On an integrable two-component Camassa-Holm shallow water system

On an in tegrable t wo-com p onen t Camassa-Holm shallo w water system Adrian Constantin 1 , 2 and Rossen I. Iv anov 3 , 2 1 F aculty of Mathematics, University of Vienna, Nor db er gstr asse 15, 1090 Vienna, Austr ia, email: adrian.c onstant in@ uni v ie. a c.at 2 Dep artment of Mathematics, Lund University, 22100 Lund, Swe den, 3 Scho ol of Mathematic al Scienc es, Dublin Institute of T e chnolo gy, Kevin Str e et, Dublin 8, Ir eland, e-mail: rivanov@dit.ie The in terest in the Camassa-Holm equ a tion inspired the searc h for v arious generalizations of this equation with in teresting prop erties and applications. In this letter we deal with suc h a t wo- compon ent integ rable system of coupled equations. First w e derive th e system in the context of shallo w w ater theory . Then w e sho w t h a t while small initial data develop in to global solutions, for some initial d a ta wa ve breaking o ccurs. W e also discuss th e solitary w av e solutions. Finally , we present an explicit construction for th e peakon solutions in th e short wa ve limit of system. P ACS n umbers: 05.45.Yv, 47.35.Bb, 47.35.Fg, 47.35.Jk In r ecent years the Camassa -Holm (CH) equa tion [1] m t + ω u x + 2 mu x + m x u = 0 , m = u − u xx , (1) ( ω b eing a n ar bitr ary constant) ha s caught a grea t deal of attention. It is a nonlinea r dispersive wav e equa - tion that models the propa g ation o f unidirectional irro- tational shallow water w av es o ver a flat bed [1–4], as well as water w av es moving ov er an under lying shear flow [5]. The CH equation also arises in the study o f a certain non-Newto nia n fluids [6] and a ls o mo dels fi- nite length, small a mplitude radial deformation wa ves in cylindrical h yp erela stic ro ds [7]. The CH equa tion has a bi-Hamiltonian struc tur e [8] (and an infinite nu mber of conserv ation laws), it is completely integrable (see [1] for the La x pair formulation and [9] for the direct/inv ers e scattering approa ch), and its solitary wa ve solutions are solitons [1 , 10–13] w ith stable pro files [11, 1 2]. The equa - tion attra cted a lot of attention in recent years due to t wo r emark able features. The first is the presence of so- lutions in the for m of pe a ked solita ry w aves or ’pe akons’ [1, 10, 14] for ω = 0: the p eakon u ( x, t ) = ce −| x − ct | trav- elling at finite speed c 6 = 0 is smo oth e xcept at its crest, where it is co ntin uous, but has a jump discontin uit y in its first deriv ative. The pe a kons replica te a characteristic of the tr avelling w av es of greatest height - exa ct travelling solutions o f the governing equations for water wa ves with a p ea k a t their cr est [16–18] whose capture b y simpler ap- proximate shallow water mo dels has eluded r esearchers un til recently [1 9]. A further r emark able prop erty of the CH eq ua tion is the pr esence o f brea k ing wav es (i.e. the equation has smo o th solutions which develop singular i- ties in finite time in the for m of breaking wa ves [1, 20, 21] - the solution remains bounded while its slop e b ecomes un b ounded in finite time [19]) a s w ell as that of smo oth solutions defined for a ll times [15]. These tw o phenomena hav e always fascinated the fluid mechanics communit y: ’Although breaking and p eaking , as well a s criteria for the o ccur rence of each, are without doubt contained in the equation of the exa ct potential theory , it is intrigu- ing to k now what kind of simpler ma thematical equa tio n could include all these phenomena’ [19]. The s hort wav e limit of the CH eq uation is the Hunter-Saxton (HS) equa- tion u xxt + 2 u x u xx + uu xxx = 0 , (2) obtained fr om (1) by taking m = − u xx . It describ es the pr opaga tion of wa ves in a massiv e director field of a nematic liquid crystal [22], with the orientation of the molecules describ ed b y the field of unit v ectors (cos u ( x, t ) , sin u ( x, t )), w he r e x is the space v a riable in a reference frame moving with the line a rized wav e velocity , and t is a ’slow time v ariable’. The equa tions (1), (2) a dmit many integrable m ul- ticomp onent generalizatio ns [23], the most p opula r o f which is m t + 2 u x m + um x + σ ρρ x = 0 , (3) ρ t + ( uρ ) x = 0 , (4) where m = σ 1 u − u xx , σ = ± 1 and σ 1 = 1 o r, in the ’short wav e’ limit, σ 1 = 0 . (The CH equation can b e 2 obtained via the obvious r eduction ρ ≡ 0 .) This sys tem app ears origina lly in [24] and its mathematical prop erties hav e bee n studied further in ma ny works, e.g. [23, 25– 28]. The system is integrable – it can b e written as a compatibility condition of tw o linear systems (Lax pair) with a sp ectral pa rameter ζ : Ψ xx =  − σ ζ 2 ρ 2 + ζ m + σ 1 4  Ψ , Ψ t =  1 2 ζ − u  Ψ x + 1 2 u x Ψ . It is bi-Hamiltonian, the firs t Poisson bracket { F 1 , F 2 } = − Z h δ F 1 δ m ( m∂ + ∂ m ) δ F 2 δ m + δ F 1 δ m ρ∂ δ F 2 δ ρ + δ F 1 δ ρ ∂ ρ δ F 2 δ m i d x, corres p o nding to the Hamiltonian H = 1 2 Z ( um + σ ρ 2 )d x and the second Poisson br a ck et { F 1 , F 2 } 2 = − Z h δ F 1 δ m ( ∂ − ∂ 3 ) δ F 2 δ m + δ F 1 δ ρ ∂ δ F 2 δ ρ i d x corres p o nding to the Hamiltonian H 2 = 1 2 Z ( σ uρ 2 + u 3 + uu 2 x )dx . There a r e tw o Cas imir s: R ρ d x and R m d x . In wha t follows, w e are go ing to demonstra te how the system (3), (4) arises in shallow water theory . W e start from the Gr een-Naghdi (GN) eq uations [3, 29], which a re derived from the Euler’s equations under certain assump- tions, as follows. Consider the motion of shallow water ov er a flat surface , which is lo ca ted at z = 0 with resp ect to the us ua l Cartesian reference frame. W e assume that the motion is in the x -dir ection and the physical v ari- ables do not dep end on y . Let h b e the mea n level of water, a – the typical amplitude o f the wav e and λ – the t ypica l wa velength of the wav e. Let us now introduce the dimensionless pa r ameters ε = a/h a nd δ = h/λ , which are suppos ed to b e small in the sha llow w ater regime. The v ariable u ( x, t ) desc rib es the horiz ontal velo city of the fluid, η ( x, t ) de s crib es the horizontal dev ia tion of the surface from equilibrium, all measured in dimensionless units. The GN equations u t + εuu x + η x = δ 2 / 3 1 + εη [(1 + εη ) 3 ( u xt + εuu xx − εu 2 x )] x , η t + [( u (1 + εη )] x = 0 . are obtained under the assumption that at leading order u is not a function of z . T he leading o rder expans ion with res pe ct to the parameters ε and δ 2 gives the system  u − δ 2 3 u xx  t + εuu x + η x = 0 , (5) η t + [( u (1 + εη )] x = 0 . (6) One can demonstra te that the system (5), (6) ca n be re- lated to the sys tem (3), (4) in the first order with r esp ect to ε and δ 2 . Indee d, let us define ρ = 1 + 1 2 εη − 1 8 ε 2 ( u 2 + η 2 ) . The expansio n of ρ 2 in the same order o f ε is ρ 2 = 1 + εη − 1 4 ε 2 u 2 . With this definition it is straightforward to write (5) in the for m  u − δ 2 3 u xx  t + 3 2 εuu x + 1 ε ( ρ 2 ) x = 0 , or, introducing the v aria ble m = u − 1 3 δ 2 u xx , at the same order (i.e. neglecting terms of order ε δ 2 ) m t + εmu x + 1 2 εum x + 1 ε ( ρ 2 ) x = 0 . (7) Next, using the fact that u t ≈ − η x , η t ≈ − u x , from the definition of ρ we get ρ t = 1 2 εη t + 1 4 ε 2 ( η u ) x . With this expression fo r ρ t and with ρ ≈ 1 + εu , equation (6) c a n be written as ρ t + ε 2 ( ρu ) x = 0 . (8) The rescaling u → 2 ε u , x → δ √ 3 x , t → δ √ 3 t in (7), (8) gives (3), (4) with σ = σ 1 = 1. The case σ = − 1, which is often considered, corr e s p o nds to the situation in which the g ravit y acceler a tion po int s up wards. W e men tion also that the Kaup - Boussinesq system [30] is another in tegra ble system matching the GN equation to the sa me order of the par a meters ε, δ [19, 31]. Notice that in the h ydro dynamical deriv ation of (3)-(4) we requir e that u ( x, t ) → 0 and ρ ( x, t ) → 1 as | x | → ∞ , at any instant t . W e will now show that for the system (3), (4) in the hydrodynamica lly relev ant ca se σ = σ 1 = 1 w av e break- ing is the only w ay that s ingularities ar ise in smo oth so- lutions. The system a dmits breaking wa ve s olutions as well as s olutions defined for all times. In particular, we will a nalyze the trav eling wa ve solutions The w ell-p os edness (existence, uniqueness, a nd con- tin uous dependence on data) follows b y Kato’s semi- group theory [32] for initial data u 0 = u ( · , 0) ∈ H 3 and ρ 0 = ρ ( · , 0) such that ( ρ 0 − 1) ∈ H 2 [33]. If T = T ( u 0 , ρ 0 ) > 0 is the maximal existence time, then the integral of motion Z [ u 2 + u 2 x + ( ρ − 1) 2 ]d x (9) ensures that u ( · , t ) is uniformly bounded (i.e. for all v al- ues of x ∈ R a nd all 0 ≤ t < T ) in view o f the inequalit y sup x ∈ R | u ( t, x ) | 2 ≤ 1 2 Z ( u 2 + u 2 x )d x. (10) 3 Consideratio ns a nalogo us to those ma de in [27 ] for a sim- ilar system show that the solution blows up in finite time (i.e. T < ∞ ) if a nd o nly if lim inf t ↑ T { u x ( t, x ) } = −∞ , (11) which, in light of the unifor m b oundednes s of u , is inter- preted as wa ve brea king. T o show that wa ve bre a king o ccurs , we introduce the family { ϕ ( · , t ) } t ∈ [0 ,T ) of diffeomorphisms ϕ ( · , t ) : R → R defined by ∂ t ϕ ( x, t ) = u ( ϕ ( x, t ) , t ) , ϕ ( x, 0) = x, (12) and we denote M ( x, t ) = u x ( ϕ ( x, t ) , t ) , γ ( x, t ) = ρ ( ϕ ( x, t ) , t ) . Consider no w initial data satisfying ρ 0 (0) = 0 and u ′ 0 (0) < − 2  || u 0 || 2 1 + || ρ − 1 || 2 0  1 / 2 . (13) Noticing tha t (1 − ∂ 2 x ) − 1 f = p ∗ f (conv olutio n) with p ( x ) = 1 2 e −| x | , and applying the o p erator (1 − ∂ 2 x ) − 1 to (3), we get u t + uu x + p ∗ ( u 2 + 1 2 u 2 x + 1 2 ρ 2 ) = 0 . Applying now ∂ x and using the identit y ∂ 2 x p ∗ f = p ∗ f − f , we o btain u tx + uu xx + 1 2 u 2 x = 1 2 ( u 2 + ρ 2 ) − p ∗ ( u 2 + 1 2 u 2 x + 1 2 ρ 2 ) . (14) This equation in co mbination with (12) yields ∂ t M ( t, x ) + 1 2 M 2 ( t, x ) ≤ 1 2  u 2 ( ϕ ( x, t ) , t ) + γ 2 ( x, t )  . (15) On the other ha nd, fro m (12) and (4), we obtain ∂ t γ = − γ M . (16) Since γ (0 , 0) = 0 we infer that γ (0 , t ) = 0 for 0 ≤ t < T . The r elation (1 3 ) together with (9), (10) ensure that 4 u 2 ( ϕ ( x, t ) , t ) ≤ M 2 (0 , 0). But then (15 ) yields ∂ t M (0 , t ) ≤ − 1 4 M 2 ( t, 0 ) for 0 ≤ t < T . As M (0 , 0) = u ′ 0 (0) < 0 , this implies M (0 , t ) ≤ 4 u ′ 0 (0) 4 + u ′ 0 (0) t → −∞ in finite time. How ever, not all so lutions develop singularities in fi- nite time. F o r example, if the initial data is suffi- ciently small, then the solutio n e volving from it is de- fined for all times. More precisely , let α ∈ (0 , 1) a nd assume that | 1 − ρ 0 ( x ) | ≤ α for all x ∈ R , while || u 0 || 2 + || ρ 0 − 1 || 0 ≤ α . Then the corr esp onding so lu- tion to (3), (4) is global in time. Indeed, if the maximal existence time were T < ∞ , then for some x 0 ∈ R we would hav e lim inf t ↑ T M ( t, x 0 ) = − ∞ . W e now s how that this is imp os sible. First, notice that u 2 ( x, t ) ≤ α 2 , ( x, t ) ∈ R × [0 , T ) . (17) by (9), (10). The in tegra l o f motion (9) also ensur es that || ρ ( · , t ) − 1 || 0 ≤ α o n [0 , T ). Using this and (4 ), (9), (17), we g et 0 ≤ p ∗ ( u 2 + 1 2 [ u 2 x + ρ 2 ]) ≤ 1 2 p ∗ u 2 + 1 2 ( u 2 + u 2 x + ( ρ − 1) 2 ) + p ∗ ( ρ − 1) + 1 2 p ∗ 1 ≤ α 2 + α + 1 on R × [0 , T ). W e now infer from (12), (14) that at x = x 0 , M t = − 1 2 M 2 + 1 2 γ 2 − f ( t ) , t ∈ [0 , T ) , (18) with the contin uous function f : [0 , T ) → [0 , ∞ ) bo unded, i.e. there is a constant k 0 > 0 suc h that 0 ≤ γ ( t ) ≤ k 0 on [0 , T ). The s olution  γ ( t ) , M ( t )  of the nonlinear system (16), (18) with initial data γ (0) = ρ 0 ( x 0 ) > 0 and M (0) = u ′ 0 ( x 0 ) supp osedly blows up in finite time as lim inf t ↑ T M ( t ) = − ∞ . How ever, notice that (16) and γ (0) > 0 ensure γ ( t ) > 0 for all t ∈ [0 , T ). Thu s w e may co nsider the p ositive function w ( t ) = γ (0) γ ( t ) + γ (0) γ ( t ) [1 + M 2 ( t )] for t ∈ [0 , T ). Using (1 6), (18) we get w ′ ( t ) = 2 γ (0) γ ( t ) M ( t )[ f ( t ) + 1 2 ] ≤ [ k 0 + 1] γ (0) γ ( t ) (1 + M 2 ) ≤ [ k 0 + 1] w ( t ) for all t ∈ (0 , T ). Th us w ( t ) ≤ w (0) exp([ k 0 + 1] t ) on [0 , T ) a nd this pr even ts blowup. The obtained con- tradiction shows that the solution is de fined glo bally in time. Global exis tence is how ever not confined to small am- plitude wa ves, as we shall see now by establishing the existence of traveling wa ves of la r ge a mplitude: solutions u ( x, t ) = ψ ( x − ct ) , ρ ( x, t ) = ξ ( x − ct ) trav eling with constant wa ve sp eed c > 0. T o find whether such solu- tions e x ist, notice tha t with the previous Ansa tz equation (4) b ecomes ξ ′ ( c − ψ ) = ξ ψ ′ and the asymptotic limits ψ ( x ) → 0 and ξ ( x ) → 1 as | x | → ∞ yield ξ = c c − ψ . 4 Thu s (3) beco mes a differential eq uation so lely for the unknown ψ . Integrating this equation o n ( −∞ , x ] and taking in to a ccount the asymptotic behaviour o f ψ , we get the equation − cψ + cψ ′′ + 3 2 ψ 2 − ψ ψ ′′ − 1 2 ( ψ ′ ) 2 + c 2 2( c − ψ ) 2 = 1 2 . Multiplication by ψ ′ and another integration on ( −∞ , x ] leads to  ( ψ ′ ) 2 − ψ 2  ( c − ψ ) + c 2 c − ψ = ψ + c, recalling the decay of ψ far out. Thus ( ψ ′ ) 2 = ψ 2 ( c − ψ ) 2 ( c − ψ − 1)( c − ψ + 1) . (19) The asymptotic behaviour ψ ( x ) → 0 as | x | → ∞ yie lds now the necessa ry condition c ≥ 1 for the exis tence of trav eling wa ves. F or c ≥ 1, a qualitative analysis of (19 ) shows that 0 ≤ ψ ≤ c − 1. Th us no nt r iv ial trav eling wa ves exist only for c > 1, in which cas e b oth ψ and ξ are smo oth wav es of elev ation with a single crest profile of maximal amplitude c − 1, resp ectively c . It is p os sible to find ex plic it formulas for the traveling wa ves in terms of elliptic functions cf. [34]. Due to the in tegra bility o f the system, w e exp ect the s olitary w aves to interact like solitons. Notice the absence of pea kons a mong the solita ry wa ve solutions. How ever, there ar e p eakon s olutions of the ’short wav e limit’ equa tion σ 1 = 0. Although this limit is not covered by the pres ented h ydro dynamical deriv ation, we will descr ib e briefly the construction o f the p eakon solutions, s ince these ar e interesting by themselves. The limit σ 1 = 0 is a t wo compo ne nt analog of the Hunter- Saxton equation. Such sy s tem is a particular case o f the Gurevich-Zybin system [35], which describ es the dynam- ics in a mo del of nondissipative da rk matter [36]. The p ea kon so lutions have the form m ( x, t ) = N X k =1 m k ( t ) δ ( x − x k ( t )) , (20) u ( x, t ) = − 1 2 N X k =1 m k ( t ) | x − x k ( t ) | , (21) ρ ( x, t ) = N X k =1 ρ k ( t ) θ ( x − x k ( t )) , (22) where θ is the Heaviside unit step function. The asymp- totic be haviour ρ ( x, t ) → 0 for x → ∞ and the condition R m d x = 0 (recall that m = − u xx ) lea d to N X l =1 m l = N X l =1 ρ l = 0 , or N X l =1 µ l = 0 in terms o f the new complex v ariable µ k ≡ m k + iρ k . The substitution o f the Ansatz (20) – (22 ) into (3), (4), under the assumption that x 1 ( t ) < x 2 ( t ) < . . . < x N ( t ) for a ll t , (a co ndition ho lding for the p eakons of (2) cf. [37 ]) gives the following dynamical s ystem fo r the time-dep endent v ar iables: d x k d t = − 1 2 N X l =1 m l | x l − x k | , (23) d µ k d t = µ k 2 N X l =1 µ l sgn( k − l ) , (24) with the conv ention sgn(0) = 0. The integrals for this system can be obtained from the integrals of (3), (4) (av ailable in [2 3]) by substituting the expr essions (20), (21), (22). It is co nvenien t to wr ite the system in terms of the new indep endent v ariables ∆ k ≡ x k +1 − x k , M k ≡ µ 1 + . . . + µ k , with k = 1 , 2 , . . . , N − 1 . The Hamiltonian of the new system is H = 1 2 N − 1 X l =1 | M k | 2 ∆ k , the eq ua tions d∆ k d t = − Re( M k ) ∆ k , d M k d t = 1 2 M 2 k , are Hamiltonian with resp ect to the br ack et { ∆ k , M l } = − M k ¯ M k δ lk , in which the bar stands for complex conjugation. 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