Journal of Geometry and Physics

Journal of Geometry and Physics 07 (206) 35 44 Contents lists available at ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp Multi-component generalization of the Camassa Holm equation Baoqiang Xia a Zhijun Qiao b a School of Mathematics and Statistics Jiangsu Normal University Xuzhou Jiangsu 226 PR China b Department of Mathematics University of Texas-Rio Grande Valley Edinburg TX 7854 USA a r t i c l e i n f o a b s t r a c t Article history: Received 20 March 205 Received in revised form 29 February 206 Accepted 27 April 206 Available online 6 May 206 Keywords: Peakon Camassa Holm equation Lax pair Bi-Hamiltonian structure In this paper we propose a multi-component system of the Camassa Holm equation denoted by CH(N H) with 2N components and an arbitrary smooth function H. This system is shown to admit Lax pair and infinitely many conservation laws. We particularly study the case N = 2 and derive the bi-hamiltonian structures and peaked soliton (peakon) solutions for some examples. 206 Elsevier B.V. All rights reserved.. Introduction In 993 Camassa and Holm derived the well-known Camassa Holm (CH) equation [] m t + 2mu x + m x u = 0 m = u u xx + k () (with k being an arbitrary constant) with the aid of an asymptotic approximation to the Hamiltonian of the Green Naghdi equations. Since the work of Camassa and Holm [] more diverse studies on this equation have remarkably been developed [2 2]. The most interesting feature of the CH equation () is that it admits peakon solutions in the case k = 0. The stability and interaction of peakons were discussed in several references [3 7]. In addition to the CH equation other similar integrable models with peakon solutions were found [89]. Recently there are two integrable peakon equations found with cubic nonlinearity. They are the following cubic equation [320 22] m t + m(u 2 u 2 x 2 x m = u u xx (2) and the Novikov s equation [2324] m t = u 2 m x + 3uu x m m = u u xx. There is also much attention in studying integrable multi-component peakon equations. For example in [25 28] multicomponent generalizations of the CH equation were derived from different points of view and in [29] multi-component extensions of the cubic nonlinear equation (2) were studied. In a previous paper [30] we proposed a two-component generalization of the CH equation () and the cubic nonlinear CH equation (2) (3) Corresponding author. E-mail addresses: xiabaoqiang@26.com (B. Xia) zhijun.qiao@utrgv.edu (Z. Qiao). http://dx.doi.org/0.06/j.geomphys.206.04.020 0393-0440/ 206 Elsevier B.V. All rights reserved.

36 B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 m t = (mh) x + mh 2 m(u u x)(v + v x ) n t = (nh) x nh + 2 n(u u x)(v + v x ) m = u u xx n = v v xx where H is an arbitrary smooth function of u v and their derivatives. Such a system is interesting since different choices of H lead to different peakon equations. We presented the Lax pair and infinitely many conservation laws of the system for the general H and discussed the bi-hamiltonian structures and peakon solutions of the system for the special choices of H. In [3] Li Liu and Popowicz proposed a four-component peakon equation which also contains an arbitrary function. They derived the Lax pair and infinite conservation laws for their four-component equation and presented a bi-hamiltonian structure for the equation in the special case that the arbitrary function is taken to be zero. In this paper we propose the following multi-component system m jt = (m j H) x + m j H + [m (N + ) 2 i (u j u jx )(v i + v ix ) + m j (u i u ix )(v i + v ix )] (5) n jt = (n j H) x n j H [n (N + ) 2 i (u i u ix )(v j + v jx ) + n j (u i u ix )(v i + v ix )] m j = u j u jxx n j = v j v jxx j N where H is an arbitrary smooth function of u j v j j N and their derivatives. The system contains 2N components and an arbitrary function H. For N = this system becomes the two-component system (4). Therefore system (5) is a kind of multi-component generalization of the two-component system (4). Due to the presence of the function H system (5) is actually a large class of multi-component equations. We show that the multi-component system (5) admits Lax representation and infinitely many conservation laws. Although having not found a unified bi-hamiltonian structure of the system (5) for the general H yet we demonstrate that for some special choices of H one may find the corresponding bi-hamiltonian structures. As examples we derive the peakon solutions in the case N = 2. In particular we obtain a new integrable model which admits stationary peakon solutions. The whole paper is organized as follows. In Section 2 the Lax pair and conservation laws of Eq. (5) are presented. In Section 3 the Hamiltonian structures and peakon solutions of Eq. (5) in the case N = 2 are discussed. Some conclusions and open problems are addressed in Section 4. 2. Lax pair and conservation laws (4) We first introduce the N-component vector potentials u v and m n as u = (u u 2... u N ) v = (v v 2... v N ) m = u u xx n = v v xx. Using this notation Eq. (5) is expressed in the following vector form m t = ( mh) x + mh + (N + ) [ m( v + v x) T ( u u 2 x ) + ( u u x )( v + v x ) T m] n t = ( nh) x nh (N + ) [ n( u u x) T ( v + v 2 x ) + ( v + v x )( u u x ) T n] m = u u xx n = v v xx where the symbol T denotes the transpose of a vector. Let us introduce a pair of (N + ) (N + ) matrix spectral problems with φ x = Uφ φ t = Vφ φ = (φ φ 2... φ 2N ) T U = N λ m N + λ n T I N V = Nλ 2 + N + ( u u x)( v + v x ) T λ ( u u x ) + λ mh N + λ ( v + v x ) T + λ n T H λ 2 I N N + ( v + v x) T ( u u x ) where λ is a spectral parameter I N is the N N identity matrix u v m and n are the vector potentials shown in (6). (6) (7) (8) (9) Proposition. (8) provides the Lax pair for the multi-component system (5).

B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 37 Proof. It is easy to check that the compatibility condition of (8) generates U t V x + [U V] = 0. From (9) we have U t = 0 λ mt N + λ n T t 0 V x = N + [ m( v + v x) T ( u u x ) n T ] λ ( u x u xx ) + λ( mh) x N + λ ( v x + v xx ) T + λ( n T H) x N + [ nt ( u u x ) ( v + v x ) T m] and where Γ Γ [U V] = UV VU = 2 (2) (N + ) 2 Γ 2 Γ 22 Γ = m( v + v x ) T ( u u x ) n T Γ 2 = (N + )[λ ( u x u xx ) λ mh] λ N + [ m( v + v x) T ( u u x ) + ( u u x )( v + v x ) T m] Γ 2 = (N + )[λ ( v x + v xx ) T + λ n T H] + λ N + [ nt ( u u x )( v + v x ) T + ( v + v x ) T ( u u x ) n T ] Γ 22 = n T ( u u x ) ( v + v x ) T m. We remark that (2) is written in the form of block matrix. As shown above the element Γ is a scalar function the element Γ 2 is a N-component row vector function the element Γ 2 is a N-component column vector function and the element Γ 22 is a N N matrix function. Substituting the expressions () and (2) into (0) we find that (0) gives rise to m t = ( mh) x + mh + (N + ) [ m( v + v x) T ( u u 2 x ) + ( u u x )( v + v x ) T m] n T t = ( n T H) x n T H (N + ) 2 [ nt ( u u x )( v + v x ) T + ( v + v x ) T ( u u x ) n T (3) ] m = u u xx n T = v T v T xx which is nothing but the vector equation (7). Hence (8) exactly gives the Lax pair of multi-component equation (5). Now let us construct the conservation laws of Eq. (5). We write the spacial part of the spectral problems (8) as φ x = Nφ + λ m i φ 2i N + j N. (4) φ 2jx = λnj φ + φ 2j N + Let Ω j = φ 2j φ j N we obtain the following system of Riccati equations Ω jx = λn j + (N + )Ω j λω j m i Ω i j N. (5) N + t Making use of the relation (ln φ ) xt = (ln φ ) tx and (8) we arrive at the conservation law N m i Ω i = λ 2 (u i u ix )Ω i + N + λ (u i u ix )(v i + v ix ) + H m i Ω i. (6) Eq. (6) means that N m iω i is a generating function of the conserved densities. To derive the explicit forms of conserved densities we expand Ω j into the negative power series of λ as Ω j = ω jk λ k j N. k=0 x (0) () (7)

38 B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 Substituting (7) into the Riccati system (5) and equating the coefficients of powers of λ we obtain ω j0 = n j N 2 m i n i ω j = (N + ) ω j0 ω j0x 2 n j N N N m i (ω i0 ω i0x ) m i n i 2 m i n i (8) and the recursion relations for ω j(k+) k ω j(k+) = (N + ) ω jk ω jkx N N N 2 2 n j m i (ω ik ω ikx ) m i n i m i n i. (9) Inserting (7) (9) into (6) we finally obtain the following infinitely many conserved densities ρ j and the associated fluxes F j : ρ 0 = N 2 m i ω i0 = m i n i F 0 = H N m i ω i0 = H ρ = m i ω i F = (u i u ix )(v i + v ix ) + H N + ρ 2 = m i ω i2 F 2 = (u i u ix )ω i0 + H m i ω i2 ρ j = m i ω ij F j = (u i u ix )ω i(j 2) + H where ω ij i N j 0 is given by (8) and (9). m i n i 2 m i ω i m i ω ij j 3 (20) Remark. The 2N-component system (5) with an arbitrary function H does possess Lax representation and infinitely many conservation laws. Such a system is interesting since different choices of H lead to different peakon equations. Let us look back why an arbitrary smooth function may be involved in system (5). System (5) is produced by the compatibility condition (0) of the spectral problems (8) where such an arbitrary function is included in V part (see the formula (9)). The appearance of this arbitrary function can be explained as that the Lax equation is an over determined system by choosing the appropriate V to match U. 3. Examples for N = 2 In the case N = 2 Eq. (5) is cast into the following four-component model m t = (m H) x + m H + 9 {m [2(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] + m 2 (u u x )(v 2 + v 2x )} m 2t = (m 2 H) x + m 2 H + 9 {m (u 2 u 2x )(v + v x ) + m 2 [(u u x )(v + v x ) + 2(u 2 u 2x )(v 2 + v 2x )]} n t = (n H) x n H 9 {n [2(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] + n 2 (u 2 u 2x )(v + v x )} n 2t = (n 2 H) x n 2 H 9 {n (u u x )(v 2 + v 2x ) + n 2 [(u u x )(v + v x ) + 2(u 2 u 2x )(v 2 + v 2x )]} m = u u xx m 2 = u 2 u 2xx n = v v xx n 2 = v 2 v 2xx where H is an arbitrary smooth function of u u 2 v v 2 and their derivatives. This system admits the following 3 3 Lax pair U = 2 λm λm 2 λn 0 V = V V 2 V 3 V 2 V 22 V 23 (22) 3 λn 2 0 3 V 3 V 32 V 33 (2)

B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 39 where V = 2λ 2 + 3 [(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] V 2 = λ (u u x ) + λm H V 3 = λ (u 2 u 2x ) + λm 2 H V 2 = λ (v + v x ) + λn H V 22 = λ 2 3 (u u x )(v + v x ) (23) V 23 = 3 (u 2 u 2x )(v + v x ) V 3 = λ (v 2 + v 2x ) + λn 2 H V 32 = 3 (u u x )(v 2 + v 2x ) V 33 = λ 2 3 (u 2 u 2x )(v 2 + v 2x ). Due to the appearance of arbitrary function H we do not know yet whether (2) is bi-hamiltonian in general. But we find that it is possible to figure out the bi-hamiltonian structures for some special cases which we will show in the following examples. Example. A new integrable model with stationary peakon solutions Taking H = 0 Eq. (2) becomes the following system m t = 9 {m [2(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] + m 2 (u u x )(v 2 + v 2x )} m 2t = 9 {m (u 2 u 2x )(v + v x ) + m 2 [(u u x )(v + v x ) + 2(u 2 u 2x )(v 2 + v 2x )]} n t = 9 {n [2(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] + n 2 (u 2 u 2x )(v + v x )} (24) n 2t = 9 {n (u u x )(v 2 + v 2x ) + n 2 [(u u x )(v + v x ) + 2(u 2 u 2x )(v 2 + v 2x )]} m = u u xx m 2 = u 2 u 2xx n = v v xx n 2 = v 2 v 2xx. Let us introduce a Hamiltonian pair 0 0 + 0 K K 2 K 3 K 4 0 0 0 + J = 0 0 0 K = K 2 K 22 K 23 K 24 9 K 3 K 32 K 33 K (25) 34 0 0 0 K 4 K 42 K 43 K 44 where K = 2m m K 2 = m 2 m m m 2 K 3 = 2m n + m 2 n 2 K 4 = m n 2 K 2 = K 2 = m m 2 m 2 m K 22 = 2m 2 m 2 K 23 = m 2 n K 24 = m n + 2m 2 n 2 K 3 = K 3 = 2n m + n 2 m 2 K 32 = K 23 = n m 2 K 33 = 2n n K 34 = n n 2 n 2 n K 4 = K 4 = n 2 m K 42 = K 24 = n m + 2n 2 m 2 K 43 = K 34 = n 2 n n n 2 K 44 = 2n 2 n 2. By direct but tedious calculations we arrive at Proposition 2. Eq. (24) can be rewritten in the following bi-hamiltonian form T δh2 mt m 2t n t n 2t = J δh 2 δh 2 δh T 2 δh = K δh δh δh T (27) δm δm 2 δn δn 2 δm δm 2 δn δn 2 where J and K are given by (25) and + H = [(u x u )n + (u 2x u 2 )n 2 ]dx H 2 = + [(u u x ) 2 (v + v x )n + (u u x )(u 2 u 2x )(v 2 + v 2x )n 9 + (u u x )(u 2 u 2x )(v + v x )n 2 + (u 2 u 2x ) 2 (v 2 + v 2x )n 2 ]dx. (26) (28)

40 B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 Suppose an N-peakon solution of (24) is in the form u = v = p j (t)e x qj(t) u 2 = r j (t)e x qj(t) s j (t)e x qj(t) v 2 = w j (t)e x qj(t). (29) Then in the distribution sense one can get u x = p j sgn(x q j )e x qj m = 2 p j δ(x q j ) u 2x = r j sgn(x q j )e x qj m 2 = 2 r j δ(x q j ) v x = s j sgn(x q j )e x qj n = 2 s j δ(x q j ) v 2x = w j sgn(x q j )e x qj n 2 = 2 w j δ(x q j ). (30) Substituting (29) and (30) into (24) and integrating against test functions with compact support we arrive at the N-peakon dynamical system as follows: q jt = 0 p jt = 2 9 3 p j(p j s j + r j w j ) pj (2p i s k + r i w k ) + r j p i w k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + pj (2p i s k + r i w k ) + r j p i w k sgn(qj q i )sgn(q j q k ) e q j q i q j q k r jt = 2 9 3 r j(r j w j + p j s j ) rj (2r i w k + p i s k ) + p j r i s k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + rj (2r i w k + p i s k ) + p j r i s k sgn(qj q i )sgn(q j q k ) e q j q i q j q k s jt = 2 9 3 s j(p j s j + r j w j ) sj (2p i s k + r i w k ) + w j r i s k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + sj (2p i s k + r i w k ) + w j r i s k sgn(qj q i )sgn(q j q k ) e q j q i q j q k w jt = 2 9 3 w j(p j s j + r j w j ) wj (2r i w k + p i s k ) + s j p i w k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + wj (2r i w k + p i s k ) + s j p i w k sgn(qj q i )sgn(q j q k ) e q j q i q j q k. (3) The formula q jt = 0 in (3) implies that the peakon position is stationary and the solution is in the form of separation of variables. Especially for N = (3) becomes

B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 4 q t = 0 p t = 4 27 p (p s + r w ) r t = 4 27 r (p s + r w ) s t = 4 27 s (p s + r w ) w t = 4 27 w (p s + r w ) which has the solution (32) q = C p = A 4 e 4 27 (A 2+A 3 )t r = p s = A 2 e 4 27 (A 2+A 3 )t w = A 3 (33) A A 4 r where C and A A 4 are integration constants. Thus the stationary single-peakon solution becomes u = A 4 e 4 27 (A 2+A 3 )t e x C u 2 = u A v = A 2 e 4 27 (A 2+A 3 )t e x C v 2 = A A 3 v. A 4 A 2 See Fig. for the stationary single-peakon of the potentials u (x t) and v (x t) with C = 0 A 2 = A 4 = and A 3 = 2. (34) Example 2. A new integrable four-component system with peakon solutions Let us choose H = 9 [(u u x )(v + v x ) + (u 2 u 2x )(v 2 + v 2x )] then Eq. (2) is cast into m t = (m H) x + 9 m (u u x )(v + v x ) + 9 m 2(u u x )(v 2 + v 2x ) m 2t = (m 2 H) x + 9 m (u 2 u 2x )(v + v x ) + 9 m 2(u 2 u 2x )(v 2 + v 2x ) n t = (n H) x 9 n (u u x )(v + v x ) 9 n 2(u 2 u 2x )(v + v x ) (35) n 2t = (n 2 H) x 9 n (u u x )(v 2 + v 2x ) 9 n 2(u 2 u 2x )(v 2 + v 2x ) m = u u xx m 2 = u 2 u 2xx n = v v xx n 2 = v 2 v 2xx. Let us set K K 2 K 3 K 4 K = K 2 K 22 K 23 K 24 9 K 3 K 32 K 33 K 34 K 4 K 42 K 43 K 44 (36) where K = m m m m K 2 = m m 2 m 2 m K 3 = m n + m n + m 2 n 2 K 4 = m n 2 K 2 = K = 2 m 2 m m m 2 K 22 = m 2 m 2 m 2 m 2 K 23 = m 2 n K 24 = m 2 n 2 + m n + m 2 n 2 K 3 = K = 3 n m + n m + n 2 m 2 K 32 = K = 23 n m 2 K 33 = n n n n K 34 = n n 2 n 2 n K 4 = K = 4 n 2 m K 42 = K = 24 n 2 m 2 + n m + n 2 m 2 K 43 = K = 34 n 2 n n n 2 K 44 = n 2 n 2 n 2 n 2. (37)

42 B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 By direct calculations we arrive at Proposition 3. Eq. (35) can be rewritten in the following Hamiltonian form T δh mt m 2t n t n 2t = K δh δh δh T (38) δm δm 2 δn δn 2 where K are given by (36) and + H = [(u x u )n + (u 2x u 2 )n 2 ]dx. (39) We believe that Eq. (35) could be cast into a bi-hamiltonian system. But we did not find another Hamiltonian operator yet that is compatible with the Hamiltonian operator (36). Suppose N-peakon solution of (35) is expressed also in the form of (29). Then we obtain the N-peakon dynamical system of (35): q jt = 9 3 (p js j + r j w j ) + (p i s k + r i w k ) sgn(q j q i ) sgn(q j q k ) e q j q i q j q k + (p i s k + r i w k ) sgn(q j q i )sgn(q j q k ) e q j q i q j q k p jt = 9 + r jt = 9 + 3 p j(p j s j + r j w j ) + pj p i s k + r j p i w k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k pj p i s k + r j p i w k sgn(qj q i )sgn(q j q k ) e q j q i q j q k 3 r j(p j s j + r j w j ) + rj r i w k + p j r i s k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k rj r i w k + p j r i s k sgn(qj q i )sgn(q j q k ) e q j q i q j q k s jt = 9 3 s j(p j s j + r j w j ) wj r i s k + s j p i s k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + sj p i s k + w j r i s k sgn(qj q i )sgn(q j q k ) e q j q i q j q k w jt = 9 3 w j(p j s j + r j w j ) sj p i w k + w j r i w k sgn(qj q i ) sgn(q j q k ) e q j q i q j q k + wj r i w k + s j p i w k sgn(qj q i )sgn(q j q k ) e q j q i q j q k. (40) For N = (40) becomes q t = 2 27 (p s + r w ) p t = 2 27 p (p s + r w ) r t = 2 27 r (p s + r w ) s t = 2 27 s (p s + r w ) w t = 2 27 w (p s + r w ). (4)

B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 43 Fig.. The stationary single-peakon solution of the potentials u (x t) and v (x t) given by (34) with C = 0 A 2 = A 4 = and A 3 = 2. Solid line: u (x t); Dashed line: v (x t); Black: t = ; Blue: t = 2. (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.) Fig. 2. The single-peakon solution of the potentials u (x t) and v (x t) given by (42) with A 4 = 0 A 2 = A 5 = and A 3 = 2. Solid line: u (x t); Dashed line: v (x t); Black: t = 2; Blue: t = 2. (For interpretation of the references to color in this figure legend the reader is referred to the web version of this article.) We may solve this equation as q = 2 27 (A 2 + A 3 )t + A 4 p = A 5 e 2 27 (A 2+A 3 )t r = p s = A 2 e 2 27 (A 2+A 3 )t w = A 3 (42) A A 5 r where A A 5 are integration constants. See Fig. 2 for the single-peakon of the potentials u (x t) and v (x t) with A 4 = 0 A 2 = A 5 = and A 3 = 2. 4. Conclusions and discussions In our paper we propose a multi-component generalization of the Camassa Holm equation and provide its Lax representation and infinitely many conservation laws. This system contains an arbitrary smooth function H thus it is actually a large class of multi-component peakon equations. We show it is possible to find the bi-hamiltonian structures for the special choices of H. In particular we study the peakon solutions of this system in the case N = 2 and obtain a new integrable system which admits stationary peakon solutions. In contrast with the usual soliton equations the peakon equations with arbitrary functions seem to be unusual. We believe that there are much investigations deserved to do for both our generalized peakon system and Li Liu Popowicz s system [3]. The following topics seem to be interesting:

44 B. Xia Z. Qiao / Journal of Geometry and Physics 07 (206) 35 44 Is there a gauge transformation that can be applied to the Lax pair to remove the arbitrary function H? Does there exist a unified (bi-)hamiltonian structure for the system (5) for the general H? Can the inverse scattering transforms be applied to solve the systems in general? Acknowledgments The authors would like to thank the reviewers for their valuable suggestions and comments which help us improve our paper. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 30229 2768 and 7295) the Natural Science Foundation of the Jiangsu Province (Grant No. BK2030224) the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 3KJB0009) the Haitian Scholar Plan of Dalian University of Technology and the China state administration of foreign experts affairs system under the affiliation of China University of Mining and Technology. References [] R. Camassa D.D. Holm Phys. Rev. Lett. 7 (993) 66. [2] R. Camassa D.D. Holm J.M. Hyman Adv. Appl. Mech. 3 (994). [3] P.J. Olver P. Rosenau Phys. Rev. E 53 (996) 900. [4] B. Fuchssteiner Physica D 95 (996) 229. [5] A.S. Fokas Q.M. Liu Phys. Rev. Lett. 77 (996) 2347. [6] R. Beals D. Sattinger J. Szmigielski Adv. Math. 40 (998) 90. [7] A. Constantin J. Funct. Anal. 55 (998) 352. [8] H.R. Dullin G.A. Gottwald D.D. Holm Phys. Rev. Lett. 87 (200) 9450. [9] F. Gesztesy H. Holden Rev. Mat. Iberoamericana 9 (2003) 73. [0] Z.J. Qiao Comm. Math. Phys. 239 (2003) 309. [] P. Lorenzoni M. Pedroni Int. Math. Res. Not. IMRN 75 (2004) 409. [2] A. Constantin V.S. Gerdjikov R.I. Ivanov Inverse Problems 22 (2006) 297. [3] A. Constantin W.A. Strauss Comm. Pure Appl. Math. 53 (2000) 603. [4] A. Constantin W.A. Strauss J. Nonlinear Sci. 2 (2002) 45. [5] R. Beals D. Sattinger J. Szmigielski Adv. Math. 54 (2000) 229. [6] M.S. Alber R. Camassa Y.N. Fedorov D.D. Holm J.E. Marsden Comm. Math. Phys. 22 (200) 97. [7] R.S. Johnson Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 459 (2003) 687. [8] A. Degasperis M. Procesi in: A. Degasperis G. Gaeta (Eds.) Asymptotic Integrability Symmetry and Perturbation Theory World Scientific Singapore 999 pp. 23 37. [9] A. Degasperis D.D. Holm A.N.W. Hone Theoret. Math. Phys. 33 (2002) 463. [20] A.S. Fokas Physica D 87 (995) 45. [2] B. Fuchssteiner Physica D 95 (996) 229. [22] Z.J. Qiao J. Math. Phys. 47 (2006) 270; J. Math. Phys. 48 (2007) 08270. [23] V. Novikov J. Phys. A 42 (2009) 342002. [24] A.N.W. Hone J.P. Wang J. Phys. A 4 (2008) 372002. [25] P. Casati G. Ortenzi J. Nonlinear Math. Phys. 3 (3) (2006) 352 364. [26] S.Q. Liu Y. Zhang J. Geom. Phys. 54 (2005) 427453. [27] G. Falqui J. Phys. A 39 (2006) 327 342. [28] D.D. Holm R.I. Ivanov J. Phys. A 43 (200) 49200. [29] C.Z. Qu J.F. Song R.X. Yao SIGMA 9 (203) 00. [30] B.Q. Xia Z.J. Qiao R.G. Zhou Stud. Appl. Math. 35 (205) 248 276. [3] N. Li Q.P. Liu Z. Popowicz J. Geom. Phys. 85 (204) 29 39.