On Camassa Holm Equation with Self-Consistent Sources and Its Solutions

Transcription

1 Commun. Theor. Phys. Beiing China pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53 No. 3 March On Camassa Holm Equation with Self-Consistent Sources and Its Solutions HUANG Ye-Hui áûï 1 YAO Yu-Qin 2 and ZENG Yun-Bo É 1 1 Department of Mathematical Science Tsinghua University Beiing China 2 Department of Applied Mathematics China Agricultural University Beiing China Received March ; revised manuscript received January Abstract Regarded as the integrable generalization of Camassa Holm CH equation the CH equation with selfconsistent sources CHESCS is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution N-soliton N-cuspon N-positon and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants. PACS numbers: Ik Yv Key words: Camassa Holm equation with self-consistent sources Lax representation conservation laws peakon soliton positon negaton 1 Introduction Camassa Holm CH equation which was implicitly contained in the class of multi-hamiltonian system introduced by Fuchssteiner and Fokas [1 and explicitly derived as a shallow water wave equation by Camassa and Holm [2 3 has the form u t + 2ωu x u xxt + 3uu x = 2u x u xx + uu xxx 1 u = ux t is the fluid velocity in the x direction and the constant 2ω is related to the critical shallow water wave speed. Let q = u u xx + ω we have the following equivalent equation [4 q t + 2u x q + uq x = 0. 2 It was shown by Camassa and Holm that this equation shares most of the properties of the integrable system of KdV type. [2 3 It possesses Lax pair formalism and the bi-hamiltonian structure. When ω > 0 the CH equation has smooth solitary wave solutions. When ω 0 these solutions become piecewise smooth and have cusps at their peaks. These kind of solutions are weak solutions of Eq. 2 with ω = 0 and are called peakons. Since the works of Camassa and Holm this equation has become a well-known example of integrable systems and has been studied from many kinds of views. [4 12 Soliton equations with self-consistent sources SESCS have attracted much attention in recent years. They are important integrable models in many fields of physics such as hydrodynamics state physics plasma physics etc. [13 25 For example the KdV equation with self-consistent sources describes the interaction of long and short capillary-gravity waves. [13 The nonlinear Schrödinger equation with self-consistent sources represents the nonlinear interaction of an electrostatic highfrequency wave with the ion acoustic wave in a two component homogeneous plasma. [18 The KP equation with selfconsistent sources describes the interaction of a long wave with a short wave packet propagating on the x-y plane at some angle to each other. [15 The SESCS were first studied by Melnikov. [13 15 A systematic way to construct the soliton equations with self-consistent sources and their zero-curvature representations was proposed. [21 24 The problem of finding soliton solutions or other specific solutions for SESCS has been considered in the past by many authors. [13 25 Positons and negatons are specific solutions with singularities. [26 27 The positon solutions of SESCS are long-range analogues of solitons and slowly decreasing oscillating solutions and possess supertransparent property. The positon and nagaton solutions were studied in Refs. [ The present paper falls in that line of the work on the CH equation concerning with establishing the many facts of its completely integrable character aiming at the integrable generalization of CH equation by deriving the Camassa Holm equation with self-consistent sources CHESCS and finding its solutions. We first construct the CHESCS by using the approach presented in Refs. [ The Lax pair of the CHESCS is obtained which means that the CHESCS is Lax integrable and can be viewed as integrable generalization of CH equation. Since the CH equation describes shallow water wave and the SESCSs in general describe the interaction of different solitary waves it is reasonable to speculate on the potential application of CHESCS that is CHESCS may describe the interaction of different solitary waves in shallow water. It was pointed out in Refs. [28 29 that SESCS can be regarded as soliton equations with non-homogeneous terms and accordingly proposed to look for explicit solutions by using the method of variation of constants. Applying this technique to CHESCS we have been able to find its peakon Supported by the National Basic Research Program of China 973 program under Grant No. 2007CB and the National Science Foundation of China under Grant Nos and Corresponding author huangyh@mails.tsinghua.edu.cn yuqinyao@mail.tsinghua.edu.cn yzeng@math.tsinghua.edu.cn

2 404 HUANG Ye-Hui YAO Yu-Qin and ZENG Yun-Bo Vol. 53 solution. In order to find other solutions of CHESCS we consider the reciprocal transformation [30 31 which relates CH equation to an alternative of the associated Camassa Holm ACH equation and propose the reciprocal transformation which relates the CHESCS to associated CHESCS ACHESCS. By using the Darboux transformation DT one can find the n-soliton and n- cuspon solution [8 9 as well as positon and negaton solution of alternative ACH equation. Then by means of the method of variation of constants we can obtain the N-soliton N-cuspon N-positon and N-negaton solution for ACHESCS. Finally using the inverse reciprocal transformation we obtain the N-soliton N-cuspon N-positon and N-negaton solution of CHESCS. This paper is organized as follows. In Sec. 2 we present how to derive the CHESCS and its Lax representation. In Sec. 3 the conservation laws of the CHESCS are constructed. In Sec. 4 the peakon solution is obtained. In Sec. 5 we consider the reciprocal transformation for CH equation and CHESCS respectively. In Sec. 6 by using the DT we find the solution for alternative ACH equation then by using the method of variation of constants and inverse reciprocal transformation we obtain the N- soliton N-cuspon N-positon and N-negaton solution for CHESCS. In Sec. 7 the conclusion is presented. 2 The CHESCS and Its Lax Pair 2.1 The CHESCS The Lax pair for CH equation 2 is given by [2 ϕ xx = λq + 1 ϕ 4 1 ϕ t = 2λ u ϕ x u xϕ. 3 It is not difficult to find that δλ δq = λϕ2. 4 The CH equation possesses bi-hamiltonian structure [2 q t = J δh 0 δq = K δh 1 δq 5 K = 3 + J = q + q H 0 = 1 u 2 + u 2 x 2 dx H 1 = 1 u 3 + uu 2 x 2 dx. According to the approach proposed in Refs. [21 24 the CHESCS is defined as follows N δh0 q t = J δq 2 δλ δq = q + q u + 2 = 2qu x uq x + λ ϕ 2 8λ qϕ ϕ x 2λ q x ϕ 2 6a ϕ xx = λ q + 1 ϕ = 1...N 6b 4 which has an equivalent form by using Eq. 6b q t = 2qu x uq x + [ϕ 2 x ϕ 2 xxx ϕ xx = 7a λ q + 1 ϕ = 1...N. 7b Lax Representation of CHESCS Based on the Lax pair of the CH equation 3 we may assume the Lax representation of the CHESCS 6 or 7 has the form ϕ xx = λq ϕ t = 1 2 B xϕ + Bϕ x B = 1 2λ u + N ϕ 8a α fϕ λ λ + β fϕ 8b 8c fϕ is undetermined function of ϕ. The compatibility condition of Eqs. 8a and 8b gives λq t = LB + λ2b x q + Bq x 9 L = 1/2 3 +1/2. Then Eqs. 8 and 9 yield λq t = 1 α [ f ϕ 3 x 2 λ λ + 3f ϕ f λ q + 1 ϕ x + λ q x f ϕ 2f 4 [ + 2qu x uq x β 2qϕ x f + q x f λ β [f ϕ 2 x + 3f ϕ + f λ q + 1 ϕ x 4 + λ f q x ϕ f ϕ + α q x f + 2qf ϕ x. 10 Here f denotes the partial derivative of the function f with respect to the variable ϕ. In order to determine f α and β we compare the coefficients of 1/λ λ λ and λ 0 respectively. We first observe the coefficients of 1/λ λ then the coefficients of ϕ 3 x ϕ x and other terms give rise to respectively f = 0 f ϕ f = 0 f ϕ 2f = 0 which leads to f = bϕ 2. Substituting f = bϕ 2 into the coefficients of λ in Eq. 10 gives q t = 2qu x uq x + 4q N β bϕ ϕ x + q x β bϕ 2. Comparing the above equation and Eq. 6a we can determine b = 2 β = λ. Substituting f = 2ϕ 2 and β = λ into the coefficients of λ 0 in Eq. 10 we obtain α = λ 2.

3 No. 3 On Camassa Holm Equation with Self-Consistent Sources and Its Solutions 405 Thus we obtain the Lax pair of the CHESCS Eq. 7 1 ϕ xx = 4 + λq ϕ 11a ϕ t = u x 2 ϕ+ 1 2λ u ϕ x +2 λλ ϕ λ λ ϕ x ϕ ϕ ϕ x 11b which means that the CHESCS 7 is Lax integrable. 3 Infinite Conservation Laws of CHESCS With the help of the Lax representation of the CHESCS we could find the conservation laws for the CHESCS by a well-known method. First we assume that q u ϕ and its derivatives tend to 0 when x. Set Γ = ϕ x ϕ 12 then the identity lnϕ = lnϕ t x x t together with Eq. 11 implies that CHESCS has the following conservation law t Γ = ϕt = [ 1 x ϕ x 2 u x λ u 2 Using Eq. 11a gives rise to Let λλ λ λ ϕ ϕ x λλ ϕ 2 Γ. 13 λ λ Γ x = qλ Γ2. 14 Γ = µ m λ 1 m/2 15 m=0 then µ m is the density of conservation laws. Define 1 2 u λλ [ 1 x + 2 ϕ ϕ x + λ λ 2λ u 2 λλ ϕ 2 Γ = λ λ F m λ 1 m/2. 16 m=0 It is found that the density of the conservation laws µ m and the flux of the conservation laws F m satisfy the following recursion relation µ 0 = q µ 1 = 1 q x 4 q µ 2 = 1 [ 4 q + q2 x 32 q 4qx 5/2 q x 3/2 µ m = µ m 1x m 1 i=1 µ iµ m 1 i 2µ 0 m 3 17 F 0 = u 2 q λ ϕ 2 F 1 = u + 2 i=0 λ ϕ 2 qx 4q u x + 2 λ ϕ ϕ x m F 2m = u i 2 λ i+1 ϕ 2 µ 2m 2i m 1 F 2m+1 = m u i + 2 λ i+1 ϕ 2 µ 2m 2i+1 i=0 + 2 λ m+1 ϕ ϕ x m 1 18 u 0 = u u 1 = 1 u i = 0 i > 1. After some calculations we can find the first few conserved quantities given by µ 0 µ 2 and µ 4 are as follows qdx H 1 = H 2 = q + q2 x dx q 5/2 1 H 3 = 32q + 5q2 x 3/2 64q + q2 xx 7/2 32q 35q4 x dx.19 7/2 512q 11/2 The corresponding flux of the conservation laws are q G 1 = u 2 λ ϕ 2 G 2 = G 3 = λ 2 ϕ 2 q + u q + q2 x 16 q 5/2 u 2 λ ϕ q 3/2 + 5q2 x 64q 7/ λ 2 ϕ 2 qx λ ϕ 2 4q 3/2 x q2 xx 32q 7/2 4 q + q2 x q 5/2 35q4 x 512q 11/2 λ 3 ϕ2 q. 20 As the space part of the Lax Pair of the CHESCS is the same as that of CH equation the densities of the conservation laws of the CHESCS are the same as those of the Camassa Holm equation. [12 As the time part of the Lax pair is different the fluxs of the conservation laws for CH equation and CHESCS are different. 4 One Peakon Solution of CHESCS When ω 0 the CH equation 2 has peakon solutions [2 u = c exp[ x ct + α 21

4 406 HUANG Ye-Hui YAO Yu-Qin and ZENG Yun-Bo Vol. 53 α is an arbitrary constant. The corresponding eigenfunction of Eq. 3 is ϕ = β exp [ 1 2 x ct + α 22 β is an arbitrary constant. Since the CHESCS 7 can be considered as the CH equation 2 with non-homogeneous terms we may use the method of variation of constants to find the peakon solution of CHESCS from the peakon solutions 21 and 22. Taking α and β in Eqs. 21 and 22 to be timedependent αt and βt and requiring that u = c exp[ x ct + αt ϕ = βtexp [ 1 2 x ct + αt 23 satisfy the CHESCS 7 for N = 1. We find that c = 1/λ αt can be an arbitrary function of t and βt = α tc. So we have the one peakon solution for Eq. 7 with N = 1 λ 1 = λ = 1/c u = c exp[ x ct + αt ϕ = α tc exp [ 1 2 x ct + αt. 24 The one peakon of the CHESCS also has a cusp at its peak located at x = ct αt. We note that for the one peakon solution of the CH equation the solution travels with speed c and has a cusp at its peak of height c for the CHESCS the cusp is still at its peak of height c but the speed c αt/t of the wave is no longer a constant. 5 Reciprocal Transformation for CHESCS Let r = q by the reciprocal transformation [ dy = rdx urdt ds = dt and denoting f = r 1/2 φ the Lax pair 3 of CH equation is transformed to the following system Q = 1 4 φ yy = λ + Q + 1 4ω φ s = 1 rφ y 1 2λ 2 r yφ ry r φ 25a 25b 2 + r yy 2r + 1 4r 2 1 4ω. 26 The compatibility condition of Eqs. 25a and 25b gives an alternative of the associated CH ACH equation Q s = r y 1 4ω r y r yyy 1 2 Q yr Qr y = We now consider the reciprocal transformation for the CHESCS 7. Equation 7a gives r t = ru x 2 λ rϕ 2 x. 28 Equation 28 shows that the 1-form ω = rdx ru + 2 λ rϕ 2 dt 29 is closed so we can define a reciprocal transformation x t y s by the relation dy = rdx ru + 2 λ rϕ 2 dt ds = dt 30 and we have x = r y t = s ru + 2 λ rϕ 2 y. 31 Denoting ϕ = r 1/2 ψ ϕ = r 1/2 ψ and using Eq. 26 the Lax pair 11 of CHESCS 7 is correspondingly rewritten as ψ yy = λ + Q + 1 4ω ψ ψ s = 1 rψ y 1 2λ 2 r yψ + 2 λ 2 ψ λ λ ψ y ψ ψ ψ y.32 The compatibility condition of Eqs. 32 leads to an associated CHESCS ACHESCS Q s = r y 8 λ 2 ψ ψ y 1 4ω r y r yyy 1 2 Q yr Qr y = 0 ψ yy = λ + Q + 1 ψ = N. 33 4ω Equations 33 can be regarded as the Eqs. 27 with selfconsistent sources. In order to obtain the solutions of the CHESCS 7 we have to get the relation of the variables y s and the variables x t. From the reciprocal transformation we have x y = 1 r x N s = u + 2 λ ϕ By making use of the compatibility of the above two equations we have 1 xy s = dy. 35 r The solutions of the CHESCS 7 with respect to the variables y s are given by q = r 2 y s ϕ y s = ψ r uy s = r 2 r ys + r yr s r xy s = N 2r 2 λ ϕ 2 yy ω 1 r dy. N 2rr y λ ϕ 2 y 36a 36b 36c We now prove Eq. 36b. From q = u u xx + ω and the reciprocal transformation 31 we have u = q + rr y u y + r 2 u yy ω. 37

5 No. 3 On Camassa Holm Equation with Self-Consistent Sources and Its Solutions 407 By using the reciprocal transformation 31 Eq. 28 gives rise to u y = r N s r 2 2 λ ϕ 2 y. 38 Substituting Eqs. 38 and 26 into Eq. 27 leads to Eq. 36b. 6 Solutions for CHESCS Notice that Q = 0 r = ω is the solution of Eqs. 26 and 27. Let the functions φ 0 y s λ Ψ 1 y s λ 1... Ψ n y s λ n be different solutions of Eq. 25 with Q = 0 r = ω and the corresponding λ and λ = λ 1... λ n respectively. We construct two Wronskian determinants from these functions = WΨ 1... Ψ m1 1 Ψ 2... Ψ m2 2...Ψ n... Ψ mn n W 2 = WΨ 1... Ψ m1 1 Ψ 2... Ψ m2 2...Ψ n... Ψ mn n φ 0 39 m i 0 are given numbers and Ψ i = i Ψ y s λ λ i λ=λ. Based on the generalized Darboux transformation for KdV hierarchy [32 and using Eq. 27a the following generalized Darboux transformation of Eq. 25 is valid [ Qy s = 2 2 y 2 log ry s = ω 2 2 y s log φy s λ = W 2 40 namely Qy s ry s and φy s λ satisfy Eqs and Multisoliton Solutions Taking Ψ i and Φ i be the solutions of Eq. 25 with Q = 0 r = ω and λ i = k 2 i 1/4ω < 0 or 4ωk2 i 1 < 0 0 < k 1 < k 2 < < k n as follows Ψ i = cosh ξ i i is an odd number Ψ i = sinhξ i i is an even number. 41 Φ i = e ξi 42 henceforth [ ξ i = k i y + 2ω3/2 s 4ωki α i. 43 By using Darboux transformation 40 with m 1 = = m n = 0 the n-soliton solution Qy s and ry s of Eq. 27 and the corresponding eigenfunction φ i y s λ i of Eq. 25 with λ i = ki 2 1/4ω is given by Qy s = 2[logWΨ 1 Ψ 2...Ψ n yy ry s = ω 2[log WΨ 1 Ψ 2...Ψ n ys φ i y s λ i = WΨ 1 Ψ 2... Ψ n Φ i WΨ 1 Ψ 2... Ψ n. 44 When n = 1 and 4k1 2 ω 1 < 0 Eq. 44 gives rise to one soliton solution for Eq. 27 and the corresponding eigenfunction of Eq. 25 with λ 1 = k1 2 1/4ω[8 9 Qy s = 2k 2 1 sech 2 ξ 1 ry s = ω 4k2 1 ω3/2 sech 2 ξ 1 4k 2 1 ω 1 φ 1 = k 1 sech ξ Since Eq. 44 can be considered to be Eq. 27 with non-homogeneous terms and φ 1 satisfies Eq. 25a with λ = λ 1 we may apply the method of variation of constant to find the solutions of the CHESCS 33 by using the solution 45 of ACH equation 27 and corresponding eigenfunction. Taking α 1 in Eq. 43 to be time-dependent functions α 1 s and requiring that Qy s = 2k 2 1 sech 2 ξ1 ry s = ω 4k2 1 ω3/2 sech 2 ξ1 4k 2 1 ω 1 ψ 1 = β 1 sk 1 sech ξ 1 46 satisfy the system 33 for N = 1 henceforth we denote [ ξ 1 = k 1 y + 2ω3/2 s 4ωk α 1s. 47 We find that α 1 s can be an arbitrary function of s and 2ω β 1 s = 1 4k1 2ω 2α 1 s. 48 So the one-soliton solution of the CHESCS 7 with N = 1 and λ 1 = k1 2 1/4ω < 0 is obtained with respect to the variables y s from Eq. 36 qy s = ω 1 4k2 1 ω sech 2 ξ1 2 4k1 2ω 1 uy s = 8k 2 1ω 2 sech 2 ξ1 1 4k 2 1 ω1 4k2 1 ω + 4k2 1 ω sech 2 ξ1 2 2α 1 ϕ 1 y s = sk 1ω sech ξ 1 ω1 4k 2 1 ω1 4k2 1 ω + 4k2 1 ω sech 2 ξ1 xy s = y 2 ln 1 2k 1 ω tanh ξ1. 49 ω 1 + 2k 1 ω tanh ξ1 The requirement 4k 2 1ω 1 < 0 guarantees the nonsingularity of solution 49. In Fig. 1 we plot the single soliton solution of u and ϕ 1. When n = 2 λ 1 = k 2 1 1/4ω < 0 λ 2 = k 2 2 1/4ω < 0 we have Ψ 1 = coshξ 1 Ψ 2 = sinhξ 2 Φ 1 = e ξ1 Φ 2 = e ξ2 Ψ 1 Ψ 2 = k 2 coshξ 2 coshξ 1 k 1 sinhξ 2 sinhξ 1

6 408 HUANG Ye-Hui YAO Yu-Qin and ZENG Yun-Bo Vol. 53 W 2 Ψ 1 Ψ 2 Φ 1 = k 2 k 2 2 k2 1 sinhξ 1 W 2 Ψ 1 Ψ 2 Φ 2 = k 1 k 2 1 k 2 2coshξ 2 50 Then Eq. 44 with n = 2 gives rise to two soliton solution for Eq. 27 and the corresponding eigenfunction of Eq. 25. In the same way as we did on the one-soliton solution we can apply the method of variation of constants to get the two soliton solution of the ACHESCS 33 which together with Eq. 36 yields to the two soliton solution for CHESCS 7 with N = 2 λ 1 = k 2 1 1/4ω and λ 2 = k 2 2 1/4ω ry s = ω 2[log Ψ 1 Ψ 2 ys ξi= ξ i ψ i = 2ω 1 i+1 2α i sw 2Ψ 1 Ψ 2 Φ i 1 4ki 2ω i k2 k2 i W 1Ψ 1 Ψ 2 ξi= ξ i i = In Fig. 2 we plot the interactions of two soliton solution for u and ϕ 1 ϕ 2 which is shown that u is elastic collision. Fig. 1 Single soliton solutions for u and the eigenfunction ϕ 1 when w = 0.01 k 1 = 1 α 1s = 4s and s = 2. Fig. 2 Two soliton solutions for u and the eigenfunction ϕ 1 ϕ 2 when w = 0.01 k 1 = 2 k 2 = 1 α 1s = 2s and α 2s = 4s.

7 No. 3 On Camassa Holm Equation with Self-Consistent Sources and Its Solutions 409 Notice that the soliton solutions of CHESCS contains arbitrary s functions α s. This implies that the insertion of sources into the CH equation may cause the variation of the speed of soliton. In the same way as in Ref. [29 we may apply the method of variation of constant to find the N-soliton solution of Eq. 7 with λ i = ki 2 1/4ω > 0 i = 1...N from Eq. 36 ry s = 2ω 1 ω 2[log Ψ 1 Ψ 2... Ψ N ys ξi= ξ i ψ i = i+1 2α i sw 2Ψ 1 Ψ 2...Ψ N Φ i 1 4ki 2ω i k2 k2 i W.52 1Ψ 1 Ψ 2... Ψ N ξi= ξ i 6.2 Multicuspon Solutions Taking Ψ i and Φ i be the solutions of Eq. 25 when Q = 0 r = ω and λ i = k 2 i 1/4ω > 0 0 < k 1 < k 2 < < k n as follows Ψ i = sinhξ i i is an odd number Ψ i = cosh ξ i i is an even number 53 Φ i = e ξi 54 ξ i is given by Eq. 43. The n-cuspon solution Qy s and ry s of Eq. 27 and the corresponding eigenfunction φ i y s λ i of Eq. 25 with λ i = ki 2 1/4ω is given by Eq. 44. When n = 1 and 4k1ω 2 1 > 0 Eq. 44 gives rise to one cuspon solution for Eq. 27 and the corresponding eigenfunction of Eq. 25 with λ 1 = k1 2 1/4ω [8 9 Qy s = 2k 2 1 csch 2 ξ 1 ry s = ω + 4k2 1 ω3/2 csch 2 ξ 1 4k 2 1 ω 1 φ 1 = k 1 cschξ Similarly we may apply the method of variation of constant to find the solutions of the ACHESCS 33 by using the solution 55 of Eq. 27 and corresponding eigenfunction. Taking α 1 in Eq. 43 to be time-dependent functions α 1 s and requiring that Qy s = 2k 2 1 csch2 ξ1 ry s = ω + 4k2 1 ω3/2 csch 2 ξ1 4k 2 1 ω 1 ψ 1 = β 1 sk 1 csch ξ 1 56 satisfy the system 33 for N = 1 we find that α 1 s can be an arbitrary function of s and 2ω β 1 s = 1 4k1 2ω 2α 1 s. 57 So the one-cuspon solution of the CHESCS 7 with N = 1 and λ 1 = k1 2 1/4ω > 0 is obtained with respect to the variables y s from Eq. 36 qy s = ω 1 + 4k2 1 ω csch2 ξ1 2 8k uy s = 1 2ω2 csch 2 ξ1 4k1 2ω 1 1 4k1 2ω 1 + 4k2 1 ω + 4k2 1 ω csch2 ξ1 2 2α 1 ϕ 1 y s = sk 1ω csch ξ 1 ω1 xy s = y + 2 ln 1 2k 1 ω coth ξ k 2 ω 1 ω 1 + 4k2 1 ω + 4k2 1 ω csch2 ξ k 1 ω coth ξ1 In Fig. 3 we plot the one-cuspon solution of u ϕ 1. Fig. 3 Single cuspon solution for u and the eigenfunction ϕ 1 when w = 1 k 1 = 1 α 1s = 2s s = 2. Similarly we can apply the method of variation of constant to find the N-cuspon solution of Eq. 7 with λ i = ki 2 1/4ω < 0 i = 1...N from Eq. 36 ry s = 2ω 1 ω 2[log Ψ 1 Ψ 2... Ψ N ys ξi= ξ i ψ i = i+1 2α i sw 2Ψ 1 Ψ 2...Ψ N Φ i ki 2ω i k2 k2 i Ψ 1 Ψ 2... Ψ N ξi= ξ i

8 410 HUANG Ye-Hui YAO Yu-Qin and ZENG Yun-Bo Vol. 53 Furthermore in the same way we can find mixed k 1 - soliton-k 2 -cuspon solution for Eq. 7 with N = k 1 + k 2 λ i = ki 2 1/4ω > 0 i = 1...k 1 and λ i = ki 2 1/4ω < 0 i = K k 1 + k 2 by using Eqs. 44 and 36. In Fig. 4 we plot the one-positon solution of u and ϕ Multipositon Solutions Let λ = k 2 1/4ω λ i = ki 2 and take 1/4ω i = 1...N Ψ i = sinξ i i is an odd number Ψ i = cosξ i i is an even number 60 Φ i = cosξ i i is an odd number Φ i = sin ξ i i is an even number 61 ξ = k y 2ω3/2 s 4k 2 + ω + 1 ξ i = ξ k=ki. For N = 1 we have i=1 N k k 2 α i k k i Ψ 1 = sinξ 1 Ψ 1 1 = γ 1 cosξ 1 ωs ξ 1 = k 1 y 2k ω. γ 1 = ξ k and k=k1 = α 1 + y + 16k2 1 ω5/2 s 1 + 4k 2 1 ω2 2ω3/2 s 1 + 4k 2 1 ω 62 Ψ 1 Ψ 1 1 = k 1γ sin 2ξ 1 W 2 Ψ 1 Ψ 1 1 Φ 1 = 2k 2 1 sin ξ Then the one-positon solution of Eq. 27 and the corresponding eigenfunction for Eq. 25 is given by Eq. 40 with N = 1 m 1 = 1 Qy s = 2[log yy ry s = ω 2[log ys ψ 1 y s λ 1 = β 1 W 2 64 α 1 and β 1 are arbitrary constants. By using the method of variation of constants which means we change α 1 and β 1 into α 1 s and β 1 s we obtain the one-positon solution for the CHESCS 7 with N = 1 λ 1 = k1 2 1/4ω from Eq. 36 ry s = ω 2[log ys γ1= γ 1 ψ 1 y s = 2ω α 1 s k k 2 1 ω W 2 γ1= γ 1 γ 1 = α 1 s + y + 16k2 1ω 5/2 s 1 + 4k 2 1 ω2 2ω3/2 s 1 + 4k 2 1 ω 65 α 1 s is an arbitrary function of s. Fig. 4 One-positon solutions for u and the eigenfunction ϕ 1 when w = 0.01 k 1 = 1 α 1s = 2s s = 2. The positon solution of CHESCS is long-range analogue of soliton and is slowly decreasing oscillating solution. [33 In the same way we can find N-positon solution for Eq. 7. For a detailed discussion on positon solution we refer to Ref. [33. For N we have Ψ 1 i = γ i cosξ i i is an odd number Ψ 1 i = γ i sin ξ i i is an even number 66 We find that γ i = ξ = i k k k=ki ik 2 α i + y + 16k2 i ω5/2 s 1 + 4k 2 i ω2 2ω3/2 s 1 + 4k 2 i ω. = WΨ 1 Ψ Ψ N Ψ 1 N φ i = WΨ 1 Ψ Ψ N Ψ 1 N Φ i 67 and the N-positon solution of Eq. 27 and the corresponding eigenfunction for Eq. 25 is given by Qy s = 2[log yy ry s = ω 2[log ys. ψ y s λ i = β i W 2 i = 1... N 68 α i and β i are arbitrary constants. By using the method of variation of constants we obtain the N-positon solution for the CHESCS 7 from

9 No. 3 On Camassa Holm Equation with Self-Consistent Sources and Its Solutions 411 Eq. 36 ry s = ω 2[log ys γi= γ i 2ω 1 ψ i y s = k i 1 + 4ki 2ω i k + k i 1α i sw 2 γi= γ W i 1 γ i = k i k 2 α i s + y + 16k2 i ω5/2 s 1 + 4k 2 i i ω2 αs is an arbitrary function of s. In Fig. 5 we plot the one-negaton solution of u and ϕ 1. 2ω3/2 s 1 + 4k 2 i ω 69 α i s are arbitrary functions of s. 6.4 Multinegaton Solutions Let λ = k 2 1/4ω > 0 λ i = k 2 i 1/4ω > 0 i = 1...N and take ξ = k Ψ i = sinhξ i i is an odd number Ψ i = cosh ξ i i is an even number. 70 Φ i = e ξi 71 ξ i = ξ k=ki then we have and y + 2ω3/2 s 4k 2 ω 1 + i=1 Ψ 1 = sinhξ 1 Ψ 1 1 = γ 1 coshξ 1 ωs ξ 1 = k 1 y + 2k1 2 1/4ω N k k 2 α i k k i γ 1 = α 1 + y + 16k2 1ω 5/2 s 4k 2 1 ω ω3/2 s 4k 2 1 ω 1 72 Ψ 1 Ψ 1 1 = k 1γ sinh 2ξ 1 W 2 Ψ 1 Ψ 1 1 Φ 1 = 2k 2 1 sinhξ Then the one-negaton solution of Eq. 27 and the corresponding eigenfunction for Eq. 25 is given by Qy s = 2[log yy ry s = ω 2[log ys φ 1 y s λ 1 = β 1 W 2 74 α and β are arbitrary constants. By using the method of variation of constants we obtain the one-negaton solution for the CHESCS 7 with N = 1 λ 1 = k 2 1 1/4ω from Eq. 36 ry s = ω 2[log ys γ1= γ 1 ψ 1 y s = 2ω α 1 s k 1 4k 2 1 ω 1 W 2 γ1= γ 1 γ 1 = α 1 s + y + 16k2 1 ω5/2 s 4k 2 1 ω ω3/2 s 4k 2 1 ω 1 75 Fig. 5 One-negaton solution for u and the eigenfunction ϕ 1 when w = 0.01 k 1 = 1 α 1s = 2s s = 2. Similarly we can find N-negaton solution for CHESCS 7 with λ i = ki 2 1/4ω i = 1... N. For N we have Ψ 1 i = γ i coshξ i i is an odd number Ψ 1 i = γ i sinhξ i i is an even number 76 γ i = ξ k k=k i = k i k 2 α i + y i + 16k2 i ω5/2 s 4ki 2ω + 2ω3/2 s 12 4ki 2ω 1. We find that = WΨ 1 Ψ Ψ N Ψ 1 N φ i = WΨ 1 Ψ Ψ N Ψ 1 N Φ i 77 and the N-negaton solution of Eq. 27 and the corresponding eigenfunction for Eq. 25 is given by Qy s = 2[log yy ry s = ω 2[log ys ψ y s λ i = β i W 2 i = 1...N 78 α i and β i are arbitrary constants. By using the method of variation of constants we obtain the N-negaton solution for the CHESCS 7 from Eq. 36 ry s = ω 2[log ys γi= γ i

10 412 HUANG Ye-Hui YAO Yu-Qin and ZENG Yun-Bo Vol. 53 2ω 1 ψ i y s = k i 4ki 2ω 1 i k + k i α i sw 2 γi= γ i γ i = k i k 2 α i s + y + 16k2 i ω5/2 s 4k 2 i i ω ω3/2 s 4k 2 i ω 1 79 α i s are arbitrary functions of s. 7 Conclusion The CHESCS and its Lax representation are derived. Conservation laws are constructed. It is reasonable to speculate on the potential application of CHESCS that is CHESCS may describe the interaction of different solitary waves in shallow water. Since SESCS can be regarded as soliton equations with non-homogeneous terms we look for explicit solutions by using the method of variation of constants. By considering a reciprocal transformation which relates CH equation to an alternative of ACH equation we propose a similar reciprocal transformation which relates the CHESCS to ACHESCS. By using the Darboux transformation one can find the n- soliton and n-cuspon solution as well as n-positon and n-negaton solution of alternative ACH equation. Then by means of the method of variation of constants we can obtain N-soliton N-cuspon N-positon and N-negaton solutions of the ACHESCS. Finally using the inverse reciprocal transformation we obtain N-soliton N-cuspon N-positon and N-negaton solutions of the CHESCS. References [1 B. Fuchssteiner and A.S. Fokas Physica D [2 R. Camassa and D. Holm Phys. Rev. Lett [3 R. Camassa D. Holm and J. Hyman Adv. Appl. Mech [4 A. Parker Proc. R. Soc. Lond. A [5 R.S. Johnson Proc. R. Soc. Lond. A [6 Z.J. Qiao Commun. Math. Phys [7 R. Beals D.H. Sattinger and J. Szmigielski Adv. Math [8 Y.S. Li and J.E. Zhang Proc. R. Soc. Lond. A [9 Y.S. Li J. Nonlinear Math. Phys [10 Z.J. Qiao and G.P. Zhang Euro. Phys. Lett [11 H. Holden J. Hyp. Diff. Equ [12 J. Lenells J. Phys. A: Math. Gen [13 V.K. Mel nikov Phys. Lett. A [14 V.K. Mel nikov Commun. Math. Phys [15 V.K. Mel nikov Commun. Math Phys [16 D.J. Kaup Phys. Rev. Lett [17 J. Leon and A. Latifi J. Phys. A: Math. Gen [18 C. Claude A. Latifi and J. Leon J. Math. Phys [19 M. Nakazawa E. Yomada and H. Kubota Phys. Rev. Lett [20 E.V. Doktorov and V.S. Shchesnovich Phys. Lett. A [21 Y.B. Zeng Physica D [22 Y.B. Zeng W.X. Ma and Y.J. Shao J. Math. Phys [23 Y.B. Zeng Y.J. Shao and W.M. Xue J. Phys. A: Math. Gen [24 T. Xiao and Y.B. Zeng J. Phys. A: Math. Gen [25 R.L. Lin Y.B. Zeng and W.X. Ma Physica A [26 V.B. Matveev Theor. Math. Phys [27 C. Rasinariu U. Sukhatme and A. Khare J. Math. Gen [28 H.X. Wu Y.B. Zeng and T.Y. Fan Inverse Problems [29 H.X. Wu X.J. Liu Y.H. Huang and Y.B. Zeng Appl. Math. Comput doi: /.amc [30 J. Schiff Physica D [31 A. Hone J. Phys. A L307. [32 V.B. Matveev and M.A. Salle Darboux Transformations and Solitons Springer-Verlag Berlin [33 R. Ivanov Phys. Lett. A