BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA-HOLM EQUATION

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1 Jornal of Alied Analsis and Comtation Volme 8, Nmber 6, December 8, Website:htt://jaac.ijornal.cn DOI:.98/8.85 BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA-HOLM EQUATION Minzhi Wei,, Xianbo Sn and Honging Zh Abstract In this aer, the traveling wave soltions for a generalized Camassa- Holm eqation t xxt = +)+) x ) 3 x x xx xxx are investigated. B sing the bifrcation method of dnamical sstems, three major reslts for this eqation are highlighted. First, there are one or two singlar straight lines in the two-dimensional sstem nder some different conditions. Second, all the bifrcations of the generalized Camassa- Holm eqation are given for either ositive or negative integer. Third, we rove that the corresonding traveling wave sstem of this eqation ossesses eakon, smooth solitar wave soltion, kink and anti-kink wave soltion, and eriodic wave soltions. Kewords Generalized Camassa-Holm eqation, bifrcation theor, eakon, solitar wave soltion, kink and anti-kink wave soltions. MSC) 35C8, 37K, 7J35, 35Q5.. Introdction Dring the ast few decades, more and more mathematicians aid attentions to certain nonlinear evoltion eqations which ossess solitar wave soltions, sch as eaked wave soltions [3, 7 9,, 3]. The well-known Camassa-Holm CH) eqation [3], which describes nonlinear disersive waves is in the following form t xxt = 3 x x xxx..) The CH eqation arises from the theor of shallow water waves [3, ] and rovides a model of wave breaking for a large class of soltions [5, 6]. Particlarl, the CH eqation.) has eakon soltion with the form x, t) = ce x ct. The name eakon, which means traveling wave with sloe discontinities, is sed to distingish them from general traveling wave soltions since the have a corner at the eak of height c, where c is the wave seed. However, it is necessar to oint ot that a eakon soltion is not a classical soltion, bt a weak soltion in the sense of satisfing a integral formlation, which is introdced in []. the corresonding athor. address:weiminzhi@cn.comm. Wei) Deartment of Alied mathematics, Gangxi Universit of Finance and E- conomics, Nanning, Gangxi, 533, China The athors were sorted b Gangxi College Enhancing Yoth s Caacit Project,KY6YB, KY6LX35) and National Natral Science Fondation of China No. 6,6).

2 85 M. Wei, X. Sn & H. Zh Recentl, some generalizations of the CH eqation admitting breaking wave soltions have been stdied, sch as the b famil eqation [9, ] t xxt = b + ) x + b x xx + xxx, b.) has been stdied for ossessing eakon soltions. Frthermore, Wazwaz [] stdied nonlinear disersive variants CHn, n) of the generalized Camassa-Holm eqation in + ), + ) and 3 + ) dimensions, resectivel. Motivated b the literatre, the resent aer focses on the following generalized Camassa-Holm eqation []: t xxt = + ) + ) x ) 3 x x xx xxx,.3) where Z and is the arameter with the nonlinear terms, and derived b Anco, Recio, Gandarias and Brzon [], which is considered b one of the related Hamiltonian strctres of the Camassa-Holm eqation..3) is exactl the Camassa-Holm eqation.) when =. In this aer, all the bifrcations of the generalized Camassa-Holm eqation.3) are given for either ositive or negative integer. The resent aer focses on the bifrcation analsis see [7, 8,,,, 5,,,5]) of eqation.3). Let x, t) = x ct) = ξ), where c is the wave seed, sbstitting it into.3) and integrating once, the artial differential eqation.3) can be transformed to the following ordinar differential eqation c + c = + )+ + g,.) where is the derivative with resect to ξ, and g is an integral constant. Eq..) is eqivalent to the following ODE sstem as follows d dξ =, with the first integral d dξ = g + c + + )+ c +,.5) H, ) = [c + + c + ) ] + g = h..6) Eq..5) is a 3-arameter lanar dnamical sstem deending on the arameter gro c, g, ). For different c, g and a fixed, the bifrcations of.5) in the hase lane, ) wold be shown as the arameters c, g) are changed. Notice that when c + =, the right hand side of the second eqation in.5) is not continos. In other words, on the straight line c + =, ξ) has no definition, it imlies that the smooth sstem.3) sometimes has non-smooth traveling wave soltions. The existence of singlar straight line for a traveling wave eqation wold lead to traveling waves losing their smoothness.. Bifrcations of Phase Portraits of.5) In this section, we std the hase ortraits for eqation.5). First, imosing the transformation dτ = c+ )dξ to Eq..5), it ields the following nonlinear sstem dφ dτ = c + ), d dτ = g + c + + )+..)

3 Bifrcations of traveling wave soltions Obviosl, Eq..) has the same toological hase ortraits as Eq..) excet for the straight lines c + =. Eqs..) and.5) are integrable becase of the same first integral 3.). For a fixed h, Eq..6) determines a set of invariant crves with different branches. As h is varied, Eq..6) defines different families of orbits with different dnamical behaviors. On the following, g = and g will be considered searatel. Case : g =. Firstl, we investigate the bifrcations of hase ortraits of.) when g =. Let f ) = c + + )+. To investigate the critical oints of the sstem, we need to find ot all zeros of the eqation f )=. It is eas to be fond that in the, ) hase lane, the abscissas of eqilibrim oints for Eq..5) on -axis are the zeros of f ). Note that f ) = c+ +)+). For an even and +)+) >, f ) has two zeros ũ ± = ± +)+) ), for an odd and + ) + ), f ) has one zero ũ + = +)+) ). Using these information, we know that all zeros of f ) lies on -axis. For an even, when + >, as f ũ ) <, f ũ + ) > or f ũ ) >, f ũ + ) < ), there exists three eqilibrim oints of.) at O, ), E,,, ), where = + ), = + ). For an odd, when +)+) >, as f ũ + ) > or f ũ + ) < ), there exists two eqilibrim oints of.6) at O, ), E 3 3, ), where 3 = + ). Otherwise, when c >, and is even or odd, on the singlar lines = ± c) there are for eqilibrim oints S i s±, Y ± ), where s± = ± c), Y± = ± ), i =. Case : g. Denote that f ) = g + c + + )+, then f ) = c + + ) + ). ) For = m, = ũ = +)+) satisfies f ±) =. Ths f ũ ) = ) ) + ) g +c +)+) + +) +)+) and f ũ ) = g c +)+) ) + + ) +)+). It imlies the following relations in the c, g)-arameter lane L a : g = + ) + ) L b : g = + ) + ) L d : g = + ) + ) L e : g = + ) + ) For = m +, = ũ = f ũ ) = g +c +)+) ) ) ) ) +)+) ) + +) c, c <, g >, >. + c, c <, g <, >. + c, c <, g >, <. + c, c <, g <, <. + ) +)+) satisfies f ) =. Then we have ) +, which imlies the following

4 85 M. Wei, X. Sn & H. Zh relations in the c, g)-arameter lane L c : g = + ) + ) L g : g = + ) + ) ) ) c +, >. c +, <. Next, the bifrcations of the hase ortraits of.) will be stdied b sing the above statements. In c, g)-arameter lane and on the line g =, the crve c + = artitions it into some regions for = m, = m +, = m + ) or = m + ) shown in Figs. -)--), resectivel. 5 8 La) g A) A) 5 A3) A) c 5 A5) A6) Lb) 5 -) Lc) 6 B3) B) g B) B) c B5) B6) -) 5 8 Ld) C) g C) 5 C3) C) c 5 C5) C6) Le) 5-3) Lg) D3) 6 g D) D) D) c D) D) -) Figre. The bifrcations set of.5) in c, g)-arameter lane, for m Z +. Let M i, i ) be the coefficient matrix of the linearized sstem of.5) at an eqilibrim oint i, i ). It is given M i, i ) = i i c + i, + ) + ) i + c ) i i i i which therefore reveals so that J i, ) = c + i ) + ) + ) i + c ). B the theor of lanar dnamical sstems, the eqilibrim oint is a saddle oint if J < ; and the eqilibrim oint is a center oint if J > and

5 Bifrcations of traveling wave soltions T racem i, ))=; the eqilibrim oint is a node if J > and T racem i, ))) J i, ) >, the eqilibrim oint is a cs if J = and the Poincaré index of the eqilibrim oint is. For or convenience, denote that h i = H i, ) = g i + c + + i, i =,..., 5. Using the qalitative analsis above, we can obtain the bifrcation crves and hase ortraits nder varios arameter conditions. Figs.-Figs. show the hase ortraits of.) in the cases g = and g, resectivel. 3 3 S.5 O.5 S -) = m, c > -) = m, c > -3) = m, c < -) = m, c < ) = m+), c < -6) = m+), c < -7) = m+), c > ) =, c < -9) =, c > -) =, c < Figre. Phase ortraits of.5) nder the arameter conditions g = and m Z +, m. When = m + ), c > and =, c >, the hase ortraits are smmetr to Figs. -5) and -8), resectivel.

6 856 M. Wei, X. Sn & H. Zh ) = m, c, g) A) 6 3-) = m, c, g) A) 6 3-3) = m, c, g) A3) ) = m,c, g) B) 6 3-5) = m, c, g) B) 6 3-6) = m, c, g) B5) Figre 3. Phase ortraits of.5) nder the arameter conditions g and m Z +, m. Similarl, for = m, c, g) A6), = m, c, g) A5), = m, c, g) A), = m +, =,c, g) B), = m, c, g) B3), = m, c, g) B6), resectivel, we can obtain similar bifrcations of hase ortraits of 3.). To save sace, we omit them. 3. Some Exact Parametric Reresentations of Traveling Wave Soltions of.3) In this section, we rovide some arametric reresentations for heteroclinic orbits, eriodic orbits and homoclinic orbits defined b.5) in different arameter conditions besides =. Obviosl,.) has the same orbits as.5) excet for c + =. The transformation of variables dτ = c + )dξ leads to some differences between the arametric reresentations of orbits for.5) and.) when c + =. Notice that H, ) = h is defined b.6), and in order to find the arametric reresentation of.5),.6) can be cast into = g + c + + h c +. 3.) ) Denote Γ h is the oen crve famil determined b H, ) = h, h h, h 3 ). A γ and B γ are called trning oints on the crve Γ h, and Cφ m, ) stands for the intersection of Γ h and the -axis see Fig. -)). The detailed discssion on the existence of eakon can be seen in []. For Fig. -), sosed that = m, c >, linking with Eq..5), sing the standard hase ortrait analtical techniqe gives when h = then =. Taking the integration on both sides of eqation.5) leads to = c) e x ct, 3.)

7 Bifrcations of traveling wave soltions ) = m + ), c, g) C) -) = m + ), c, g) C) -3) = m + ), c, g) C3) 3 -) = m + ), c, g) D) ) = m + ), c, g) D).8-6) =, c, g) C) -7) =, c, g) C3) -8) =, c, g) C6) ) =, c, g) D) -) =, c, g) D) -) =, c, g) D) Figre. Phase ortraits of.5) nder the arameter conditions g 6= and m Z +, m. Similarl, for = m + ), c, g) C6), = m + ), c, g) C5), = m + ), c, g) C), = m + ), c, g) D), = m + ), c, g) D3), =, c, g) C5), =, c, g) C), =, c, g) C), =, c, g) D3), resectivel, we can obtain similar bifrcations of hase ortraits of 3.). To save sace, we omit them here. which gives rise to a eakon soltion of.3) see Fig. 5-)). Similarl, -3) gives another eakon soltion see Fig. 5-)), which were also roved in []. ) Sose that =, c <, g = see Fig. -8)). Corresonding to the eriodic orbit defined b H, ) = h, h3 ) enclosing the center oint 3, ). It 3 h ) t ), with < t < t < c. Ths, b = t+ imlies that = c + c+ c the first eqation in.5) and from the formlas 58., 58., 3. in [], we obtain the following arametric reresentation of eriodic orbit see Fig. 6-)): + ct ) ξ = Πϕ,, k ) + F ϕ, k ), 3.3) ct ct ct ) q t where ϕ = arcsin t tc+), k = t t+ct, and Πϕ, ct, k ), F ϕ, k ) are elli ) tic integral fnctions see []). The rofile of the eriodic soltions If h, the eriodic orbit is closing to the triangle. This fact tells s that the wave rofile of ξ) determined b the eriodic orbit is a eakon soltion, that is the triangle gives rise to a eakon wave soltion see Fig. 7-)).

8 858 M. Wei, X. Sn & H. Zh xi ) xi 5-) Figre 5. Peakon soltions corresonding to Figs. -) and -3), resectivel xi 6-). hi zeta 6-) Figre 6. Profiles of eriodic wave soltions and smooth solitar wave soltions. 3) Sose that = m, c, g) A3). In this case, we have the hase ortrait of.) shown in Fig. 3-3). For concrete vale, taking =, Eq..6) can be changed to = + c + g h c + = t ) t 3 ) t ) c +, 3.) where t < < t < t 3. Setting G) = c+ t ) t ) t 3) Ths, b the first eqation of.5) and [], we obtain the following arametric reresentation: ξ = t3 G)d = t c t3 t )t t ) Φ ) Φ 3 ) + β, 3.5) where t t 3 ) t )+t t ) t 3 ) t t 3 )t t ) t 3 ) t ) Φ )=ln t, Φ 3 )=ln t3 ) t )+ t +t 3, β =ln t 3 t t c t3 t )t t ) ln t 3 t.

9 Bifrcations of traveling wave soltions ) gives rise to a smooth solitar soltion of sstem.5) see Fig. 6-)). ) Sose that =, c, g) C). In this case, we have the hase ortrait of.) shown in Fig. -6). Corresonding to the orbits defined b H, ) = h h s, ) and Eq..), there are a famil of homoclinic orbits inside the triangle. Then Eq..6) can be rewritten as = c + g + h) c + = c t 5 ) t 6 ) c, 3.6) + with t 6 < < t 5. Hence, b the first eqation of.5), we have the following arametric reresentation: ξ = ln t 5 t t 5 ) t 6 ) Φ ), 3.7) ct5 t 6 where Φ ) = ln t5t6 t5+t6)+ t5t 6 t5) t 6). 3.7) gives rises to a smooth solitar soltionsee Fig. 6-)). 5) Sose that =, c, g) C6). In this case, we have the hase ortrait of.) shown in Fig. -8). I) Corresonding H, ) = h defined b 9) and Eq..), there are two heteroclinic orbits connecting to two eqilibrim oints O, ) and E, ). Then Eq..6) can be rewritten as = c + g + h ) c + = c t 7 ) c. 3.8) + Ths, b the first eqation of.5), we obtain the following arametric reresentation of a kink and anti-kink wave soltions see Fig. 7-)): c ξ = + c t 7 ) d = + ) ln t 7 ct7 t 7 + ln ct7. 3.9) II) Corresonding H, ) = h h, ) defined b.6) and.) there are a famil of homoclinic orbits ass throgh the original oint O, ), which inside two heteroclinic orbits mentioned in 5)I). Ths Eq..6) can be rewritten as = c +g+ h) c + = c t 8) t 9) c + where < t 9 < t 8. It has the same arametric reresentation as Eq. 3.7). 6) Sose that =, c, g) D). In this case, we have the hase ortrait of.) shown in Fig. -9). I) Corresonding to the crve triangle defined b H, ) = h s, b 3.3) there are an arch heteroclinic orbits connecting to two eqilibrim oints E s+, Y + ) and E s+, Y ). Eq..6) can be rewritten as = c3 + + g) h s c + = t 8 ), 3.) where t 8 < Ths, we obtain the following arametric reresentation of the eriodic eakon wave soltion is obtained: ) ) t8 ξ) = t 8 coshξ), ξ, cosh, 3.) t8

10 86 M. Wei, X. Sn & H. Zh which gives rise to eriodic eakon soltion of.3) see Fig. 7-)), which is introdced in detail b Li and Zh in Theorem A of [6]. Particlarl, when > c, the direction of the vector field defined b.5) is oosite to the direction of the vector field defined b.), the gives rise to an nbonded soltion of sstem.5). II) Corresonding to the orbits defined b H, ) = h h, h s ), b Eq..) there are a famil of eriodic orbits enclosing the eqilibrim oint E, ). Eq..6) can be rewritten as = c + + g) h) c + = c t 9) t ), 3.) c + with c < t 9 < t <. Ths, b the first eqation of.5) and from the formlas 58., 58., 3. in [], we have the following arametric reresentation of the eriodic wave soltions see Fig. 7-3)): ct + ξ = ct 9 ct ct + Πϕ 5, ) ct, k 5) + F ϕ 5, k 5 ), 3.3) where k5 = t t9 t, ϕ 9ct +) 5 = arcsin ct+) t. ) If h h s, the eriodic is going to the crve triangle. This fact indicates that the wave rofile of ξ) determined b the eriodic orbit is a eriodic eakon wave, which will be solved on the following, and the crve triangle gives rise to a eriodic eakon wave soltion sch like Eq. 3.)..7.6 xi hi 5 5 xi 7-) 7-) ) xi Figre 7. Profiles of kink soltion, eriodic eakon soltions and eriodic wave soltions... Conclsion In smmar, the bifrcations of the generalized Camassa-Holm eqation are investigated b sing the bifrcation theor and hase ortraits of.6). It shows that the sstem.3) has eakon, eakon eriodic soltions, solitar wave soltion, kink and anti-kink wave soltions, and eriodic wave soltions nder some arameter conditions. Finall, we describe the dnamical behaviors of.3) and exlain the reason for the existence of non-smooth traveling wave. From the above analsis, it can be known that this method can be sitable for other nonlinear wave eqations.

11 Bifrcations of traveling wave soltions Acknowledgments One of the athors M. Z. Wei) is gratefl to Professor Zhijn Qiao for his enthsiastic gidance and hel dring her visit to Universit of Texas-Rio Grande Valle. All the athors thank the referee for the valable sggestions which hel imrove the aer. References [] S. C. Anco, E. Recio, M. L. Gandarias and M. S. Brzon, A nonlinear generalization of the Camassa-Holm eqation with eakon soltions, Dnamical Sstems. Diff. Eqat. Al. AIMS Proceedings, 5 Secial, 97. [] P. F. Brd and M. D. Friedman, Handbook of ellitic integrals for engineers and scientists, New York: Sringer-Verlag, 97. [3] R. Camassa and D. D. Holm, An integrable shallow water eqation with eaked solitons, Phs. Rev. Lett., 993, 7, 66 66,. [] R. Camassa, D. D. Holm and J. M. Hman, A new integrable shallow water eqation, Adv. Al. Mech., 99, 3, 3. [5] A. Constantin, Existence of ermanent and breaking waves for a shallow water eqation: a geometric aroach, Ans. Inst. Forier Grenoble),, 5, [6] A. Constantin and J. Escher, On the blow- rate and the blow- set of breaking waves for a shallow water eqation, Math. Z.,, 33, [7] A. Y. Chen, J. B. Li, X. J. Deng and W.T. Hang, Traveling wave soltions of the Fornberg-Whitham eqation, Al. Math. Comt., 9, 5, [8] A. Y. Chen and J. B. Li, Single eak solitar wave soltions for the osmosis K, ) eqation nder inhomogeneos bondar condition, J. Math. Anal. Al.,, 369, [9] A. Degaseris, D. D. Holm and A. N. W. Hone, A new integrable eqation with eakon soltions, Theor. Math. Phs.,, 33, [] D. D. Holm and A. N. W. Hone, A class of eqations with eakon and lson soltions, J. Non. Math. Phs., 5,, 389. [] M. A. Han, Bifrcation theor of limit ccles of lanar sstems, in: A. Canada, P. Drabek, A. Fonda Eds.), Handbook of differential eqations, Ordinar differential eqations, vol. 3, Elsevier, 6. [] M. A. Han, J. M. Yang, A. A. Tarta and Y. Gao, Limit ccles near homoclinic and heteroclinic loos, J. Dn. Diff. Eqat., 8,, [3] J. B. Li and Y. Zhang, Exact loo soltions, cs soltions, solitar wave soltions and eriodic wave soltions for the secial CH-DP eqation, Nonlinear Anal.: Real World Al., 9,, [] J. B. Li and H. H. Dai, On the std of singlar nonlinear traveling wave eqations: Dnamical Sstem Aroach, Beijing: Science Press, 7. [5] J. B. Li and F. J. Chen, Exact traveling wave soltions and bifrcations of the dal Ito eqation, Nonlinear Dn., 5, 8,

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