Negative order KdV equation with both solitons and kink wave solutions

Negativ e order KdV equation with b oth solitons and kink w a v e solutions Zhijun Qiao 1 and Jibin Li 2 1 Departmen t of Mathematics, The Univ ersit y of T exas-P an American 1201 W Univ ersit y Drive, Edin burg, TX 78539 , U.S. 2 Departmen t of Mathematics, Zhejiang Normal Univ ersit y , Jinh ua, Zhejiang, 321 004, China Email: qiao@utpa.edu Abstract In this pap er, w e rep ort an in teresting inte grable equation that has b oth solitons and kink solutions. The int egrable equation we stu dy is ( − u xx u ) t = 2 uu x , w hic h actually comes from the negativ e KdV hierarc hy and could b e transformed to the Camassa- Holm equation through a gauge transform. The Lax p air of th e equation is deriv ed to guaran tee its integ rabilit y , and furtherm ore the equation is sho wn to ha v e classical solitons, p erio dic soliton and kink solutions. P A CS. 02.30 .Ik C Integrable systems. P A CS. 05.45 .Yv C Solit o ns. P A CS. 03.75.Lm C T unneling, Jose phson effect, Bose-Eins tein condensates in p erio dic p o- ten tials, solitons, v o rtices, and to p ological excitations. 1 In t ro duction Soliton theory and in tegrable systems play an imp ortant role in the study of nonlinear water w a v e equations. They hav e man y significan t applications in fluid mec ha nics, nonlinear optics, classical and quan tum fields theories etc. P articularly in recen t y ears, more fo cuses ha v e b een pulled to in tegrable systems with non-smo o th solitons, suc h as p eak ons, cuspons, sinc e the study of the remark able Camassa-Holm (CH) equation with p eak on solutions [2]. Henceforth, m uc h prog ress hav e b een made in the study of non-smo oth solitons for integrable equations [4-20, 22, 27 -33]. In this pap er, w e consider the follow ing in tegrable equation ( − u xx u ) t = 2 u u x , (1) 1 whic h is actually the first mem b er in t he negative KdV hierar ch y [29]. Equation (1) is pro v en equiv alen t to the Camassa-Holm (CH) equation: m t + m x u + 2 mu x = 0 , m = u − u xx through a gauge transform (see Remark 1 in the pap er). Therefore, w e find a simpler reduced form of the CH equation. The Lax pair of the equation (1) is deriv ed to guarantee its in tegrability , and furthermore the equation is sho wn to hav e classical solitons, p erio dic solitons and kink solutions. 2 Deriv ation of Equation (1) and Lax Representation Let us consider the Sc hr¨ odinger-KdV sp ectral problem: Lψ ≡ ψ xx + v ψ = λψ , (2) where λ is an eigen v alue, ψ is the eigenfunction correspo nding to the eigen v alue λ , and v is a p o t en t ia l f unction. O ne can easily get the following Lenard op erator relation: K ∇ λ = λJ ∇ λ, (3) where ∇ λ ≡ δλ δv = ψ 2 is the functional gradien t of the sp ectral problem (2) with respect to v , K = 1 4 ∂ 3 + 1 2 ( v ∂ + ∂ v ) and J = ∂ are t w o Hamiltonian op erators as kno wn in t he literature [1]. By setting v = − u xx u , w e hav e the pro duct form o f op erat o rs K , L , L a nd their inv erses K = 1 4 u − 2 ∂ u 2 ∂ u 2 ∂ u − 2 , K − 1 = 4 u 2 ∂ − 1 u − 2 ∂ − 1 u − 2 ∂ − 1 u 2 , L = 1 4 ∂ − 1 u − 2 ∂ u 2 ∂ u 2 ∂ u − 2 , L − 1 = 4 u 2 ∂ − 1 u − 2 ∂ − 1 u − 2 ∂ − 1 u 2 ∂ , L = ∂ 2 + v = u − 1 ∂ u 2 ∂ u − 1 , L − 1 = u ∂ − 1 u − 2 ∂ − 1 u , where L = J − 1 K and its in v erse L − = K − 1 J are the recursion operat o rs for the p ositive order and negativ e order KdV hierarc h y that w e study b elo w. No w, according to the Lenard’s op era t ors K and J , w e construct the en tire KdV hier- arc h y , and then we show the in tegrabilit y of the hierarch y through solving a k ey op erator equation. Let G 0 ∈ K er J = { G ∈ C ∞ ( R ) | J G = 0 } and G − 1 ∈ K er K = { G ∈ C ∞ ( R ) | K G = 0 } . W e define the Lenard’s sequence G j = ( L j G 0 , j ∈ Z , L j +1 G − 1 , j ∈ Z , (4) where L , L − 1 are defined by Eq. (2). Therefore w e generate a hierarch y of nonlinear ev olution equations (NLEEs): v t k = J G k = K G k − 1 , ∀ k ∈ Z , (5) 2 whic h is called the entire KdV hierarch y . W e will see b elo w that the p ositive or der ( k ≥ 0) giv es the regular KdV hierarc h y usu ally mentioned in the literature [1], while the negative order ( k < 0) pro duces some in teresting equations ga uge-equiv alen t to the Camas sa-Holm equation [2 ]. Apparen tly , this hierarc h y p ossesses the bi-Hamiltonian structure b ecause of the Hamiltonian pro p erties of K , J . Let us now giv e sp ecial equations in the entire KdV hierarc h y (5). • Cho o sing G 0 = 2 ∈ K er J (therefore G 1 = u ) leads to the second p ositiv e mem b er of the hierarch y (5): v t 2 = 1 2 v xxx + 3 2 v v x , (6) whic h is exactly the we ll-know n KdV equation. Here there is nothing new. Therefore, the p ositiv e order ( k ≥ 0) in the hierarc h y (5) yields the regular KdV hierarch y usually studied in the literature [1]. • Now, let us find kernel elemen ts G − 1 ∈ K er K in order to get the negativ e mem b er of the hierarch y (5). Due to the pro duct form of K and K − 1 , G − 1 = K − 1 0 has the follo wing three seed solutions: G 1 − 1 = f ( t n ) u 2 , G 2 − 1 = g ( t n ) u 2 ∂ − 1 u − 2 , G 3 − 1 = h ( t n ) u 2 ∂ − 1 u − 2 ∂ − 1 u − 2 , where f ( t n ) , g ( t n ) , h ( t n ) are three arbitra r ily give n functions with resp ect to the time v ariables t n , but indep enden t of x . They pro duce three iso-sp ectral ( λ t k = 0) negative order KdV hierarc hies of Eq. (5) v t k = J L k +1 · G l − 1 , l = 1 , 2 , 3 , k = − 1 , − 2 , .... (7) When k = − 1, their represen tative equations a re:  − u xx u  t − 1 = 2 f ( t n ) uu x , (8)  − u xx u  t − 1 = g ( t n )  2 uu x ∂ − 1 u − 2 + 1  , (9)  − u xx u  t − 1 = h ( t n )  2 uu x ∂ − 1 u − 2 ∂ − 1 u − 2 + ∂ − 1 u − 2  . (10) Remark 1. A pp ar ently, the firs t one is diffe r ential and simpler and exactly r e c overs the e quation (1 ) after setting f ( t n ) = 1 , which we fo cus on i n the curr e n t p ap er. A ctual ly, these thr e e r epr esentative e quations (8), (9), and (10) c ome fr om v t − 1 = J G − 1 = J K − 1 0 with v = − u xx u . Cle arly, v t − 1 = J K − 1 0 i s e quivalent to K J − 1 v t = 0 ( t = t − 1 ), that is,  v txx v x  x + 4  v v t v x  x + 2 v t = 0 . (11) This e quation is exactly the one studie d by F uchssteiner [16] (se e e q uations (7.1) and (7 .22) ther e. Equations (7. 1 ) in [1 6] has a typ o an d should b e same as (11). F r om [16], the 3 Camassa-Holm (CH) e quation is gauge-e quiva lent to e quation (11) thr ough s ome ho do gr aph tr ansformations (7.11 ) and ( 7.12) in [16]. In our p ap er, thr o ugh using v = − u xx u we further r e duc e e quation (11) to a mor e sim p le form (i.e. e quation (1 ) ):  − u xx u  t = 2 u u x . (12) In other wor ds, we f ound a very inter esting fact that e quation (12) c an b e viewe d as a r e duc- tion form of the CH e quation due to the ab ove gauge-e quivalenc e. In next se ction, we wil l solve this form. In t he pap er [19], the author dealt with equation ( ∂ 2 + 4 v + 2 v x ∂ − 1 ) v t = 0 by using the p ositiv e KdV hierarch y approach, and all soliton solutio ns w ere giv en implicitly . This equation could b e transformed t o ( − u xx /u ) t = 2 uu x through v = − u xx /u , lik e we men tioned earlier in our pap er, but, this is only one of three reductions. So, solutions of this equation can not g iv e a ll solitons of our equation ( − u xx /u ) t = 2 uu x . In our pap er, w e presen t all solitons and kink solutions in a explicit for m. Also, there is t he connection of the first negativ e KdV equation ( ∂ 2 + 4 v + 2 v x ∂ − 1 ) v t = 0 with sine-Gordon [20]. But, the equation ( − u xx /u ) t = 2 uu x w e prop ose in the current pap er is not equiv alen t to the sine-Gordon equation, because the sine-Gordon equation has only kink solution while our equation has b oth kink solutions a nd classical solitons. Of course, we ma y generate higher order nonlinear equations b y selecting different mem- b ers in the hierarc h y . In the follo wing, w e will see that a ll equations in the KdV hierarch y (5) are integrable. P articularly , the ab ove three equations (8), (9), and (10) a re integrable . Let us return to the sp ectral problem (2). Apparen tly , the Gateaux deriv ativ e matrix L ∗ ( ξ ) of the sp ectral op erator L in t he direction ξ ∈ C ∞ ( R ) at p o in t v is L ∗ ( ξ ) △ = d d ǫ     ǫ =0 U ( u + ǫξ ) = ξ (13) whic h is obvious ly an injectiv e homomorphism, i.e. U ∗ ( ξ ) = 0 ⇔ ξ = 0. F or any give n C ∞ -function G , one ma y consider the following op erato r equation [3 ] with resp ect to V = V ( G ) [ V , L ] = L ∗ ( K G ) − L ∗ ( J G ) L. (14) Theorem 1. F or the sp e ctr a l pr ob lem (2 ) and an arbitr ary C ∞ -function G , the op er ator e quation (14) has the fol lowing solution V = − 1 4 G x + 1 2 G∂ , (15) wher e ∂ = ∂ x = ∂ ∂ x , and subscripts stand for the p artial derivatives in x . Pro of: A direct substitution will complete the pro of. 4 Theorem 2. L et G 0 ∈ Ker J , G − 1 ∈ Ker K , and let e ach G j b e g i v en thr ough the L enar d se quenc e (4). Then, 1. e ach new ve ctor field X k = J G k , k ∈ Z satisfies the fol lowing c ommutator r epr es e nta- tion L ∗ ( X k ) = [ V k , L ] , ∀ k ∈ Z ; (16) 2. the entir e KdV hier ar chy (5), i.e. v t k = X k = J G k , ∀ k ∈ Z , (17) p ossesses the L ax r epr esentation L t k = [ V k , L ] , ∀ k ∈ Z , (18) wher e V k = X V ( G j ) L ( k − j − 1) , X =      P k − 1 j =0 , k > 0 , 0 , k = 0 , − P − 1 j = k , k < 0 , (19) and V ( G j ) is given by Eq. (15) with G = G j . Pro of: Let us only prov e t he case for k < 0. W e ha v e [ V k , L ] = − − 1 X j = k [ V ( G j ) , L ] L k − j − 1 = − − 1 X j = k ( L ∗ ( K G j ) − L ∗ ( J G j ) L ) L k − j − 1 = − − 1 X j = k L ∗ ( K G j ) L ( k − j − 1 − L ∗ ( K G j − 1 ) L k − j = L ∗ ( K G k − 1 ) − L ∗ ( K G − 1 ) L k = L ∗ ( K G k − 1 ) = L ∗ ( J G k ) = L ∗ ( X k ) . Noticing L t k = L ∗ ( v t k ), w e hav e L t k − [ V k , L ] = L ∗ ( v t k − X k ) . The injectiv eness of L ∗ implies the second result holds. So, the en t ire K dV hierarc h y (5) has Lax pair and all equations in the hierarch y are therefore inte grable. In particular, the KdV equation (6) has the Lax pair L t 1 = [ W 1 , L ] with L = ∂ 2 + v and W 1 = ∂ 3 + 3 2 v ∂ + 3 4 v x , whic h w as w ell-kno wn in the literature [1]. 5 In teresting thing is that the first member ( k = − 1) in the negat ive order KdV hierarch y (7) has the standard Lax represen tation L t − 1 = [ V l − 1 , L ] with V l − 1 =  1 4 G l − 1 ,x − 1 2 G l − 1 ∂  L − 1 , l = 1 , 2 , 3 , L = ∂ 2 + v = u − 1 ∂ u 2 ∂ u − 1 , and L − 1 = u∂ − 1 u − 2 ∂ − 1 u . In particular, the negativ e KdV equation (8) p ossesses the f ollo wing Lax f orm: L t − 1 = [ V 1 − 1 , L ] with V 1 − 1 =  1 4 G 1 − 1 ,x − 1 2 G 1 − 1 ∂  L − 1 =  1 2 uu x − 1 2 u 2 ∂  L − 1 = − 1 2 u∂ − 1 u. All o f those negativ e mem b ers in the hierarchies (7) are in tegrable. 3 All tra v e ling w a v e solu tions of (1) Let us no w consider the tr a v eling w av e solution of equation (1) throug h a generic setting u ( x, t ) = U ( x − ct ), where c is the wa v e sp eed. Let ξ = x − ct , then u ( x, t ) = U ( ξ ). Substituting it into equation (1) yields c ( U ′′ U ) ′ = 2 U U ′ . (20) In tegrating it once, we o btain the fo llowing standard cubic Ha milto nian system for c 6 = 0 : dU dξ = y = ∂ H ∂ y , dy dξ = g U + 1 c U 3 = − ∂ H ∂ U , (21) where g is an integral constan t, and the Ha milto nian f unction is H ( U, y ) = 1 2 y 2 − 1 2 g U 2 − 1 4 c U 4 . (22) When g c ≥ 0 , (21) has only one equilibrium p oin t O (0 , 0). When g c < 0, (2 1) has three equilibrium p oints O (0 , 0) and E 1 , 2 ( ± p | cg | , 0). W rite that h 0 = H (0 , 0) = 0 , h 1 = H ( ± p | cg | , 0) = 1 4 cg 2 . By qualitative analysis, w e hav e the bifurcations of phase p o rtraits of (21) in t he ( c, g ) − parametric plan show n in Fig.1 (1-1) -(1-6). –1.5 –1 –0.5 0 0.5 1 1.5 y –1.5 –1 –0.5 0.5 1 1.5 x –2 –1 0 1 2 y –1 –0.5 0.5 1 x –1 –0.5 0 0.5 1 y –1.5 –1 –0.5 0.5 1 1.5 x (1-1) (1-2) (1-3) 6 –0.4 –0.2 0 0.2 0.4 y –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 phi –0.6 –0.4 –0.2 0 0.2 0.4 0.6 y –0.4 –0.2 0.2 0.4 x –2 –1 1 2 y –1.5 –1 –0.5 0.5 1 1.5 x (1-4) (1-5) (1-6) Fig. 1 The c hange of phase p ortraits of (2 1) in the ( c, g ) − parameter plan. (1-1) g = 0 , c > 0 . (1-2) g > 0 , c > 0 . (1 -3) g > 0 , c < 0 . (1-4) g = 0 , c < 0 . (1- 5) g < 0 , c < 0 . (1 -6) g < 0 , c > 0. Next, w e presen t t he exact trav eling w a v e solutions of (1) in an explicit form. Case 1: c > 0 , g = 0 (see Fig.1 ( 1 -1)). In this case, corresp onding to the saddle p oin t O (0 , 0) (21) reads a s y = ± U 2 √ 2 c . Using the first equation of (21) and t aking integration, we o btain U ( ξ ) = ∓ √ 2 c ξ + ξ 0 , ξ = x − ct, (23) where ξ 0 is an initial v alue of ξ . Clearly , when ξ → − ξ 0 , U ( ξ ) → ∞ . i.e., U ( ξ ) is un b ounded at ξ = ξ 0 . Th us, we ha v e t w o un b ounded breaking wa v e solutions show n in Fig.2. 0 2 4 6 8 10 y –2 –1 1 2 3 4 xi Fig. 2 The pr o files o f the functions (23) where ξ 0 = − 1. Case 2: c > 0 , g > 0 (see Fig.1 ( 1 -2)). Corresp onding to the saddle O (0 , 0) (2 1) reads as y 2 = g U 2 + 1 2 c U 4 . Using the first equation of (21) to tak e in tegration, w e obtain U ( ξ ) = ± 8 Acg A 2 e √ g ( ξ + ξ 0 ) − 8 c g e − √ g ( ξ + ξ 0 ) , (24) where A is an in tegran t constan t. When A 2 = 8 cg , the f unctions defined b y (24) are discon tinuous at ξ = − ξ 0 . The pro files of (24) like Fig.2 . Case 3: g > 0 , c < 0 (see Fig.1 ( 1 -3)). There exist t hree equilibrium p o in ts of (21) at E 1 , 2 and O (0 , 0). O is a saddle p oin t, E 1 , 2 are cen ter p o ints. 7 Corresp onding to t w o ho mo clinic orbits, defined b y H ( U, y ) = 0, we hav e the parametric represen tations U ( ξ ) = ± p 2 | c | g sec h r | c | g 2 ξ . (25) They give t w o soliton solutions of (1 ). F or h ∈ ( 1 4 cg 2 , 0) , corresp o nding to t w o families of p erio dic orbits of (21), defined by H ( U, y ) = h , i.e., y 2 = 1 2 | c | (4 | c | h + 2 g | c | U 2 − U 4 ) = 1 2 | c | ( r 2 1 − U 2 )( U 2 − r 2 2 ), where r 2 1 = g | c | + p g 2 c 2 − 4 | c | h, r 2 2 = g | c | − p g 2 c 2 − 4 | c | h, we obtain the parametric represen tations of p erio dic w a v e solutions of (1) as follows: U ( ξ ) = ± r 1 dn r 1 p 2 | c | ξ , p r 2 1 − r 2 2 r 1 ! . ( 2 6) F or h ∈ (0 , ∞ ) , corresp onding to the family of p erio dic orbits of (21), enclosing three equilibrium p oin ts defined by H ( U, y ) = h , w e hav e the fo llo wing pa rametric represen tation of p erio dic w av e solutions o f (1): u ( ξ ) = r 1 cn s ( r 2 1 − r 2 2 ) 2 | c | ξ , r 1 p r 2 1 − r 2 2 ! . (27) Case 4: g ≤ 0 , c < 0 (see Fig.1 (1-4), (1-5)). In this case, the orig in O (0 , 0) of (21) is an unique equilibrium p o in t, whic h is a cen ter. There exists a family of p erio dic orbits of (21), enclosing the origin. (1) has the same parametric r epresen tation o f p erio dic w a v e solutions as (2 7). Case 5: g < 0 , c > 0 (see Fig.1 ( 1 -6)). In this case, t here exist three equilibrium p o in ts of (21) at E 1 , 2 and O (0 , 0). O is a cen ter, E 1 , 2 are saddle p oints. The hetero clinic o r bit s, defined by H ( u, y ) = h 1 , ha v e the pa r a metric represen tations U ( ξ ) = ± p c | g | tanh ξ p 2 | g | ! , (28) whic h giv es a kink w a v e solution and an an ti-kink w av e solution of (1). W e see from (22 ) that y 2 = 1 2 c (4 ch + 2 cg U 2 + U 4 . F or h ∈ (0 , h 1 ) , it can b e written as y 2 = 1 2 c (( z 2 1 − U 2)( z 2 2 − U 2 ) , where z 2 1 = | g | c + p g 2 c 2 − 4 c h, z 2 2 = | g | c − p g 2 c 2 − 4 c h, Th us, the fa mily of p erio dic orbits, defined b y H ( u, y ) = h , ha s the parametric represen tat ion u ( ξ ) = Z 2 sn  z 1 ξ √ 2 c , z 2 z 1  , (29) whic h giv es a family of p erio dic w a v e solutions of (1). 4 Conclus ions In this pap er, w e repo r ted an intere sting prop erty of in tegrable system: solitons and kink solutions can o ccur in the same in tegrable equation, and t ho se solutions are give n explicitly . 8 Within our knowle dge, this is probably the first integrable example p ossessing suc h prop ert y . W e found this equation in the negat iv e order KdV hierarc h y , whic h is gauge-equiv alent to the CH equation. Since equation (1) has the Lax pair, w e may try to g et r - mat rix structure of the constrained system of Lax equations, a nd parametric a nd algebro-g eometric solutions [28], but that is b ey o nd the scop e of this pap er. The symmetry of equations (8), (9), and (10) w ere already discussed in [23]. Recen tly , a t w o-fold in tegrable hierarc h y asso ciated with the KdV equation was giv en in [21]. Ab o ut other negativ e o r der in tegrable hierarc hies, suc h as the AKNS, the Ka up-New ell, the Harry-Dym , the T oda etc, one ma y see the literature [29]. Ac kn o w ledgmen t The Qiao’s w ork is partially supp o r ted b y the U. S. Army Researc h Office under con- tract/grant num b er W911NF-08- 1 -0511 a nd b y the T exas Norman Hac k erman Adv anced Researc h Program under G r a n t No. 003599-000 1-2009. References [1] M.J. Ablowitz and H. 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