′ k ) + w k + 1 + w k 2 + 1 c 1 − 4 a 2 c 2 δ 2 k 1 − a 2 c 2 = 0 , ∂ t δ k =  1 − δ 2 k 4  ( w k + 1 − w k ) . Eliminati ng r ′ k , we hav e                          − 2  w k + 1 − w k δ k − w k − w k − 1 δ k − 1  + δ k w k + 1 + w k 2 + δ k c 1 − 4 a 2 c 2 δ 2 k 1 − a 2 c 2 + δ k − 1 w k + w k − 1 2 + δ k − 1 c 1 − 4 a 2 c 2 δ 2 k − 1 1 − a 2 c 2 = 0 , ∂ t δ k =  1 − δ 2 k 4  ( w k + 1 − w k ) . (3.32) This is a semi-discrete Camassa-Holm equation. N ote that the lattice parameter depends on the time and space. Diffe rentiating the first equation in (3.32) with respect to t , we obtain − 2  w k + 1 − w k δ k − w k − w k − 1 δ k − 1  t + δ k ∂ t w k + 1 + ∂ t w k 2 + δ k − 1 ∂ t w k + ∂ t w k − 1 2 +  1 − δ 2 k 4  ( w k + 1 − w k )      w k + 1 + w k 2 + 1 c 1 + 4 a 2 c 2 δ 2 k 1 − a 2 c 2      + 1 − δ 2 k − 1 4 ! ( w k − w k − 1 )      w k + w k − 1 2 + 1 c 1 + 4 a 2 c 2 δ 2 k − 1 1 − a 2 c 2      = 0 , T aking a continuous limit, this leads to the CH equation. Then we hav e the follo wing theorem. Corollary 3.4. The semi-discr ete CH equation h as a determinant form of N -soliton solutions . Pr oof. From T heorem 3.2 and 3.3, the proof is obvious. 4. Peakons, solitons and cuspons in the semi-discrete CH equ ation 4.1. P eakon limit in the semi-discr ete Camassa-Holm equation It is kn own that the CH equation has a peakon s olution in the limit κ → 0. Here we consider a peakon limit in the semi-discrete CH equation (3.27). An inte grable semi-discr eti zation of the C amassa-Holm equation 22 In the semi-discrete CH equation (3.27), consider the limit κ → 0 , κ 2 δ k → δ ( k ) , where δ ( k ) is a di rac delta function. In this limit, the first equati on of the semi-discrete CH equation (3.27) leads to ∂ 2 X w − w = δ ( X − cT ) . This diffe rential equation has a solution w = N ∑ i = 1 A i ( T ) exp | X − cT | , which is a form of the peakon solution. Thus if we consider v ery small κ with very small δ k at some points k , the solutions of the semi-discrete CH equation tend to the peakon solutions of the CH equation. 4.2. One solit on/cuspon solution From the determinant formu la with a i , 1 / a i , 2 = ± 1, the τ -functions for one solit on/cuspon solution are g ∝ 1 ±  c − p c + p  e θ , h ∝ 1 ±  c + p c − p  e θ , (4.1) with θ = 2 p ( x − vt − x 0 ) , v = 1 / ( c 2 − p 2 ) where c = 1 / κ 2 > 0. Thi s leads to a solution w ( x , t ) = 4 p 2 cv ( c 2 + p 2 ) ± ( c 2 − p 2 ) cos h θ , (4.2) X = 2 cx + log  g h  , T = t , (4.3) where th e posit iv e case in Eq .(4.2) stands for one smooth soliton s olution when p < c , while the negativ e case in E q.(4.2) stands for one-cuspon solution when p > c . Otherwise, th e solution is singular . Thus Eq.(4.2) for nonsingul ar cases can be e xpressed to w ( x , t ) = 4 p 2 cv ( c 2 + p 2 ) + | c 2 − p 2 | cos h θ . (4.4) Similarly , for the semi-discrete ca se, we have g k ∝ 1 +     c − p c + p      1 + a p 1 − a p  k e θ , h k ∝ 1 +     c + p c − p      1 + a p 1 − a p  k e θ , (4.5) with θ = − 2 pv ( t + x 0 ) , resulting in a solution of the form w k ( t ) = 4 p 2 cv ( c 2 + p 2 ) + | c 2 − p 2 |   1 + a p 1 − a p  − k e − θ +  1 + a p 1 − a p  k e θ  , (4.6) in conjunction wit h a transform between an equidistance mesh ( a ) and a non-equidistance mesh δ k = 2 ( 1 + ac ) g k + 1 h k − ( 1 − ac ) g k h k + 1 ( 1 + ac ) g k + 1 h k + ( 1 − ac ) g k h k + 1 . (4.7) Eq. (4.6) corresponds to 1-soli ton solution when p < c , 1-cuspon solution when p > c . An inte grable semi-discr eti zation of the C amassa-Holm equation 23 4.3. T wo soliton/ cuspon solut ions From the determinant formula with a i , 1 / a i , 2 = ± 1, the τ -functions for two soliton/ cuspon solution are g ∝ 1 +     c 1 − p 1 c 1 + p 1     e θ 1 +     c 2 − p 2 c 2 + p 2     e θ 2 +     ( c 1 − p 1 )( c 2 − p 2 ) ( c 1 + p 1 )( c 2 + p 2 )      p 1 − p 2 p 1 + p 2  2 e θ 1 + θ 2 , h ∝ 1 +     c 1 + p 1 c 1 − p 1     e θ 1 +     c 2 + p 2 c 2 − p 2     e θ 2 +     ( c 1 + p 1 )( c 2 + p 2 ) ( c 1 − p 1 )( c 2 − p 2 )      p 1 − p 2 p 1 + p 2  2 e θ 1 + θ 2 , with θ 1 = 2 p 1 ( x − v 1 t − x 10 ) , θ 2 = 2 p 2 ( x − v 2 t − x 20 ) , v 1 = 1 / ( c 2 1 − p 2 1 ) , v 2 = 1 / ( c 2 2 − p 2 2 ) . The parametric solution can be calculated through w ( x , t ) =  log g h  t , X = 2 cx + log  g h  , T = t , (4.8) whose form is complicated and is omitt ed her e. Note th at the abov e e xpression includes two- soliton solution ( p 1 < c 1 , p 2 < c 2 ), two-cuspon soluti on ( p 1 > c 1 , p 2 > c 2 ), or soliton -cuspon solution ( p 1 < c 1 , p 2 > c 2 ). Similarly , for the semi-discrete ca se, we have g k ∝ 1 +     c 1 − p 1 c 1 + p 1      1 + a p 1 1 − a p 1  k e θ 1 +     c 2 − p 2 c 2 + p 2      1 + a p 2 1 − a p 2  k e θ 2 +     ( c 1 − p 1 )( c 2 − p 2 ) ( c 1 + p 1 )( c 2 + p 2 )      p 1 − p 2 p 1 + p 2  2  1 + a p 1 1 − a p 1  k  1 + a p 2 1 − a p 2  k e θ 1 + θ 2 , h k ∝ 1 +     c 1 + p 1 c 1 − p 1      1 + a p 1 1 − a p 1  k e θ 1 +     c 2 + p 2 c 2 − p 2      1 + a p 2 1 − a p 2  k e θ 2 +     ( c 1 + p 1 )( c 2 + p 2 ) ( c 1 − p 1 )( c 2 − p 2 )      p 1 − p 2 p 1 + p 2  2  1 + a p 1 1 − a p 1  k  1 + a p 2 1 − a p 2  k e θ 1 + θ 2 , with θ 1 = 2 p 1 ( − v 1 t − x 10 ) , θ 2 = 2 p 2 ( − v 2 t − x 20 ) . The soluti on can be calculated through w k ( t ) =  log g k h k  t , (4.9) with a transform δ k = 2 ( 1 + ac ) g k + 1 h k − ( 1 − ac ) g k h k + 1 ( 1 + ac ) g k + 1 h k + ( 1 − ac ) g k h k + 1 . (4.10) The form is complicated and is omitted here. 5. Numerical computations In this section, severa l examples will be ill ustrated to show the integrable semi-discretization of the CH equation is a powerful scheme for the numerical s olutions of the CH equation. They include (1) Propagatio n of one-cuspon solutio n; (2) Interaction of two-cuspon solutions; (3) An inte grable semi-discr eti zation of the C amassa-Holm equation 24 Head-on collis ion of soli ton-cuspon. In actual computatio ns, the initial mesh spacing δ k , is assigned by δ k = 2 ( 1 + ac ) φ k + 1 − ( 1 − ac ) φ k ( 1 + ac ) φ k + 1 + ( 1 − ac ) φ k , (5.1) where φ k = g k / h k is obt ainable from the corresponding determi nant sol utions. At each tim e step, from the second equation in (3.27), the ev olut ion of δ k can be exactly ca lculated by δ n + 1 k = 2 c n k e ( w n k + 1 − w n k ) − 1 c n k e ( w n k + 1 − w n k ) + 1 , (5.2) with c n k = ( 2 + δ n k ) / ( 2 − δ n k ) . Then, the solu tion w n + 1 k is easil y comp uted by s olving a tridiagonal linear syst em based on the first equation of the scheme. Therefore, the computation cost is l ess than other traditional numerical methods . Furthermore, we com ment that the mesh spacing δ k is drive n automatically by the solution during the numerical computation. Therefore, we would like to call it the self-adaptive method . Example 1: 1-cuspon propagation. The p arameters taken for the 1-cuspon solut ion are p = 10 . 98, c = 10 . 0. The number of grid is t aken as 100 in an interval of widt h of 4 in the x -domain, which impli es a mesh si ze of h = 0 . 04. Howe ver , through the hodograph transformation, this correspon ds to an interval o f width 74 . 34 in the X -domain, which implies an av erage mesh size of 0 . 74 34. In subsequent examples except for Example 4, t he grid number is fixed to be 100. The t ime step size is taken as ∆ t = 0 . 0004. Fig. 1(a) sho ws the initial conditi on. Figs. 1 (b)-(c) display t he numerical sol utions (solid line) and exact solut ions (dotted l ine) at t = 2 , 4, respectively . The L ∞ norm are 0 . 0 365 at t = 2, and 0 . 0985 at t = 4. It is no ted that the numerical error is mainly due to the nu merical disp ersion. In other words, e ven after a fairly long time, the num erical solution of a cus pon preserves its shape very well except for a phase shift. The reason for the numerical dispersion is thought due to the e xpl icit method to solve the equation for the time e volution of mesh size. The detailed analysis is left for our another paper focusing on numerical solutions of the CH equation. Example 2: 2-cuspon interaction. The parameters taken for the t wo-cuspon solut ion are p 1 = 11 . 0, p 2 = 10 . 5, c = 10 . 0. Fig. 2(a) sh ows the i nitial condition, and Figs. 2(b)- 2(e) display the process o f collisi on at sev eral dif ferent times. As far as we know , what is shown here is the first numerical demonstrati on for the cusp on-cuspon interaction d ue to the singularities of cuspon solut ions. As shown in Fig. 2(e), the 2-cuspon soluti on regain th eir shapes after the collision , only resulting in a phase shift . As mentio ned in [18], the two cuspon points are alwa ys present during the collision . Example 3: Soliton-cuspon interaction. Here we s how two examples for the solito n- cuspon interaction with c = 10 . 0. In Fig. 3, we plot the interaction process b etween a solito n of p 1 = 9 . 12 and a cuspon of p 2 = 10 . 98 at se veral different times where the so liton and the cus pon hav e almost the s ame amp litude. It can b e seen t hat another si ngularity point with infinite deri vati ve ( w x ) occurs when the col lision starts ( t = 12 . 0). As col lision goes on ( t = 14 . 4 , 14 . 6 , 14 . 8), the solit on seems ’eats up’ the cuspon, and the profile looks like a complete elev ation. H owe ver , the cuspon p oint is present at all times, especially , at t = 14 . 6, the profile becomes one symmetrical hump with a cuspon point in the middle of the hump. An inte grable semi-discr eti zation of the C amassa-Holm equation 25 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 X w(X,t) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 X w(X,t) (a) (b) 0 20 40 60 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 X w(X,t) (c) Figure 1. Num erical so lution of on e sing le cuspon solutio n: ( a) t = 0 . 0; (b ) t = 2 . 0; (c) t = 4 . 0. In Figure 4, we present another example of a collisi on between a soli ton ( p 1 = 9 . 12) and a cuspon ( p 2 = 10 . 5) where t he cuspon has a larger amplitude (2 . 0) th an the solit on (1 . 0). Again, when the colli sion starts, ano ther sing ularity po int appears. As collision goes on, the soliton is gradually absorbed by the cuspon. At t = 10 . 3, the whole profile looks like a single cuspon when the soliton is com pletely absorbed. Later on, the soliton emerges from the right until t = 16, the soliton and cuspon recover t heir original shapes except for a phase shift when the collision is complete. Example 4: Initial condition of non exact soliton s olutions. W e show that the integrable scheme can be also appl ied for the ini tial va lue problem starting wit h a non- exact soluti on. W e choose an initial conditio n in the following procedure. The mesh size is determined by δ k = 2 ac ( 1 − 0 . 8s ech ( 2 ka − W x / 3 )) , (5.3) where W x ( = 8) is the widt h of com putation, N = 201 is the number of grid in x -domain , k = 1 , · · · , N − 1, and a = W x / ( N − 1 ) = 0 . 04. Then the in itial profile can be c alculated t hrough the second e quation of the semi-discretization (3.2 7). The initial profile is plotted in Fig. 5(a). Figs. 5(b), (c) and (d) show the e volution at t = 10 , 20 , 30, respectively . It can be seen that An inte grable semi-discr eti zation of the C amassa-Holm equation 26 0 10 20 30 40 50 60 70 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 X w(X,t) 0 10 20 30 40 50 60 70 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 X w(X,t) (a) (b) 0 10 20 30 40 50 60 70 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 X w(X,t) 0 10 20 30 40 50 60 70 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 X w(X,t) (c) (d) 0 10 20 30 40 50 60 70 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 X w(X,t) (e) Figure 2. Num erical solution f or the collision of two-cuspo n solu tion with p 1 = 11 . 0, p 2 = 10 . 5, c = 1 0 . 0: (a) t = 0 . 0 ; (b) t = 13 . 0; (c) t = 14 . 8; (d) t = 16 . 6; (e) t = 25 . 0. An inte grable semi-discr eti zation of the C amassa-Holm equation 27 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) (a) (b) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) (c) (d) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) (e) (f) 0 10 20 30 40 50 60 70 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 X w(X,t) (g) Figure 3 . Numerical solution f or soliton-cuspon collision with p 1 = 9 . 12 , p 2 = 10 . 9 8 and c = 10 . 0: (a) t = 0 . 0; (b) t = 12 . 0 ; (c) t = 14 . 4; ( d) t = 14 . 6 ; (e) t = 14 . 8; ( f) t = 17 . 0; (g ) t = 25 . 0; An inte grable semi-discr eti zation of the C amassa-Holm equation 28 0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) 0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) (a) (b) 0 10 20 30 40 50 60 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) 0 10 20 30 40 50 60 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) (c) (d) 0 10 20 30 40 50 60 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) 0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) (e) (f) 0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 X w(X,t) (g) Figure 4. Numerical solu tion for soliton-cu spon collision with p 1 = 9 . 12, p 2 = 10 . 5 and c = 10 . 0: (a) t = 0 . 0; (b) t = 9 . 0; (c ) t = 10 . 0; (d) t = 10 . 3; (e) t = 10 . 6; (f) t = 11 . 5; (g) t = 16 . 0. An inte grable semi-discr eti zation of the C amassa-Holm equation 29 −60 −40 −20 0 20 40 60 0 0.5 1 1.5 2 2.5 x w(x,t) −60 −40 −20 0 20 40 60 0 0.5 1 1.5 2 2.5 x w(x,t) (a) (b) −60 −40 −20 0 20 40 60 0 0.5 1 1.5 2 2.5 x w(x,t) −60 −40 −20 0 20 40 60 0 0.5 1 1.5 2 2.5 x w(x,t) (c) (d) Figure 5. Numerica l solutio n startin g f rom an initial condition : (a) t = 0 . 0; (b) t = 10 . 0; (c) t = 20 . 0; (d) t = 30 . 0 . a solit on with large amplit ude is developed first, and m oving fast to th e right. By t = 30, a second soliton with small amplitud e is dev eloped and a th ird soli ton i s born from the second soliton. 6. Concluding remarks An integrable semi -discretization o f the CH equation has been presented in this paper . Determinant f ormulas of the N -soliton solutions of both the continuous and semi-discrete C H equations have been derived. Multi-solit on, multi-cuspon and mult i-soliton-cusp on soluti ons can be gener ated from abov e determinant formulas. As f urther topics, we attempt to construct integrable s emi-discretizations of the Degasperis-Proce si equation and other solito n equati ons possessing non-smo oth soluti ons s uch as peakon, cuspon, or loop-so liton solution s. W e will address these issues in forthcoming papers. Applying integrable discretizations o f soliton equations to numerical computations remains a promising but not thoroughl y explored subject. In the present paper , even for relativ ely large mesh sizes, very accurate numerical so lutions of cus pon-cuspon and soliton - cuspon interactions for the CH equ ations are achiev ed through our proposed integrable semi- discrete s cheme. In addition, a numerical compu tation starting with an non-exact ini tial An inte grable semi-discr eti zation of the C amassa-Holm equation 30 condition is performed with a sati sfactory result. It is worth noting that the integrable semi - discrete scheme of the CH equation is also a self-adaptive method, w hich is of great interest in the area of num erical partial differential equations. W e intend to extend this new id ea of self-adaptiv e metho d to other PDEs in the near future. 7. Append ix In this appendix, we prove t he dif ferential formulas for τ n , (2.9)-(2.16). Let us introduce a simplified notation, | ψ ( n 1 ) , ψ ( n 2 ) , · · · , ψ ( n N ) | =          ψ ( n 1 ) 1 ψ ( n 2 ) 1 · · · ψ ( n N ) 1 ψ ( n 1 ) 2 ψ ( n 2 ) 2 · · · ψ ( n N ) 2 . . . . . . . . . ψ ( n 1 ) N ψ ( n 2 ) N · · · ψ ( n N ) N          . (7.1) In this notation , w e have τ n = | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 1 ) | , thus differentiating τ n by x and using (2.3) we obtain ∂ x τ n = N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , ∂ x ψ ( n + j ) , · · · , ψ ( n + N − 1 ) | = N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + j + 1 ) + c ψ ( n + j ) , · · · , ψ ( n + N − 1 ) | = N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + j + 1 ) , · · · , ψ ( n + N − 1 ) | + N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , c ψ ( n + j ) , · · · , ψ ( n + N − 1 ) | = | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 2 ) , ψ ( n + N ) | + N c | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 1 ) | , which giv es (2.9) . Simi larly dif ferentiating τ n by y and using (2.4) we get ∂ y τ n = N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , ∂ y ψ ( n + j ) , · · · , ψ ( n + N − 1 ) | = N − 1 ∑ j = 0 | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + j + 2 ) + 2 c ψ ( n + j + 1 ) + c 2 ψ ( n + j ) , · · · , ψ ( n + N − 1 ) | = | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 2 ) , ψ ( n + N + 1 ) | − | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 3 ) , ψ ( n + N − 1 ) , ψ ( n + N ) | + 2 c | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 2 ) , ψ ( n + N ) | + N c 2 | ψ ( n ) , ψ ( n + 1 ) , · · · , ψ ( n + N − 1 ) | . W e obtain (2.10) from t he above equation and (2.9). Eqs. (2.11) and (2.12) can be proved by using (2.5) and (2.6) in a similar way . Finally eqs. (2.13 )-(2.16) can be verified by diffe rentiating (2.9) and (2.10) by t and s through similar calculations. Refer ences [1] Camassa R and Holm D 199 3 Phys. Rev . 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