On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations

ITP-UH-17/08 On the N =2 Sup ersymmetric Camassa-Hol m and Hun ter-Saxton E quations J. Lenells a and O. Lec h tenfeld b a Dep artment of Applie d Mathematics and The or etic al Physics, University of Cambridge, Cambridge CB3 0W A, Unite d Kingdom b Institut f¨ ur The or etische Physik, L eibniz Universit¨ at Hannover, App elstr aße 2, 30167 Hannover, Germany j.lenell s@damtp. cam.ac.u k, lechtenf @itp.uni -hannover.de Abstract W e consider N =2 sup ersy mmetric extensio ns of the Camassa -Holm and Hunter- Saxton equations. W e show that they a dmit geometric interpretations a s Eu- ler equations on the sup erconfor mal algebr a of c ontact vector fields on the 1 | 2 - dimensional sup ercir cle. W e use the bi-Hamiltonia n formulation to derive Lax pairs. Mo reov er , w e present some simple exa mples of explicit solutions . As a by- pro duct of our a nalysis we obtain a de scription of the b ounded traveling-w ave solutions for the tw o-comp onent Hun ter- Saxton e quation. P A CS: 02.30.Ik, 11.30.Pb Keyw ord s: Integ r able system; sup ersymmetry; Camassa -Holm equation; bi-Hamiltonian structure. 1 In tro duction The Camassa-Holm (CH) equation u t − u txx + 3 uu x = 2 u x u xx + uu xxx , x ∈ R , t > 0 , (CH) and the Hun ter-Saxton (HS) equatio n u txx = − 2 u x u xx − uu xxx , x ∈ R , t > 0 , (HS) where u ( x, t ) is a real-v alued function, are integrable mo dels for the propagat ion of nonlinear wa v es in 1 + 1-dimension. Equation (CH) m o dels the propagation of sh al- lo w w ater wa ves o ver a flat b ottom, u ( x, t ) represent in g the wate r ’s free su rface in non-dimensional v ariables. It w as first obtained mathematically [20] as an abstract equation with t wo distinct, but compatible, Hamiltonian formulati ons, and wa s sub - sequen tly d eriv ed from physic al principles [3, 10, 19, 27]. Among its m ost notable prop erties is the existence of p eak ed s olitons [3]. Equation (HS) describ es the ev o- lution of nonlinear oriten tation w av es in liqu id crystals, u ( x, t ) b eing related to th e deviation of th e av erage orien tation of the molecules from an equilibr ium p osition [24]. Both (CH) and (HS) are completely integ rable sys tems with an infi nite num b er of conserv ation la ws (see e.g. [6, 9, 11, 21, 25, 28]). Moreo v er , b oth equ ations admit geometric in terpretations as E uler equations for geod esic flo w on the diffeomorphism group Diff ( S 1 ) of orienta tion-preserving diffeomorphisms of the unit circle S 1 . More precisely , the geo desic motion on Diff( S 1 ) endo wed with the righ t-inv arian t metric giv en at the iden tity by h u, v i H 1 = Z S 1  uv + u x v x  dx, (1.1) is describ ed b y th e Camassa-Holm equation [34] (see also [7, 8] ), whereas (HS) de- scrib es the geod esic flo w on the quotien t space Diff ( S 1 ) /S 1 equipp ed with the righ t- in v ariant metric given at the identit y by [28] h u, v i ˙ H 1 = Z S 1 u x v x dx. (1.2) W e w ill consider an N = 2 sup er symmetric generalization of equations (CH) and (HS), whic h was first int ro duced in [35] b y means of bi-Hamiltonian considerations. In this pap er w e: (a) Sho w that this sup ersymmetric generalization adm its a geometric in terp retation as an Eu ler equation on the sup erconformal alge bra of con tact vect or fields on the 1 | 2-dimensional sup ercircle; 1 (b) C onsider the bi-Hamiltonian stru cture; (c) Use the bi-Hamiltonian formulation to d eriv e a Lax pair; (d) Pr esen t some simple examples of explicit solutions. 1 The interpretation of the N = 2 sup ersy mm etric Camassa-Holm equation as an Euler equation related to the sup erconformal algebra was already described in [1]. 1 1.1 The super symmetric equation In ord er to simulta n eously consid er s up ers ymmetric generalizations of b oth the CH and the HS equation, it is con ve n ien t to in tro d uce th e follo win g notatio n: • γ ∈ { 0 , 1 } is a parameter wh ic h satisfies γ = 1 in the case of CH and γ = 0 in the case of HS. • Λ = γ − ∂ 2 x . • m = Λ u . • θ 1 and θ 2 are antico m m u ting v ariables. • D j = ∂ θ j + θ j ∂ x for j = 1 , 2. • U = u + θ 1 ϕ 1 + θ 2 ϕ 2 + θ 2 θ 1 v is a sup erfield. • u ( x, t ) and v ( x, t ) are b osonic fields. • ϕ 1 ( x, t ) and ϕ 2 ( x, t ) are fermionic fields. • A = iD 1 D 2 − γ . • M = AU . Equations (CH) and (HS) can then b e combined int o the single equation m t = − 2 u x m − um x , x ∈ R , t > 0 . (1.3) Although there exist sev eral N = 2 sup ersymmetric extensions of (1.3), some gen- eralizatio n s hav e the particular prop ert y that their b osonic sectors are equiv alen t to the most p op ular tw o-compon en t generalizations of (CH) and (HS) given by (see e.g. [4, 16, 26]) ( m t + 2 u x m + um x + σ ρρ x = 0 , σ = ± 1 , ρ t + ( uρ ) x = 0 . (1.4) The N = 2 su p ersy mmetric generalization of equation (1.3) that we will consider in this pap er shares this prop ert y and is give n b y M t = − ( M U ) x + 1 2 [( D 1 M )( D 1 U ) + ( D 2 M )( D 2 U )] . (1.5) Defining ρ b y ρ = ( v + iγ u if σ = 1 , iv − γ u if σ = − 1 , (1.6) the b osonic sector of (1.5) is exactly the t w o-comp onent equatio n (1.4). If u and ρ are allo w ed to b e complex-v alued fu nctions, the t wo versions of (1.4) corresp ondin g to σ = 1 and σ = − 1 are equiv alen t, b ecause the s ubstitution ρ → iρ con ve rts one in to the other. Ho we v er, in th e usual con text of real-v alued fi elds, the equations are distinct. The discussion in [35] fo cused atten tion on the t wo- comp onent generalization with σ = − 1. The observ ation that the N = 2 sup er- symmetric Camassa-Holm equatio n arises as an Eu ler equation was already m ade in [1]. 2 2 Geo desic flo w As noted abov e, e q uations (CH) and (HS) allo w geometric interpretatio ns as equatio ns for geo desic flo w r elated to the group of diffeomorphisms of the circle S 1 endo wed with a righ t-in v arian t metric. 2 More precisely , us ing the righ t-in v ariance of the metric, the full geo desic equations can b e reduced by sym metry to a so-called Euler equation in the Lie algebra of ve ctor fields on the circle together with a reconstruction equation [33] (see also [29, 30]). Th is is ho w equations (CH) and (HS) arise as Eu ler equations related to the algebra V e ct( S 1 ). In this section we describ e how equation (1.5) s imilarly arises as an Euler equa- tion related to the sup erconformal algebra K ( S 1 | 2 ) of con tact ve ctor fields on the 1 | 2-dimensional sup ercircle S 1 | 2 . Since K ( S 1 | 2 ) is related to the group of su p erd if- feomorphisms of S 1 | 2 , this lea ds (at least form ally) to a geometric interpretatio n of equation (1.5) as an equation for geodesic flow. 2.1 The super c onformal algebra K ( S 1 | 2 ) The Lie sup eralgebra K ( S 1 | 2 ) is defin ed as follo ws cf. [15, 23]. T he sup ercircle S 1 | 2 admits lo cal co ord inates x, θ 1 , θ 2 , wh ere x is a lo cal co ordinate on S 1 and θ 1 , θ 2 are o dd co ordin ates. Let V ect( S 1 | 2 ) d enote the set of vec tor fields on S 1 | 2 . An elemen t X ∈ V ect( S 1 | 2 ) can b e wr itten as X = f ( x, θ 1 , θ 2 ) ∂ ∂ x + f 1 ( x, θ 1 , θ 2 ) ∂ ∂ θ 1 + f 2 ( x, θ 1 , θ 2 ) ∂ ∂ θ 2 , where f , f 1 , f 2 are f unctions on S 1 | 2 . Let α = dx + θ 1 dθ 1 + θ 2 dθ 2 , b e the conta ct form on S 1 | 2 . T he sup erconformal algebra K ( S 1 | 2 ) consists of all con tact v ector fields on S 1 | 2 , i.e. K ( S 1 | 2 ) = n X ∈ V ec t( S 1 | 2 )   L X α = f X α for some fun ction f X on S 1 | 2 o , where L X denotes th e Lie deriv ativ e in the direction of X . A con venien t description of K ( S 1 | 2 ) is obtained b y viewing its elemen ts as Hamil- tonian v ector fi elds corresp onding to fun ctions on S 1 | 2 . I ndeed, defi ne the Hamilto- nian vect or field X f asso ciated with a fu nction f ( x, θ 1 , θ 2 ) by X f = ( − 1) | f | +1  ∂ f ∂ θ 1 ∂ ∂ θ 1 + ∂ f ∂ θ 2 ∂ ∂ θ 2  , where | f | denotes the parit y of f . The Euler v ector field is defined by E = θ 1 ∂ ∂ θ 1 + θ 2 ∂ ∂ θ 2 , 2 In this section we consider all eq uations within t he spatially p eriod ic setting—although formally the same arguments ap ply to t h e case on the line, further technical complications arise du e to the need of imp osing b ou n dary conditions at infinity cf. [5]. 3 and, for eac h function f on S 1 | 2 , we let D ( f ) = 2 f − E ( f ). Then the map f 7→ K f := D ( f ) ∂ ∂ x − X f + ∂ f ∂ x E , satisfies [ K f , K g ] = K { f ,g } where { f , g } = D ( f ) ∂ g ∂ x − ∂ f ∂ x D ( g ) + ( − 1) | f |  ∂ f ∂ θ 1 ∂ g ∂ θ 1 + ∂ f ∂ θ 2 ∂ g ∂ θ 2  . (2.1) Th us, the map f → K f is a h omomorphism f rom the Lie sup eralgebra of fun ctions f on S 1 | 2 endo wed with the b rac ke t (2.1) to K ( S 1 | 2 ). 2.2 Euler equation on K ( S 1 | 2 ) F or t wo ev en functions U an d V on S 1 | 2 (w e usu ally refer to U and V as ‘sup erfields’), w e define a Lie br ac ke t [ · , · ] by [ U, V ] = U V x − U x V + 1 2 [( D 1 U )( D 1 V ) + ( D 2 U )( D 2 V )] . (2.2) It is easily v erified that [ U, V ] = 1 2 { U, V } , where {· , ·} is the br ac k et in (2 .1). The Euler equ ation with resp ect to a metric h· , ·i is giv en by [2] U t = B ( U, U ) , (2.3) where the bilinear m ap B ( U, V ) is defined by the r elation h B ( U, V ) , W i = h U, [ V , W ] i , for any three ev en sup erfields U, V , W . Recall that A = iD 1 D 2 − γ . Letting h U, V i = − i Z dxdθ 1 dθ 2 U AV , (2 .4) a computation sho ws that B ( U, U ) = A − 1  − ( M U ) x + 1 2 [( D 1 M )( D 1 U ) + ( D 2 M )( D 2 U )]  . (2.5) It follo ws from (2.3 ) and (2.5) that (1.5 ) is the Euler equ ation corresp onding to h· , ·i giv en b y (2.4). Remark 2.1 W e make the follo w ing observ ations: 1. Th e action of the in v erse A − 1 of the op erator A = iD 1 D 2 − γ in equation (2.5) is well -defined. Indeed, the map A can b e expressed as A :     u ϕ 1 ϕ 2 v     7→     iv − γ u iϕ 2 x − γ ϕ 1 − iϕ 1 x − γ ϕ 2 − iu xx − γ v     , (2.6) 4 where w e iden tify a sup erfield w ith the col umn v ector m ade up of its four comp o- nen t fields (e.g. U = u + θ 1 ϕ 1 + θ 2 ϕ 2 + θ 2 θ 1 v is iden tified with the column v ector ( u, ϕ 1 , ϕ 2 , v ) T ). L et C ∞ ( S 1 ; C ) denote the space of smo oth p erio dic complex-v alued functions. Since the op erator 1 − ∂ 2 x is an isomorphism from C ∞ ( S 1 ; C ) to itself, w e dedu ce from (2. 6) that the op eration of A − 1 on smo oth p erio d ic complex-v alued sup erfields is w ell-defined when γ = 1. Consider no w the ca s e of γ = 0. In this case it fol lo ws from (2 .6) that A − 1 in volv es the in verses of the op erators ∂ 2 x and ∂ x . In order to m ak e sense of these inv erses w e restrict th e domain of A to the set E = { U = u + θ 1 ϕ 1 + θ 2 ϕ 2 + θ 2 θ 1 v is smo oth and p erio d ic | u (0) = ϕ 1 (0) = ϕ 2 (0) = 0 } . The op erator A = iD 1 D 2 maps this restricted d omain E bijectiv ely on to the set F =  M = i ( n + θ 1 ψ 1 + θ 2 ψ 2 + θ 2 θ 1 m ) is smo oth and p erio d ic     Z S 1 mdx = Z S 1 ψ 1 dx = Z S 1 ψ 2 dx = 0  . Hence the inv erse A − 1 is a well- defined map F → E . W riting M = i ( n + θ 1 ψ 1 + θ 2 ψ 2 + θ 2 θ 1 m ), A − 1 M is giv en explicitly b y A − 1 M =     − R x 0 R y 0 m ( z ) dz dy + x R S 1 R y 0 m ( z ) dz dy − R x 0 ψ 2 ( y ) dy R x 0 ψ 1 ( y ) dy n ( x )     . Note that the expr ession − ( M U ) x + 1 2 [( D 1 M )( D 1 U ) + ( D 2 M )( D 2 U )] = − i  ( D 1 D 2 U ) U + 1 2 ( D 1 U )( D 2 U )  x acted on by A − 1 in equation (2.5 ) b elongs to F for any even s up erfi eld U . Hence the op eration of A − 1 in equation (2.5) is we ll-defined also wh en γ = 0. Th e r estriction of the domain of A to E is r elated to the fact that th e t wo- comp onen t equation (1.4) with γ = 0 is in v arian t un der the symmetry u ( x, t ) → u ( x − c ( t ) , t ) + c ′ ( t ) , ρ ( x, t ) → ρ ( x − c ( t ) , t ) , for any s ufficien tly regular function c ( t ). Hence, by enforcing the condition u (0) = 0 w e remo v e obvious non-un iqueness of th e s olutions to the equatio n . 2. If w e r estrict atten tion to the b osonic sector and let U = u 1 + θ 2 θ 1 v 1 , V = u 2 + θ 2 θ 1 v 2 , the Lie b rac ke t (2.2) induces on t wo pairs of functions ( u 1 , v 1 ) and ( u 2 , v 2 ) the b rac ke t [( u 1 , v 1 ) , ( u 2 , v 2 )] = ( u 1 u 2 x − u 1 x u 2 , u 1 v 2 x − u 2 v 1 x ) . (2.7) 5 W e recognize (2.7) as the comm utation relation for the semidirect pro duct Lie algebra V ec t( S 1 ) ⋉ C ∞ ( S 1 ), where V e ct( S 1 ) d enotes the sp ace of sm o oth v ector fields on S 1 , see [22]. 3. The restriction of the metric (2.4) to the b osonic s ector in duces on pair of functions ( u, v ) the b ilinear form h ( u 1 , v 1 ) , ( u 2 , v 2 ) i = Z ( iγ ( u 1 v 2 + u 2 v 1 ) + u 1 x u 2 x + v 1 v 2 ) dx. (2.8) Changing v ariables from ( u j , v j ) to ( u j , ρ j ), j = 1 , 2, according to (1.6), equation (2.8) b ecomes h ( u 1 , ρ 1 ) , ( u 2 , ρ 2 ) i = Z ( γ u 1 u 2 + u 1 x u 2 x + σ ρ 1 ρ 2 ) dx. (2.9) Hence, as exp ected, when ρ 1 = ρ 2 = 0 th e metric r educes to the H 1 metric (1.1) in the case of CH, while it reduces to the ˙ H 1 metric (1.2 ) in the case of HS. 4. Our discussion freely used complex-v alued expressions and took place only at the Lie alge b ra lev el. The exten t to wh ic h there actually exists a corresp ond ing geod esic flo w on the sup erdiffeomorphism group of the su p ercircle S 1 | 2 has to b e further in v estigated. A fir st step to wards dev eloping suc h a geometric picture inv olv es dealing with th e presence of imaginary num b ers in th e definition of the m etric (2.4) when γ = 1. Although these imaginary f actors disapp ear in th e b osonic sector when c hanging v ariables to ( u, ρ ) (see (2.9)), they are still presen t in the fermionic sector. It seems una voidable to encoun ter complex-v alued expressions at one p oin t or another of the present construction if one insists on the system b eing an extension of equation (CH) (in the references [1] and [35] imaginary factors app ear when the co efficien ts are c h osen in suc h a w ay that the system is an extension of (CH)). 3 Bi-Hamiltonian form ulation Equation (1.5) admits the bi-Hamiltonian str ucture 3 M t = J 1 δ H 1 δ M = J 2 δ H 2 δ M , (3.1) where the Hamiltonian op erators J 1 and J 2 are d efined by J 1 = i  − ∂ x M − M ∂ x + 1 2 ( D 1 M D 1 + D 2 M D 2 )  , J 2 = − i∂ x A, (3.2) and the Hamiltonian fu nctionals H 1 and H 2 are defin ed by H 1 = − i 2 Z dxdθ 1 dθ 2 M U, H 2 = − i 4 Z dxdθ 1 dθ 2  M U 2 − γ 3 U 3  . (3.3) 3 By definition the v ariational deriv ative δ H /δ M of a functional H [ M ] is requ ired t o satisfy d dǫ H [ M + ǫδ M ] ˛ ˛ ˛ ˛ ǫ =0 = Z dxdθ 1 dθ 2 δ H δ M δ M , for an y smo oth v ariation δ M of M . 6 δ δM  − i 4 R dxdθ 1 dθ 2  M U 2 − γ 3 U 3  = δH 2 δM J 2 s s g g g g g g g g g g g g g g g g g g g g Equation (1.5) δ δM  − i 2 R dxdθ 1 dθ 2 M U  = δH 1 δM J 1 k k W W W W W W W W W W W W W W W W W W W W J 2 s s g g g g g g g g g g g g g g g g g g g g M t = − M x δ δM  − i R dxdθ 1 dθ 2 M  = δH 0 δM J 1 k k W W W W W W W W W W W W W W W W W W W W J 2 s s g g g g g g g g g g g g g g g g g g g g g g M t = 0 Figure 1 R e cursion scheme for th e op e r ators J 1 and J 2 asso ciate d with the sup er- symmetric e quation (1.5). The bi-Hamil tonian form u lation (3.1) is a particular case of a construction in [35], where it w as v erified that the op erators J 1 and J 2 are compatible. The first few conserv ation la w s in the hierarc h y generated by J 1 and J 2 are presented in Figure 1. Restricting attent ion to the purely b osonic sector of the bi-Hamiltonian stru cture of (1.5), we reco v er bi-Hamiltonian form ulations for the t w o-comp onent generali za- tions of (CH ) and (HS). More pr ecisely , w e find that equation (1.4) can b e pu t in th e bi-Hamiltonian form 4  m ρ  t = K 1 grad G 1 = K 2 grad G 2 , (3.4) 4 The gradien t of a functional F [ m, ρ ] is defined by grad F = „ δF δm δF δρ « , provided that there exist functions δF δm and δF δρ such that d dǫ F [ m + ǫδ m, ρ + ǫδ ρ ] ˛ ˛ ˛ ˛ ǫ =0 = Z „ δ F δ m δ m + δ F δ ρ δ ρ « dx, for an y smo oth v ariations δ m and δ ρ . 7 where the Hamiltonian op erators are defined by K 1 =  − m∂ x − ∂ x m − ρ∂ − ∂ ρ 0  , K 2 =  − ∂ Λ 0 0 − σ ∂  , (3.5) and the Hamiltonians G 1 and G 2 are giv en by G 1 = 1 2 Z ( um + σ ρ 2 ) dx, G 2 = 1 2 Z ( σ uρ 2 + γ u 3 + uu 2 x ) dx. F or j = 1 , 2, G j are the restrictions to the purely bosonic sector of the functionals H j . The hierarc hy of conserved quantiti es G n for the tw o-comp onen t system (1.4) can b e obtained from the recursive r elations K 1 grad G n = K 2 grad G n +1 , n ∈ Z . The asso ciated comm uting Hamiltonian fl o ws are give n b y  m ρ  t = K 1 grad G n = K 2 grad G n +1 , n ∈ Z . Since K − 1 2 is a nonlo cal op erator the expressions for mem b ers G n with n ≥ 3 are nonlo cal w hen wr itten as functionals of ( u, ρ ). On th e other hand, since the op erator K 1 can b e explicitly in ve r ted as K − 1 1 = 0 − 1 ρ ∂ − 1 x − ∂ − 1 x 1 ρ ∂ − 1 x 1 ρ ( m∂ x + ∂ x m ) 1 ρ ∂ − 1 x ! , it is p ossible to implement on a computer a recursiv e algorithm for fi nding the con- serv ation laws G n for n ≤ 2. The first few of these conserv ed quant ities and their asso ciated Hamiltonian flo ws are presen ted in Figure 2. Note that G 0 is a C asimir for the p ositiv e hierarch y , G − 2 is a Casimir for the negativ e hierarch y , and G − 1 is a Casimir for b oth the p ositiv e and negat ive hierarc hies. This discussion illustrates that eve n though many p rop erties of the t wo-co m p onent system (1.4) carry ov er to its sup ersymmetric extension (1.5) (su c h as the prop ert y of b eing a geo desic equation as sh o wn ab o ve , an d the Lax pair formulat ion as shown b elo w ), the construction of a negativ e h ierarc hy do es not app ear to generalize. Cer- tainly , since the conserv ation la ws G − 1 , G − 2 , ..., co n tain negativ e p ow ers of ρ , these functionals cannot b e the restriction to the b osonic sector of functionals H − 1 , H − 2 , ..., whic h inv olv e only p olynomials of the fi elds U and M and d eriv ative s of these fields. 4 Lax pair Equation (1.5) is the condition of compatibilit y of th e lin ear system ( iD 1 D 2 G =  1 2 λ M + γ 2  G, G t = 1 2 U x G − 1 2 [( D 1 U )( D 1 G ) + ( D 2 U )( D 2 G )] − ( λ + U ) G x , (4.1) 8 grad 1 2 R  σ uρ 2 + γ u 3 + uu 2 x  dx = grad G 2 K 2 s s g g g g g g g g g g g g g g g g g g g g g  m ρ  t =  − 2 u x m − um x − σ ρρ x − ( uρ ) x  grad 1 2 R  um + σ ρ 2  dx = grad G 1 K 1 k k W W W W W W W W W W W W W W W W W W W W W K 2 s s g g g g g g g g g g g g g g g g g g g g g g g g g g g  m ρ  t = −  m x ρ x  grad R mdx = grad G 0 K 1 k k X X X X X X X X X X X X X X X X X X X X X X X X X X X K 2 s s f f f f f f f f f f f f f f f f f f f f f f f f f f f f f  m ρ  t =  0 0  grad R ρdx = grad G − 1 K 1 k k X X X X X X X X X X X X X X X X X X X X X X X X X X X X X K 2 s s f f f f f f f f f f f f f f f f f f f f f f f f f f f f f  m ρ  t =  0 0  grad R m ρ dx = grad G − 2 K 1 k k W W W W W W W W W W W W W W W W W W W W W W W W W W W W K 2 s s h h h h h h h h h h h h h h h h h h h h h h h  m ρ  t =   γ ρ x ρ 2 +  1 ρ  xxx σ  m ρ 2  x   grad R  − σm 2 2 ρ 3 − γ 2 ρ − ρ xx 4 ρ 2  dx = grad G − 3 K 1 k k V V V V V V V V V V V V V V V V V V V V V V Figure 2 R e cursion scheme for two-c omp onent gener alization (1.4) of the CH (c or- r esp onding to m = u − u xx ) and the H S (c orr esp onding to m = − u xx ) e qu ations. 9 where the ev en sup erfield G serves as an eigenfunction and λ ∈ C is a sp ectral parameter. T he Lax pair (4.1) can b e deriv ed from the b i-Hamiltonian formulation present ed in Section 3 as f ollo ws. In view of the general theory of recurs ion op erators [17], we exp ect J 1 F = λJ 2 F , (4.2) to b e the x -part of a L ax pair for (1.5). This Lax p air inv olv es the ev en sup erfield F , whic h serves as a ‘squared eigenfun ction’ cf. [18]. Letting F = G 2 , a computation sho ws that equation (4.2) holds w henev er G satisfies the x -part of (4.1). In order to find the corresp onding t -part, w e m ak e the Ans atz G t = A 1 G + B 1 D 1 G + B 2 D 2 G + A 2 G x , (4.3) where A 1 , A 2 ( B 1 , B 2 ) are ev en (o dd ) sup erfi elds. Long but straigh tforward compu- tations sho w that th e compatibilit y of (4.3) with the x -part of (4.1) is equ iv alent to the follo win g four equations: M t =2 λiD 1 D 2 A 1 + ( D 1 B 1 + D 2 B 2 )( M + λγ ) (4.4) + ( D 1 A 2 − B 1 ) D 1 M + ( D 2 A 2 − B 2 ) D 2 M + A 2 M x ,  1 2 λ M + γ 2  B 1 = iD 1 D 2 B 1 − iD 2 A 1 + ( D 1 A 2 − B 1 )  1 2 λ M + γ 2  , (4.5)  1 2 λ M + γ 2  B 2 = iD 1 D 2 B 2 + iD 1 A 1 + ( D 2 A 2 − B 2 )  1 2 λ M + γ 2  , (4.6) 0 = iD 1 D 2 A 2 − iD 1 B 2 + iD 2 B 1 . (4.7) Equations (4.4)-( 4.7) ascertain the e qualit y of the c o efficien ts of G, D 1 G, D 2 G, G x , re- sp ectiv ely , on the left- and r igh t-hand sides of the compatibilit y equati on iD 1 D 2 ( G t ) = ( iD 1 D 2 G ) t . Letting A 1 = 1 2 U x , B 1 = − 1 2 D 1 U, B 2 = − 1 2 D 2 U, A 2 = − ( λ + U ) , equations (4.5)-(4 .7) are ident ically satisfied, wh ile equation (4.4) redu ces to the su- p ersymm etric system (1.5). 5 Explicit solutions: First deconstruction In order to obtain explicit solutions of the su p ersymm etric system (1.5), w e co nsider the simplest case of fields taking v alues in the Gr assmann algebra consisting of tw o elemen ts 1 and τ sati sfying the follo w ing relations: 1 2 = 1 , τ · 1 = τ = 1 · τ , τ 2 = 0 . (5.1) In the case of this so-called first deconstru ction the fields can b e w ritten as u = u · 1 , ρ = ρ · 1 , ϕ 1 = τ · f 1 , ϕ 2 = τ · f 2 , 10 where f 1 ( x, t ) and f 2 ( x, t ) are ordinary real-v alued functions. The comp onent equa- tions of (1.5) consist of the b osonic t w o-comp on en t system (1.4) together with the follo wing equations for the ferm ionic fi elds: ( ( f 1 ρ + 2 f 2 t + 2 uf 2 x + f 2 u x ) x + iγ (2 f 1 t + 3 uf 1 x + 3 f 1 u x ) = 0 , ( − f 2 ρ + 2 f 1 t + 2 uf 1 x + f 1 u x ) x − iγ (2 f 2 t + 3 uf 2 x + 3 f 2 u x ) = 0 , σ = 1 , (5.2) ( ( − if 1 ρ + 2 f 2 t + 2 uf 2 x + f 2 u x ) x + iγ (2 f 1 t + 3 uf 1 x + 3 f 1 u x ) = 0 , ( if 2 ρ + 2 f 1 t + 2 uf 1 x + f 1 u x ) x − iγ (2 f 2 t + 3 uf 2 x + 3 f 2 u x ) = 0 , σ = − 1 . (5.3) W e seek tra veling-w a v e solutions for which u and ρ are of the form u = u ( y ) , ρ = ρ ( y ) , y = x − ct, (5.4) where c ∈ R den otes the v elo cit y of the wa v e. Substituting (5.4) in to (1.4), w e find ( − cm y + 2 u y m + um y + σ ρρ y = 0 , σ = ± 1 , − cρ y + ( uρ ) y = 0 . (5.5) The second of these equations yields, for some constan t of integ ration a , ρ ( y ) = a c − u ( y ) . (5.6) A t this p oint it is con v en ien t to r estrict atten tion to the case of either CH or HS. W e c ho ose to consider solutions of the HS system. 5.1 T ra veli ng w av es for the supersymmetric HS equation W e first derive solutions for the b osonic fields b efore considering the extension to the fermionic s ector. S ince the b osonic sector coincides with equation (1.4), this analysis yields as a b y-pr o duct a classificatio n of the b ounded tra veling-w a v e solutions of the t wo- comp onent HS equation. 5.1.1 Bosonic fields Substituting expression (5.6 ) for ρ into the fi rst equation of (5.5) and in tegrating the resulting equation with resp ect to y , we fin d cu y y − 1 2 u 2 y − uu y y + σ a 2 2( c − u ) 2 = b 2 , (5.7) where b is an arbitrary constant of in tegration. W e m ultiply this equ ation by 2 u y and in tegrate the resulting equati on with resp ect to y . After simplification we arriv e at the ordinary differen tial equation u 2 y = ( bu + d )( c − u ) − σ a 2 ( c − u ) 2 , (5.8) 11 where d is yet another in tegration constan t. Int ro ducing z and Z s o that ( bu + d )( c − u ) − σ a 2 = b ( Z − u )( u − z ) , (5.9) an an alysis of (5.8) rev eals that b ounded trav eling w av es arise when z , Z, c, b satisfy 5 b > 0 and either z < Z < c or c < z < Z . (5 .10 ) If (5 .10 ) is fulfilled, then there exists a smo oth p erio dic solution u ( y ) of equation (5.8) suc h that min y ∈ R u ( y ) = z , max y ∈ R u ( y ) = Z . W e obtain a trav eling wa v e of the tw o-comp onen t HS equation b y constructing ρ ( y ) from u ( y ) according to (5.6 ). It can b e c hec ke d b y considerin g all p ossible distrib u- tions of th e p arameters b, z , Z , c that th ere are no ot her boun ded conti n u ous tra v eling- w av e solutions of the t w o-comp onent HS equation. 6 The r elations b ( z + Z ) = bc − d, − bz Z = dc − σ a 2 , obtained from (5.9), sh o w that b ( c − z )( c − Z ) = σ a 2 . (5.11) There are t w o cases to consider: a = 0 and a 6 = 0. If a = 0, then either b = 0, z = c or Z = c , so that (5.10) d o es not hold. Hence no tr a ve lin g wa ves exist in th is case. If a 6 = 0, then a 2 is a strictly p ositiv e n u m b er and relation (5.11) requires that b ( c − z )( c − Z ) ≷ 0 for σ = ± 1 . The left-hand sid e of this equation is p ositiv e wh enev er (5.10) holds. This sh o ws that no tra veli ng w a ves exist when σ = − 1. On the other hand, if σ = 1 any com bination of b, z , Z , c ∈ R satisfying (5.10) corresp onds to exactly t wo uniqu e smo oth p eriod ic tra ve ling-w a v e solutions. There are two solutions for eac h admissible co m b ination of b, z , Z, c b ecause these p arameters determine a , and hence ρ , only up to sign. I n general, equation (1.4) is in v ariant und er the sy mmetry u → u, ρ → − ρ. Let us men tion that equation (HS) admits no b ounded tra ve lin g wa v es. This can b e seen from the ab o v e analysis b ecause ρ = 0 if an d only if a = 0 and we ha ve seen that n o trav eling wa v es exist in this case. Hence, in this regard the t w o- comp onent generalization with σ = 1 p resen ts a qualitativ ely r ic her structure than its one-comp onent analogue. 5 See [32] for more details of a similar analysis in the case of eq u ation (CH). 6 If b > 0 and z < c < Z there exist p erio dic solutions u ( y ) of (5.8) with cusps, where we say that a con tinuous funct ion u ( y ) has a cusp at y = y 0 if u is smo oth on either side of y 0 but u y → ±∞ as y → y 0 . How ever, the existence of a cusp of u at y = y 0 implies that u ( y 0 ) = c so that th e correspondin g function ρ ( y ) given by ( 5.6) is unbounded near y 0 . This argumen t exp lains wh y these solutions are excluded from the list of b ou n ded trav eling w av es. Similarly , the cusp ed solutions of (5.8) which exist when b < 0 and either z < Z < c or c < z < Z d o not give rise to b ounded trav eling w aves . 12 5.1.2 F ermionic fields W e no w consider the fermionic fields corresp onding to the p erio dic tra veling wa v es found in the p revious sub section. Hence let σ = 1 and let ( u, ρ ) b e a smooth p eri- o dic tra ve ling-w a v e solution of th e tw o-comp onen t HS equation (1.4). The relev an t fermionic equations obtained fr om (5.2) and (5.6) are ( af 1 c − u + 2 f 2 t + 2 uf 2 x + f 2 u x = B , − af 2 c − u + 2 f 1 t + 2 uf 1 x + f 1 u x = C, (5.12) where B and C are arbitrary constan ts of in tegration. One w ay to obtain explicit (complex-v alued) solutions of (5.12) is to seek solutions of the form B = C = 0 and f 1 = if 2 =: f ( y ), wher e, as ab o ve , y = x − ct . In this case the sys tem (5.12) reduces to the single equation iaf c − u − 2( c − u ) f y + f u y = 0 , (5.13) This equation can b e solv ed explicitly for f . Equ ations (5.8), (5.9), and (5.13) yield ia + sgn(( c − u ) u y ) p b ( Z − u )( u − z ) c − u ! f − 2( c − u ) f y = 0 . Using that du = sgn(( c − u ) u y ) √ b ( Z − u )( u − z ) c − u dy , w e can write the solution to this equation as log f = Z u u 0 ia + sgn(( c − u ) u y ) p b ( Z − u )( u − z ) 2( c − u ) du sgn(( c − u ) u y ) p b ( Z − u )( u − z ) . (5.14) W e find f by computing the in tegral in (5.14) and usin g the r elation a = sgn( a ) p b ( c − z )( c − Z ) to eliminate a . L etting f 0 denote th e v alue of f at a p oin t where u = z , we obtain f ( y ) = f 0 r c − z c − u ( y ) e i 2 sgn( a ( c − u ) u y ) arctan „ 2 √ ( c − z )( c − Z )( Z − u ( y ))( u ( y ) − z ) − 2 zZ + c ( z + Z − 2 u ( y ))+( z + Z ) u ( y ) « . (5.15) Note that the identit y e ± i 2 arctan t =  1 + it 1 − it  ± 1 / 4 , implies that apart from the presence of fractional p o wers the righ t-hand side of (5.15 ) is a rational fun ction of u . 5.1.3 Summarized result W e can su mmarize our discussion as follo ws. Consider the sup ersymmetric Hunter- Saxton equation for σ = 1 with fields taking v alues in th e Grassmann algebra { 1 , τ } : M t = − ( M U ) x + 1 2 [( D 1 M )( D 1 U ) + ( D 2 M )( D 2 U )] , (5.16) U = u + θ 1 τ f 1 + θ 2 τ f 2 + θ 2 θ 1 ρ, M = iD 1 D 2 U. 13 5 10 15 20 y - 2 - 1 0 1 2 u 5 10 15 20 y - 1 0 1 2 3 Ρ Figure 3 A tr aveling- wave solution ( u, ρ ) of the two-c omp onent Hunter-Saxton e qua- tion (1 .4) with σ = 1 and the p ar ameter values b = 1 , z = − 1 , Z = 1 , c = 2 . The top and b ottom gr aphs show u and ρ as functions of y = x − ct , r esp e ctively. F or an y α = ± 1, f 0 ∈ C , and b, z , Z, c ∈ R satisfying (5.10 ), equation (5.16) admits the smo oth p erio dic solution u ( x, t ) = ˜ u ( y ) where ˜ u solv es ˜ u 2 y = b ( Z − ˜ u )( ˜ u − z ) ( c − ˜ u ) 2 , y = x − ct, (5.17) ρ ( x, t ) = α p b ( c − z )( c − Z ) | c − u ( x, t ) | , (5.18) f 1 ( x, t ) = if 2 ( x, t ) = f 0 r c − z c − u e α sgn( u x ) i 2 arctan „ 2 √ ( c − z )( c − Z )( Z − u )( u − z ) − 2 zZ + c ( z + Z − 2 u )+( z + Z ) u « , ( 5.19) where we hav e su ppressed the ( x, t )-dep endence of u in the last equ ation. I n p articu- lar, restriction to the b osonic sector yields that ( u, ρ ) giv en by (5.17)-(5.18 ) co nstitute a tra veling-w a v e solution of the t wo-c omp onen t Hunter-Saxton equation (1 .4) with σ = 1, see Figure 3. Let us p oin t out that although w e cannot present an explicit form u la f or u ( x, t ), in tegration of the ODE for ˜ u sho ws that u is giv en imp licitly u p to a translation in x b y x − ct = ± ( u − z )( Z − u ) + p ( u − z )( Z − u )(2 c − z − Z ) tan − 1 q u − z Z − u  p b ( u − z )( Z − u ) . The stabilit y of these tra veling-w a v e solutions is of interest, esp ecially since in the case of the Camassa-Ho lm equation it is kn o wn that the smo oth and p eak ed tra veling w av es are orbitally stable cf. [12, 13 , 14, 31]. 14 6 Explicit solutions: Second deconstruction In this section w e extend the analysis of the pr evious section to the case of the second deconstruction, that is, to the case of fields taking v alues in the Grassmann algebra { 1 , τ 1 , τ 2 , τ 1 τ 2 } where τ 1 and τ 2 are o dd v ariables. Under this assumption w e ma y write u = u 1 + τ 2 τ 1 u 2 , ρ = ρ 1 + τ 2 τ 1 ρ 2 , ϕ 1 = τ 1 f 1 + τ 2 g 1 , ϕ 2 = τ 1 f 2 + τ 2 g 2 , (6.1) where u 1 , u 2 , ρ 1 , ρ 2 , f 1 , f 2 , g 1 , g 2 are real- v alued fun ctions of x and t . F or simplicit y w e henceforth consider only the case of HS with σ = 1. In view of th e expressions in (6.1) for u, ρ, ϕ 1 , ϕ 2 , w e fi nd that equation (1. 5) is equiv alen t to th e follo wing list of equations: ρ 1 t + ( ρ 1 u 1 ) x = 0 , (6.2) − ρ 1 ρ 1 x + 2 u 1 x u 1 xx + u 1 txx + u 1 u 1 xxx = 0 , (6.3) ( f 1 ρ 1 + 2 f 2 t + 2 u 1 f 2 x + f 2 u 1 x ) x = 0 , (6.4) ( g 1 ρ 1 + 2 g 2 t + 2 u 1 g 2 x + g 2 u 1 x ) x = 0 , (6.5) ( − f 2 ρ 1 + 2 f 1 t + 2 u 1 f 1 x + f 1 u 1 x ) x = 0 , (6.6) ( − g 2 ρ 1 + 2 g 1 t + 2 u 1 g 1 x + g 1 u 1 x ) x = 0 , (6.7) 2 ρ 2 t + ( f 2 g 1 − f 1 g 2 + 2 u 2 ρ 1 + 2 u 1 ρ 2 ) x = 0 , (6.8)  2 u 2 tx + g 1 f 1 x + g 2 f 2 x − f 1 g 1 x − f 2 g 2 x +2 u 1 x u 2 x + 2 u 2 u 1 xx + 2 u 1 u 2 xx − 2 ρ 1 ρ 2  x = 0 . (6.9) Equations (6.2 )-(6.9) ascertain the equalit y of the co efficien ts of 1, θ 1 θ 2 , θ 1 τ 1 , θ 1 τ 2 , θ 2 τ 1 , θ 2 τ 2 , τ 1 τ 2 , θ 1 θ 2 τ 1 τ 2 , resp ectiv ely , on the left- and right-hand sides of equation (1.5). W e m ak e the follo wing observ ations: • Equations (6.2) and (6.3) mak e up the t wo-co m p onent HS equation (1.4) for σ = 1 with ( u, ρ ) replaced by ( u 1 , ρ 1 ). • Equations (6.4) and (6.6) are the fermionic equatio ns (5.2) encounte r ed in the case of the firs t deconstruction with ( u, ρ ) r eplaced by ( u 1 , ρ 1 ). • Equations (6.5) and (6.7) are the fermionic equatio ns (5.2) encounte r ed in the case of the first deconstruction with ( u, ρ ) replaced by ( u 1 , ρ 1 ) and ( f 1 , f 2 ) replaced by ( g 1 , g 2 ). • Equations (6.8 ) and (6.9) are the only equations inv olving u 2 and ρ 2 . In view of these observ ations, ( u 1 , ρ 1 , f 1 , f 2 ) and ( u 1 , ρ 1 , g 1 , g 2 ) constitute tw o sets of solutions to the equ ations considered in the ca se of the first deconstruction. W e can therefore apply the analysis of Section 5 to obtain explicit expressions for these fields wh ic h fulfill equ ations (6.2)-(6.7); equatio ns (6. 8) an d (6.9) can then b e used 15 to determine u 2 and ρ 2 . Note th at for solutions of the form f := f 1 = if 2 and g := g 1 = ig 2 , equations (6.8) an d (6.9) reduce to ρ 2 t + ( u 2 ρ 1 + u 1 ρ 2 ) x = 0 , (6.10)  u 2 tx + u 1 x u 2 x + u 2 u 1 xx + u 1 u 2 xx − ρ 1 ρ 2  x = 0 . (6.11) Assuming that ( u 1 , ρ 1 ) is a trav eling-wa ve solution of (1 .4) of the f orm constructed in Sectio n 5, we can obtain solutions u 2 = u 2 ( y ) and ρ 2 = ρ 2 ( y ), y = x − ct , of equations (6.10) and (6.11) as follo ws . In view of (5.6), we ha v e ρ 1 = a/ ( c − u 1 ) for some constan t a . Hence equatio n (6.10) yields ρ 2 = au 2 ( c − u 1 ) 2 − E c − u 1 , where the constan t E arose from an in tegration with resp ect to y . Substituing th is expression f or ρ 2 in to equation (6.11), w e find  − cu 2 yy + u 1 y u 2 y + u 2 u 1 yy + u 1 u 2 yy − a 2 u 2 ( c − u 1 ) 3 + E a ( c − u 1 ) 2  x = 0 . Letting u 2 = c − u 1 , this equation b ecomes  cu 1 yy − 1 2 u 2 1 y − u 1 u 1 yy − a ( a − E ) 2( c − u 1 ) 2  x = 0 . A comparison with (5.7) sh o ws that this equ ation is fulfilled pr o vided that E = 2 a . W e conclude that (6.10) and (6.1 1 ) are satisfied for u 2 = c − u 1 , ρ 2 = − a c − u 1 . W e note that ρ 2 = − ρ 1 for this solution. 6.1 Summarized result The discu ssion in this s ection can b e summ arized as follo ws. Consider the su p ersym - metric Hunter-Saxto n equation f or σ = 1 with fields taking v alues in the Grassmann algebra { 1 , τ 1 , τ 2 , τ 1 τ 2 } : M t = − ( M U ) x + 1 2 [( D 1 M )( D 1 U ) + ( D 2 M )( D 2 U )] , (6.12) U =( u 1 + τ 2 τ 1 u 2 ) + θ 1 ( τ 1 f 1 + τ 2 g 1 ) + θ 2 ( τ 1 f 2 + τ 2 g 2 ) + θ 2 θ 1 ( ρ 1 + τ 2 τ 1 ρ 2 ) , M = iD 1 D 2 U. 16 F or an y α = ± 1, f 0 , g 0 ∈ C , and b, z , Z, c ∈ R satisfying (5.10), equation (6.12) admits the smo oth p erio dic solution u 1 ( x, t ) = ˜ u ( y ) where ˜ u solv es ˜ u 2 y = b ( Z − ˜ u )( ˜ u − z ) ( c − ˜ u ) 2 , y = x − ct, ρ 1 ( x, t ) = α p b ( c − z )( c − Z ) | c − u 1 ( x, t ) | , f 1 ( x, t ) = if 2 ( x, t ) = f 0 r c − z c − u 1 e α sgn( u 1 x ) i 2 arctan „ 2 √ ( c − z )( c − Z )( Z − u 1 )( u 1 − z ) − 2 zZ + c ( z + Z − 2 u 1 )+( z + Z ) u 1 « , g 1 ( x, t ) = ig 2 ( x, t ) = g 0 r c − z c − u 1 e α sgn( u 1 x ) i 2 arctan „ 2 √ ( c − z )( c − Z )( Z − u 1 )( u 1 − z ) − 2 zZ + c ( z + Z − 2 u 1 )+( z + Z ) u 1 « , u 2 ( x, t ) = c − u 1 ( x, t ) , ρ 2 ( x, t ) = − ρ 1 ( x, t ) . 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