Integrable peakon equations with cubic nonlinearity

In tegrable p eak on equations with cubic nonlinearity Andrew N.W. Hone and Jing P i ng W ang † † Institute of Mathema tics, Statistics & Actuarial Science, Universit y o f Ke nt , Canterbury CT2 7NF, UK E-mail: anwh@ kent.a c.uk j w83@k ent.ac .uk Abstract. W e present a new in tegr able partial differential equation found by Vladimir Noviko v. Like the Camas sa-Holm and Degasp eris- P ro cesi equations, this new equation admits p eaked soliton (pe akon) solutions, but it has nonlinear terms that are cubic, ra ther than quadra tic. W e give a matrix Lax pair for V. Novik ov’s equation, and show how it is related by a recipro c a l transfor mation to a negative flow in the Saw ada- K otera hierarch y . Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to o btain the Hamiltonian for m of the finite-dimensional system for the interaction of N p eakons, and the t wo-bo dy dynamics ( N = 2) is explicitly int egr ated. Finally , all of this is compared with so me analogo us results for a nother cubic p ea kon derived by Zhijun Qiao. Submitted to: J. Phys. A: Math. Gen. Pe akon e quations with cubic nonl i n e arity 2 1. In tro duction The sub ject of this pap er is the partial differen tial equation (PDE) u t − u xxt + 4 u 2 u x = 3 uu x u xx + u 2 u xxx , (1) whic h w as discov ered v ery recen tly b y Vladimir Novik o v in a symmetry classification o f nonlo cal PDEs with cubic nonlinearity [18]. The p erturbative symmetry approac h [17] yields necessary conditions for a PDE to admit infinitely man y symmetries. Using this approach, No vik o v w as able to isolate the equation (1) and find its fir st few symmetries , and he subseque ntly found a scalar La x pair fo r it, pro ving that the equation is in tegrable. D ue to the u xxt term on the left hand side of (1), this equation is not an ev olutionary PDE for u . How ev e r, taking the con v olution with the Green’s function g ( x ) = exp( −| x | ) / 2 for the Helmholtz op erator (1 − ∂ 2 x ) giv es the nonlo cal (in tegro- differen tial) equation u t + u 2 u x + g ∗ [3 uu x u xx + 2( u x ) 3 + 3 u 2 u x ] = 0. It is con v enien t to define a new dep enden t v ariable m to b e the Helmholtz o p erator acting on u , in whic h case the equation (1) can b e more concisely written as m t + u 2 m x + 3 uu x m = 0 , m = u − u xx . (2) Henceforth w e w ork with the ab o ve form of the equation. The w ork of Camassa and Holm [3], who deriv ed the equation m t + um x + 2 u x m = 0 , m = u − u xx (3) from an asymptotic appro ximation to the Hamiltonian for the Green-Naghdi equations in shallow water theory , has a ttracted a lot of in terest in the past fif t een y ears, fo r v arious reasons. T o b egin with, it is remark able that the Camassa-Holm equation (3) appro ximates unidirectional fluid flo w in Euler’s equations at the next order b ey ond the KdV equation, and y et preserv es the prop ert y of b eing integrable, fitting as it do es in to the hereditary symmetry framew ork of F ok as and F uc hssteine r [7], with a bi-Hamilto nian structure and a Lax pair based o n a linear sp ectral problem of second order. Also, while there are smo oth soliton solutions of (3) on a non-zero constan t bac kground (or equiv alently , with the additio n of linear disp ersion terms), the Camassa-Holm equation has p eak on solutions, which a re p eak ed solitons of the form u ( x, t ) = N X j =1 p j ( t ) exp( −| x − q j ( t ) | ) , (4) where the p ositions q j and amplitudes p j satisfy the system of ODEs ˙ q j = N X k =1 p k e −| q j − q k | , ˙ p j = p j N X k =1 p k sgn( q j − q k ) e −| q j − q k | (5) for j = 1 , . . . , N . The p eak ons are smo oth solutions of (3) except at the p eak p ositions x = q j , where the deriv a tiv e of u is discon tin uous. The equations ( 5) form an in tegrable finite- dimensional Hamiltonian system, corresp onding to geo desic flow on an N -dimensional Pe akon e quations with cubic nonl i n e arity 3 manifold with inv erse metric g j k = exp ( −| q j − q k | ). The p o sitions q j and momen ta p j satisfy the canonical P oisson brack et { q j , p k } = δ j k . The dynamics of tw o p eakons ( N = 2) w as solv ed explicitly in the original paper b y Camass a and Holm [3], while the explicit solutio n for ar bitrary N w as found b y Beals, Sattinger and Szmigielski [2]. F uc hssteiner also show ed that the equation (3) is related via a recipro cal transformation to the first negativ e flo w in the hierarch y of the Kortew eg–de V ries equation. One might w onder whether the Camassa-Holm equation is the only in tegrable PDE of its kind, b eing a shallo w w ater equation whose disp ersionless v ersion has w eak soliton solutions. This turns out not t o b e the case. Degasp eris and Pro cesi used an asymptotic in tegrability approach to isolate in tegrable third order equations, and disco v ered a new equation with the disp ersionless form m t + um x + 3 u x m = 0 , m = u − u xx . (6) The D egasp eris-Pro cesi equation turns out to b e in tegrable, with a bi- Hamiltonian structure and a Lax pair based on a third order sp ectral problem [4], and it also arises in shallow w ater theory [6]. The equation (6) is related b y a recipro cal transformatio n to a negativ e flow in the hierar ch y of the Kaup-Kup ershmidt equation [4, 13], and it also has p eakon solutions of the fo rm (4) whose dynamics is describ ed by an in tegrable finite-dimensional Hamiltonian system with a non-canonical P oisson bra c k et (see [1 4], and section 4 b elow ). The explicit solution of the N -p eak on dynamics was deriv ed by Lundmark and Szmigiels ki [1 6 ]. There are at least t wo distinct inte gra ble analogues of the Camassa-Holm equation in 2+ 1 dimensions [12, 15], while the Euler-P oincar ´ e equation on the diffeomorphism group (EPDiff ) prov ides a g eometrical generalisation of the Camassa-Holm equation in ar bit r a ry dimension [11], and admits w eak solutions with supp or t on lo w er-dimensional submanifolds. Rosenau also found v arious PDEs with nonlinear disp ersion, whic h ha v e solutions with compact supp ort [21], s ome of whic h are relativ es o f the Camassa-Holm equation [19]. In what follows we presen t a bi-Hamiltonian structure for the integrable hierarc h y of PDEs o f whic h (2) is a mem b er, presen t a matrix Lax pair corresp onding to a zero curv a ture represen tation fo r this equation, and sho w how it is related via a recipro cal transformation to a negative flow in the Sa wada-Kotera hierarch y . W e also presen t a system of Hamiltonian O DEs fo r the dynamics of p eak on solutions of (2), and explicitly in tegrate the equations for the in teraction of t w o p eak ons. Finally , we compare our results with analogous prop erties of the in tegrable PDE m t +  m ( u 2 − u 2 x )  x = 0 , m = u − u xx , (7) whic h w as recen tly obtained b y Zhijun Qiao [20]. Q ia o’s equation w as the original starting p oin t for our study , since it ha s cubic (rather than quadratic) nonlinear terms, and this is what led us to ask Vladimir Nov iko v to seek o ther integrable equations o f this kind. Pe akon e quations with cubic nonl i n e arity 4 2. La x pair and recipro cal transformation The equation (2) arises as a zero curv ature equation F t − G x + [ F , G ] = 0, this b eing the compatibilit y condition fo r the linear system Ψ x = FΨ , Ψ t = GΨ , (8) where F =    0 mλ 1 0 0 mλ 1 0 0    , G =    1 3 λ 2 − uu x u x λ − u 2 mλ u 2 x u λ − 2 3 λ 2 − u x λ − u 2 mλ − u 2 u λ 1 3 λ 2 + uu x    . (9) W e found the linear system (8) directly b y applying the prolongation algebra metho d of W ahlquist and Estabro o k (see [22], and also [8 ]) , but the details of t his deriv ation will not b e giv en here ‡ . In any case, once a Lax pair is giv en one can use it to deriv e most of the imp ortan t pro p erties of an in tegrable PDE. The first imp ortant observ ation w e wish to make ab out Vladimir Novik o v’s equation is that it is connected to a negativ e flo w in the Saw ada-Kotera hierarc hy via a recipro cal transformation. Up on r ewriting the PDE (2 ) in the form ( m 2 / 3 ) t + ( m 2 / 3 u 2 ) x = 0 , (10) it is immediately clear that m 2 / 3 is a conserv ed densit y . Since eac h of the equations (3) and (6 ) has a conserv ed densit y of the form m 1 /b , for b = 2 , 3 resp ectiv ely , and these densities yield recipro cal transformatio ns to negativ e flo ws in more familia r hierarc hies, this suggests that w e should define the new indep enden t v ariables X and T b y dX = m 2 / 3 dx − m 2 / 3 u 2 dt, dT = dt. (11) The closure condition d 2 X = 0 for the exact o ne-form dX in the recipro cal transformation (11) is just the conserv a tion la w ( 10). T ransforming the time evolution of m in (2), together with the definition m = u − u xx , leads to the equations  1 V  T =  W 2 V  X , W X X −  V X X 2 V − ( V X ) 2 4 V 2 + 1 V 2  W + 1 = 0 , (12) where V = m 2 / 3 and W = um 1 / 3 . The evolution equation f or 1 /V in the new indep enden t v ariables X, T is the recipro cal transformation of the equation (2). Ho w ev er, in o rder to recognise (12) as a mem b er of the Sa w ada-Kotera hierarc h y w e need to apply the recipro cal transformation to the La x pair. (F or details of t he Saw ada- Kotera hierarc h y and its extensions w e refer the reader to [10].) By writing the column v ec tor Ψ in components as Ψ = ( ψ 1 , ψ 2 , ψ 3 ) T , w e can eliminate ψ 1 and ψ 3 from Ψ x = FΨ to get a single scalar equation for ψ = ψ 2 , na mely ψ xxx − 2 m x m − 1 ψ xx − ( m xx m − 1 − 2( m x ) 2 m − 2 + 1) ψ x = m 2 λ 2 ψ . (13) ‡ V. Novik ov told us that he earlie r found a scalar Lax pair for the PDE (2) based on a third order sp ectral pr oblem, b y applying a recipro c al transforma tion to a symmetry of fif th order. An y sca lar linear problem for (2) should be equiv alen t to the matrix system (8), p os sibly after a gauge transformatio n. Pe akon e quations with cubic nonl i n e arity 5 When the reciprocal transformation (11) is used to t r a nsform the x deriv ative s as ∂ x = V ∂ X , the equation (13) b ecomes ψ X X X + U ψ X = λ 2 ψ , with U = − V X X 2 V + ( V X ) 2 4 V 2 − 1 V 2 , (14) so that the se cond equation in (12 ) has the form W X X + U W + 1 = 0 for the same p oten tial U . The third order o p erator ∂ 3 X + U ∂ X in ( 14) is the standard Lax op erator for the Sa w ada- Kotera hierarch y , and by tra nsfor ming the t deriv ativ es in Ψ t = GΨ according to ∂ t = ∂ T − W 2 ∂ X w e find that the T ev olution of ψ is given by ψ T = 1 λ 2  W ψ X X − W X ψ X  − 2 3 λ 2 ψ . (15) After gauging ψ b y a factor of e 2 T / (3 λ 2 ) to r emo v e the final term ab ov e, a nd then replacing λ 2 b y λ and setting φ = − 3 W , w e see that (14) and ( 1 5) are respectiv ely equiv alent to equations (2 .25) and (2.26) in [13], and the compatibilit y requiremen t ψ T X X X = ψ X X X T for this pair of scalar equations g iv es t w o conditions, namely that W X X + U W is independent of X , and U T + 3 W X = 0. The latter tw o conditions follow from (12) pro vided that U is giv en in t erms of V as in (14). 3. Conserv ed densities and bi-Hamiltonian str uct ure The Lax pair ( 8 ) can b e used to find infinitely man y conserv ed densities for (2). Up on setting ρ = (log ψ ) x in ( 1 3) it is clear that ρ satisfies the equation ρ xx + 3 ρρ x + ρ 3 − 2 m x m − 1 ( ρ x + ρ 2 ) +  m ( m − 1 ) xx − 1  ρ = m 2 λ 2 . (16) The corresp o nding t ev olution of ψ implies that ρ t = F x for some flux F , and so b y expanding ρ in p ow ers of λ one finds co efficien ts that are conserv ed densities. The asymptotic expansion for λ → ∞ has ρ 3 ∼ m 2 λ 2 , so ρ ∼ m 2 / 3 λ 2 / 3 , whic h extends to an infinite series ρ ∼ m 2 / 3 λ 2 / 3 + P ∞ j =1 µ j λ − 2 j / 3 . The densities µ j are all determined recursiv ely from (16) as lo cal functions of m ; fo r example µ 1 = m − 5 / 3 m xx − 4 3 m − 8 / 3 m 2 x + 3 m − 2 / 3 . An expansion in p ositiv e p o w ers of λ for λ → 0 can consisten tly b egin with ρ ∼ − muλ 2 , but one must solv e a second order differen tial equation to obtain eac h subseque nt term, whic h leads to increasingly nonlo cal expressions in m and u . Since w e kno w that (2) is recipro cally related t o a negat iv e Saw ada-Kotera flo w, it is natural to regard the µ j as densities for Hamiltonians t hat generate a p ositiv e hierarc h y of flows, with the expansion around λ = 0 pro ducing Hamiltonian densities for negativ e flows. Ha ving found these conserv ed densities, w e r equire a pair of Hamiltonian op erators B 1 , B 2 whic h are compatible (in the sense t ha t B 1 + B 2 , o r any linear com bination of them, is Hamiltonian) and can b e used t o generate the hierarc h y of flow s that commute with (2). F rom earlier studies on the Camassa-Holm and Degasp eris-Pro cesi equations [13, 14], w e kno w that all nonlo cal op erators of the form B = m 1 − 1 /b D x m 1 /b ˆ Gm 1 /b D x m 1 − 1 /b , (17) Pe akon e quations with cubic nonl i n e arity 6 with ˆ G = ( c 1 D x + c 2 D 3 x ) − 1 for constants b, c 1 , c 2 , are Hamiltonian, and hav e Casimir R m 1 /b dx . In fact, the case b = 2 giv es the third Hamiltonia n structure for t he Camassa- Holm equation, and b = 3 giv es the second Hamilto nian structure for the Degasp eris- Pro cesi equation. Since R m 2 / 3 dx is a conserv ed quan tity for (2), this suggests w e should consider the op erato r (17) with b = 3 / 2 , and indeed w e find that the equation can b e written in Hamiltonian form as m t = B 1 δ ˜ H δ m , ˜ H = 1 4 Z mu dx, (18) for B 1 = − 18 B | b =3 / 2 in the case c 1 = 4, c 2 = − 1. Some other conserv ed quan tities are H 1 = R 1 8  u 4 + 2 u 2 u 2 x − u 4 x 3  dx , H 5 = R m 2 / 3 dx , H 7 = R 1 3 ( m − 8 / 3 m 2 x + 9 m − 2 / 3 ) dx = R µ 1 dx , and the next one has leading term H 11 = R ( m − 16 / 3 m 2 xxx + . . . ) dx (up to rescaling). These are the first few in the sequence of Ha milto nians that generate lo cal symmetries of w eigh t k ≡ ± 1 mo d 6 according to m t k = B 2 δ H k δ m = B 1 δ H k +6 δ m , (19) where B 2 = (1 − D 2 x ) m − 1 D x m − 1 (1 − D 2 x ). The recursion op erato r is R = B 2 B − 1 1 , and it g enerates the flo ws R n m x of w eigh t 6 n + 1 and the flows R n m t 5 of w eigh t 6 n + 5. Ho w ev er, when k = 5 or 7 t he rightmost part of the iden tit y (19 ) fails, since b oth H 5 and H 7 are Casimirs for B 1 ; and the Hamiltonian ˜ H is a Casimir for B 2 . The pro of of the follo wing theorem will b e presen ted in a forthcoming article. Theorem 1 T he op er a tors B 1 = − 2(3 mD x + 2 m x )(4 D x − D 3 x ) − 1 (3 mD x + m x ) and B 2 = (1 − D 2 x ) m − 1 D x m − 1 (1 − D 2 x ) pr ovide a bi-Hamiltoni a n structur e for the hier ar chy of symmetries o f the e quation (2). 4. P eak on solutions F rom (10 ) the trav elling w av e s u = u ( z ), z = x − ct of (2) satisfy ( u 2 − c ) m 2 / 3 = const. In the g eneral case this gives m = 1 2 c 2 D ( u 2 − c ) − 3 / 2 for constan t D 6 = 0, whic h in tegrates further to ( u ′ ) 2 = u 2 + c D u ( u 2 − c ) − 1 / 2 + cE , fo r another constant E . This can b e reduced to a quadrature whic h is the sum of elliptic integrals o f the third kind, namely dz = ( 1 w − 1 − 1 w +1 ) dw 2 p ( D w + E )( w 2 − 1) + w 2 , w = u ( u 2 − c ) − 1 / 2 . (20) Ho w ev er, if w e require w av es that v anish at spatial infinit y , then D = 0, whic h implies that m = 0 whenev er u 2 6 = c . No smo oth solution can satisfy the latter requiremen t, but this observ at io n suggests that there should b e a w eak solution of the form u ( x, t ) = ± √ c e −| x − ct − x 0 | , c > 0 , x 0 constan t , (21) whic h has the same form as the p eak on for the Camassa-Holm and Degasp eris-Pro cesi equations, except that the amplitude is the square ro o t of the sp eed rat her than b eing equal to the sp eed, as is the case for the p eak on solutions of (3 ) a nd (6). The expression (21) ha s m = 0 a w ay from the p eak, and u 2 = c at the p eak, but to regard it as a w eak Pe akon e quations with cubic nonl i n e arity 7 solution of (2) it is necessary t o substitute it in to the equation and integrate a g ainst suitable test functions with suppor t ar ound the p eak. F or the single p eak on (21 ) w e ha v e m = ± 2 √ c δ ( x − ct − x 0 ), but t here is some subtlet y in in terpreting this as a solution, b ecause u x = ∓ √ c sgn( x − ct − x 0 ) e −| x − ct − x 0 | and m are distributions, while the equation (2 ) includes the pro duct u x m . The integrals can b e regularised b y taking the conv en tion sgn(0) = 0, but a more rigorous a lternativ e is to construct the p eak on distribution as a limit of smo o t h solutions of the PD E. F or t he Camassa-Holm equation it is kno wn that the single peakon arises as a w eak solution in this wa y (see [19] fo r a v ery detailed treatmen t), and multi-p eakons arise similarly as a degenerate limit of algebro-geometric solutions [1]. If w e ta k e u to b e a linear sup erp osition o f N p eakons , a s in (4), so that m = 2 P N j =1 p j ( t ) δ ( x − q j ( t )), then substituting in to t he eq uation (2) and integrating against test functions supp orted at x = q j giv es the equations of motion fo r t he p eak p ositions and amplitudes. Prop osition 1 The e quation (2) has p e akon solutions of the form (4), whose p ositions q j ( t ) and am plitudes p j ( t ) evolve ac c o r ding to the dynamic al system ˙ q j = P N k ,ℓ =1 p k p ℓ e −| q j − q k |−| q j − q ℓ | , ˙ p j = p j P N k ,ℓ =1 p k p ℓ sgn( q j − q k ) e −| q j − q k |−| q j − q ℓ | . (22) The ab o v e equations are not in canonical Hamiltonian form. Ho w ev er, in [1 4] one of us sho w ed how Hamiltonian op erators of the for m (17) are reduced to Poiss on structures on the finite-dimensional submanifold of N p eaks or pulses, resulting in the Poiss on brac k et { q j , q k } = G ( q j − q k ) , { q j , p k } = ( b − 1) G ′ ( q j − q k ) p k , { p j , p k } = − ( b − 1) 2 G ′′ ( q j − q k ) p j p k , (23) where G is the sk ew-symmetric G r een’s function for the op erato r ˆ G . F or N > 2 , the Jacobi iden tit y ho lds fo r this brac k et if and only if G satisfies the functional equation G ′ ( α )( G ( β ) + G ( γ )) + cyclic = 0 for α + β + γ = 0 . (24) This functional equation is a lso a sufficien t condition f or the op erator (17) to b e Hamiltonian, and Braden and Byatt-Smith prov ed in the app endix t o [14] that the unique con tinuously differen tiable, o dd solution of equation (24) is G ( x ) = A sgn( x )(1 − e − B | x | ) for arbitrary constan ts A, B . Up to rescaling x , this is the Green’s function for the o p erator ˆ G = ( D x − D 3 x ) − 1 (or ˆ G = D − 1 x in the degenerate case B → ∞ ). In the case at hand, the op erator B in Theorem 1 has ˆ G = (4 D x − D 3 x ) − 1 , and the Hamiltonian ˜ H reduces to a conserv ed quantit y h for the equations of motion (22), whic h is quadratic in the amplitudes p j . Theorem 2 T he e quations (22) f o r the motion of N p e akons in the PDE (2) ar e a n Hamiltonian v e ctor field ˙ q j = { q j , h } , ˙ p j = { p j , h } Pe akon e quations with cubic nonl i n e arity 8 for the Hamiltonian h = 1 2 P N j,k =1 p j p k exp( −| q j − q k | ) , wi th the Poisson br acket s p e cifie d by { q j , q k } = sgn( q j − q k )(1 − e − 2 | q j − q k | ) , { q j , p k } = e − 2 | q j − q k | p k , { p j , p k } = sgn( q j − q k ) e − 2 | q j − q k | p j p k . (25) W e conjecture that t he equations (22) constitute a Liouville in tegrable Hamiltonian system with N degrees of freedom. F o r N = 1 this is trivial, and for N = 2 the r esult follo ws from the existence of a second indep enden t in tegral in in volution with h , namely k = p 2 1 p 2 2 (1 − e − 2 | q 1 − q 2 | ) , { k , h } = 0 . (26) The inv ariant k is degree four in the amplitudes, a nd fo r all N there is an ana lo gous in tegral, quartic in p j , obtained b y restricting the Hamiltonian H 1 to t he p eakon submanifold. Indeed, the conserv ed densities for the negative flows in the hierarch y of the PDE (2) should all reduce to integrals for the N -p eak on dynamics, but the explicit construction of N indep enden t P oisson-comm uting in tegrals for (22) is still in pro g ress. It is also w orth mentioning that the Lax pair (8) can b e used to obtain a n N × N Lax matrix for the finite-dimensional system, satisfying LΦ = − λ − 2 Φ , L = SPEP , (27) where S j k = sgn ( q j − q k ), P = diag( p 1 , . . . , p N ), E j k = exp( −| q j − q k | ). The j th comp onen t of the vec tor Φ is just ψ 2 ( q j ( t ) , t ), where ψ 2 ( x, t ) is the second comp onen t of Ψ in (8), and the corresp onding time evolution ˙ Φ = MΦ yie lds the Lax equation ˙ L = [ M , L ] fo r the system (22 ). Ho w ev er, unfortunately the sp ectral inv ariants of L , whic h a r e the co efficien ts of the c haracteristic p olynomial det( L + λ − 2 I ) (a p o lynomial in λ − 2 ), do no t pro vide enough in tegrals. F or instance, when N = 2 w e find that the trace of L v anishes, while the trace of L 2 giv es k , but h do es not app ear. F or higher v alues of N w e ha v e found that the sp ectral inv ariants o f L ha ve degrees 4 , 8 , 12 , . . . but the in tegrals of degrees 2 , 6 , 10 , . . . are missing. This leads us to exp ect that there should b e another Lax represen tation fo r this system whic h would pro vide the correct n um b er of in tegrals fo r Liouville’s theorem. F or the t w o-p eakon dynamics, the equations of motion are ˙ q 1 = ( p 1 + p 2 e −| q 1 − q 2 | ) 2 ˙ q 2 = ( p 2 + p 1 e −| q 1 − q 2 | ) 2 ˙ p 1 = sgn( q 1 − q 2 ) e −| q 1 − q 2 | ( p 1 + p 2 e −| q 1 − q 2 | ) p 1 p 2 , ˙ p 2 = − sgn( q 1 − q 2 ) e −| q 1 − q 2 | ( p 2 + p 1 e −| q 1 − q 2 | ) p 1 p 2 , (28) and without loss of generalit y w e consider the case where the peaks are initially well separated, so that q 1 << q 2 with q 1 ∼ c 1 t , q 2 ∼ c 2 t (f or c 1 > c 2 > 0), and we assume that b oth amplitudes are p ositiv e, so p 1 → √ c 1 and p 2 → √ c 2 as t → −∞ . In terms of these asymptotic sp eeds the Hamiltonian is h = 1 2 ( c 1 + c 2 ) and the quartic inv arian t is k = c 1 c 2 . Up on in tegrat ing the equations (28) w e find elemen tary formulae for p 2 2 − p 2 1 , p 1 p 2 and e −| q 1 − q 2 | , leading to the express ions p 2 2 − p 2 1 = ( c 1 − c 2 ) tanh T p 1 p 2 = q c 1 c 2 + ( c 1 − c 2 ) 4 16( c 1 + c 2 ) 2 sec h 4 T q 2 − q 1 = 1 2 log  1 + 16 c 1 c 2 ( c 1 + c 2 ) 2 ( c 1 − c 2 ) 4 cosh 4 T  , (29) Pe akon e quations with cubic nonl i n e arity 9 where T = ( c 1 − c 2 )( t − t 0 ) / 2, with t 0 b eing an arbitrary constan t. The form ula fo r q 1 + q 2 is somewhat more formidable, b eing give n in terms o f a certain quadratur e as q 1 + q 2 = ( c 1 + c 2 )( t − t 0 ) + Z f ( T ) d T + const . (30) The in tegrand f is f ( T ) = 2( c 2 1 − c 2 2 )  ( c 1 − c 2 ) 2 + 8 c 1 c 2 cosh 2 T  ( c 1 − c 2 ) 4 + 16 c 1 c 2 ( c 1 + c 2 ) 2 cosh 4 T , and the quadrature can b e p erformed explicitly b y partial fra ctions in tanh( T ), but the answ er is omitted here. F rom (2 9 ) and (30) it is apparen t tha t t he p eak ons exc hange sp eeds under the interaction, without a head-on collision, so that q 1 ∼ c 2 t , q 2 ∼ c 1 t a s t → ∞ . They also undergo a phase shift, whic h is describ ed b y t he asymptotics of the term R f ( T ) d T in (30), but the precise form ula is rather un wieldy and will b e presen ted elsewhere . 5. Qiao’ s equation As w e already men tioned, o ur interes t in p eakon equations with cubic nonlinearity b egan with Qiao’s equation (7), whic h can also b e written as m t + ( u 2 − u 2 x ) m x + 2 u x m 2 = 0 . (31) Qiao presen ted a 2 × 2 Lax pair for this equation giv en b y the linear system Ψ x = UΨ , Ψ t = V Ψ with U = − 1 2 1 2 mλ − 1 2 mλ 1 2 ! , V = λ − 2 + 1 2 ( u 2 − u 2 x ) − λ − 1 ( u − u x ) − 1 2 mλ ( u 2 − u 2 x ) λ − 1 ( u + u x ) + 1 2 mλ ( u 2 − u 2 x ) − λ − 2 − 1 2 ( u 2 − u 2 x ) ! . (32) Qiao also found a bi-Hamiltonian structure for his equation, namely m t = ˜ B 1 δ ˜ H δ m = ˜ B 2 δ H 1 δ m (33) where ˜ B 1 = − 4 D x mD − 1 x mD x , ˜ B 2 = − 2( D x − D 3 x ) , (34) and ˜ H , H 1 are the same as the conserv ed quantities for (2) giv en in section 3 ab ov e. (In Qia o’s original pap ers the quan tit y H 0 , prop o rtional to ˜ H here, is out by a facto r of 2, while the quan tity denoted H 1 in [2 0] is missing the u 4 x term.) Note that the first op erator in (34) is of the form (17) with b = 1, and the compatibility of these Hamiltonian structures can b e prov ed b y a sligh t extension of a result in [13]. If w e apply the recipro cal tra nsformation dX = m 2 dx − 1 2 m ( u 2 − u 2 x ) dt, dT = d t Pe akon e quations with cubic nonl i n e arity 10 to Qiao’s equation (7) then w e find the pair of equations ( m − 2 ) T = − 2 u X , ( mu X ) X = 4( u/ m − 1) . (35) By transforming the Lax pair give n b y (32) and writing a scalar linear problem for ψ 1 , the first comp onen t o f Ψ , w e find that the X part is ψ 1 ,X X + ( v X − v 2 ) ψ 1 = − λ 2 ψ , v = m − 1 , whic h is the Sc hr¨ odinger equation corresp onding to the sp ectral problem for KdV, and the expression v X − v 2 is t he standard Miura map from mo dified KdV. The corresp onding time ev olution is ψ 1 ,T = − 1 λ 2  aψ 1 ,X − 1 2 a x ψ 1  , a = u − mu X / 2 , from whic h it is clear that the pair of equations (35) corresp onds t o a negativ e flow in the (mo dified) KdV hierarc hy . Qiao ha s noted that the equation (7) do es not hav e standard p eak ons of the form u = ce −| x − ct | . The general tra v elling w av e solution for this equation can b e solv ed in terms of a n elliptic inte gral, and some in teresting w av e shap es ha v e b een found in [20] in cases where this in tegral reduces to expressions in h yp erb olic f unctions. How ev er, here w e should lik e to p o in t o ut that, at least f ormally , p eak ons of t he form u = ± √ ce −| x − ct | (just as found for (2) ab ov e) do prov ide solutions of Q ia o’s equation. F rom the equation in the f orm (31) it is clear that if m is given by a delta function then the m 2 terms do not mak e sense. Ho w ev er, if w e tak e tra v elling w a ve s u = u ( z ), z = x − ct and in tegrate (7) along the z axis aga inst an a r bit r a ry t est function ϕ , and then p erform an integration b y parts, w e find Z m  u 2 − ( u ′ ) 2 − c  ϕ ′ ( z ) dz = 0 . (36) F or the p eakon u ( z ) = √ c e −| z | w e hav e u ′ ( z ) = − √ c sgn( z ) e −| z | and m ( z ) = 2 √ c δ ( z ), and this satisfies (36) a s long as w e assume t he usual conv en tion tha t sgn(0 ) = 0 . A more careful deriv ation could b e carried o ut along the lines of [19 ]. The equations for N p eak ons should b e extremely degenerate, since b = 1 and G ( x ) is prop o r tional to sgn( x ) in the brack et (23), so p j are constan t and the amplitudes of the p eak ons do not c hange. The same conclusion is reac hed b y integrating (7 ) against a test function and p erforming integration by part s. It seems that p eak on equations with cubic nonlinearit y hav e sev eral no v el features compared with the Camassa-Holm and Degasp eris-Pro cesi eq uations, and there are man y more things to b e reve aled b y further study . Ac kno wledgemen ts. W e are v ery grateful to Vladimir No vik ov for sharing his latest classification results with us. AH thanks Darryl Holm for suggesting in tegrating b y parts with Q iao’s equation, and is v ery grateful to Alexander Strohmaier for his remarks ab out distributional solutions. Pe akon e quations with cubic nonl i n e arity 11 [1] Alb er M S, Camassa R, Holm D D and Mar sden J E 1 994 L ett. Math. Phys. 3 2 13 7 -51; Alb er M S, Camass a R, F e dorov Y, Holm D D, Marsden J E 2001 Commun.Math.Phys. 2 21 197- 227 [2] B eals R, Satinger D H and Szmigiels ki J 199 9 Inverse Pr oblems 15 L1-4 ; Beals R, Satinger D H and Szmigielski J 2000 Ad v. Math. 154 2 2 9-57 . [3] C a massa R and Holm D D 1993 P hys. Rev. Lett. 7 1 16 61-4; Camass a R, Holm D D and Hyman J M 199 4 Adv ances in Applied Mechanics 31 1- 33. [4] Deg asp eris A, Holm D D and Hone A N W 2002 Theo retical and Mathematica l Physics 133 1461- 72 [5] Deg asp eris A and Pro cesi M 199 9 Asymptotic integrability S ymmetry and Pertu rb ation The ory eds A Degasp er is and G Gaeta (W o r ld Scientific) pp 2 3-37. [6] Dullin H R, Gottw ald G A and Ho lm D D 2 004 Physic a D 190 1-1 4 [7] F ok a s A S and F uc hssteiner B 19 8 1 Physic a D 4 4 7-66 [8] F ordy A P 19 90 P rolonga tion structures of nonlinear evolution eq ua tions Soliton The ory: a Su rvey of Results ed A P F or dy (Manchester: Manchester Universit y Pr ess) pp 403-25 . [9] F uc hssteiner B 19 96 Physic a D 95 229- 2 43. [10] Gordoa P R and Pick ering A 199 9 J. Math. Phys 40 5749- 86. [11] Holm D D and Mars de n J E 20 04 Mo ment um maps and mea sure-v alued solutio ns (p eakons, filaments a nd sheets) for the E PDiff equation The Br e adth of Symple ctic and Poisson Ge ometry eds J E Ma rsden and T S Ra tiu Pr o gr ess in Mathematics 232 (Birkhauser ). [12] Hone A N W 2000 Applied Ma thematics Letters 1 3 37-42 . [13] Hone A N W and W ang J P 200 3 Inverse Pr oblems 19 129-1 45. [14] Hone A N W and Holm D D 200 5 Jo urnal o f Nonlinear Mathema tica l Ph ysic s 12 Supplement 1 380-9 4 [15] Kraenkel R A and Zench uk A I 19 99 P hys. Lett. A 260 218; Kra enkel R A, Sen thilvelan M and Zench uk A I 2000 Phys. Lett. A 273 183-9 3. [16] Lundmark H and Szmigie ls ki J 20 03 Inverse Pr oblems 19 12 41-5; Lundmark H and Szmigielski J 2005 In t. Math. Res. Pap ers 2005 Issue 2 53 -116; K ohlenberg J, Lundmark H and Szmigielski J 2007 Inverse Pr oblems 23 99 - 121 [17] Mikhailov A V and Novik ov V S 2 002 J . Phys. A 35 47 75-47 90. [18] No viko v V S 2007 priv ate communication. [19] Li Y A and Olver P J 1997 Discrete Cont. Dyn. Syst. 3 4 19-32 [20] Qiao Z 2 006 J. Math. Phys 47 1 12701 -9; Qiao Z 200 7 J . Math. Phys 48 082 701 [21] Rosenau P 1994 Phys. Re v. L ett . 73 173 7; Ros enau P and Hyman J M 1993 Phys. R ev. L ett. 70 564 [22] W a hlquist H D and E stabro ok F B 1975 J. Math. Phys 16 1-7 ; W ahlquis t H D and Estabr o ok F B 1976 J. Math. Phys 17 1293-7 .