Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa-Holm equations

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1 JOURNAL OF MATHEMATICAL PHYSICS (013) Peakon pseudo-peakon and uspon soluions for wo generalized Camassa-Holm equaions Jibin Li 1a) and Zhijun Qiao 3a) 1 Deparmen of Mahemais Zhejiang Normal Universiy Jinhua Zhejiang People s Republi of China Cener for Nonlinear Siene Sudies Kunming Universiy of Siene and Tehnology Kunming Yunnan People s Republi of China 3 Deparmen of Mahemais The Universiy of Texas Pan-Amerian 101 Wes Universiy Drive Edinburg Texas USA (Reeived 17 May 013; aeped 14 November 013; published online 3 Deember 013) In his paper we sudy peakon uspon and pseudo-peakon soluions for wo generalized Camassa-Holm equaions. Based on he mehod of dynamial sysems he wo generalized Camassa-Holm equaions are shown o have he parameri represenaions of he soliary wave soluions suh as peakon uspon pseudo-peakons and periodi usp soluions. In pariular he pseudo-peakon soluion is for he firs ime proposed in our paper. Moreover when a raveling sysem has a singular sraigh line and a heerolini loop under some parameer ondiions here mus be peaked soliary wave soluions appearing. C 013 AIP Publishing LLC. [hp://dx.doi.org/ / I. INTRODUCTION In reen years nonlinear wave equaions wih non-smooh soliary wave soluions suh as peaked solions (peakons) and usped solions (uspons) ara muh aenion in he lieraure. Peakon was firs proposed by Cammasa and Holm 1 and hereafer oher peakon equaions were developed (see Degasperis and Proesi 3 Degasperis Holm and Hone 4 Qiao 5 6 Li and Dai 7 Novikov 8 and ied referenes herein). Peakons are he so-alled peaked solions i.e. solions wih disoninuous firs order derivaive a he peak poin. Usually he profile of a wave funion is alled peakon if a a oninuous poin is lef and righ derivaives are finie and have differen sign. 9 Bu if is boh lef and righ derivaives are posiive and negaive infiniies hen he wave profile is alled uspon. In his paper we shall show ha here exiss pseudo-peakon soluion for nonlinear wave equaions. The so alled pseudo-peakon means ha he wave profile looks like peakon bu he soluion sill has oninuous firs order derivaive. By using he dynamial sysem approah i has heoreially been proved ha here exiss a leas one singular sraigh line in he raveling wave sysem orresponding o some nonlinear wave equaion suh ha he raveling wave soluions have peaked profiles and lose heir smoohness. In fa he exisene of a singular sraigh line leads o a dynamial behavior wih wo sale variables. For a singular nonlinear raveling wave sysem of he firs lass he following wo resuls hold (see Li and Dai 7 Li and Chen 10 and more reenly Li 11 ). Theorem A ( The rapid-jump propery of he derivaive near he singular sraigh line). Suppose ha in a lef (or righ) neighborhood of a singular sraigh line here exiss a family of periodi orbis. Then along a segmen of every orbi near he sraigh line he derivaive of he wave funion jumps down rapidly on a very shor ime inerval. a) Eleroni addresses: jibinli@gmail.om; lijb@zjnu.n; andqiao@upa.edu /013/54(1)/13501/14/$ C 013 AIP Publishing LLC

2 J. Li and Z. Qiao J. Mah. Phys (013) Theorem B ( Exisene of finie ime inerval of soluion wih respe o wave variable in he posiive or negaive direion). For a singular nonlinear raveling wave sysem of he firs lass wih possible hange of he wave variable if an orbi ransversely inerses wih a singular sraigh line a a poin or i approahes a singular sraigh line bu he derivaive ends o infiniy hen i only akes a finie ime inerval o make moved poin of he orbi arrive on he singular sraigh line. In order o undersand rigorously he ourrene of peaked raveling wave soluions and he hange of wave profiles we hope o obain exa parameri represenaions of raveling wave soluions for a given nonlinear parial differenial sysem. Using exa soluion formulas we an see he hange of wave profiles. In his paper we ake he following wo nonlinear wave equaions as examples o ahieve his goal. 1. The generalized Camassa-Holm (CH) equaion wih real parameers k α: u + ku x u xx + αuu x = u x u xx + uu xxx. (1) Equaion (1) wih α = 3 is exaly he sandard CH equaion 1 as a shallow waer model.. The wo-omponen Camassa-Holm sysem wih real parameers k α e 0 =±1 (see Olver and Rosenau 1 Chen Liu and Zhang 13 and Chen Liu and Qiao 14 ): m + um x Au xx + mu x + 3(1 )uu x + e 0 ρρ x = 0 ρ + (ρu) x = 0 () where m = u α u xx k. The orresponding raveling wave sysems of equaions (1) and () have one and wo singular sraigh lines respeively (see Ses. II and III). Under some pariular parameer ondiions here exiss a leas one family of periodi orbis suh ha he boundary urves of he period annulus are a homolini orbi or a heerolini loop (see he phase porrais in Ses. II and III). Applying he lassial analysis mehod we an obain he parameri represenaions for hese boundary urves. When we ake hese homolini orbis and heerolini loops ino aoun as he limi urves of period annulus hese exa parameri represenaions provide very good undersanding for he ourrene of peaked raveling wave soluions. Namely he homolini urve gives rise o a soliary peakon-like wave soluion (alled pseudo-peakon) while he urve riangle (heerolini loop) gives rise o a soliary wave soluion wih some peak (peakon). How o lassify raveling wave soluions seems kind of ineres for a given raveling sysem. Lenells sudied raveling wave soluions for some nonlinear shallow waer wave models admiing smooh peaked and usped soluions as well as sumpons. Qiao and Zhang invesigaed all possible single solion soluions of he CH equaion hrough he proedure of funional analysis. 17 Our mehod is differen from hose wo approahes Adoping he mehod of dynamial sysems wih Theorem A and Theorem B we an obain dynamial behavior of all raveling wave soluions o inegrable PDE models. Therefore we know whih orbi gives rise o wha wave profiles and how he wave profiles are hanged depending on he parameers. In addiion applying he firs inegrals of he inegrable raveling wave sysems we are able o ge some explii soluions (see Li e al. and Li 11 ). This paper is organized as follows. In Ses. II and III we disuss he exa soluions of equaion (1) and equaion () respeively. II. PEAKON PSEUDO-PEAKONS CUSPON AND PERIODIC CUSP WAVE SOLUTIONS OF EQUATION (1) Le u(x ) = (x ) = (ξ) where is he wave speed. Subsiuing i ino (1) and inegraing one we have ( ) = 1 ( ) + 1 α + (k ) g

3 J. Li and Z. Qiao J. Mah. Phys (013) where g is an inegral onsan and is he derivaive wih respe o ξ. The above equaion wih g = 0 is equivalen o he following differenial sysem d dξ = y whih has he following firs inegral: dy dξ = y + (k ) + α (3) ( ) H( y) = ( )y [(k ) α3 = h. (4) ( Sysem (3) has wo riial poins E 0 (0 0) and E ( k) 1 0 ) in he -axis. Making he ransformaion dξ = ( )dζ sysem(3) beomes is regular assoiaed sysem α 10 d dζ = y( ) dy dζ = y + (k ) + α. (5) Apparenly he sraigh line = is a soluion of sysem (5). On his sraigh line sysem (5) has wo riial poins S 1 ( ± Y 0 ) when Y 0 = α + (k ) > 0. By he resuls 10 and Theorem A and Theorem B in Se. I we know ha he dynamis of sysems (3) and (5) are differen in he neighborhood of he sraigh line =. Speially he variable ζ is a fas variable while he variable ξ is a slow variable in he sense of he geomeri singular perurbaion heory. Le h 0 = H(0 0) = 0 h 1 = H ( ( k) 0 ) = 4( k)3 h α 3α s = H( ± Y 0 ) = [ ( k) 1 3 α. Under he parameer ondiion of > 0 k < sysem(5) has he phase porrais shown below in Figs. 1(a) 1(). Le us firs onsider he parameri represenaions of he bounded orbis given by Fig. 1(a). (i) For he family of periodi orbis of Eq. (3) defined by H( y) = h h (0 h 1 )in(4) we see ha y = α( 3h α + 3 α ( k) 3 ). Thus from he firs equaion of sysem (3) we have = α(r 1 )( r )( r 3 ) 3( ) 3( ) 3 r1 ( )d ξ = α ( )(r1 )( r )( r 3 ) whih leads o he parameri represenaion (χ) = r 1 α 0 ( r 1)sn (χk) 1 α0 sn (χk) ξ(χ) = ( r 1) α 3( r )(r 1 r 3 ) (arsin(sn(χk))α 0 k) (6) where k = (r 1 r )( r 3 ) ( r )(r 1 r 3 ) α 0 = r 1 r r ( α k) is he ellipi inegral of he hird kind sn(u k) n(u k) dn(u k) are he Jaobian ellipi funions (see Byrd and Fridman 0 ). We noie from Fig. 1(a) ha when parameer α is very lose o 3( k) wehave0< h s and h s is arbirarily small. I implies ha a segmen of he homolini orbi defined by a branh of he level urve H( y) = 0 ompleely lies in a lef neighborhood of he singular sraigh line =. By using Theorem A he sae oordinae y of he poins in his segmen rapidly jumps (following ξ varies) from a posiive number o negaive number suh ha his homolini orbi gives rise o a soliary usp wave. In addiion from (6) one an easily see ha d 3α dξ = 0 sn(χk)n(χk)dn(χk). (7) α(1 α 0 sn (χk)) When h h s 1 i.e. 1 k 1( 1 means very lose o 1) he graph of d dξ Fig. (a). Clearly when χ = 4nK(k) n = 0 ± 1 ± d is shown in = 0 where K(k) is he omplee dξ ellipi inegral of he firs kind wih he modulo k. By Theorem A when χ passes hrough 4nK(k) is sign hanges from + o and is value jumps rapidly from a posiive maximum o a negaive maximum. This fa implies ha he wave profile of (ξ) deermined by he periodi orbi losing o he homolini orbi is a smooh periodi usp wave (see Fig. (b). Finally when h 0 k 1

4 J. Li and Z. Qiao J. Mah. Phys (013) (a) alpha > 3( k)/ (b) alpha = 3( k)/ () ( k)/ < alpha < 3( k)/ FIG. 1. The phase porrais of (5)for > 0 k <. he period of he periodi usp wave soluions ends o hus he homolini orbi in Fig. 1(a) gives rise o a smooh soliary usp wave soluion of (3) (seefig.()). Beause for (ξ) given by (8) we have d dξ ξ=0 = 0 herefore he soliary usp wave soluion defined by (8) is no a peakon i is a pseudo-peakon. (ii) For he homolini orbi defined by H( y) = 0 le P M ( M 0) be he inerseion poin of he homolini orbi wih he -axis. We know ha M = 3( k). When h = h α 0 = 0 Eq. (4) beomes α y =± ( M ). Using he firs equaion of sysem (3) and aking iniial value (0) = 3( ) M we obain he parameri represenaion of he homolini orbi o he riial poin E 0 (0 0) defined by H( y) = 0 as follows: M ( M ) osh( M χ)+(+ M ) 3 α (χ) = [ ( ξ(χ) = χ ln (M )( ) + 1 ( + M) ln ( 1 ( M) )) for χ ( 0 and for χ [0 ) respeively. (8) In a shor we have he following onlusion.

5 J. Li and Z. Qiao J. Mah. Phys (013) hi 0.5 y() (a) (b) () FIG.. The hange of wave profiles of (ξ). Theorem 1. (1) When he parameer group (α k ) ofsysem(3) saisfy he ondiion α> 3( k) wih > 0 k < here exiss a homolini orbi of sysem (5) given by a branh of he urves H( y) = 0. The homolini orbi has he exa parameri represenaion given by (8). () When α 3( k) 0( 0means very lose o zero) i.e. 0 < h s 0 (namely h s is srily posiive and arbirarily small) as a limi urve of a family of periodi orbis of sysem (3) defined by he losed branh of he urves H( y) = h h (h 1 0)) in Fig. 1(a) he homolini orbi gives rise o a smooh soliary usp-like wave soluion (a pseudo-peakon) of equaion (1). (3) When h varies from h 1 o h 0 = 0 periodi wave soluions of equaion (1) deermined by periodi orbis of sysem (3) will gradually beome peaked periodi wave and evolve from nonpeaked periodi waves o he smooh periodi usp-like waves and finally onverge o a smooh soliary usp-like wave (a pseudo-peakon). Seond le us onsider he parameri represenaions of he orbis given by Fig. 1(b).Nowwe have h 0 = h s = 0. (iii) For he family of periodi orbis of Eq. (3) defined by H( y) = h h (0 h 1 )in(4) we have he same parameri represenaion as (6). (iv) For wo sraigh line orbis onneing he equilibrium poins (0 0) and ( Y ± )of(4) defined by H( y) = 0 we have y = 1 3 α. Thus by Theorem B in Se. I we an ake iniial

6 J. Li and Z. Qiao J. Mah. Phys (013) value as (0) =. Then we have whih is a real peakon soluion o equaion (1). Therefore we have u(x ) = (ξ) = e α 3 ξ (9) Theorem. (1) When he parameer group (α k ) ofsysem(3) saisfy he ondiion α = 3( k) wih > 0 k < here exiss a heerolini loop of sysem (5) given by hree branhes of he urves H( y) = 0. () As he limi urves of a family of periodi orbis of sysem (3) he urve riangle (i.e. heerolini loop) in Fig. 1(b) gives rise o a soliary peaked wave soluion (a peakon) of equaion (1) whih has he exa parameri represenaion given by (9). (3) When h varies from h 1 o 0 periodi wave soluions of equaion (1) deermined by periodi orbis of sysem (3) will gradually beome peaked periodi wave and evolve from non-peaked periodi waves o he smooh periodi usp-like waves and finally onverge o a soliary peaked wave soluion (a peakon). Third we disuss he exa soluions for he orbis shown in Fig. 1(). (v) For he family of periodi orbis of (3) defined by H( y) = h h (h s h 1 )in(4) we see ha y = α( 3h α + 3 α ( k) 3 ). Thus we have from he firs equaion of (3) ha = α(r 1 )( r )( r 3 ) 3( ) 3( ) ξ = 3 α r whih implies he parameri represenaion ( )d ( )(r1 )( r )( r 3 ) (χ) = r + (r r 3 )sn (χk) 1 α1 sn (χk) [ ξ(χ) = ( r3 )χ (r r 3 ) (arsin(sn(χk))α1 k) 4α 3( r )(r 1 r 3 ) (10) where k = (r 1 r )( r 3 ) ( r )(r 1 r 3 ) α 1 = r 1 r r 1 r 3. (vi) For he arh orbi defined by H( y) = h s in (4) we have y = α 3 (m m + ) where m = 3 ( k). Hene we obain he parameri represenaion of he smooh periodi usp-like α wave soluion of (1) asfollows: (ξ) = 1 [ ( ) ( ) α m m(m 4) osh 3 ξ + m 0 ξ osh 1. (11) m(m 4) (vii) For he sable and unsable manifolds in he righ phase plane of he riial poin E 0 (0 0) defined by H( y) = 0in(4) we have y = α( 3 α ( k) ) α(e 1 ). 3( ) 3( ) On he basis of Theorem B in Se. I we an ake iniial value (0) =. Using he firs equaion of (3) we have he following uspon soluion of equaion (1): (ξ) = ξ(χ) = e 1 (e 1 ) osh(χ)+(e 1 +) 3 α χ ( 0 and χ [0 ) respeively [ e 1 χ ( ln (e 1 )( ) + 1 (e 1 + ) ln ( 1 (e 1 ) )). (1) Aording o (1) we may plo he graph of uspon soluion o equaion (1) shown in Fig. 3. Theorem 3. When he parameer group (α k ) ofsysem(3) saisfy he ondiion 0 <α< 3( k) wih > 0 k < orresponding o he sable and unsable manifolds in he righ phase plane of he riial poin E 0 (0 0) in Fig. 1() defined by H( y) = 0 equaion (1) has a uspon soluion given by (1) (beause is lef and righ derivaives equal o posiive infiniy and negaive infiniy respeively).

7 J. Li and Z. Qiao J. Mah. Phys (013) FIG. 3. The uspon wave of equaion (1). III. PEAKON AND PSEUDO-PEAKON SOLUTIONS OF EQUATION () Le u(x ) = (x ) = (ξ) ρ(x ) = v(x ) = v(ξ) where is he wave speed. Then he seond equaion of () beomes v + (v) = 0 where sands for he derivaive wih respe o ξ. Inegraing his equaion one and seing he inegraion onsan as B B 0 i follows ha The firs equaion of () reads as v(ξ) = B. (13) Inegraing his equaion yields = (A + ) + 3 [ 1 ( ) + + e 0 vv. ( ) = 1 ( ) (A + ) e 0 B ( ) 1 g (14) where g is an inegraion onsan. Equaion (14) is equivalen o he following wo-dimensional sysem: d dξ = y dy dξ = y ( ) + ( ) [3 (A + ) g + e 0 B (15) ( ) ( ) whih admis he following firs inegral: H( y) = y ( ) 3 + (A + ) + g + e 0 B = h. (16) ( ) Wihou loss of generaliy he wave speed > 0 is given. Then sysem (15) is a four-parameer planar dynamial sysem wih he parameer uple (A B g ).

8 J. Li and Z. Qiao J. Mah. Phys (013) Assume A > 0. Imposing he ransformaion dξ = ( ) ( )dζ for on sysem (15) wih e 0 =±1 leads o he following regular sysem: d dζ = y( ) ( ) dy dζ = 1 y ( ) + 1 [( ) (3 (A + ) g) + e 0 B. Apparenly wo singular lines = and = are wo invarian sraigh line soluions of (17). Near hese wo sraigh lines he variable ζ is a fas variable while he variable ξ is a slow variable in he sense of he geomeri singular perurbaion heory. To see he equilibrium poins of (17) le us mark and alulae he following (17) f () = ( ) (3 (A + ) g) + e 0 B (18) f () = ( )[6 3(A + ) + (A + ) g (19) f () = (18 6(A + 4) + (4A + 7) g. (0) Apparenly f () has one zero a = s1 =. When = 9A + 1A g > 0 f () has wo zeros a = 1 = 1 1 [3(A + ). So we have f() = e 0 B f () = 0 and f () = ( A g) f(0) = e 0 B g. In he -axis he equilibrium poins E j ( j 0)of(17) saisfy f( j ) = 0. Geomerially for a fixed > 0 he real zeros j (j = 1 or j = 1 3 4) of he funion f() an be deermined by he inerseion poins of he quadrai urve y = 3 (A + ) g and he hyperbola y = e 0 B. Obviously sysem (17) has a mos 4 equilibrium poins a E ( ) j ( j 0)j = On he sraigh line = here is no equilibrium poin of (17) ifb 0. On he sraigh line = ( here exis wo equilibrium poins S Y f ( s) of (17) wih Ys = ) if f ( ( ) ) > 0. Nex we assume ha e 0 = 1. Le h i = H( i 0) and h s = H ( Y s) where H is given by (16). For a given wave speed > 0 assume ha one of he following wo ondiions holds: (1) g > 0 < A + A + g. For given A and g f ( 1 ) < 0 f ( ) < 0. () g < 0 A + 4g > 0 A A + g < < A + A + g. For given A and g f ( 1 ) < 0 f ( ) < 0. Then Eq. (17) has four simple equilibrium poins E j ( j 0)j = saisfying 1 < 1 < < < 3 < < 4. Noie ha for every j = j does no depend on he parameer. Suppose ha < 1. Then we have he following differen opologial phase porrais of Eq. (15) shown in Figs. 4(a) 4(). Le us firs onsider exa soluions of he orbis shown in Fig. 4(a). From (16) for a given inegral onsan h wehave y = ( )[3 (A + ) g + h e 0 B ( )( ) G() ( )( ) = 4 (A + ) 3 + ( + A g) + (h + g) (h + e 0 B ). (1) ( )( ) In ligh of he firs equaion of (15) and aking inegraion on a branh of he invarian urve H( y) = h wih iniial value (ξ 0 ) = 0 one an obain ξ ξ 0 =± 0 ( )( ) d. () G() (i) The homolini orbi of sysem (15) o he saddle poin E 3 ( 3 0) is a losed branh of he level se H( y) = h 3 whih is around he ener E 4 ( 4 0)inFig.4(a). In his ase funion G()in(1)

9 J. Li and Z. Qiao J. Mah. Phys (013) (a) (b) () FIG. 4. The bifuraions of phase porrais of sysem (15)when<1and 4 <. an be wrien as G() = ( M )( 3 ) ( m ) where ( M 0) is he inerseion poin of he homolini orbi wih he -axis. Thus he righ-hand side of () reads as M ( )( )d ( 3 ) ( M )( m ) = M [ F1 () + A 11 F1 () + A 1 ( 3 ) F 1 () where F 1 () = ( )( M )( )( m ) A 11 = 3 ( 1 + 1) A 1 = [ 3 ( 1 + 1). So we have he following parameri represenaion of he soliary wave soluion of (): (3) (χ) = M α 1 sn (χk) 1 α1 sn (χk) ξ(χ) = g [ ± ( A A 1 ( 3 ) ) χ +( M ) (arsin(sn(χk))α 1 k) + A 1( M ) ( M 3 )( 3 ) (arsin(sn(χk))α k) (4)

10 J. Li and Z. Qiao J. Mah. Phys (013) where g = ( )( M m ) α 1 = M α = α 1 ( 3) M 3 k = ( M )( m) ( )( M m ) ( α k) is he ellipi inegral of he hird kind sn(u k) is he Jaobian ellipi funion (see Byrd and Fridman 0 ). (ii) The homolini orbi of sysem (15) o saddle poin E 1 ( 1 0) is a losed branh of he level se H( y) = h 1 whih is around he ener E ( 0)inFigs.4(a) 4(). In his ase funion G() in (1) an be wrien as G() = ( L )( M1 )( 1 ) where ( M1 0) is he inerseion poin of he homolini orbi wih he -axis. Hene he inegral on he righ side of () leads o M1 ( )( )d ( 1 ) ( L )( M1 ) = M1 [ F () + A 1 F () + A ( 1 ) F () where F () = ( L )( )( )( M 1 ) A 1 = 1 ( 1 + 1) A = [ 1 ( 1 + 1). Therefore we obain he following parameri represenaion of soliary wave soluion of (): (5) (χ) = ( L ) M1 ( L M1 )sn (χk) ξ(χ) = ĝ ( L ) ( L M1 )sn (χk) [( ) A 1 + M1 + A L 1 χ + ( M 1 ) M1 (arsin(sn(χk))α3 k) + A ( L ) ( 1)( L 1 ) (arsin(sn(χk))α 4 k) (6) where ĝ = (L )( M 1 ) α 3 = L M1 M1 α 4 = ( 1)α M1 1 k = ( )( L M1 ) ( L )( M 1 ). Seond we invesigae exa parameri represenaions of he wo heerolini orbis of (15) defined hrough H( y) = h 3 = h s in Fig. 4(b). (iii) In his ase funion G()in(1) an be wrien as G() = ( )( 3) ( l ). Hene aking inegrals along he heerolini orbis E 3 S + and E 3 S hoosing iniial value (0) = wearrivea ± ξ = d ( )( l ) + ( 3 ) Thus we obain a new peakon soluion of () asfollows: (χ) = B 0 ξ(χ) = [ ( ) e χ + l B 0 e χ + + l [ χ 3 3 l ln B 0 ( X((χ) 3 )+ X( 3 )) (χ) 3 d ( 3 ) ( )( l ). (7) χ ( l X( 3 ) ) + B 1 (8) and (χ) = B 0 ξ(χ) = [ ( ) e χ + l B 0 e χ + + l [ χ l ln χ [0 ) B 0 ( X((χ) 3 )+ X( 3 )) (χ) l X( 3 ) ) B 1 (9) where X() = ( )( l ) B 0 = X( ) + 1 ( + l) ( B 1 = 3 X( 3 l ln 3)+ ) X( 3 ) + 3 l 3. X( 3 ) Third we onsider exa uspon soluions of () shown in Fig. 4().

11 J. Li and Z. Qiao J. Mah. Phys (013) (iv) The sable and unsable manifolds in he righ phase plane of he riial poin E 3 ( 3 0) defined by H( y) = h 3 in (16) approah he singular sraigh line =. The funion G() in (1) an be wrien as G() = ( L )( 3 ) ( l ). On he basis of Theorem B in Se. I we an ake iniial value (0) =. Thus he righ-hand side of () reads as ( )( )d ( 3 ) ( L )( l ) = [ F3 () + A 11 F3 () + A 1 ( 3 ) F 3 () (30) where F 3 () = ( L )( )( )( l) A 11 = 3 ( 1 + 1) A 1 = [ 3 ( 1 + 1). So we have he following parameri represenaion of he uspon soluion o (): (χ) = lα5 sn (χk) ξ(χ) = ǧ 1 α 1 sn (χk) [( A 11 + L + A 1 ( L 3 ) ) χ +( L) (arsin(sn(χk))α5 k) + A 1( L ) ( L 3 )( 3) (arsin(sn(χk))α 6 k) (31) where ǧ = ( l)( L ) α 5 = L α 6 = α 5 ( L 3 ) ( k = ( L l )( ) 3) ( l)( L ). As an example we ake he following parameer values: A =.5 B = 0.6 g = = 0.55 and = 3. Then we have 1 = = = = s = Y s = and h 1 = h = h 3 = h s = h 4 = Under his parameer ondiion he phase porrai of sysem (15) is shown in Fig. 5(a). Corresponding o he urve defined by H( y) = h 3 = h s we obain a peaked soliary wave soluion o equaion () shown in Fig. 5(b). Nex we ake A =.5 B = 0.6 g = = 0.55 and = 3. Then we have a similar phase porrai as Fig. 4(a). In his ase we know ha M = whih is lose o = s1 = = In addiion M1 = is lose o = s = = y x (a) (b) FIG. 5. A peaked soliary wave soluion defined by formulas (8)and(9).

12 J. Li and Z. Qiao J. Mah. Phys (013) y() 1.5 y() (a) (b) y() 1.5 y() () (d) FIG. 6. The hange of wave profiles of (ξ). By aking iniial values (0) = M (0) = M (0) = M (0) = M1 respeively we obain hree profiles of periodi usped soluions and a pseudo-peakon soluion of equaion () shown in Figs. 6(a) 6(d). By aking oher iniial values (0) = M 0.1 (0) = M (0) = M (0) = M respeively we may obain hree profiles of periodi usped soluions and a pseudopeakon soluion of equaion () shown in Figs. 7(a) 7(d). In a summary we obain he following resuls. Theorem 4. Suppose ha he ravelling wave sysem (15) of equaions () saisfies he parameer ondiion <0 g > 0 < A + A + g and for given A and g f ( 1 ) < 0 f ( ) < 0. Then we have he following resuls: (1) If he parameer group (A B g ) make he numbers M and M1 very lose o = and = respeively hen orresponding o wo homolini orbis of sysem (17) defined by H( y) = h 3 and H( y) = h 1 in (16) respeively he formulas (4) and (6) giveriseowo pseudo-peakon soluions of equaion (). () Corresponding o he heerolini loop of sysem (17) defined by H( y) = h s in (16) formulas (8) and (9) givesriseoa peakon soluion of equaion (). (3) Corresponding o he sable and unsable manifolds in he righ phase plane of he riial poin E 3 ( 3 0) defined by H( y) = h 3 in (16) formulas (31) gives rise o a uspon soluion of equaion ().

13 J. Li and Z. Qiao J. Mah. Phys (013) y() y() (a) (b) y() 4.5 y() () (d) FIG. 7. The hange of wave profiles of (ξ). ACKNOWLEDGMENTS This work was parially suppored by he Naional Naural Siene Foundaion of China (Gran Nos and ) and he China sae adminisraion of foreign expers affairs sysem under he affiliaion of China Universiy of Mining and Tehnology. The auhors would like o express heir sinere hanks o he reviewers for heir valuable suggesions o improve heir paper. 1 R. Cammasa and D. D. Holm An inegrable shallow waer equaion wih peaked soluion Phys. Rev. Le (1993). R. Cammasa D. D. Holm and J. M. Hyman A new inegrable shallow waer equaion Adv. Appl. Meh (1994). 3 A. Degasperis and M. Proesi Asympoi inegrabiliy in Symmery and Perurbaion Theory edied by A. Degasperis and G. Gaea (World Sienifi Singapore 1999). pp A. Degasperis D. D. Holm and A. N. W. Hone A new inegrable equaion wih peakon soluions Theor. Mah. Phys (00). 5 Z. J. Qiao A new inegrable equaion wih uspons and W/M-shape-peaks solions J. Mah. Phys (006). 6 Z. J. Qiao New inegrable hierarhy parameri soluions uspons one-peak solions and M/W-shape peak soluions J. Mah. Phys (007).

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