Camassa-Holm cuspons, solitons and their interactions via the dressing method
aa r X i v : . [ n li n . S I] A ug CAMASSA-HOLM CUSPONS, SOLITONS AND THEIRINTERACTIONS VIA THE DRESSING METHOD
ROSSEN IVANOV, TONY LYONS, AND NIGEL ORR
Abstract.
A dressing method is applied to a matrix Lax pair for the Camassa-Holm equation, thereby allowing for the construction of several global solutionsof the system. In particular solutions of system of soliton and cuspon type areconstructed explicitly. The interactions between soliton and cuspon solutions ofthe system are investigated. The geometric aspects of the Camassa-Holm equa-tion ar re-examined in terms of quantities which can be explicitly constructedvia the inverse scattering method. Introduction
This paper aims to explai how the dressing method, well known in the solitontheory, can be applied to one of the most famous integrable equations of the last20 years - the Camassa-Holm (or CH) equation. In particular, the method allowsfor the explicit construction of the soliton and cuspon solutions and for furtherinvestigation of the interactions between them. The Camassa-Holm equation isgiven by(CH) (cid:26) q t + 2 u x q + uq x = 0 q = u − u xx and in the following we impose the boundary conditions lim | x |→∞ u ( x, t ) = u , where u > u ( x, t ) representthe fluid particle velocity induced by the passing wave, or alternatively as thesurface elevation associated with the wave. Moreover, the system also constitutesa model for the propagation of nonlinear waves in cylindrical hyper-elastic rods,in which the solutions u ( x, t ) represent the radial stretching of a rod relative tothe undisturbed state, see [19].For the past number of decades the Camassa-Holm equation has proven to be aremarkably fertile field of mathematical research, with the volume of research pa-pers dedicated to various aspects of the system most likely measured in thousands,and as such our bibliography is by no means exhaustive. A particularly striking feature of the Camassa-Holm equation relates to the existence of peaked solutionsfor the system, which are solutions of the form(1.1) u ( x, t ) = q e −| x − p t | where lim | x |→∞ u ( x, t ) = 0 , with q and p being constants. These peaked solutions (or peakons) are weaksolutions whose wave crests appear as peaks, see [3, 4, 18, 43]. In addition theCamassa-Holm equation also allows for the existence of breaking wave solutions,which are realised as solutions which remain bounded but whose gradient becomesunbounded in a finite time, cf. [3, 4, 9, 10, 7, 53]. The presence of both peaked andbreaking wave solutions for the system (CH) ensures the Camassa-Holm equationis a highly interesting physical model. To compliment the utility of the system inmodelling a diversity of physical phenomena, the Camassa-Holm equation exhibitsa rich mathematical structure. The equation is a member of a bi-Hamiltonianhierarchy of equations [25] and it is integrable with a Lax pair representation [3].A notable property of the system is its formulation as a geodesic flow on theBott-Virasoro group [44, 30, 31, 15].Soliton solutions of the Camassa-Holm equation have been derived by manifoldmethods, including but not restricted to Hirota’s method [42, 43, 46, 47, 48], viathe B¨acklund transform method [52, 41], along with the inverse scattering method[11, 2]. In the current work we develop a modified version of the inverse scatteringmethod, namely the dressing method, to construct the cuspon and soliton solu-tions and cuspon-soliton interactions of the Camassa-Holm system. The dressingmethod is an efficient variation of the inverse scattering transform which allows fora very direct construction of soliton solutions of integrable PDE [54, 55, 45, 26].The essential procedure behind this dressing method is the construction of a non-trivial (dressed) eigenfunction of an associated spectral problem from the known(bare) eigenfunction, by means of the so-called dressing factor. This dressing fac-tor is analytic in the entire complex plane (of values of the spectral parameter),except for a collection simple poles at pre-assigned discrete eigenvalues. This barespectral problem is obtained for some trivial solution, e.g. u ( x, t ) = u , where u as indicated above is the asymptotic value of the solution, which we require tobe constant and strictly positive. Since the potential terms of this bare spectralproblem are simply constant, this means the spectral problem is readily solved toyield the bare eigenfunction. In the following we will outline the construction ofthe solutions of the CH equation associated with the discrete spectrum of the Laxoperator, i.e. the solitons and cuspons.The Camassa-Holm equation has many similarities with the integrable Degasperis-Procesi (or DP) equation [20, 21]. The inverse scattering transform of the DPequation is studied in [14, 1], and in particular the dressing method for the DPequation is presented in [13]. In the following we will reformulate equation (CH) inthe form of a matrix Lax pair, and impose an appropriate gauge transformation on DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 3 this matrix equation, which reduces the spectral problem to the familiar Zakharov-Shabat spectral problem. We deduce several important reduction symmetries ofthe spectral problem, which are then utilised in constructing the dressing factor.This in turn allows for the construction of solutions of the dressed spectral prob-lem, from solutions of the bare spectral problem which are readily solved. Finally,using these dressed eigenfunctions, we obtain the physical solutions we seek bysolving a straight forward differential equation.2.
The Spectral Problem for the Camassa-Holm Equation
From the scalar to the matrix Lax pair.
The following spectral problem(2.1) φ xx = (cid:18)
14 + λ q (cid:19) φφ t = (cid:18) λ − u (cid:19) φ x + u x φ. may be seen to represent the Camassa-Holm equation (cf. equation (CH)), byimposing the compatibility condition φ xxt ≡ φ txx on the spectral function φ andcomparing terms of equal order in λ, [3]. The constant λ appearing in equation(2.1) is the time-independent spectral parameter, while the potential u ( x, t ) corre-sponds to a solution of the CH equation. Solutions of the Camassa-Holm equationmay be obtained from the spectral problem above by means of the Inverse Scatter-ing Transform, and the reader is referred to the works [11, 12] for further discussionin this regard. A discussion of the Inverse Scattering Transform applied to con-structing periodic solutions of the system (CH) may be found in [16, 6, 27]. Wenote that in contrast to some previous works we shall omit the dispersion term u x in the CH equation, and instead we will allow for a constant asymptotic value u ( x, t ) → u > | x | → ∞ . This is the setup adopted in [52].Suppose that u ( · , t ) − u is a Schwartz class function for all t , while the initialdata is chosen such that q ( x, >
0. Symmetry of the Camassa-Holm equationthen ensures that q ( x, t ) > t , cf. [8]. Letting k = − − λ u , the spectralparameter may be written as(2.2) λ ( k ) = − u (cid:18) k + 14 (cid:19) , and the reader is referred to [8] for a discussion of the spectrum of the problemformed by equations (2.1)–(2.2). Then the continuous spectrum in terms of k corresponds to k ∈ R . The discrete spectrum (corresponding to k ∈ C + –theupper half-plane) consists of a finite number of points k n = iκ n , n = 1 , . . . , N where κ n is real and 0 < κ n < /
2, with the corresponding spectral parameter λ n = λ ( iκ n ) being purely imaginary. Moreover for any κ n there are two sucheigenvalues, denoted by λ n = ± iω n where ω n > . ROSSEN IVANOV, TONY LYONS, AND NIGEL ORR
We note that a discrete eigenvalue with κ n > / u ( · , t ) − u from the Schwartz class. In such case λ n is real. Laterwe will find out that indeed this choice corresponds to solutions with a cusp at thecrest (cuspons), which are clearly outside of the Schwartz class functions.To implement the dressing method it is first necessary to reformulate the spectralproblem in (2.1) as a matrix Lax pair . To achieve this we let φ denote an eigen-function of equation (2.1), and we observe that the first member of this spectralproblem may be reformulated as(2.3) (cid:18) ∂ x − (cid:19) (cid:18) ∂ x + 12 (cid:19) φ = λ qφ . The second member of equation (2.1) may be re-written in terms of the followingauxiliary spectral function(2.4) φ := 1 λ (cid:18) ∂ + 12 (cid:19) φ from which we immediately deduce that (cid:18) ∂ x − (cid:19) φ = λqφ , having imposed equation (2.3). Defining the eigenvectorΦ = (cid:18) φ φ (cid:19) , we reformulate the spectral problem (2.1) according to(2.5) Φ x = L Φ L := (cid:18) − λλq (cid:19) Φ t = M Φ M := (cid:18) ( u + u x ) − λ λ − λu λ ( q + u x + u xx ) − λuq λ − ( u + u x ) (cid:19) which constitutes a matrix Lax pair for the Camassa-Holm equation. Moreover, L , M take values in the sl (2) algebra, thus Φ belongs to the corresponding group, SL (2).The compatibility condition Φ tx ≡ Φ xt for every eigenvector Φ immediatelyimplies the zero-curvature condition , namely(2.6) L t − M x + [ L , M ] = 0 , where the bi-linear operator [ · , · ] denotes the usual matrix commutator. As withthe scalar formulation of the spectral problem, comparison of terms of equal orderin the spectral parameter λ within the zero-curvature condition yields O ( λ ) : u − u xx = q O ( λ ) : q t + 2 u x q + uq x = 0,which is precisely the Camassa-Holm equation. DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 5
The gauge transformed SL (2) spectral problem. In the following, thedressing method will be implemented on a gauge equivalent matrix-valued eigen-function Ψ ∈ SL (2), defined as follows(2.7) Φ =: G Ψ , where the gauge transformation G ∈ SL (2) is given by(2.8) G = (cid:18) q − q (cid:19) . Making this replacement in the spectral problem (2.5), the gauge equivalent spec-tral problem for Ψ is given by(2.9) Ψ x = ˜ L Ψ Ψ t = ˜ M Ψ , where we denote(2.10) ˜ L := G − L G − G − G x ˜ M := G − M G − G − G t . In particular we find that the equation for Ψ may be written as(2.11) Ψ x + (˜ hσ − λ √ qJ )Ψ( x, t, λ ) = 0˜ h = 12 − q x q , J = (cid:18) (cid:19) σ = (cid:18) − (cid:19) . We note that this spectral problem appears to be “energy dependent” since thepotential appears in combination with the spectral parameter in the off diagonalterms. However, introducing the re-parameterisation(2.12) dy = √ qdx, y = y ( x, t ) , the spectral problem acquires the form of the standard Zakharov-Shabat spectralproblem(2.13) Ψ y + L ( λ )Ψ( y, t, λ ) = 0 ,L ( λ ) = hσ − λJh = √ q − q y q , and the reader is referred to [54, 55, 45, 26] for further discussion concerning suchspectral problems. Since L ( λ ) takes values in the Lie algebra sl (2) it follows thatthe eigenfunctions take values in the corresponding Lie group - SL (2) . From (2.12)one can write x = X ( y, t ) , which gives the parametric representation of x for given t . This is a very important object in what follows due to the fact that the solutioncan be expressed through X ( y, t ) . We note that when x → ∞ , asymptotically q → u . Then in the view of (2.12) it is natural to expect that when y → ∞ ,X → y √ u + const . Theorem 2.1.
Suppose that when y → ∞ we have X y → √ u and X t → u . Thenthe solution in parametric form can be represented as u ( X ( y, t ) , t ) = X t ( y, t ) . ROSSEN IVANOV, TONY LYONS, AND NIGEL ORR
Proof.
We can write (CH) in the form ∂ t p q ( X ( y, t ) , t ) + ∂ X ( p q ( X, t ) u ( X, t )) = 0 , where X depends on y and t . We reformulate these derivative in terms of ( y, t )-varibles and noting that ddt = ∂∂t + X t ∂∂X , ∂∂X = 1 X y ∂∂y , p q ( X ( y, t ) , t ) = 1 X y ( y, t )we find that ddt p q ( X ( y, t ) , t ) − X t ∂ X p q ( X ( y, t ) , t ) + ∂ X ( p q ( X, t ) u ( X, t )) = 0 , and with some algebra this gives (cid:16) u ( X,t ) − X t X y (cid:17) y = 0 . Thus u ( X, t ) = X t + F ( t ) X y for some function F ( t ). The boundary conditions when y → ∞ give F ( t ) ≡ (cid:3) Diagonalisation.
Imposing the trivial solution u ( x, t ) ≡ u on the re-parameterisedspectral problem, the the so-called bare spectral problem emerges, given by(2.14) Ψ ,y + ( h σ − λJ )Ψ = 0 , Ψ ,t − h (cid:0) u − λ (cid:1) ( h σ − λJ )Ψ = 0 ,h = √ u . Since dy = √ u dx then y is simply a re-scaling of x for this bare spectral problem.The solutions of this linear system are readily obtained, and found to be of theform(2.15) Ψ ( y, t, λ ) = V ( λ ) e − σ Ω( y,t,λ ) V T ( λ ) C, where C is an arbitrary constant matrix and(2.16) Ω( y, t, λ ) = Λ( λ ) (cid:18) y − h (cid:18) u − λ (cid:19) t (cid:19) Λ( λ ) = q h + λ V ( λ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) cos θ = r Λ + h , sin θ = r Λ − h . In the following, the spectral parameter λ will be restricted to ensure Λ is alwaysreal and positive, however this in turn will mean θ may be either real or imaginary. DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 7
Symmetry reductions of the spectral problem.
It is easily verified thatthe spectral operator L ( λ ) = hσ − λJ appearing in equation (2.13) possesses thefollowing Z -symmetry reductions: σ ¯ L ( − ¯ λ ) σ = L ( λ )¯ L (¯ λ ) = L ( λ )(2.17)where ¯ λ denotes the complex-conjugate of λ. This reduction is found to arise due tothe symmetry relation σ J σ = − J . Moreover, the same Z -symmetry reductionis observed by the associated operator M ( λ ). A crucial result of the symmetryrelation (2.17) is that the potential h ( y, t ) must be real. Furthermore, since allaspects of the spectral problem (2.16) obey the reductions in (2.17), the associatedsolutions Ψ( y, t, λ ) and the dressing factor g ( y, t, λ ) (cf. Section 3) observe thefollowing reductions: σ ¯Ψ( y, t, − ¯ λ ) σ = Ψ( y, t, λ ) , ¯Ψ( y, t, ¯ λ ) = Ψ( y, t, λ ) . (2.18)Moreover, noting that(2.19) J = , J σ J = − σ , while also using Ψ − ( x, t, λ )Ψ( x, t, λ ) = , we observe that(2.20) Ψ − y ( λ ) = Ψ − ( λ ) L ( λ ) ⇒ Ψ − y ( λ ) T = L ( λ )Ψ − ( λ ) T , having used L T ( λ ) = L ( λ ) in the last equation. Hence with equation (2.19) wededuce(2.21) (cid:0) J Ψ − ( λ ) T J (cid:1) y + L ( − λ ) (cid:0) J Ψ − ( λ ) T J (cid:1) = 0 , that is to say Ψ( − λ ) and J Ψ − ( λ ) T J satisfy the same spectral problem, the solu-tions of which are unique (when fixed by the corresponding asymptotics in y and λ ), and thus(2.22) Ψ − ( y, t, λ ) = J Ψ( y, t, − λ ) T J. The Dressing Method
The dressing factor.
The soliton, cuspon and soliton-cuspon solutions haveassociated discrete spectra containing a finite number distinct eigenvalues { λ n } Nn =1 ,with the eigenfunctions of the spectral problem (2.13) being singular at thesediscrete eigenvalues. Starting from a trivial (or bare) solution u ( x, t ) = u , where u is constant, with its associated eigenfunction Ψ ( x, t, λ ), we may obtain aneigenfunction Ψ( x, t, λ ) corresponding to soliton solutions, via the dressing factor g ( x, t, λ ), defined by the following(3.1) Ψ( x, t, λ ) = g ( x, t, λ )Ψ ( x, t, λ ) . ROSSEN IVANOV, TONY LYONS, AND NIGEL ORR
The dressing factor g ( x, t, λ ) is singular at each λ = λ n belonging to the discretespectrum, and is otherwise analytic for λ ∈ C + .In the following we shall work with the y -representation introduced in equa-tion (2.13), which we implement via the diffeomorphism x = X ( y, t ). Under thisrepresentation the dressing factor then satisfies(3.2) ∂ y g + hσ g − gh σ − λ [ J, g ] = 0 , where g ( y, t, λ ) is to be interpreted as g ( X ( y, t ) , t, λ ). Moreover, since the solutionΨ( X ( y, t ) , t, λ ) ≡ Ψ( y, t, λ ) belongs to the Lie group SL (2), the factor g ∈ SL (2)and also satisfies the reductions given by equations (2.18) and (2.22), namely σ ¯ g ( y, t, − ¯ λ ) σ = g ( y, t, λ ) or ¯ g ( y, t, ¯ λ ) = g ( y, t, λ ) ,g − ( y, t, λ ) = J g ( y, t, − λ ) T J. (3.3)The physical solutions u ( x, t ) are extracted from the associated spectral func-tions Ψ( x, t, λ = 0), which we evaluate via the spectral problem (2.11) at λ = 0.These solutions of the spectral problem are of the form(3.4) Ψ( X ( y, t ) , t,
0) = e − ( X − ln √ q ) σ . Meanwhile, re-parameterising in terms of the y -variable, the eigenfunction of thedressed spectral problem at λ = 0 (cf. equation (2.13)) may be written as(3.5) Ψ( X ( y, t ) , t,
0) = g ( y, t, ( y, t, K , where Ψ ( y, t,
0) is a solution of the bare spectral problem when λ = 0. Hencethe matrix K ∈ SL (2) is an arbitrary constant matrix, a consequence of equa-tion (3.1). We note however that the t -dependence of the bare spectral functionΨ ( y, t, λ ) becomes singular when λ = 0, an observation which is immediatelyobvious when we refer to the second equation of the spectral problem (2.14). Assuch, in equation (3.5) we only consider the time-independent solution, namely,the solution which satisfies the spectral problem corresponding to the L ( λ = 0)-operator.To circumvent singular behaviour of Ψ t ( x, t, λ ) at λ = 0, we note from equation(2.12) that as x → ∞ then y → ∞ since q > t . Specifically we find y → √ u x as x → ∞ , in which caseΨ( X, t,
0) = Ψ ( y = X √ u , t, K as X → ∞ . Hence, we conclude that Ψ( X → ∞ , t,
0) should be time-independent. Addition-ally, differentiating equation (3.4) with respect to t , we findΨ t ( X, t,
0) = − σ (cid:18) X t − q t q (cid:19) Ψ( X, t, , and Ψ t ( X → ∞ , t,
0) = − σ u Ψ( X → ∞ , t, , having imposed X t = u ( X, t ) → u as X → ∞ . Hence Ψ( X, t,
0) must be of the formΨ(
X, t,
0) = e − ( X − ln √ q − u t ) σ DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 9 in order to have the appropriate asymptotic behaviour (the correction being inde-pendent of y ). It follows that(3.6) e − ( X ( y,t ) − ln √ q − u t ) σ = g ( y, t, e − √ u σ y K , which gives a differential equation for X since ∂ y X = q − / , cf. equation (2.12).Thus it provides the change of variables x = X ( y, t ) in parametric form, where y serves as the parameter. This of course is valid only in cases where the dressingfactor is known and in what follows we shall explain how to construct it.3.2. The dressing factor with a real simple pole.
In the SL (2) Zakharov-Shabat spectral problems, the simplest form of g possesses one simple pole [45, 26],which leads to the following: Proposition 3.1.
The dressing factor g ( y, t, λ ) is assumed to be of the form (3.7) g = + 2 λ B ( y, t ) λ − λ , where λ ∈ R and B is a matrix-valued residue of rank 1. By virtue of equation (3.3) and Proposition 3.1, we deduce that the dressingfactor must satisfy(3.8) (cid:18) + 2 λ Bλ − λ (cid:19) (cid:18) − λ J B T Jλ + λ (cid:19) = , and taking residues as λ → ± λ we observe(3.9) ( B (cid:0) − J B T J (cid:1) = 0( − B ) J B T J = 0 . Rewriting the matrix B as(3.10) B = | n i h m | , with | n i = (cid:18) n n (cid:19) and h m | = (cid:0) m m (cid:1) , equations (3.9)–(3.10) combined with the symmetry relation (3.3) now ensure(3.11) h n | = h m | J h m | J | m i ⇒ B = | m i h m | J h m | J | m i , in which case B = B meaning B is a projector. Moreover, equation (3.11)combined with the symmetry relation ¯ g ( y, t, ¯ λ ) = g ( y, t, λ ) of (3.3) also yields that B should be real.Replacing equation (3.7) in equation (3.2) and taking residues as λ → λ and λ → ∞ , we have( h − h ) σ = 2 λ [ J, B ] ,B y + ( h σ + λ J ) B − B ( h σ − λ J ) − λ BJ B = 0 . (3.12) Replacing equation (3.11) in equation (3.12), multiplying everywhere by J fromthe right and using J σ J = − σ we have(3.13)( | m y i + ( h σ + λ J ) | m i ) h m |h m | J | m i + | m ih m | J | m i ( h m y | + h m | ( h σ + λ J )) − h m y | J | m i | m i h m |h m | J | m i − λ h m | J | m i | m i h m |h m | J | m i = 0 . Assuming(3.14) | m y i + ( h σ + λ J ) | m i = 0 , we also observe that equation (3.13) is satisfied identically provided (3.14) holds.Thus | m i is an eigenvector of the bare spectral problem (evaluated at λ = − λ ),in which case | m i is known explicitly.4. Cuspons, solitons and the cuspon-soliton interaction
The one-cuspon solution.
Equation (3.14) suggest that | m i satisfies thebare spectral problem (2.14) with λ = − λ , i.e. with spectral operator L ( y, t, − λ ) = h σ + λ J Furthermore equation (3.11) allows us to solve for | n i explicitly, thereby providingan explicit formula for the dressing factor g ( y, t, λ ). We can write(4.1) | m i = Ψ( y, t, − λ ) | m i where | m i is a constant vector, and Ψ( y, t, − λ ) ∈ SL (2) satisfies the bare spectralproblem(4.2) ( Ψ y + L ( y, t, − λ )Ψ = 0Ψ t − h (cid:16) u − λ (cid:17) L ( y, t, − λ )Ψ = 0 . With λ = − λ , we have Λ = p h + λ > h , (4.3) sin( θ ) = r Λ − h
2Λ cos( θ ) = r h + Λ2Λ , thus we conclude that θ is real.It follows from equation (2.16) that(4.4) Ψ( y, t, − λ ) = V e − σ Ω( y,t ) V − Ω( y, t ) = Ω( y, t, λ ) = Λ (cid:18) y − h (cid:18) u − λ (cid:19) t (cid:19) DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 11 while equation (4.1) now ensures(4.5) | m i = µ e − Ω( y,t ) q h +Λ2Λ − µ e Ω( y,t ) q Λ − h µ e − Ω( y,t ) q Λ − h + µ e Ω( y,t ) q h +Λ2Λ where the real coefficients µ and µ are given by(4.6) V − | m i = (cid:18) µ µ (cid:19) . Making the replacement ν = µ √ and ν = µ √ we simplify | m i = ( m , m ) T according to(4.7) | m i = (cid:18) ν e − Ω( y,t ) √ h + Λ − ν e Ω( y,t ) √ Λ − h ν e − Ω( y,t ) √ Λ − h + ν e Ω( y,t ) √ h + Λ (cid:19) . It follows from equations (3.7), (3.11) and (4.7) that(4.8) g ( y, t ; 0) = (cid:18) − m m − m m (cid:19) , while letting y → ∞ we also deduce the form of K , namely(4.9) K = g − ( y → −∞ , t ; 0) = − q Λ+ h Λ − h − q Λ − h Λ+ h . Equations (3.6) and (4.7)–(4.9) now yield the following differential equation ∂X∂y e X − u t = e − y √ u m (Λ + h ) m (Λ − h ) = e − y √ u (cid:20) ν | λ | e − Ω( y,t ) + ν (Λ + h ) e Ω( y,t ) ν | λ | e − Ω( y,t ) − ν (Λ − h ) e Ω( y,t ) (cid:21) . (4.10)Implementing the change of variables ( y, t ) → ( − y, − t ) and observing thatΩ( − y, − t ) = − Ω( y, t ), the differential equation for X becomes dXdy e X − u t − y √ u = (cid:20) se Ω( y,t ) + (Λ + h ) e − Ω( y,t ) se Ω( y,t ) − (Λ − h ) e − Ω( y,t ) (cid:21) , s := ν | λ | ν (4.11)The reason for doing so is the following: The Camassa-Holm equation written interms of the ( y, t )-variables yields the so-called ACH equation (see for instance [52,33, 34]), which is invariant under ( y, t ) → ( − y, − t ). Thus choosing any solutionof the ACH equation and imposing the change of variables ( y, t ) → ( − y, − t ), weobtain another solution once we determine x = X ( y, t ). In other words, if X ( y, t ) isa solution of the Camassa-Holm equation then so too is x = ˜ X ( y, t ) = X ( − y, − t ).Moreover, with this change of variables we also have x → y √ u as y → ±∞ , cf.equation (2.12). Explicitly this change of variables imposes the following transformation on ourdifferential equation for Xd ˜ X ( − y, − t ) d ( − y ) e ˜ X ( − y, − t ) − u ( − t ) = e − y √ u (cid:20) ν | λ | e − Ω( y,t ) + ν (Λ + h ) e Ω( y,t ) ν | λ | e − Ω( y,t ) − ν (Λ − h ) e Ω( y,t ) (cid:21) ,dX ( y, t ) dy e X ( y,t ) − u t = e y √ u (cid:20) ν | λ | e − Ω( − y, − t ) + ν (Λ + h ) e Ω( − y, − t ) ν | λ | e − Ω( − y, − t ) − ν (Λ − h ) e Ω( − y, − t ) (cid:21) . (4.12)Formally this may be integrated by separation of variables, however we may alsolook for a solution in the form(4.13) X ( y, t ) = y √ u + u t + ln (cid:12)(cid:12)(cid:12)(cid:12) A C B C (cid:12)(cid:12)(cid:12)(cid:12) with(4.14) A C = a e Ω( y,t ) + a e − Ω( y,t ) B C = se Ω( y,t ) − (Λ − h ) e − Ω( y,t ) . Replacing equations (4.13)–(4.14) in equation (4.10), we conclude that a = s , a = − ( h + Λ ) / (Λ − h ) and thus(4.15) X ( y, t ) = y √ u + u t +ln (cid:12)(cid:12)(cid:12)(cid:12) ( h + Λ) e − Ω( y,t ) + γe Ω( y,t ) ( h − Λ) e − Ω( y,t ) + γe Ω( y,t ) (cid:12)(cid:12)(cid:12)(cid:12) , γ = − s (Λ − h ) . When γ > X ( y, t ) = y √ u + u t + ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ +( h +Λ) γ − ( h +Λ) coth Ω( y, t ) + 1 γ +( h − Λ) ,γ − ( h − Λ) coth Ω( y, t ) + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + const . We introduce the constant U = Λ h > { ν , ν } such that γ = λ , thereby simplifying the expressionfor X ( y, t ), which is now given by(4.17) X ( y, t ) = y √ u + u t + ln (cid:12)(cid:12)(cid:12)(cid:12) U coth Ω( y, t ) − U coth Ω( y, t ) + 1 (cid:12)(cid:12)(cid:12)(cid:12) . having ignored a trivial additive constant.We recall that λ ( k ) = − u (cid:0) k + (cid:1) , while the discrete spectrum correspondsto k = iκ n with κ n ∈ ( , ∞ ) , thus ensuring the discrete spectral parameter λ n isreal. Written in terms of the parameter U = Λ( λ ) h >
0, the one-cuspon solution is
DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 13 now given by u ( X ( y, t ) , t ) = X t ( y, t ) with(4.18) X ( y, t ) = y √ u + u t + ln (cid:12)(cid:12)(cid:12) U coth Ω( y,t ) − U coth Ω( y,t )+1 (cid:12)(cid:12)(cid:12) u ( X ( y, t ) , t ) = u + U (cid:18) u − λ (cid:19) (1 − coth Ω( y,t ))1 − U coth Ω( y,t ) Ω( y, t, λ ) = Λ( λ ) (cid:16) y − h (cid:16) u − λ (cid:17) t (cid:17) Λ( λ ) = p h + λ . We can rewrite the second line of equation (4.18) as u − u = (cid:18) u − λ (cid:19) U (1 − tanh Ω) U − tanh Ωand we note that u > u when u > λ with the corresponding profile calledan a cuspon , left-panel Figure 1. The case u < u when u < λ is called an anti-cuspon , right-panel Figure 1. - x u ( x , 0 ) - - x u ( x , 0 ) Figure 1.
One-cuspon and one-anticuspon solution profiles of theCamassa-Holm equation, λ = 1 . . On the left panel u = 1 . , onthe right panel u = 0 . . Let us now evaluate the slope u X of the cuspon profile to investigate the dis-continuity at the cusp. We have X y = ( U −
1) sinh Ω √ u [ U + ( U −
1) sinh Ω]along with Λ =
U h and 2 h √ u = 1 and so we find u X ( X ( y, t ) , t ) = 1 X y ∂u∂y = − U (cid:16) u − λ (cid:17) U + ( U −
1) sinh Ω coth Ω . Thus, in the cuspon case ( u > λ ) we have u X → ∞ when Ω → − left ofthe cusp at Ω = 0, and u X → −∞ when Ω → + right of the cusp. For the anti-cuspon the signs change in an obvious way. The cusp is located at Ω =Λ( λ ) (cid:16) y − h (cid:16) u − λ (cid:17) t (cid:17) = 0 and moves with a constant velocity dydt = 12 h (cid:18) u − λ (cid:19) with respect to the y -axis. This can be both positive (for the cuspons) and neg-ative (for the anti-cuspons). However since y is merely a parameter, the velocityshould be measured with respect to the physical x -axis. Noting that when Ω = 0 ,X ( y, t ) = y √ u + u t +const, we find that the cuspon (anti-cuspon) velocity is dxdt = 2 (cid:18) u − λ (cid:19) thereby indicating that the threshold velocity is u = λ . Thus, the cuspon solu-tion is always right-moving, since u > λ > λ , while the anti-cuspon solutionis either right-moving, when λ > u > λ or left-moving, when u < λ . Thespecial case u = λ therefore corresponds to a “standing” anti-cuspon.4.2. The one-soliton solution.
The one soliton solution can be obtained in asimilar way by a dressing factor with a simple imaginary pole iω :(4.19) g ( y, t, λ ) = + 2 iω A ( y, t ) λ − iω . The details can be found in [35], and the solution is(4.20) X ( y, t ) = 2 h y + u t + ln (cid:12)(cid:12)(cid:12) U tanh Ω( y,t ) − U tanh Ω( y,t )+1 (cid:12)(cid:12)(cid:12) u ( X, t ) = u + U ( u + ω ) (1 − tanh Ω)1 − U tanh Ω Ω( y, t ) = Λ y − U (cid:0) u + ω (cid:1) t Λ = p h − ω , where as in the previous case the solution is obtained via X t ( y, t ) = u ( X ( y, t ) , t ).We note that as y → ±∞ , then tanh( y ) → ± u → u , which we observein the soliton profile shown in Figure 2. Interestingly, choosing constants suchthat c ( h − Λ) = − γ < X ( y, t ) is no longer a monotonic function for all y ∈ R meaning the solution u ( x, t ) becomes a function with discontinuities, andthe reader is referred to [52] for further discussion where such solutions are termed unphysical .4.3. The two-cuspon solution.
The dressing factor associated with the two-cuspon solution has two real simple poles λ and λ , with residues 2 λ k B k ( k =1 , . ). Extending Proposition 3.1, this dressing factor is of the form(4.21) g ( y, t, λ ) = + 2 λ B ( y, t ) λ − λ + 2 λ B ( y, t ) λ − λ , DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 15 - x u ( x , 0 ) Figure 2.
The one-soliton solution at t = 0 where u = 1 and ω = 0 . B and B are two matrix valued residues. The Z reduction ¯ g ( y, t, ¯ λ ) = g ( y, t, λ ) necessitates B k to be real.Applying equation (3.2) to the dressing factor g as given by equation (4.21)ensures the corresponding equations for the residues, namely(4.22) B k,y + hσ B k − B k h σ − λ k [ J, B k ] = 0 for k = 1 , . The matrix valued residues B k are of the form(4.23) B = | n i h m | , B = | N i h M | , where the vectors | n i , | N i , h m | and h M | are found to satisfy(4.24) ∂ y | n i + ( hσ − λ J ) | n i = 0 , ∂ y h m | = h m | ( h σ − λ J ) ∂ y | N i + ( hσ − λ J ) | N i = 0 , ∂ y h M | = h M | ( h σ − λ J ) . The vectors h m | , h M | satisfy the bare equations and therefore are known in prin-ciple, while the reality condition can be satisfied by assuming | m i , | M i , | n i , and | N i are all real. The reduction given in equation (3.3) leads to(4.25) (cid:20) + 2 λ B λ − λ + 2 λ B λ − λ (cid:21) (cid:20) − λ J B T Jλ + λ − λ J B T Jλ + λ (cid:21) = , which is identically satisfied for all λ. Thus, the residues obtained at λ and λ , ensure B (cid:0) − J B T J − η J B T J (cid:1) = 0 , η = 2 λ λ + λ ,B (cid:0) − η J B T J − J B T J (cid:1) = 0 , η = 2 λ λ + λ . (4.26)Using equations (4.23)-(4.26) we obtain the following system relating the bare anddressed eigenvectors:(4.27) h m | = h m | J | m i h n | J + η h m | J | M i h N | J, h M | = η h M | J | m i h n | J + h M | J | M i h N | J. Hence, the dressed vectors | n i and | N i may be written explicitly in terms of thebare vectors | m i and | M i as follows(4.28) | n i = 1∆ (cid:0) h M | J | M i J | m i − η h m | J | M i J | M i (cid:1) , | N i = 1∆ (cid:0) h m | J | m i J | M i − η h M | J | m i J | m i (cid:1) , with(4.29) ∆ = h M | J | M i h m | J | m i − η η h m | J | M i = ( η m M − η m M )( η m M − η m M ) . Thus, the residues B k can be expressed in terms of components of the knownvectors(4.30) h m | = (cid:10) m (0) (cid:12)(cid:12) Ψ − ( y, t, λ ) , h M | = (cid:10) M (0) (cid:12)(cid:12) Ψ − ( y, t, λ ) , where (cid:10) m (0) (cid:12)(cid:12) , (cid:10) M (0) (cid:12)(cid:12) are arbitrary constant vectors.The dressing factor g ( λ ) ∈ SL (2) (cf. equation (4.21)), when evaluated at λ = 0,is given by(4.31) g ( y, t ; 0) = − B + B )= diag( g , g )= diag (cid:18) λ M m − λ M m λ M m − λ M m , λ M m − λ M m λ M m − λ M m (cid:19) , while the differential equation for X ( y, t ) is of the form(4.32) ( ∂ y X ) e X − h y − u t = g = (cid:18) λ M m − λ M m λ M m − λ M m (cid:19) . In the case of the two-cuspon solution, equations (2.15)-(2.16) now become(4.33) Ψ( y, t, λ k ) = V k e − σ Ω k V − k with(4.34) Ω k ( y, t ) = Λ k (cid:18) y − h (cid:18) u − λ k (cid:19) t (cid:19) , Λ k = q h + λ k > h ,V k = (cid:18) cos( θ k ) − sin( θ k )sin( θ k ) cos( θ k ) (cid:19) , cos( θ k ) = r Λ k + h k sin( θ k ) = r Λ k − h k for k = 1 ,
2. Using equation (4.30) and noticing that (cid:10) m (0) (cid:12)(cid:12) V = ( µ , µ ) is aconstant vector, upon choosing µ , µ to be real and positive then explicitly we DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 17 have m = p h + Λ r µ µ e Ω ( y,t ) − p Λ − h r µ µ e − Ω ( y,t ) ,m = p Λ − h r µ µ e Ω ( y,t ) + p Λ + h r µ µ e − Ω ( y,t ) , up to an overall constant foactor √ µ µ (2Λ ) − / , which we may neglect since itultimately cancels, cf. equation (4.38).We redefine Ω ( y, t ) by an additive constant, which becomes(4.35) Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ln r µ µ , whereupon the vector components m and m become m = p Λ + h e Ω ( y,t ) − p Λ − h e − Ω ( y,t ) m = p Λ − h e Ω ( y,t ) + p Λ + h e − Ω ( y,t ) . (4.36)Similarly, taking the constant vector (cid:10) M (0) (cid:12)(cid:12) V = ( ν , − ν ) with ν , ν real andpositive, we have M = p Λ + h e Ω ( y,t ) + p Λ − h e − Ω ( y,t ) ,M = p Λ − h e Ω ( y,t ) − p Λ + h e − Ω ( y,t ) , Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ln r ν ν . (4.37)The dressing factor component g now takes the form(4.38) g = λ M m − λ M m λ M m − λ M m = T CC B CC whose explicit form we deduce from equations (4.36)–(4.37). We note in particularthat the denominator of this expression is given by B CC = λ λ Λ − Λ p (Λ − h )(Λ − h ) e Ω +Ω + Λ − Λ p ( h + Λ )( h + Λ ) e − Ω − Ω + Λ + Λ p (Λ − h )(Λ + h ) e Ω − Ω + Λ + Λ p (Λ + h )(Λ − h ) e − Ω +Ω ! (4.39)and introducing the constants(4.40) n k = r Λ k + h Λ k − h for k = 1 , , we obtain(4.41) B CC = p λ λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ + 1 n n e − Ω − Ω Λ + Λ + n n e Ω − Ω Λ − Λ + n n e − Ω +Ω Λ − Λ (cid:19) . Similarly it is found that(4.42) T CC = p λ λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ + n n e − Ω − Ω Λ + Λ − n n e Ω − Ω Λ − Λ − n n e − Ω +Ω Λ − Λ (cid:19) . We seek a solution of equation (4.32) in the form of equation (4.13) X ( y, t ) = y √ u + u t + ln A CC B CC with(4.43) A CC = α e Ω +Ω + α e − Ω − Ω + α e Ω − Ω + α e − Ω +Ω , for some as yet unknown constants { α l } l =1 . Equation (4.32) ensures that A CC must satisfy the following(4.44) 2 h A CC B CC + B CC ∂ y A CC − A CC ∂ y B CC = 2 h T CC whose solution is given by(4.45) A CC = p λ λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ + n n e − Ω − Ω Λ + Λ + n n e Ω − Ω Λ − Λ + n n e − Ω +Ω Λ − Λ (cid:19) . The ratio A CC / B CC may be written as(4.46) A CC B CC = n n n Λ +Λ Λ − Λ e + n Λ +Λ Λ − Λ e + e n n n
21 Λ +Λ Λ − Λ e + n
22 Λ +Λ Λ − Λ e + n n e +2Ω which we simplify by means of the following re-definitions:Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ln r µ µ − ln n + 12 ln Λ + Λ Λ − Λ , Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ln r ν ν − ln n + 12 ln Λ + Λ Λ − Λ . (4.47)Alternatively, these may be simply written as(4.48) Ω k ( y, t ) = Λ k (cid:18) y − t h (cid:18) u − λ k (cid:19)(cid:19) + ξ k for k = 1 , , where the constants ξ k relate to the initial sepparation of the solitons. It followsthat DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 19 A CC B CC = n n n e + n e + (cid:16) Λ − Λ Λ +Λ (cid:17) e n n n e + n e + (cid:16) Λ − Λ Λ +Λ (cid:17) n n e +2Ω and thus(4.49) X ( y, t ) = y √ u + u t + ln n e + n e + (cid:16) Λ1 − Λ2Λ1+Λ2 (cid:17) e n n n e + n e + (cid:16) Λ1 − Λ2Λ1+Λ2 (cid:17) n n e ! + X u ( X ( y, t ) , t ) = ∂X∂t , which yields a solution of a form similar to that found in [42]. Here X is an overalladditive constant that appears due to the translational invariance of the problem.Given the expression (4.48), we see that the phase velocity Λ k (cid:16) u − λ k (cid:17) maybe both positive and negative, thereby ensuring cuspon solutions may be left-moving or right-moving. In Figure 3 we show snapshots of a two cuspon solutionwith both cuspons right moving. Such a solution arises when u > λ where λ min = min { λ , λ } . Conversely, we may have a two-cuspon solution with bothcuspons left-moving when u < λ with λ max = max { λ , λ } , and an example ofsuch a solution is shown in Figure 4. The third category of two-cuspon solution isa mixture of both, that is to say, when one cuspon is right moving while the othercuspon is left moving. Such a solution occurs when the asymptotic value u isconstrained by λ max < u < λ min and an example of such a solution is illustratedin Figure 5.4.4. The two-soliton solution.
As with the one-soliton solution, the dressingfactor for the two-soliton solution has simple poles at the imaginary discrete eigen-values which we denote iω and iω , with residues 2 iω k A k ( k = 1 , g ( y, t, λ ) = + 2 iω A ( y, t ) λ − iω + 2 iω A ( y, t ) λ − iω . The Z reduction σ ¯ g ( y, t, − ¯ λ ) σ = g ( y, t, λ ) necessitates(4.51) σ ¯ A k ( y, t ) σ = A k ( y, t ) , k = 1 , . The detailed computations of the two-soliton solution for the system (CH) by thedressing method outlined here can be found in [35]. The solution formally has thesame form (4.49) where this time the parameters n and n are given by(4.52) n k = r h + Λ k h − Λ k , for k = 1 , . with Λ k = p h − ω k for k = 1 ,
2. The spectral parameters ω and ω belong tothe discrete spectrum of the spectral problem (2.13), and to ensure each Λ k is real, - - - - - - - x u ( x , - ) x u ( x , 11 )
36 38 40 42 44 x u ( x , 25 ) Figure 3.
Snapshots of a two-cuspon solution. Both cuspons areright-moving, λ = 1 . , λ = 2 . , u = 1 . , ξ = 0 . , ξ = 5 . . we must have ω k ∈ (0 , h ). Moreover, the phase of each soliton is given by(4.53) Ω k ( y, t ) = Λ k (cid:18) y − t h (cid:18) u + 12 ω k (cid:19)(cid:19) + ξ k for k = 1 , , where the constants ξ k are related to the initial separation of the solitons. Thetwo-soliton interaction is illustrated in Figure 6 below.4.5. The cuspon-soliton interaction.
The dressing factor associated with thecuspon-soliton interaction has two simple poles, one imaginary pole at λ = iω and one real pole at λ = λ , as follows(4.54) g ( y, t, λ ) = + 2 iω A ( y, t ) λ − iω + 2 λ B ( y, t ) λ − λ . Following our previous results, the residue B is required to be real.The details are provided in the Appendix. The solution is formally given by theexpression (4.49) where Λ = p h − ω and Λ = p h + λ along with(4.55) n = r h + Λ h − Λ , n = r Λ + h Λ − h . DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 21 - - x - - - - u ( x , - ) - - x - - - - u ( x , 15 ) - - - x - - - - u ( x , 50 ) Figure 4.
Snapshots of a two-cuspon solution. Both cuspons areleft moving, λ = 1 . , λ = 2 . , u = 0 . , ξ = 0 . , ξ = 10 . . The associated phases are given byΩ ( y, t ) = Λ (cid:18) y − t h (cid:18) u + 12 ω (cid:19)(cid:19) + ξ Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ξ . (4.56)Again, we note that the cuspon phase velocity may be either positive or nega-tive, while the soliton phase velocity is strictly positive. Thus, the cuspon-solitoninteraction may arise in two forms, namely a right moving soliton interacting witheither left or right moving cuspon. Both scenarios are presented in the following:In Figure 7 we present a cuspon-soliton interaction wherein both the cuspon andsoliton are moving to the right. In Figure 8 a cuspon-soliton interaction is shownin which the solition is right moving while the cuspon moves to the left.4.6. The general solution with multiple solitons and cuspons.
Now it isclear that the dressing factor for a solution with N solitons and N cuspons( N = N + N ) has the form - - x u ( x , - ) x u ( x , 25 ) x u ( x , 50 ) Figure 5.
Snapshots of a two-cuspon solution. One cuspon is leftmoving and the other cuspon is right moving, λ = 1 . , λ = 2 . ,u = 0 . , ξ = 0 . , ξ = 10 . .g ( y, t, λ ) = + N X j =1 iω j A j ( y, t ) λ − iω j + N X j =1 λ j B j ( y, t ) λ − λ j . Formally these solutions are always of the form of the N -soliton solution(4.57) X ( y, t ) = y √ u + u t + ln (cid:12)(cid:12)(cid:12)(cid:12) f + f − (cid:12)(cid:12)(cid:12)(cid:12) with(4.58) f ± ≡ X σ =0 , exp " N X i =1 σ i (2Ω i ∓ φ i ) + X ≤ i 30 40 50 60 x u ( x , 5 ) Figure 6. Snapshots of the two soliton solution of the Camassa-Holm equation (CH), for three values of t ∈ {− , , } . The otherparameters are u = 1, ω = 0 . 35 and ω = 0 . 25. The constants ofintegration were chosen as ξ = 0 and ξ = 2 . j = q h − ω j for a soliton , q h + λ j for a cuspon . - - - - 10 0 x u ( x , - ) - x u ( x , 6 ) 30 40 50 60 70 80 x u ( x , 20 ) Figure 7. The soliton-cuspon solution with parameters λ = 1, ω = 0 . u = λ = 1 . ξ = 5 and ξ = − n j = s h + Λ j h − Λ j for a soliton , s Λ j + h Λ j − h for a cuspon . along with(4.62) φ j = ln( n j ) γ ij = ln (cid:18) Λ i − Λ j Λ i + Λ j (cid:19) . Example. To illustrate this generalisation we construct the soliton-cuspon-anticusponsolution as showin in Figure 9. DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 25 - - 10 0 10 x u ( x , - ) x u ( x , 14 ) 10 15 20 25 30 35 x u ( x , 30 ) Figure 8. The soliton-anticuspon solution with parameters λ = 1, ω = 0 . u = λ = 0 . ξ = 5 and ξ = − N = 3and we find that f + = 1 + X k =1 n k e k ! + (cid:18) Λ − Λ Λ + Λ (cid:19) e +2Ω n n + (cid:18) Λ − Λ Λ + Λ (cid:19) e +2Ω n n + (cid:18) Λ − Λ Λ + Λ (cid:19) e +2Ω n n + (cid:18) Λ − Λ Λ + Λ (cid:19) (cid:18) Λ − Λ Λ + Λ (cid:19) (cid:18) Λ − Λ Λ + Λ (cid:19) e +2Ω +2Ω n n n f − = 1 + X k =1 n k e k ! + (cid:18) Λ − Λ Λ + Λ (cid:19) n n e +2Ω + (cid:18) Λ − Λ Λ + Λ (cid:19) n n e +2Ω + (cid:18) Λ − Λ Λ + Λ (cid:19) n n e +2Ω + (cid:18) Λ − Λ Λ + Λ (cid:19) (cid:18) Λ − Λ Λ + Λ (cid:19) (cid:18) Λ − Λ Λ + Λ (cid:19) n n n e +2Ω +2Ω . (4.63) - 50 0 500.10.20.30.40.50.6 30 40 50 60 70 x u ( x , 60 ) 50 60 70 80 x u ( x , ) Figure 9. The soliton-cuspon-anticuspon soltion with ω = 0 . λ = 1 . λ = 1 . u = λ = 0 . ξ = 15, ξ = − 45 and ξ = − The geometry of the soliton and cuspon solutions It is well known that the CH equation is closely related to the group of thediffeomorphisms of the real line, cf. [30, 15]. We are going to explore this relationin some more detail, making use of the variables that we have already evaluatedexplicitly for the soliton and cuspon solutions. Let us consider a local chart pa-rameterisation of the Lie group G ≃ Diff( R ) given by the coordinate X. Then dX is the basis for the co-tangent bundle T ∗ Diff( R ), L = P dX is a 1-form, and ω = d L = dP ∧ dX is the standard symplectic form. Thus, if ( X, P ) are the canon-ical Hamiltonian variables, then ( dX, dP ) are the canonical local coordinates onthe phase space T ∗ G . The action of G in coordinate form is g ( t ) y = X t ( y, t ) = u ( X ( y, t ) , t ) = u ◦ X ( y, t ) = ( u ◦ g ) y. Thus u = g t g − ∈ g , where g = V ect ( R ), is the Lie algebra of vector fields of theform u∂ x .Now we recall the following result: DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 27 Theorem 5.1. [39, 40] The dual space of g is a space of distributions but thesubspace of local functionals, called the regular dual g ∗ , is naturally identified withthe space of quadratic differentials q ( x ) dx on R . The pairing is given for anyvector field u∂ x ∈ Vect ( R ) by h qdx , u∂ x i = Z R q ( x ) u ( x ) dx . The coadjoint action coincides with the action of a diffeomorphism on the qua-dratic differential:Ad ∗ g : q ( y, dy q ( X, t ) dX = q ( X ( y, t ) , t ) X y dy . We therefore have(5.1)dd t Ad ∗ g ( t ) q (0) = dd t (cid:0) X y q ( X ( y, t ) , t ) (cid:1) = X y (2 u X q ( X, t ) + uq X + q t ) = X y [(2 u x q + uq x + q t ) ◦ g ] ( y ) = 0 , iff q satisfies the Camassa-Holm equation (CH). In order to establish the relation-ship with the Hamiltonian variables, first we notice that ∂u ( X ( y, t ) , t ) ∂y = u X ( X ( y, t ) , t ) X y and also ∂u ( X ( y, t ) , t ) ∂y = ∂X t ∂y = X ty . Thus u X ( X ( y, t ) , t ) = X ty /X y . Similarly, u XX ( X ( y, t ) , t ) = 1 X y (cid:18) X ty X y (cid:19) y and q ( X ( t, y ) , t ) = u ( X ( t, y ) , t ) − u XX ( X ( t, y ) , t ) = X t − X y (cid:18) X ty X y (cid:19) y ;(5.2) q ( x, t ) = Z R P ( y, t ) δ ( x − X ( y, t )) dy with(5.3) P ( y, t ) = X t X y − (cid:18) X ty X y (cid:19) y , and(5.4) u ( x, t ) = 12 Z R G ( x − X ( y, t )) P ( y, t ) dy. where G ( x ) ≡ e −| x | is the Green function of the operator 1 − ∂ x and ( X ( y, t ) , P ( y, t ))are quantities well defined in terms of the scattering data.With a substitution of (5.2) and (5.4) into the Camassa-Holm equation (CH)and using the fact that f ( x ) δ ′ ( x − x ) = f ( x ) δ ′ ( x − x ) − f ′ ( x ) δ ( x − x )we derive a system of integral equations for X and P , namely X t ( y, t ) = Z R G ( X ( y, t ) − X ( y, t )) P ( y, t )d y, (5.5) P t ( y, t ) = − Z R G ′ ( X ( y, t ) − X ( y, t )) P ( y, t ) P ( y, t )d y. (5.6)Moreover, from equations (5.2) and (5.4) a Hamiltonian H can be identified(5.7) H [ X, P ] = 12 Z R G ( X ( y , t ) − X ( y , t )) P ( y , t ) P ( y , t )d y d y in which case equations (5.5) and (5.6) can be written as(5.8) X t ( y, t ) = δH δP ( y, t ) , P t ( y, t ) = − δH δX ( y, t ) , that is to say, these equations are Hamiltonian with respect to the canonical Pois-son bracket(5.9) { A, B } c = Z R (cid:18) δAδX ( y, t ) δBδP ( y, t ) − δBδX ( y, t ) δAδP ( y, t ) (cid:19) d y. where the canonical variables are X ( y, t ), P ( y, t ), with { X ( y , t ) , P ( y , t ) } c = δ ( y − y ) , (5.10) { P ( y , t ) , P ( y , t ) } c = { X ( y , t ) , X ( y , t ) } c = 0 . (5.11)Furthermore, using the canonical Poisson brackets (5.10), (5.11) and the definingintegral (5.2) one can compute { q ( x , t ) , q ( x , t ) } c = − (cid:18) q ( x , t ) ∂∂x + ∂∂x q ( x , t ) (cid:19) δ ( x − x ) ≡ J ( x ) δ ( x − x ) . Now it is straightforward to check that (CH) can be written in a Hamiltonian formas q t = { q, H } c , with the Poisson bracket, generated by J :(5.12) { A, B } c = Z R δAδq ( x ) J ( x ) δBδq ( x ) d x = − Z R q ( x ) (cid:18) δAδq ( x ) ∂∂x δBδq ( x ) − δBδq ( x ) ∂∂x δAδq ( x ) (cid:19) d x DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 29 ( dX, dP ) t =0 ∈ T ∗ G ( dX, dP ) t ∈ T ∗ G q (0) ∈ g ∗ q ( t ) ∈ g ∗ ❄ J ✲ g ( t ) ∈G ❄ J ✲ Ad ∗ g ( t ) Figure 10. Equivariant Momentum Map: quantities related to theCamassa-Holm equation. The subindex t indicates that the corre-sponding variables are evaluated at time t .and the Hamiltonian H given in equation (5.7), which can be written also as H [ q ] = 12 Z R (cid:0) q ( x, t ) u ( x, t ) − u (cid:1) dx. Thus we have an equivariant momentum map J : T ∗ G → g ∗ for the co-adjointaction of G as shown on Figure 10. This means that the values of the corresponding g ∗ quantities produced by the co-adjoined action of the group G are conserved bythe momentum map J in the sense of (5.1).6. Discussion Since the emphasis of this study was on the Zakharov-Shabat dressing methodmany important additional questions have been overlooked - such as the phaseshifts after the interaction and the peakon (antipeakon) limit when u → . Theseissues have been studied previously, for instance in [49, 50, 51]. We mention onlythat the cuspon behavior is very similar to the peakon behaviour, especially thepeakon-antipeakon interactions.Another interesting aspect of the momentum map obtained here is that in thepeakon limit equation (5.2) becomes the well known singular momentum map usedfor the construction of peakon, filament and sheet singular solutions for higher di-mensional EPDiff equations [29]. Holm and Staley [32] introduced the followingmeasure-valued singular momentum solution ansatz for the n − dimensional solu-tions of the EPDiff equation(6.1) q ( x , t ) = N X a =1 Z P a ( s, t ) δ ( x − Q a ( s, t ) ) ds. These singular momentum solutions, called “diffeons,” are vector density functionssupported in R n on a set of N surfaces (or curves) of co-dimension ( n − k ) for s ∈ R k with k < n . They may, for example, be supported on sets of points (vectorpeakons, k = 0), one-dimensional filaments (strings, k = 1), or two-dimensionalsurfaces (sheets, k = 2) in three dimensions. These solutions represent smooth This should not be confused with J from the ZS spectral problem (2.11). embeddings Emb( R k , R n ) with k < n . In contrast, the similar expression (5.2) forthe soliton solutions represent smooth functions R → R . Acknowledgements R.I. is grateful to Prof. D.D. Holm for many discussions on the problems treatedin this paper. 7. Appendix In this appendinx we provide some details on the derivation of the soliton-cusponsolution. Applying equation (3.2) to the dressing factor g as given by equation(4.54) ensures the matrix valued residues satisfy the following A ,y + hσ A − A h σ − iω [ J, A ] = 0 ,B ,y + hσ B − B h σ − λ [ J, B ] = 0 , (7.1)Writing the rank one matrix solutions A and B in the form A = | n i h m | , B = | N i h M | , (7.2)we deduce(7.3) ( ∂ y | n i + ( hσ − iω J ) | n i = 0 , ∂ y h m | = h m | ( h σ − iω J ) ∂ y | N i + ( hσ − λ J ) | N i = 0 , ∂ y h M | = h M | ( h σ − λ J ) . The vectors h m | , h M | satisfy the bare equations and therefore are known in prin-ciple and have been obtained previously (see sections §§ λ = 0 is g ( y, t ; 0) = − A + B ) = diag( g , g )(7.4) = diag (cid:18) iω M m − λ M m iω M m − λ M m , iω M m − λ M m iω M m − λ M m (cid:19) , while the differential equation for X ( y, t ) is(7.5) ( ∂ y X ) e X − h y − u t = g = (cid:18) λ M m − λ M m λ M m − λ M m (cid:19) . Choosing m , m as per the soliton solution cf. [35], and recalling Λ = p h − ω ,we then have m = µ r h + Λ e Ω ( y,t ) + µ r h − Λ e − Ω ( y,t ) ,m = i µ r h − Λ e Ω ( y,t ) + µ r h + Λ e − Ω ( y,t ) ! , DRESSING METHOD FOR THE CAMASSA-HOLM EQUATION 31 where µ k are positive constants. Ignoring an irrelevant overall constant of √ µ µ (2Λ ) − / (see § ( y, t ) by an additive constant, as givenby(7.6) Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u + 12 ω (cid:19)(cid:19) + ln r µ µ , we obtain the simplified expressions m = p h + Λ e Ω ( y,t ) + p h − Λ e − Ω ( y,t ) ,m = i (cid:16)p h − Λ e Ω ( y,t ) + p h + Λ e − Ω ( y,t ) (cid:17) . (7.7)Similarly, as per the cuspon solution, we define the constant vector (cid:10) M (0) (cid:12)(cid:12) V =( ν , ν ) with ν , ν real and positive, thereby ensuring(7.8) M = √ Λ + h e Ω ( y,t ) − √ Λ − h e − Ω ( y,t ) ,M = √ Λ − h e Ω ( y,t ) + √ Λ + h e − Ω ( y,t ) , Ω ( y, t ) = Λ (cid:16) y − t h (cid:16) u − λ (cid:17)(cid:17) + ln q ν ν , Λ = p h + λ . The expression(7.9) g = iω M m − λ M m iω M m − λ M m = T CS B CS , whose explicit form may be deduced frome equations equations (7.7)–(7.8), hasdenominator B CS = − ω λ Λ − Λ p ( h − Λ )(Λ − h ) e Ω +Ω + Λ − Λ p ( h + Λ )(Λ + h ) e − Ω − Ω + Λ + Λ p ( h − Λ )(Λ + h ) e Ω − Ω + Λ + Λ p ( h + Λ )(Λ − h ) e − Ω +Ω ! , (7.10)where we note that Λ > h > Λ thus ensuring Λ − Λ > n = r h + Λ h − Λ , n = r Λ + h Λ − h , we re-write this denominator according to(7.12) B CS = − p ω λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ + 1 n n e − Ω − Ω Λ + Λ + n n e Ω − Ω Λ − Λ + n n e − Ω +Ω Λ − Λ (cid:19) . Similarly it is found that the numerator assumes the form(7.13) T CS = i p ω λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ − n n e − Ω − Ω Λ + Λ − n n e Ω − Ω Λ − Λ + n n e − Ω +Ω Λ − Λ (cid:19) . As with the two-cupson solutions we seek a solution of equation (4.32) in the formof equation (4.13) with(7.14) A CS = α e Ω +Ω + α e − Ω − Ω + α e Ω − Ω + α e − Ω +Ω , where the constants { α l } l =1 are as yet unknown. Equation (4.32) ensures that(7.15) 2 h A CS B CS + B CS ∂ y A CS − A CS ∂ y A CS = 2 h T CS has a solution given by(7.16) A CS = p ω λ (Λ − Λ ) (cid:18) n n e Ω +Ω Λ + Λ + n n e − Ω − Ω Λ + Λ + n n e Ω − Ω Λ − Λ + n n e − Ω +Ω Λ − Λ (cid:19) . The ratio A CS / B CS may now be written as(7.17) A CS B CS = n n n Λ +Λ Λ − Λ e + n Λ +Λ Λ − Λ e + e n n n 21 Λ +Λ Λ − Λ e + n 22 Λ +Λ Λ − Λ e + n n e +2Ω which we simplify by means of the following re-definitions:Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u + 12 ω (cid:19)(cid:19) + ln r µ µ − ln n + 12 ln Λ + Λ b Λ − Λ , Ω ( y, t ) = Λ (cid:18) y − t h (cid:18) u − λ (cid:19)(cid:19) + ln r ν ν − ln n + 12 ln Λ + Λ Λ − Λ . (7.18)These re-dfinitions are valid since Λ > Λ , as was previously noted. 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