BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES OF MULTI-COMPONENT INTEGRABLE SYSTEMS

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1 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES OF MULTI-COMPONENT INTEGRABLE SYSTEMS JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU Abstract. In this paper the Bäcklund transformation based-approach is explored to obtain Hamiltonian operators of multi-component integrable systems which are governed by a compatible tri-hamiltonian dual structures. The resulting Hamiltonian operators are used not only to derive multi-component bi- Hamiltonian integrable hierarchies and their dual integrable versions but also to serve as a criterion to verify the compatibility for the corresponding dual bi-hamiltonian operators. The approach is illustrated through the construction of two families of integrable bi-hamiltonian hierarchies associated with certain two-component and three-component dispersive water wave systems and their modifications. The corresponding dual bi-hamiltonian systems are also derived. Some of them are new integrable systems. Furthermore we classify the analytic and nonanalytic traveling wave solutions to the two-component dual nonlinearly dispersive water wave system. Key words: bi-hamiltonian structure; Bäcklund transformation; tri-hamiltonian duality; dispersive water wave system; nonanalytic traveling wave. 000 Mathematics Subject Classification. Primary: 37K10 37K35 35Q Introduction In this paper we develop the Bäcklund transformation based-method in the multicomponent setting to derive Hamiltonian operators which admit compatible tri-hamiltonian structures and consequently lead to the associated bi-hamiltonian integrable hierarchies and their dual counterparts endowed with nonlinear dispersion. The algebraic theory of integrable bi-hamiltonian systems of evolution equations was developed in detail over the last three decades. The innovative work due to Magri [37] establishes for a bi-hamiltonian system the existence of an infinite hierarchy of mutually commuting conservation laws and bi-hamiltonian flows. On the one hand the recursion operator [38 39] see also [41] can be derived from the bi-hamiltonian structure of such integrable systems and thus used to generate integrable hierarchies and higher-order symmetries. On the other hand the bi-hamiltonian property is closely related to the existence of a Lax pair representation. It has been found that a notable number of integrable systems are in fact bi-hamiltonian. Those include a variety of well-known classical integrable systems such as the Korteweg de Vries KdV and modified Korteweg de Vries mkdv equations [41] as well as integrable systems endowed with nonlinear dispersion such as the Camassa- Holm CH equation the modified Camassa-Holm mch equation etc. [5 15 4]. Several methods have been employed to obtain integrable bi-hamiltonian systems endowed with nonlinear dispersion. In particular a theory of tri-hamiltonian duality was developed systematically in the references [ ]. This approach starts from the basic observation that most of integrable soliton equations which are known to possess a bi- Hamiltonian structure actually support a compatible trio of Hamiltonian structures through 1

2 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU an elementary scaling argument. Rearranging the three Hamiltonian operators provides an algorithmic method for systematically constructing dual integrable systems. Some typical dual integrable systems including the CH equation the mch equation and certain two-component CH equations etc. can be obtained in this manner. The resulting dual systems are endowed with nonlinear dispersion and thus admit non-smooth solitons including compactons cuspons peakons and more exotic species [35]. More precisely applying tri-hamiltonian duality to the bi-hamiltonian representation of the KdV equation and the mkdv equation the resulting dual integrable systems are the well-studied CH equation [ ] and the mch equation [ ]. As two prototypical models in the class of the bi-hamiltonian integrable equations with quadratic and cubic nonlinearity respectively the CH equation and the mch equation have attracted enormous attention in recent years because of their remarkable properties: complete integrability [ ] physical relevance of the nonlinear shallow-water waves [ ] non-smooth soliton structures of peakons and multi-peakons [3 5 6 ] delicate geometric formulations [9 30] and the presence of breaking waves [ ]. In addition to the scalar setting the approach of tri-hamiltonian duality can also be applied to multi-component bi-hamiltonian systems with matrix-valued Hamiltonian operators. For instance it was proved in [4] that the bi-hamiltonian structure of the integrable Ito system [6] u t = u xxx + 3uu x + vv x v t = uv x 1.1 supports the required tri-hamiltonian dual structure. The corresponding dual bi-hamiltonian system takes the form m t + u x m + um x + ρρ x = 0 ρ t + u ρ x = 0 m = u u xx 1. which is the so-called two-component CH system [7 1]. It together with several generalizatons has recently been extensively studied from a variety of perspectives [ ]. It is well-known that Bäcklund transformations play an important role in soliton theory and integrable systems and are a useful tool to obtain new integrable systems from some known integrable ones and to construct new solutions from known ones [44]. A Bäcklund transformation is a system of first-order partial differential equations relating solutions of two equations under consideration. The approach we adopt here generalizes the Bäcklund transformation method and is based on the tri-hamiltonian dual structure. We shall focus our attention on the two-component and three-component systems. Our analysis is based on generalized Miura-type Bäcklund transformations depending on several arbitrary parameters which on the one hand can be utilized to give rise to the Hamiltonian operators supporting tri-hamiltonian structures and on the other hand can serve as an alternative method to verify the compatibility of the desired Hamiltonian pair. In such situations the rearrangement of the Hamiltonian triple will produce two pairs of compatible Hamiltonian operators which lead to the multi-component integrable systems and their corresponding dual counterparts respectively. In particular we exploit this approach to derive Hamiltonian operators with appropriate parameters which in view of the intrinsic tri-hamiltonian structure produce two classes of integrable hierarchies involving the socalled dispersive water wave DWW system as well as the corresponding dual integrable version. The DWW system takes the following form [4 31] q t = q x + qr x r t = r x + r + q x 1.3 which is an integrable bi-hamiltonian system and is recognized to be in the form of a bidirectional system of Boussinesq type modeling the propagation of shallow water waves [40]. In light of the tri-hamiltonian characterization of the bi-hamiltonian structure of system

3 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES we derive the corresponding dual integrable system in the form g t νg xt = g x + fg ν fg x x f t + νf xt = f x + g + f + ν ff x x 1.4 where ν = ±1. The bi-hamiltonian structure of system 1.4 naturally follows from the tri- Hamiltonian structure of system 1.3. In addition Lax formulations admitted by systems 1.3 and 1.4 are also obtained. In [31] it was shown that via the Miura-type Bäcklund transformation q = u x u + uv r = v 1.5 the DWW system 1.3 is mapped to the modified DWW mdww system u t = u x + uv u x v t = v x u x u + uv + v x. 1.6 Making use of the recursion operator of the DWW system and the Bäcklund transformation 1.5 we establish a bi-hamiltonian representation of the mdww system 1.6 and its dual counterpart. Furthermore the following three-component integrable system s t = sr x q t = q x + qr + 1 s x r t = r x + r + q x 1.7 was proposed in [1] as a model of a quadri-hamiltonian system. It is noted that 1.7 reduces to the DWW system 1.3 when s = 0 and hence 1.7 can be regarded as an integrable three-component generalization. In this paper we introduce a general Bäcklund transformation that enables us to derive a 3 3 Hamiltonian operator that induces a compatible tri-hamiltonian structure; this allows us to construct two families of three-component integrable hierarchies involving the system 1.7 and its dual. In addition again using corresponding Bäcklund transformations several modified versions of these three-component integrable systems and their related dual systems are also derived. The existence of peaked solitons is one of the non-trivial properties of nonlinearly dispersive wave equations of CH type [5] which helps explain why these systems have attracted so much attention in the last thirty years. Recently it was found that the mch equation as the dual equation of the mkdv equation also admits peaked solitons []. However the two-component CH system 1. which is the dual system of the Ito system does not admit non-trivial peaked solitons [7 1 5]. A natural question remains: what types of dual integrable systems obtained through the tri-hamiltonian duality approach admit peaked solitons. To this end it is of interest to study and classify the traveling wave solutions of the dual system of the DDW system and its modified versions leading to some new types of nonanalytic solitary wave solutions. We refer the reader to [9] for further analysis of the existence non-existence stability and other properties of solitary wave solutions to higher order wave models. The remainder of this paper is organized as follows. In Section we first formalize some notations and definitions in the multi-component setting of Hamiltonian operators bi-hamiltonian structures as well as Bäcklund transformations and recall some basic results required throughout this paper. Next we present two theorems Theorem. and Theorem.3 which demonstrate the relationship not only between the matrix Hamiltonian operators but also between the matrix recursion operators subject to multi-component Bäcklund transformations. In Section 3 we combine the Bäcklund transformations with tri-hamiltonian duality to study two-component bi-hamiltonian integrable hierarchies involving the DWW and mdww systems as well as their dual systems. We use the same approach to propose several three-component bi-hamiltonian integrable systems and the

4 4 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU corresponding dual versions in Section 4. Finally in Section 5 the analytic and nonanalytic traveling wave solutions to the two-component dual DWW system are fully classified.. Preliminaries Throughout this paper we consider evolution equations involving a single spacial variable x R and time t R. We let A denote the space of differential functions and understand the functions in A depending on the indicated dependent variables and their spatial derivatives only. We further define A n to be the space of n-component differential functions. Consider an n-component system of evolution equations u t = Ku u = u 1 t x... u n t x T.1 where Ku = K 1 u... K n u T A n is an n-component differential function depending on the components of u and their x-derivatives up to a given order. The system.1 is called Hamiltonian if it can be written in the form u t = Ku = J δhu where Hu is the Hamiltonian functional δhu = δh/δu 1... δh/δu n T is the variational derivative of H and the n n matrix operator J is a Hamiltonian operator [41]. For a candidate Hamiltonian operator J its corresponding Poisson bracket is defined by L inner product for the functions in the Schwartz space } P L = δp J δl = δp J δl dx J which is required to be both skew-symmetric: P Q } = Q P } J J and satisfy the Jacobi identity: } P Q R} + Q R } P} + R P } Q} = 0 J J J J J J. for all functionals P Q and R cf. [41]. The system.1 is said to be bi-hamiltonian if it can be written in the form u t = Ku = J 1 δh 1 u = J δh 0 u.3 where H 0 u and H 1 u are the Hamiltonian functionals J 1 and J are independent n n Hamiltonian operators satisfying the compatibility condition that every linear combination c 1 J 1 +c J is Hamiltonian i.e. satisfies the Jacobi identity.. Assume that the operator J 1 of the Hamiltonian pair is nondegenerate then the operator R = J J1 1 is a recursion operator of the bi-hamiltonian system.3 [37 41]. The following theorem due to Magri [37] see also [41] summarizes the basic properties of bi-hamiltonian systems. Theorem.1. Consider a bi-hamiltonian system of evolution equations.3. that the Hamiltonian operator J 1 is nondegenerate. Let R = J J1 1 and K 0 u = K 1 0u... K n 0 u T = J1 δh 0. For each m = 1... define K m u = R K m 1 u. functionals H 0 H 1 H... such that i for each m 1 the evolution system Assume Then there exists a sequence of u t = K m u = J 1 δh m u = J δh m 1 u.4 is a bi-hamiltonian system; ii the Hamiltonian functionals H m are all in involution with respect to either Poisson bracket: H l H m } J1 = 0 = H l H m } J l m 0

5 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 5 and hence provide an infinite collection of conservation laws for each of the bi-hamiltonian systems.4. The method of tri-hamiltonian duality [19 4] can be applied to the bi-hamiltonian systems.4 whose associated Hamiltonian operator J can be written as the sum of two distinct Hamiltonian operators K and K 3 that scale independently under x λx and/or u µu. The fact that under the scaling transformations J = K + K 3 is mapped into J = λ α K + λ β K 3 with α β immediately implies that K and K 3 form a compatible Hamiltonian pair. We thus conclude that each linear combination of J 1 = K 1 K and K 3 is a Hamiltonian operator. In particular each possible pair of K 1 K and K 3 is compatible. As a consequence by introducing an alternative Hamiltonian pair Ĵ1 = K 1 ±K and Ĵ = K 3 one can produce a hierarchy of integrable equations which can be viewed as the dual counterpart called the tri-hamiltonian dual in [4] of the original hierarchy.4 that is generated by J 1 and J. To illustrate the method more precisely let us consider the scalar setting so n = 1. We assume that K 1 = x and K are constant coefficient skew-adjoint differential operators and further that Ĵ1 factorizes into a product of x with a self-adjoint constant coefficient differential operator A i.e. Ĵ 1 = x A. One usually introduce the new variable ρ = Au to replace u while Ĵ is obtained from K 3 by replacing u by ρ. Then the dual integrable system is found by applying the recursion operator R 1 = ĴĴ1 = ĴA 1 x 1 to the seed equation ρ t = ρ x the resulting dual counterpart will thus take the form ρ t = ĴA 1 ρ. Hence the corresponding dual integrable system can be written in local form in terms of the dependent variable u. Furthermore applying the recursion operator R succesively to ρ t = ρ x produces the dual hierarchy of integrable systems and each flow in the hierarchy will be a bi-hamiltonian system ρ t = K m ρ = Ĵ1δĤm = ĴδĤm 1 which is governed by the dual Hamiltonian pair Ĵ1 and Ĵ. We now consider the effect of a Bäcklund transformation relating n-component systems.1 involving u and a similar system involving the transformed dependent variables ũ: where Gũ = G 1 ũ... G n ũ T A n. ũ t = Gũ ũ = ũ 1 t x... ũ n t x T.5 Definition.1. An n-component implicit equation of the form Bu ũ = B 1 u ũ... B n u ũ T = 0.6 where each B i u ũ i = 1... n is a differential function depending on u ũ and their firstorder x-derivatives is called a Bäcklund transformation between systems.1 and.5 if whenever ut x ũt x are any two solutions of.1 and.5 respectively such that.6 holds at one time t = t 0 then.6 holds identically for all t x with t > t 0. Assume that systems.1 and.5 are both Hamiltonian systems. Given a Bäcklund transformation.6 relating.1 and.5 the relationship between their respective Hamiltonian operators can be established. Theorem.. Given a Bäcklund transformation.6 between the n-component systems.1 and.5 let B u and Bũ denote the n n matrix differential operators given by their Fréchet derivatives with respect to u ũ respectively and assume B u and Bũ are invertible. Set T = B 1 ũ B u.7

6 6 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU and let T denote its formal adjoint. If.1 is Hamiltonian with Hamiltonian operator J then.5 is also Hamiltonian with Hamiltonian operator J ũ = T J u T..8 The effect of a Bäcklund transformation on recursion operators also involves the operator.7. Theorem.3. Consider a Bäcklund transformation.6 between the n-component systems.1 and.5 and let T be the n n matrix differential operator given in.7. If Ru is a recursion operator for.1 then is a recursion operator admitted by.5. Rũ = T Ru T 1 The proofs for Theorems. and.3 in the scalar case n = 1 were given respectively in [0] and [17]. We now prove Theorems. and.3 in multi-component case. Let us begin with some notations and preliminary remarks. Taking the t-derivative of each equation in system.6 and using systems.1 and.5 we arrive at the following identities n Biuj K j + B iũj G j = 0 i = 1... n. j=1 For convenience we write the above expression using vectorial notation: B u K + Bũ G = 0..9 Since Bũ is invertible.9 together with.7 immediately leads to G = TK..10 Next the nondegeneracy assumption on Bũ ensures that the Bäcklund transformation.6 uniquely determines for each i = 1... n which implies that u i = U i ũ 1... ũ n.11 B i U 1 ũ 1... ũ n... U n ũ 1... ũ n ũ 1... ũ n = 0 i = 1... n.1 hold identically for ũ = ũ 1... ũ n T. Then we denote the total derivative of the differential function B = B 1... B n T A n with respect to ũ by dũb which is a n n matrix differential operator with entries dũb ij = dũj B i i j = 1... n defined for α = α 1... α n T by n dũb α = dũj B 1 αj... j=1 n dũj B n αj where in view of.11 and.1 for each i = 1... n n n n dũj B i αj = B iũj + B iuk U kũj α j j=1 j=1 with U kũj being the Fréchet derivative of differential function U k with respect to ũ j for j k = 1... n. Therefore we arrive at j=1 k=1 dũb = Bũ + B u U 1... U n ũ 1... ũ n T

7 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 7 where the last term denotes the n n matrix differential operator with entries U1... U n = U iũj i j = 1... n. ũ 1... ũ n Since the relation.1 implies dũb 0 it then follows that ij U 1... U n ũ 1... ũ n = T Finally for the n n matrix differential operator T denote its partial and total derivatives with respect to ũ along a direction α by ũt [α] and dũt [α] respectively. To be precise using entries T ij i j = 1... n the partial derivative ũt [α] along the direction α = α 1... α n T is a n n matrix differential operator with the corresponding entries ũt [α] ij n = T ijũk [α k ] i j = 1... n k=1 where T ijũk [ ] is the partial derivative of T ij with respect to ũ k along some direction. While the total derivative dũt [α] is also a n n matrix differential operator which has entries n dũt [α] = dũk T ij [α k ] i j = 1... n ij where k=1 dũk T ij [α k ] = T ijũk [α k ] + n T ijul [U lũk α k ] and T ijul [ ] is the partial derivative of T ij with respect to u l along some direction. Consequently using.13 we deduce that l=1 dũt [α] = ũt [α] u T [ T 1 α ]..14 Proof of Theorem.. It is obvious that the skew-symmetric property of J is preserved subject to the transformation.8. So it remains to verify the Jacobi identity for J which in the framework of n-component systems is equivalent to P ; dũ J [ J L ] Q + Q ; dũ J [ J P ] L + L ; [ ] dũ J J Q P = 0.15 where P Q L A n are the variational derivatives of the indicated Hamiltonian functionals P Q and L respectively. In view of the expressions.7 and.8 we find dũ J [ J P ] L = dũt [ J P ] J T L + T dũj [ J P ] T L + TJ dũt [ J P ] L..16 Using the fact that J is skew-symmetric and the symmetry property of the operator T say dũt [ α ] P ; f = dũt [ T f ] P ; T 1 α = P ; dũt [ T f ] T 1 α.17 for f = f 1... f n T and α = α 1... α n T we compute the inner product of Q and the third term in the expression.16 as Q ; TJ dũt [ ] J P L = T Q ; J dũt [ ] J P L dũt = [ ] J P L ; J T Q = L ; dũt [ ] J Q J T P. Now denote L ; dũt [ ] J Q J T P = L Q P J.

8 8 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU Then we have Q ; TJ dũt [ ] J P L = L Q P J and Q ; dũt [ ] J P J T L = Q P L J. Set for simplicity P = T P Q = T Q and L = T L. A direct computation shows that Q ; T dũj [ ] J P T L = Q ; T u J [ T 1 J P ] L = Q ; u J [ J P ] L. Hence we deduce that the second term in the Jacobi identity.15 becomes Q ; [ ] dũ J J P L = Q P L J L Q P J Q ; u J [ J P ] L which implies that the Jacobi identity associated to J is equivalent to that corresponding to J. We thus complete the proof of Theorem.. Turning to the proof of Theorem.3 the following lemma is useful. Lemma.1. Let T be the matrix differential operator given by.7. Then subject to the Bäcklund transformation.6 there holds dũt [ α ] f = dũt [ Tf ] T 1 α.18 for f = f 1... f n T and α = α 1... α n T. Proof. We start with some operator identities required. The first two formulae present the symmetry property of the corresponding second-order derivatives. More precisely for α = α 1... α n T and β = β 1... β n T [ ] [ ] [ ] [ ] u Bũ α f = ũb u f α and ũbũ α β = ũbũ β α. The third formula is associated with the partial derivative of the inverse operator: ũb 1 [ ] ] ũ α = B 1 ũ ũbũ[ α B 1 ũ..19 We are now in a position to prove the lemma. By.14 [ dũt α ] f = ũt [ α ] f u T [ α ][ T 1 α ] f. In view of.19 we obtain [ ũt α ] f = 1 [ ] ũ α Bu f + B 1 [ ] ũ ũb u α f = B 1 ] ũ ũbũ[ α B 1 ũ B uf + B 1 [ ] ] [ ] ũ ũb u α f = B 1 ũ ũbũ[ α Tf + B 1 ũ ũb u α f and u T [ T 1 α ] f = u B 1 [ ũ T 1 α ] B u f + B 1 [ ũ u B u T 1 α ] f = B 1 ũ u Bũ[ T 1 α ] B 1 ũ B uf + B 1 [ ũ u B u T 1 α ] f = B 1 ũ u Bũ[ T 1 α ] Tf + B 1 [ ũ u B u T 1 α ] f. On the other hand using the symmetry formulae of the second-order derivatives.13 and.19 the right hand side of.18 becomes [ dũt Tf ] T 1 α = ũt [ Tf ] T 1 α u T [ f ] T 1 α = B 1 ũ + B 1 ũ which suffices to prove the lemma. = ũb 1 [ ] ũ Tf Bu T 1 α + B 1 [ ] ũ ũb u Tf T 1 α u B 1 [ ] f Bu T 1 α B 1 [ ] u B u f T 1 α ũ ũbũ[ α ] Tf + B 1 ũ ũ u Bũ [ T 1 α ] Tf [ ] [ ũb u α f B 1 u B u T 1 α ] f ũ

9 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 9 Proof of Theorem.3. First since Ru is a recursion operator of system.1 we deduce that the differential function K = K 1 u... K n u T together with Ru satisfies the following operator identity u R [ K ] = [ K u R ] = K u R RK u where the n n matrix differential operator K u with entries K u ij = K iuj i j = 1... n is the Fréchet derivative of K and u R u [ K ] or u R [ K ] for brevity is the partial derivative of operator R along the direction K. Next according to Lemma.1 and using the formulae.10 and.13 we have for α = α 1... α n T ũ R ũ [ G ] α = dũtrt 1[ G ] α = dũt [ G ] RT 1 α + T dũru v [ G ] T 1 α + TR dũt 1[ G ] α = dũt [ TRT 1 α ] T 1 G T u R [ T 1 G ] T 1 α TRT 1 dũt [ G ] T 1 α = dũt [ Rα ] K + T u R [ K ] T 1 α + R dũt [ α ] K. Since Ru is a recursion operator of system.1 we have On the other hand T u R [ K ] T 1 α = TK u T 1 Rα RTKu T 1 α. Gũ Rα RGũα = dũtk [ Rα ] + R dũtk [ α ] Hence the identity = dũt [ Rα ] K T dũk Rα + R dũt [ α ] K + RT dũk α = dũt [ Rα ] K + TKu T 1 Rα + R dũt [ α ] K RTK u T 1 α. ũ R[ G ] = Gũ R RGũ = [Gũ R] holds completing the proof of the theorem. In view of Theorem. if we introduce a generalized Bäcklund transformation that involves certain arbitrary parameters and apply it to a given Hamiltonian operator then.8 implies that the transformed Hamiltonian operator will depend on the associated parameters. This motivates us to analyze families of n-component Bäcklund transformations depending on for example three constant parameters α β γ to construct a new n n matrix Hamiltonian operator J from a given n n matrix Hamiltonian operator J hoping that the resulting operator takes the form J = α K 1 + β K + γ K 3.0 where α β and γ depend on α β and γ. Assuming the three operators K 1 K and K 3 appearing in.0 scale independently under scaling transformations of the dependent variables they then form a compatible Hamiltonian triple meaning that each linear combination.0 is a Hamiltonian operator. As a consequence two pairs of compatible Hamiltonian operators are readily constructed by taking distinct recombinations of the Hamiltonian triple in which one pair will typically generate a classical soliton system while the other dual counterpart is an integrable bi-hamiltonian system which usually is endowed with nonlinear dispersion [4]. Moreover it is worth mentioning that since such linear combinations of the recombined pairs are members of the 3-parameter family.0 this argument also serves to automatically verify the compatibility of two Hamiltonian operators.

10 10 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU In the next two sections we shall demonstrate the efficacy of this approach through examples of two-component and three-component systems which arise from the models in the shallow water wave propagation. 3. The two-component DWW and mdww systems Consider the following dispersive water wave DWW system [31] q t = q x + qr x r t = r x + q + r x 3.1 which is one of the dispersive generalizations of the classical dispersiveless long wave equation and belongs to the family of the bi-directional Boussinesq-type systems modeling the propagation of shallow water waves [4 40]. The DWW system is related to the modified dispersive water wave mdww system u t = u x + uv u x v t = v x u x u + uv + v x 3. via the Miura-type transformation [ 31] q = u x u + uv r = v. 3.3 Moreover the mdww system admits the following constant coefficient Hamiltonian operator 0 x J u v =. 3.4 x x We introduce a family of Miura-type transformations q = γ u x + u uv + ᾱ β u r = γ β β u + βv 3.5 where β = β/γ and α β and γ βγ 0 are arbitrary constants. Applying 3.5 and using 3.4 as the original Hamiltonian operator we can obtain a 3-parameter family of Hamiltonian operators γq x + x q β x J + α x + γr x q r =. 3.6 β x + α x + γ x r γ x Obviously J is a linear combination of the following triple of Hamiltonian operators: 0 x 0 K 1 = K = x q x + x 0 x K3 = x q r x. 0 x r x We now define the operators J 1 = K 1 + ν K J = J where ν is a constant. Since γ in J should be nonzero we set γ = 1 without loss of generality. The fact that the linear combinations c 1 J1 + c J are members of the 3-parameter family of Hamiltonian operators 3.6 justifies their compatibility and thus Magri s Theorem [37] establishes the formal existence of a hierarchy q r t = G n = J 1 δ H n = J δ H n 1 n = 1... δ H n = δ H n δq δ H T n 3.7 δr of higher-order commuting bi-hamiltonian systems with associated higher-order Hamiltonian functionals H n n = which are conservation laws common to all members of

11 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 11 the hierarchy. The members in the hierarchy 3.7 are obtained by applying successively the recursion operator R = J 1 J 1 to the seed system q q = G r 1 = = r J 1 δ H 1. t For the second flow in this hierarchy q = G r = RG 1 = J 1 δ H = J δ H we consider two cases: t x Case 1. When ν = becomes q t = βq x + αq + qr x r t = βr x + αr + q + r x 3.9 with the associated Hamiltonian functionals βrx H 1 = q r dx H = q + r q + q + αqr dx. The system 3.9 admits the following Lax pair with spectral parameter λ: ψ1 ψ1 ψ1 ψ1 = U = V 3.10 ψ where ψ x ψ U = 1 λ + 1 r + α β V = 1 β λ + r x + 1 β β λ + r + α λ + 1 β β q ψ r + α t r + α λ β q + β q x β λ r x 1 β r β + α q r + α Case. When ν 0 without loss of generality we set ν = ±1. Setting q = g νg x r = f + νf x allows us to write the resulting bi-hamiltonian system 3.8 in the local form q t = βg x + αg + q + gf x r t = βf x + αf + g + rf x 3.1 or in full detail g t νg xt = βg x + αg + g νg x f x f t + νf xt = βf x + αf + g + f + νf x f x where the Hamiltonian functionals are H 1 = f νf x g dx H = βgx + αg + rg f dx.

12 1 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU The system 3.1 admits the Lax pair 3.10 with α U = 1 β + 1 β f + ν f x λ β g ν g x λ α β 1 β f + ν f x 1 β fx + α f + f 1 + ν ff x ν fgx V = β λ β gf + β g x λ f 1 β fx + α f + f. + ν ff x β The tri-hamiltonian duality argument implies that the dual integrable bi-hamiltonian system is obtained by rearranging the Hamiltonian triple to form the dual Hamiltonian pair 0 x x Ĵ 1 = q x + x + x Ĵ 0 = x q r x x r x The dual counterpart of the DWW system 3.1 is thus q t = q + gf x q = g g x r t = g + rf x r = f + f x 3.14 which in fact belongs to the integrable family 3.1 derived in Case. Observe that the second Hamiltonian operator Ĵ in 3.13 admits the Casimir functional Ĥ C = 4 q r dx with variational derivative T T δĥc δĥc = δ q δĥc = δ r r. 4 q r 4 q r On the one hand the functional Ĥ C is an additional conservation law admitted by 3.14; on the other hand it leads to the associated Casimir system q r = Ĝ 1 = Ĵ1δĤC t which takes the explicit form q t = x + x r r t = x + 4 q r x q r Starting from the Casimir system 3.15 one formally constructs an infinite hierarchy of higher-order commuting bi-hamiltonian systems and corresponding Hamiltonian functionals Ĥ n} in the negative direction: q r = Ĝ n = Ĵ1δĤ n = ĴδĤ n+1 n = 1... with Ĥ 1 = ĤC and δĥ n = t δĥ n/δ q δĥ n/δ r T. In the case of α = 0 β = γ = 1 the Bäcklund transformation 3.5 reduces to 3.3 which connects the DWW system 3.1 and the mdww system 3.. Now let us focus our attention on the mdww system 3.. First of all in view of Theorem.3 and transformation 3.3 we deduce that the recursion operator Ru v of mdww system satisfies x + u v u 0 1 Ru v = Rq r x + u v u 0 1

13 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 13 where Rq r = J 1 J 1 is the recursion operator for the DWW system. It follows that v u x u + Ru v = x u x 1 v u x u + x + x v x 1. The second Hamiltonian operator admitted by the mdww system 3. is thereby obtained u x + J u v = RJ 1 = x u x + x u + v x x + u x + x v v x + x v where J 1 = J u v given by 3.4 is the first Hamiltonian operator of system 3.. Therefore the mdww system 3. possesses the bi-hamiltonian structure u = J v 1 δh = J δh 1 t with Hamiltonian functionals H 1 = uv u dx H = uv x + v uv dx. In addition the Lax formulation 3.10 for the mdww system 3. is based on U = 1 λ + v ux + u uv λ v V = 1 λ + v x + v x v + λ u x + u uv v λ λ v x v. Next note that the mdww system 3. also admits the tri-hamiltonian structure and its dual version relies on the recombined dual Hamiltonian pair 0 x x J 1 = ū x + x + x J = x ū x ū + v x x ū x + x v v x + x v Applying the dual bi-hamiltonian structure 3.16 to the seed system ū ū = K v 1 = v t we obtain the integrable bi-hamiltonian hierarchy ūt = K v n = J 1 δh n = J δh n 1 n Z 3.17 t with δh n = δh n /δū δh n /δ v T. The initial members of the sequence of Hamiltonian functionals are H 0 = 1 ū + v dx H 1 = g + g x v g + g 4 x dx H = 1 gx g f f xx + g + 3g x g xxx f + g + g x g xx fx dx with ū = 1 xg and v = 1 xf. In particular the case n = in 3.17 corresponds to the dual mdww system ū t = g + g x g x g f x + f + g x f x + fū x v t = g + g x g x g f x + f + g x f x + f v x. Again the second Hamiltonian operator J admits a Casimir functional ū H C = ū v dx x

14 14 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU leading to the following Casimir system ū t = x x ū ū v v t = x 1 ū v x v ū v 3.18 for the dual mdww system which is the first member of the dual hierarchy 3.17 in the negative direction. Interestingly when ū = reduces to 1 v t = x + x v Using the reciprocal transformation x y = vξ t dξ τ = t Qy τ = 1 vx t equation 3.19 is mapped into Burgers equation On the other hand for the mdww system 3. if we take u = 0 Q τ = Q yy + QQ y. 3.0 vx t = Qy τ the resulting equation is also Burgers equation 3.0. Motivated by this argument it is anticipated that there exists such an analogous reciprocal correspondence between the mdww system 3. and the associated Casimir system 3.18 for the dual mdww system and thereby leading to a generalization for their entire hierarchies. 4. Three-component integrable generalizations In this section we shift our attention to three-component systems. Motivated by a system studied in [1] see 4.7a therein we consider the following pair of three-component systems: and w t = λu x + w + bcλu + 3 λw + wv + 1 abλu + bλuv x u t = 1 b u x + cu + 1 b w + λwu + 1 au + uv x v t = λ b w x + a b u x + 1 b v x + cλw ac u + c v a b w + λwv 1 a u auv + v x 4.1 s t = s + λq ds + sr x q t = 1 b q x + cq + qr dsq + 1 s x r t = d b s x + 1 b r x + d + λ cd s + λ + dλ a b q + c r + r dsr 4. x where a 0 b 0 c d λ are constants. Systems 4.1 and 4. are related by the following Miura-type transformation s = w q = u x + 1 w + ab u + b uv + bc u r = d + λw + v. Note that system 4.1 can be written in a Hamiltonian form w u v t δh/δw = J w u v δh/δu δh/δv with the Hamiltonian operator x 0 0 J w u v = b x b x a b x

15 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 15 and the associated Hamiltonian functional Hw u v = 1 λ w 3 + w v + abλ wu + bλ wuv + w + bcλ wu λwu x dx + 1 u v + b uv + bc uv u x v dx. Consider the following Miura-type Bäcklund transformations B 1 w u v s q r = s c 1 w = 0 B w u v s q r = q c w + c 3 u x + c 4 u + c 5 uv + c 6 u = 0 B 3 w u v s q r = r c 7 w + c 8 u + c 9 v = where c 1... c 9 are constant parameters. Plugging the original Hamiltonian operator 4.3 into formula.8 under the transformations 4.4 yields the Hamiltonian operator J s q r = α x γ x s κ x γ s x γq x + x q ɛ x β x + γ r x 4.5 κ x ɛ x + β x + γ x r ξ x where α = c 1 β = 1 b c 3c 9 γ = c = ab c 4 = 1 b c 5 ɛ = 1 b c 6c 9 ξ = c 7 + b c 8c 9 a b c 9 κ = c 1 c 7. Using a scaling argument we extract a compatible Hamiltonian triple.0 from the operator 4.5. Consider the compatible Hamiltonian pair where J 1 = K 1 + ν K J = J 4.6 α 1 0 κ K 1 = x K = x 4.7 κ 1 1 ξ depending upon the parameters α 1 0 κ 1 ξ 1. Applying successively the resulting recursion operator R = J 1 J 1 to the seed system s q r t = G 1 s q r = s q r produces a hierarchy of three-component higher-order commuting bi-hamiltonian systems namely s q r = G n s q r = J 1 δ H n = J δ H n 1 n = 1... t where the variational derivative of the Hamiltonian functionals is given by δ H δ n = H n δs δ H n δq δ H T n. δr Focussing our attention on the second flow of this hierarchy s q = G s q r = RG 1 s q r = R s q r r t we set the parameters γ = α 1 = 1 for simplicity and distinguish two cases. x x 4.8

16 16 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU Case 1. When ν = 0 1 κ 1 0 J 1 1 = κ 1 ξ 1 x where ξ = κ 1 ξ 1. The flow 4.8 takes the bi-hamiltonian form s q r t = J δ H 1 = J 1 δ H = J s κ 1q κ 1 s + ξ q + r q where H 1 = 1 s + ξ q + r κ 1 s q dx H = 1 ξ q 3 κ 1 s 3 + ξ + κ 1s 3κ 1 ξ q sq + s 4κ 1 q sr + 3ξ q + r qr + αs κ 1 q + ɛξ κκ 1 + ξ q + κ ɛκ 1 sq + ɛ qr + β q x κ 1 s r dx. Explicitly s t = αs + κ ακ 1 q + ξ q κ 1 s + rs x q t = β q x + ɛ q + 3ξ 4 q κ 1s + r q + 1 s 4.9 x r t = βκ 1 s x + βξ q x + β r x + κ ɛκ 1 s + ɛ ξ + ξ κκ 1 q + ɛ r + ξ q κ 1 s + r r. Case. J 1 1 = When ν 0 without loss of generality we set ν = ±1 and so 1 κ 1 1 ν x 1 0 κ ν x 1 ξ 1 x ν x ν x 1 x 1 ξ = κ 1 ξ 1. 0 Thus system 4.8 becomes s q = J s κ 1 1 ν x 1 q 1 + ν x 1 r κ 1 s + ξ 1 x 1 q r 1 ν x 1 q t Two subcases are considered: Case.1. Setting κ 1 = ξ = 0 so that ξ 1 = 0 in 4.7 we introduce new variables g = 1 ν x 1 q f = 1 + ν x 1 r. Then 4.10 becomes s t = αs + κg + sf x g t νg xt = βg x + ɛg + g + qf + 1 s x f t + νf xt = βf x + κs + ξg + ɛf + r f x and the associated Hamiltonian functionals are H 1 = 1 s + g + νg x f dx H = 1 s f + rfg + αs + ξg + κsg + ɛ f + βf x g dx. x Case.. When κ 1 + ξ 0 we define the variables h f and g by s = 1 xh q = 1 xg r = 1 xf

17 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 17 respectively. We then arrive at the following bi-hamiltonian system h t h xxt = α s + κ ακ 1 νg x + g + f νf x + ξ g κ 1 h νh x s x g t g xxt = βνg xx + ɛν βg x + ɛg + ξ 3g gx gg xx + f νf x κ 1 h νh x g + νg x g xx + 1 s βκ1 f t f xxt = ν + κ h xx βνf xx κ 1 β ɛνh x + βξ + ξν κκ 1 νg x + β ɛνf x + κ ɛκ 1 h + ɛξ + ξ κκ 1 g + ɛf + f νf x κ 1 h νh x + ξ g r and the associated Hamiltonian functionals are H 1 = 1 s + ξ g + g x + q + νq x f κ 1 h dx x 4.11 x H = 1 s f νf x κ 1 h νh x + ξ g + q κ 1 h νh x f + νf x + g + νgx + ξ κ 1 h νh x + f νf x 3g gx gg xx + ξ g 3 + ggx + αs + κ ακ 1 g + νg x s + κ 1 h νh x + f νf x ɛg + ɛν βgx βνg xx + κ 1 ακ 1 κ + ɛξ g + κ 1 ακ 1 κ + βνξ g x dx. Next we focus our attention on the three-component DWW system s t = sr x q t = q x + qr + 1 s x r t = 4.1 r x + q + r x which is a special case of system 4.9 corresponding to the choice of α = κ = κ 1 = ɛ = ξ = 0 β = 1 and ξ = in 4.9. This system was proposed in [1] as an example of a threecomponent quadri-hamiltonian system and it appears as a three-component generalization of the DWW system 3.1. According to the preceding analysis the dual integrable counterpart of system 4.1 is the system arising from Case.1 i.e. the bi-hamiltonian system admitting the Hamiltonian operators J 1 and J defined in 4.6 and 4.7 with α 1 = γ = 1 ν = ±1 and κ 1 = ξ 1 = 0. More precisely the following dual Hamiltonian pair Ĵ 1 = x x ν x Ĵ = 0 xŝ 0 ŝ x q x + x q r x x + ν x 0 0 x r x with ν = ±1 gives rise to the three-component bi-hamiltonian system ŝ t = ŝf x q t = q + g f + 1 ŝ x q = g νg x r t = g + rf x r = f + νf x 4.14 which turns out to be the dual version of 4.1. In particular when ŝ = reduces to the dual counterpart 3.14 of the DWW system 3.1. The Hamiltonian operators Ĵ1 and Ĵ given in 4.13 when projected to the q r subspace yield the Hamiltonian pair 3.13 admitted by the two-component dual system Finally it is easy to see that setting b = 1 and a = in 4. produces the system s t = s + λq ds + sr x q t = q x + c q + qr dsq + 1 s x r t = ds x + r x + d + λ cds + λ + dλ + q + c r + r dsr x 4.15

18 18 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU which belongs to the integrable family 4.9 corresponding to α = β = 1 ɛ = c κ = d + λ ξ = d + λ + κ 1 = d ξ = 0 i.e. ξ 1 = d. Therefore following from the Hamiltonian operators J 1 and J of system 4.9 defined in and 4.7 with ν = admits the Hamiltonian pair 1 0 d x x s d + λ x J 1 = x J = s x q x + x q d 1 d x + c x + r x d + λ x x + c x + x r d + λ +. x In addition 4.15 admits the following Lax pair with spectral parameter µ: ψ1 µ 1 ψ1 ψ1 A B ψ1 = = L µ C A ψ x ψ where L = d λ + d + s λ 4 r λ + dλ + q + 1 cd λ r 1 4 r x + 1 λ λ 4 d + c d c µ A = Bµ + B x B = ds + r + λ d + 1 C = Bµ B x µ + d λ λ + d + 1 Bs 1 Br + λ + dλ + Bq λ + d + 1 c Br + Br x + 1 λ λ B xx B d + c d c On the other hand 4.15 is related to w t = λu x + w + cλu + wv λu + λuv + 3 λw x u t = u x + c u u + λwu + uv + 1 w x v t = λw x u x + v x + cλw + c u + c v + v + λwv + uv u + w x via the Miura-type transformation s = w q = u x u + 1 w + uv + cu r = d + λw + v. According to Theorem.3 the recursion operator Rw u v of 4.16 satisfies where and Rs q r = J = ψ T Rw u v = Rs q r T t ψ B T = w Φ u Φ = x u + v + c 4.17 d + λ 0 1 J 1 1 Rw u v = 1 d x s x 1 λ x s x 1 s dq d x q x 1 x ds + r + c q + x q x 1 d x d x r x 1 + λ + d cd λ + λd + x + x r x 1 + c is the recursion operator of system Consequently the recursion operator of 4.16 is 1 + λw + λ x w x 1 λφ λu + x w x 1 w + λ x u x 1 Φ u + x u x 1 λc + w + λ x + λ x v x 1 Φ c + u + x + x v x 1

19 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 19 where the operator Φ is defined in Furthermore in the case b = 1 and a = the first Hamiltonian operator J 1 w u v of system 4.16 is J w u v given by 4.3 which along with Rw u v gives rise to the second Hamiltonian operator J w u v = Rw u vj 1 w u v x + λ w x + x w λu x + x w λ x + λc + v x + x w = w x + λ x u x u + u x x + c + v x + x u. λ x + cλ + w x + λ x v x + c + u x + x v 4c x + v x + x v With the Hamiltonian pair J 1 and J in hand the dual integrable bi-hamiltonian hierarchy for system 4.16 can also be readily constructed. For instance if we set c = λ = 0 the system 4.16 reduces to w t = w + wv x u t = u x u + uv + 1 w x v t = u x + v x + v + uv u + w x 4.18 which can be viewed as a three-component generalization of the mdww system 3.. The dual version of system 4.18 is obtained by recombining the Hamiltonian pair x 0 0 x x w x w J 1 = 0 0 x J = w x x u + u x x + v x + x u 0 x x w x x + x v + u x v x + x v admitted by 4.18 to define the dual Hamiltonian pair 1/ x w x w J 1 = x x J = w x x ū + ū x v x + x ū x w x ū x + x v v x + x v The dual of the original integrable system 4.18 will thus take the form w t = g x f x + f w x ū t = g + g x g x g f x + f + g x f x + fū + w x v t = g + g x g x g f x + f + g x f x + f v + w x where f = 1 x 1 v and g = 1 x 1 ū and H 1 = w g + gx + g + g x v dx H = 1 w g x + f f x + gx g f f xx + g + 3g x g xxx f + g + g x g xx fx dx are the required Hamiltonian functionals. 5. Solitary wave solutions of the dual DWW system 3.1 In this section we analyze the solitary wave solutions of the following dual DWW system g t νg xt = βg x + gf νg x f x f t + νf xt = βf x + g + f + νf x f x ν = ±1 5.1 which is equivalent to system 3.1 under the Galilean transformation x x αt. We consider the solitary wave solutions which take the form gt x ft x = ψx c t φx c t c R

20 0 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU for some ψ φ : R R such that ψ a a 0 φ 0 as x. In view of the asymptotic behavior of ψ we set g = a + h with h 0 as x and rewrite system 5.1 as h t νh xt = βh x + af + hf νh x f x f t + νf xt = βf x + h + f + νf x f x ν = ±1. 5. Definition 5.1. Let 0 < T. A function h f C [0 T X is called a solution of 5. on [0 T if it satisfies 5. in the distribution sense on [0 T. Definition 5.. A solitary wave of 5. is a nontrivial traveling wave solution of 5. of the form Φ c t x = ηx c t φx c t with the constant wave speed c R where η φ Hloc 1 R H1 loc R and η φ vanish at infinity. Definition 5.3. We say that a continuous function φ : R R has a peak at x if φ is smooth locally on either side of x and 0 lim x x φ x x = lim x x φ x x ±. Wave profiles with peaks are called peaked waves or peakons. Definition 5.4. We say that a continuous function φ : R R has a cusp at x if φ is smooth locally on either side of x and lim φ x x = lim φ x x = ±. x x x x Wave profiles with cusps are called cusped waves or cuspons. Definition 5.5. We say that a traveling wave solution φ : R R is a kink solution if φ satisfies the boundary conditions k l = lim φx and k r = lim φx x x + where < k l < k r < + or k l > k r. The function φ sometimes also satisfies the additional asymptotic condition lim x φj x x = 0 j = Substituting the traveling wave ansatz ht x ft x = ηx c t φx c t where the wave speed c is constant into 5. one obtains the following system of ordinary differential equations for η φ c η x + c νη xx = βη x + aφ + η νη x φ x c φ x c νφ xx = βφ x + η + φ + νφ x φ x. 5.3 Integrating both equations produces νφ + cν + βη x cη a + ηφ = 0 η = 1 νφ + cν + βφ x 1 φ 1 cφ. 5.4 Then we substitute the second equation in system 5.4 into the first to derive a single second order equation for φx: φ c φ xx + φ cφ x φ 3 3cφ c 4aφ = 0 in D R 5.5 where c = c + νβ. Further calculation gives [ φ c 3 ] xx = 3φ cφ x + 3 [ φ 3 + 3cφ + c 4aφ ] in D R. 5.6 The following lemma deals with the regularity of the solitary waves. The idea is inspired by the study of the traveling waves of the CH equation [3]; see also [8] and [4].

21 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 1 Lemma 5.1. Let η φ Hloc 1 R H1 loc R be a solitary wave of 5.. Then Therefore φ C R \ φ 1 c. Proof. We set v = φ c and denote φ c k C j R \ φ 1 c for k j. 5.7 Gv = v c + cv + 6 c + c + 6c c 4av + [ c + c c + c 4a] c. Since φ Hloc 1 R solves equation 5.6 in the distributional sense v satisfies v 3 xx = 3vv x + 3 Gv in D R. We then deduce that v 3 xx L 1 loc R which indicates that v3 x is absolutely continuous and v v as well as v 3 belong to C 1 R \ v 1 0. Moreover for k 4 v k xx = kv k 1 v x x = 1 3 k [ v k 3 v 3 x ]x = kk 3vk vx k vk 3 v 3 xx = kk v k vx + 3kv k 3 Gv 5.8 = 1 4 kk vk 4 [ v x ] + 3 k v k 3 Gv C R \ v 1 0. It follows that v k C R \ v 1 0 for k 4. In the case of k 8 based on the previous result we have v 4 and v k 4 belong to C R \ v 1 0 while v k 3 Gv C 1 R \ v 1 0. On the other hand since v k v x = formula 5.8 allows us to deduce that 1 v 4 4k 4 x v k 4 x C1 R \ v 1 0 v k kk xx = v 4 k 4 x v k 4 + x 3kvk 3 Gv C 1 R \ v 1 0 which immediately leads to v k C 3 R \ v 1 0 k 8. Extending these arguments to higher values of k concludes that v k C j R \ v 1 0 for k j and then 5.7 follows. This completes the proof. Denote x = minx: φx = c} while if φ c for all x we set x = + so x +. From Lemma 5.1 a solitary wave φ is smooth on x and hence 5.5 holds pointwise on x. Therefore we may multiply 5.5 by φ x and integrate on x] for x < x to obtain φ x = φ φ A 1 φ A φ c := F φ 5.9 where A 1 = c a and A = c + a. Next we will explore the qualitative behavior of solutions to 5.9 near points where F φ has zero or a pole. Applying the similar arguments as introduced in [9 3] we arrive at the following conclusions. 1. When F φ has a simple zero at m = A 1 or m = A so that F m = 0 and F m 0 the solution φ of 5.9 satisfies which gives rise to where φx 0 = m. φ x = φ mf m + O φ m as φ m φx = m x x 0 F m + O x x 0 4 as x x

22 JING KANG XIAOCHUAN LIU PETER J. OLVER AND CHANGZHENG QU. When F φ has a double zero at φ = m so that F m = F m = 0 F m > 0 we have φ x = F mφ m + O φ m 3 as φ m and thus φx m γ 1 exp x F m as x 5.11 for some constant γ 1. Thus φ m exponentially as x. Analogous computations allow us to draw the following conclusions for cases of the third and fourth order zeros: 3. When F φ has k th order zero k = 3 4 at φ = m so that F i m = 0 for i = 0... k 1 while F k m 0 and Hence we have φ x = F k mφ m k + O φ m k+1 as φ m. φx m x k as x. Therefore when k = 3 or k = 4 φ m algebraically at the rate O 1/x and O 1/x respectively as x. 4. Peaked solitary waves occur when φ changes direction: φ x φ x according to Suppose φ approaches a simple pole φ x = c such that 1/F φ has a simple zero. Without loss of generality we assume that φx < c and then φ x c = γ x x /3 + O x x 4/3 as x x 5.1 and φ x = 3 γ x x 1/3 + O x x 1/3 x x 3 γ x x 1/3 + O x x 1/3 x x 5.13 for some constant γ < 0. Furthermore if φ approaches a pole of order at φ x = c such that 1/F φ has a double zero then φ x c = γ 3 x x 1/ + Ox x as x x We thus have 1 φ x = γ 3 x x 1/ + O 1 x x 1 γ 3 x x 1/ + O 1 x x for some constant γ 3 < 0. Altogether it follows from 5.13 and 5.15 that lim x x φ x = lim x x φ x = ± 5.15 which means that whenever F has a pole the continuous solution φ will have a cusp at x. We conclude that nonanalytic behavior will only arise at the singular points which are precisely the points of genuine nonlinearity of the original systems. Now let us consider the classification of traveling waves of 5.. As in [9] our approach is based on the configuration of the zeros and poles of F φ in formula 5.9. This allows to determine several precise parameter regimes in which 5. admits solitary wave solutions. In contrast to the dual counterpart of the KdV equation whose qualitative properties change considerably as the coefficient ν in potential changes sign [33] the sign of ν involved in system 5. just affect the parameter regimes that govern the corresponding wave patterns. Thus from here on we only consider the case when ν = 1. The qualitative behavior of the component φx of the solitary wave solution to 5. in the case of a > 0 is summarized in the following theorem.

23 BÄCKLUND TRANSFORMATIONS FOR TRI-HAMILTONIAN DUAL STRUCTURES 3 Theorem 5.1. When a > 0 the necessary condition for the existence of the solitary wave solutions is c a or c a. Moreover we have Case 1. c < a i.e. 0 < A 1 < A If β < a i.e. c > A 1 then there is a smooth solitary wave φ > 0 with max x R φx = A 1 ; 1.. If a β < c i.e. 0 < c A 1 then there is a cusped solitary wave φ > 0 with max x R φx = c; 1.3. If β > c i.e. c < 0 then there is a smooth solitary wave φ > 0 with max x R φx = A 1 and an anticusped solitary wave the solution profile has a cusp pointing downward φ < 0 with min x R φx = c. Case. When c = a i.e. A 1 = 0 < A a solitary wave exists if and only if β > a i.e. c < 0 and is anticusped with min x R φx = c. Case 3. When c = a i.e. A 1 < 0 = A a solitary wave exists if and only if β < a i.e. c > 0 and is cusped with max x R φx = c. Case 4. c > a i.e. A 1 < A < If β > a i.e. c < A then there is a smooth solitary wave φ < 0 with min x R φx = A ; 4.. If c < β a i.e. A c < 0 then there is an anticusped solitary wave φ < 0 with min x R φx = c; 4.3. If β < c i.e. c > 0 then there is a smooth solitary wave φ < 0 with min x R φx = A and a cusped solitary wave φ > 0 with max x R φx = c. Moreover each of the above solitary waves is unique and even up to translations. When A 1 = 0 or A = 0 the corresponding anticusped or cusped waves decay algebraically at the rate of O 1/x while when A 1 > 0 or A < 0 all solitary waves decay exponentially to zero at infinity. When c = A 1 > 0 or c = A < 0 the corresponding cusped or anticusped solitary waves have the singular behavior in 5.1 and 5.13 while the other cases satisfy 5.14 and Proof. First of all formula 5.9 together with the fact that φx decays at infinity leads to the necessary condition for the existence of solitary wave: A 1 0 or A If A 1 = 0 then 5.9 becomes φ x = φ3 φ A φ c := F 1 φ 5.16 which indicates that φx < 0 near. Since φx 0 as x there exists some x 0 sufficiently negative satisfying φx 0 = ɛ < 0 with ɛ > 0 sufficiently small and φ x x 0 < 0. According to the standard ODE theory one can generate a unique local solution φx on [x 0 δ x 0 + δ] for some δ > 0. Furthermore since A 1 < A we deduce that F 1φ = φ φ A + 4 cφ + 3A c φ c 3 = φ φ µ 1 φ µ φ c 3 µ 1 = A + 4 c A 16A c + 16 c 4 µ = A + 4 c + A 16A c + 16 c. 4 If c > 0 then 0 < µ 1 < µ so F 1φ < 0 for φ < 0. While if c < 0 we have µ 1 < 0 < µ and then F 1φ < 0 for µ 1 < φ < 0. In view of this we can choose x 0 so that in the case of c < 0 µ 1 < φ < 0 on [x 0 δ x 0 + δ]. Summarizing one has F 1φ < 0 on [x 0 δ x 0 + δ] which together with the fact that φ xx = F 1φ/ implies φ x x decreases on [x 0 δ x 0 + δ]. Hence φ x x < 0 on [x 0 x 0 + δ] which further implies that φx decreases on [x 0 x 0 + δ]. If c < 0 since F 1 φ is locally Lipschitz in φ for c < φ 0 one can easily continue the local solution to x 0 δ] with φx 0 as x. As for x > x 0 + δ we can solve