Peaked Periodic Wave Solutions to the Broer Kaup Equation

Transcription

1 Commun. Theor. Phys. 67 (2017) Vol. 67, No. 1, January 1, 2017 Peake Perioic Wave Solutions to the Broer Kaup Equation Bo Jiang ( 江波 ) 1, an Qin-Sheng Bi ( 毕勤胜 ) 2 1 Department o Applie Mathematics, Jiangsu University o Technology, Changzhou , China 2 Faculty o Civil Engineering an Mechanics, Jiangsu University, Zhenjiang , China (Receive Septemer 22, 2016; revise manuscript receive Decemer 1, 2016) Astract By qualitative analysis metho, a suicient conition or the existence o peake perioic wave solutions to the Broer Kaup equation is given. Some exact explicit expressions o peake perioic wave solutions are also presente. PACS numers: Jr, Ik, Ge DOI: / /67/1/22 Key wors: qualitative analysis, Broer Kaup equation, perioic peake wave solution 1 Introuction In recent years, nonlinear wave equations with peake wave solutions attracte much attention (see Res. [1 9] an the reerences cite in). It is known that the wave type o peake wave solutions may e solitary or perioic as in the case o Camassa Holm equation. Peake solitary wave, also calle peakon, has a unique peak at crest or trough. Peake perioic wave, however, is a type o perioic traveling wave with a peak at each crest or trough, which was also calle perioic peakon y Lenells, [2 3] coshoial wave y Boy [4] an perioic cusp wave y Li an Liu. [5] Usually, we say a continuous unction has a peak at some point i at this point its let an right erivatives are inite an have ierent sign, an naturally its wave proile is calle a peake wave solution. The ollowing nonlinear wave equation u t = (u 2 + 2v u x ) x, v t = (v x + 2uv) x (1) was propose y Broer an Kaup (BK) as a moel escriing the i-irectional propagation o long waves in shallow water, where u(x, t) is relate to the horizontal velocity, an v(x, t) represents the height o the water surace aove a horizontal ottom. [10 11] It turns out that this equation was also erive rom the Kaomtsev Petviashvili equation. [12] Various aspects o BK equation (1) have een stuie. [13 19] It was shown in Re. [13] that Eq. (1) is integrale an possesses tri-hamiltonian structure an an ininite numer o conservation laws. The geometric properties o non-noether symmetries as well as their applications were iscusse in Re. [14]. A Daroux transormation an some exact solutions were presente in Re. [15]. Satsuma et al. otaine the soliton solutions an reveale ission an usion phenomena. [16] The author o Re. [17] gave a amily o traveling wave solution an its higher version. The interaction solutions etween the solitons an other ierent types o nonlinear waves were given using a consistent tanh expansion metho in Re. [18]. Very recently, y the iurcation metho o ynamical system, Meng et al. [19] constructe some smooth an peake solitary wave solutions. However whether there are the peake perioic waves to Eq. (1) remains unknown. In the present paper, we employ the qualitative analysis metho or ierential equations, which was irst introuce y Lenells, [2 3] to prove the existence o peake perioic waves to Eq. (1) an otain some exact expressions o peake perioic wave solutions. To the est o our knowlege, those otaine solutions have not een reporte in the literature. 2 Existence o Peake Perioic Waves o Eq. (1) In this section we irst introuce some notations. C n (X) enotes the set o all n times continuously ierentiale unctions on the open set X. C0 (R) represents the space o smooth unctions with compact support. L p loc (R) reers to the set o all unctions whose restriction on any compact suset is L p integrale. Hloc 1 (R) stans or Hloc 1 (R) = {u L2 loc (R) u ξ L 2 loc (R)}. Sustituting u(x, t) = u(ξ) an v(x, t) = v(ξ) with ξ = x ct into Eq. (1) leas to cu ξ = (u 2 + 2v u ξ ) ξ, cv ξ = (2uv + v ξ ) ξ. (2) It can e oserve that Eq. (2) is vali in the sense o istriutions i u, v Hloc 1 (R). Thereore the ollowing einition is natural. Deinition 1 A pair o unctions (u, v) where u, v Hloc 1 (R), is calle a traveling wave solution o Eq. (1) i u an v satisy Eq. (2) in the sense o istriutions. Since every istriution has a primitive which is a istriution, we may integrate Eq. (2) to get α cu = u 2 + 2v + u ξ, β cv = 2uv + v ξ, (3a) (3) Supporte y National Nature Science Founation o China uner Grant No an Natural Science Fun or Colleges an Universities in Jiangsu Province uner Grant No. 15KJB Corresponing author, j@jsut.eu.cn c 2017 Chinese Physical Society an IOP Pulishing Lt

2 No. 1 Communications in Theoretical Physics 23 with two integration constants α an β. By Eq. (3), u can e solve as u = β cv v ξ, (4) 2v or v 0. Sustituting Eq. (4) into Eq. (3a) we can otain an equation or the unknown v only 2vv ξξ v 2 ξ + 8v 3 µv 2 + β 2 = 0, (5) where µ = c 2 + 4α. To eal with the regularity o the traveling wave solutions, we give the ollowing lemma, which is inspire y the stuy o traveling waves o Camassa Holm equation. [2] Lemma 1 Let (u, v) e a traveling wave solution o Eq. (1). Then we have Thereore v k C j (R) or k 2 j, j 1. (6) v C (R \ v 1 (0)). (7) Proo Denote p(v) = 8v 3 + µv 2 β 2. Thus p(v) is a polynomial in v an then Eq. (5) can e written as (v 2 ) ξξ = 3v 2 ξ + p(v). (8) Since v Hloc 1 (R), Eq. (8) implies that (v2 ) ξξ L 1 loc (R). Thereore (v 2 ) ξ is asolutely continuous an v 2 C 1 (R). Note that v Hloc 1 (R) C(R). Moreover, (v k ) ξξ = (kv k 1 v ξ ) ξ = k 2 (vk 2 (v 2 ) ξ ) ξ = k(k 2)v k 2 v 2 ξ + k 2 vk 2 (v 2 ) ξξ = k(k 2)v k 2 v 2 ξ + k 2 vk 2 (3v 2 ξ + p(v)) ( = k 2 k ) v k 2 vξ 2 + k 2 2 vk 2 p(v). (9) For k 3 the right-han sie o (9) is in L 1 loc (R). Thereore v k C 1 (R) or k 2. (10) Thus Eq. (6) hols or j = 1. Next, we assume that v k C j 1 (R) or k 2 j 1 an j 2. Then or k 2 j we have v k 2 vξ 2 = 1 2 j 1 (2j 1 v 2 j 1 1 v ξ ) 1 k 2 j 1 ((k 2j 1 )v k 2 j 1 1 v ξ ) 1 j 1 = 2 j 1 (k 2 j 1 ) (v2 ) ξ (v k 2j 1 ) ξ C j 2 (R). Also we have v k 2 p(v) C j 1 (R). Thereore the righthan sie o Eq. (9) is in C j 2 (R). Hence, y inuction on j, we know Eq. (6) hols. Furthermore, it ollows rom Eq. (10) that kv k 1 v ξ = (v k ) ξ C(R), which implies that v ξ C(R \ v 1 (0)). Thereore, v C 1 (R \ v 1 (0)). Now, we assume that v C j (R \ v 1 (0)) or j 1. Then or k 2 j+1, we have v k C j+1 (R). Thus kv k 1 v ξ = (v k ) ξ C j (R), which shows that v ξ C j (R \ v 1 (0)). Hence, v C j+1 (R \ v 1 (0)). Thus, y inuction on j, we know Eq. (7) hols. Remark 1 In view o Eq. (4), it ollows rom Lemma 1 that u C (R \ v 1 (0)). From this act an Eq. (7), we know that the traveling wave solutions (u, v) o Eq. (1) are smooth except at points where v = 0. Since v is continuous on R, then v 1 (0) is a close set. This implies that R \ v 1 (0) is an open set. Since every open set is a countale union o isjoint open intervals, then there are isjoint open intervals (a i, i ), i 1, such that R \ v 1 (0) = i=1 (a i, i ). Then it ollows rom Lemma 1 that v is smooth on every interval (a i, i ) an hence Eq. (5) hols pointwise on (a i, i ). Thereore, we may multiply oth sies o Eq. (5) y v 2 v ξ an integrate on (a i, i ) to get v 2 ξ = 4v 3 + µv 2 + hv + β 2 : = F (v), (11) with a new integration constant h. Remark 2 Notice that F (v) 0 i v is a solution o Eq. (11). Moreover, rom the continuity o v on R, we know that v 0 at the inite enpoints o (a i, i ). To estalish the existence o perioic peake wave solutions o Eq. (1), we nee the ollowing lemma. Lemma 2 The solution o Eq. (11) has the ollowing asymptotic properties: (i) I v approaches 0, then we have 0 lim ξ ξ0 v ξ = lim ξ ξ0 v ξ ± or β 0, (12) where v(ξ 0 ) = 0. (ii) I v approaches a simple zero m o F (v), then we have v(ξ)=m (ξ ξ 0) 2 F (m)+ O((ξ ξ 0 ) 4 ) as ξ ξ 0, (13) where v(ξ 0 ) = m an (ξ) = O(g(ξ)) as ξ a means that (ξ)/g(ξ) is oune in some neighorhoo o a. Proo Since the proo o (ii) can e oun in Re. [2], here we only consier the proo o (i). In a small neighourhoo o v = 0, Eq. (11) can e expane as v ξ = ± F (0) + O(v), (14) where F (0) = β 2 > 0. For ξ close enough to ξ 0, integration o Eq. (14) yiels ξ ξ 0 = ± v F (0) + O(v 2 ), (15) which implies O( ξ ξ 0 2 ) = O(v 2 ). Thus we have v(ξ) = ± F (0) ξ ξ 0 + O( ξ ξ 0 2 ) as ξ ξ 0, (16) { ± F (0) + O( ξ ξ0 ), ξ ξ 0, v ξ = (17) F (0) + O( ξ ξ 0 ), ξ ξ 0. From Eq. (17) we otain Eq. (12). Remark 3 I the solution v o Eq. (11) approaches a oule zero or a triple zero m o F (v), y similar analysis

3 24 Communications in Theoretical Physics Vol. 67 to the proo o Lemma 2, we can prove that v(ξ) m as ξ ±. Using Lemma 2, we can otain the ollowing result which gives a suicient conition or the existence o peake perioic wave solutions o Eq. (1). Theorem 1 I β 0 an F (v) has a simply zero at v s such that v s > 0 (or v s < 0) an F (v) > 0 or v (0, v s ) (or v (v s, 0)), then there exists a perioic peake wave solution v(ξ) o Eq. (1) satisying min ξ R v(ξ) = 0 an max ξ R v(ξ) = v s (or min ξ R v(ξ) = v s an max ξ R v(ξ) = 0). Proo Here we only consier the case v s > 0 since similar analysis can e employe or the case v s < 0. It ollows rom Eq. (11) that v 2 ξ = (v s v)λ(v), (18) where λ(v) is a secon-orer actor o F (v) such that λ(v) > 0 or v (0, v s ). Assume that v is the solution in this interval. I v increases an approaches v s, y Eq. (13), we get that v is symmetric with respect to ξ 1, where v(ξ 1 ) = v s, i.e. v(ξ) = v(ξ 1 (ξ ξ 1 )), which means that v will reach v s an immeiately turn ack own. From Lemma 1, we know that v will not stop or turn ack anywhere ecause that woul yiel a singularity o v at a point where v 0. When v ecreases an approaches the point v = 0, accoring to Eq. (12), v will suenly change its irection at ξ 0 = 0, where v(ξ 0 ) = 0, i.e. v ξ v ξ, so that v will yiel a peak at v = 0. Hence, we euce that there exists a peake perioic wave solution o Eq. (1) with min ξ R v(ξ) = 0, max ξ R v(ξ) = v s. Remark 4 For a solution v(ξ) o Eq. (11), i there exists a oule zero or a triple zero v s 0 o F (v) such that F (v) > 0 or v (0, v s ) (or v (v s, 0)), in view o (i) o Lemma 2 an Remark 3, employing a similar analysis to the proo o Theorem 1, we can iner that v(ξ) gives a peake solitary wave solution o Eq. (1) with a single peak at ξ 0 such that v(ξ 0 ) = 0, which satisies min ξ R v(ξ) = 0 (or max ξ R v(ξ) = 0) an lim ξ = v s. 3 Exact Peake Perioic Wave Solutions o Eq. (1) To etermine the peake perioic wave solutions o Eq. (1) in speciie parameter region o parameter space, we nee to iscuss the istriution o zero points o F (v) or β 0. Dierentiating F (v) with respect to v yiels F (v) = 12v 2 + 2µv + h. (19) Let = µ h. I < 0, then F (v) < 0 hols or v R an thus F (v) is strictly monotonically ecreasing on R. Moreover, in view o the act that F (0) = β 2 > 0, we can euce that there exists a unique simple zero v a (1) or F (v) such that v a (1) > 0. I = 0, solving the equation F (v) = 0 gives v = v = µ/12 such that F (v ) = F (v ) = F (v ) = 0 an F (v ) = 24 0, which means that v is a triple zero o F (v). Moreover, ue to the act that F (0) > 0 an F (v) < 0 or v R \ {v }, we can iner that v > 0. I > 0, setting F (v) to zero leas to v1 = µ µ h, v2 = µ + µ h, (20) with v1 < v2. From Eqs. (19) an (20), it ollows that F (v 1) = 2 µ h > 0, F (v 2) = 2 µ h < 0. This shows that F (v) has a minimum at v 1 an has a maximum at v 2. Further, we can check that F (v 1) < F (v 2). Base on the aove analysis, it can e checke that there exist in total nine qualitatively cases or F (v) when β 0 (see Fig. 1). Fig. 1 The graph o F (v) or β 0. (a) < 0, () = 0, (c) > 0 an F (v1) > 0, () > 0, v1 < 0 an F (v1) = 0, (e) > 0, v1 > 0 an F (v1) = 0, () > 0, v1 < 0 an F (v1) < 0 < F (v2), (g) > 0, v1 > 0 an F (v1) < 0 < F (v2), (h) > 0, v1 > 0 an F (v1) < 0 = F (v2), (i) > 0, v1 > 0 an F (v1) < F (v2) < 0.

4 No. 1 Communications in Theoretical Physics 25 Accoring to Theorem 1 an Fig. 1, exact peake perioic wave solutions o Eq. (1) in ierent parameter regions o parameter space can e presente. We will use some symols on the elliptic unctions an elliptic integrals (see Re. [20]). sn(u) an cn(u) are Jacoian elliptic unctions with the moulus k. sn 1 (u) an cn 1 (u) are the inverse unctions o sn(u) an cn(u), respectively. Proposition 1 (The expression o peake perioic wave pointing upwars) I > 0, v1 < 0 an F (v1) < 0 < F (v2) or β 0, Eq. (1) has a peake perioic wave solution pointing upwars with perio 2T 1, which on the interval ( T 1, T 1 ) has the explicit expression v 1 (ξ) = v (1) + 1 k1 2sn2 (ω 1 (T 1 ξ ) 1 ), u 1 (ξ) = β cv 1(ξ) v 1(ξ) 2v 1 (ξ), (21), v(2) an v (3), satisying v(1) < < 0 < v (3) are three istinct simple zeros o F (v) (see Fig. 1()), ω 1 = v (3) 1 = v(3), v (3) T 1 = 1 ( sn 1 (v(3) ) )v(2) ω 1 (v (3) 1. )v(1), Proo It can e oserve rom Fig. 1() that F (v) > 0 or v I = (, 0), where v(2) < 0 is a simple zero o F (v). By Theorem 1, we know that there exists a peake perioic wave solution with min ξ R v(ξ) = an max ξ R v(ξ) = 0. For the solution v 1 (ξ) in the interval I, it ollows rom Eq. (11) that v 1 ξ = ±2 (v 1 )(v 1 )(v(3) v 1 ). (22) Integration o Eq. (22) leas to v1 s = 2(T 1 ξ ), (23) (s )(s v(1) )(v(3) s) where T 1 = s (s )(s v(1). (24) )(v(3) s) In view o Eq. (4), completing the integrals in Eqs. (23) an (24) gives Eq. (21). Employing a similar analysis as aove, we have the ollowing results. Proposition 2 (The expressions o peake perioic waves pointing ownwars) (i) Uner the same assumptions as in Proposition 1, Eq. (1) has a peake perioic wave solution pointing ownwars with perio 2T 2, which on the interval ( T 2, T 2 ) has the explicit expression v 2 (ξ) = v (3) (v (3) )sn2 (ω 2 (T 2 ξ ) 2 ), u 2 (ξ) = β cv 2(ξ) v 2(ξ) 2v 2 (ξ), (25), v(2) an v (3) are the same as escrie in Proposition 1, ω 2 = ω 1 2 = k 1 an T 2 = 1 ( ) sn 1 v(3) ω 2 v (3) 2. (ii) I > 0, v 1 < 0 an F (v 1) = 0 or β 0, Eq. (1) has a peake perioic wave solution pointing ownwars with perio 2T 3, which on the interval ( T 3, T 3 ) has the explicit expression v 3 (ξ) = v (1) (v (1) v 1) tanh 2 (ω 3 (T 3 ξ )), u 3 (ξ) = β cv 3(ξ) v 3(ξ) 2v 3 (ξ), (26) where v1 an v (1), satisying v1 < 0 < v (1), are a oule zero an a simple zero o F (v) (see Fig. 1()), ω 3 = v (1) v1, an T 3 = 1 arctanh v(1). ω 3 v (1) v1 (iii) I > 0, v 1 > 0 an F (v 1) < 0 < F (v 2) or β 0, Eq. (1) has a peake perioic wave solution pointing ownwars with perio 2T 4, which on the interval ( T 4, T 4 ) has the explicit expression v 4 (ξ) = e u 4 (ξ) = β cv 4(ξ) v 4(ξ) 2v 4 (ξ) v e (2) v e (1) 1 sn 2 (ω 4 (T 4 ξ ) 4 ),, (27) where v e (1), v e (2) an v e (3), satisying 0 < v e (1) < v e (2) < v e (3), are three istinct simple zeros o F (v) (see Fig. 1(g)), ω 4 = v (3) e e 4 = (v (3) e e )/(v (3) T 4 = 1 ( sn 1 v e (1) ω 4 e 4 ). e v e (1) ) an (iv) I > 0, v 1 > 0 an F (v 1) < 0 = F (v 2) or β 0, Eq. (1) has a peake perioic wave solution pointing ownwars with perio 2T 5, which on the interval ( T 5, T 5 ) has the explicit expression v 5 (ξ) = v (1) (v 2 ) tan2 (ω 5 (T 5 ξ )), u 5 (ξ) = β cv 5(ξ) v 5(ξ) 2v 5 (ξ), (28) an v2, satisying 0 < v (1) < v2, are a simple zero an a oule zero o F (v) (see Fig. 1(h)), ω 5 = v2 v(1) an T 5 = 1 arctan v(1) ω 5 v2. v(1) (v) Uner one o the parameter conitions: (c1) < 0, (c2) > 0 an F (v 1) > 0, (c3) > 0, v 1 > 0, an F (v 1) < F (v 2) < 0, Eq. (1) has a peake perioic

5 26 Communications in Theoretical Physics Vol. 67 wave solution pointing ownwars with perio 2T 6, which on the interval ( T 6, T 6 ) has the explicit expression v 6 (ξ) = v a (j) 2A + A 1 + cn(ω 6 (T 6 ξ ) 6 ), u 6 (ξ) = β cv 6(ξ) v 6(ξ) 2v 6 (ξ), j = 1, 2, 3, (29) where v a (j) > 0 an m j ±n j i (m j, n j R), are a unique real root an a pair o conjugate complex roots o F (v) = 0 (see Figs. 1(a), 1(c), 1(i)), A = (v a (j) m j ) 2 + n 2 j, ω 6 = 2 A 6 = (A m j + v a (j) )/2A an T 6 = 1 ( (j) ) A v cn 1 a ω 6 A + v a (j) 6. To show the correctness o our results, we select the peake perioic wave solutions or v given y Eqs. (21) an (25) as two examples an plot their planar graphs in Fig. 2. In such two cases we take c = 2, α = 6 an β = 4, so that the parameter conitions o Proposition 1 are satisie. Fig. 2 Perioic peake wave solutions o Eq. (1). Remark 5 It is worth noting that, or each solution (u, v) given aove, the expression or u contains v an v, in which v (ξ) represents the weak erivative o v(ξ) in istriution sense, i.e. v (ξ) has to satisy R v φξ = R vφ ξ or any test unction φ(ξ) C0 (R). For instance, in the case o (iv) o Proposition 2, v 5(ξ) can e compute as v 5(ξ) = 2ω 5 (v2 )sign (ξ) tan(ω 5(T 5 ξ ))sec 2 (ω 5 (T 5 ξ )) an hence u 5 can e expresse explicitly. Reerences [1] R. Camassa an D.D. Holm, Phys. Rev. Lett. 71 (1993) [2] J. Lenells, J. Di. Equ. 217 (2005) 393. [3] J. Lenells, J. Math. Anal. Appl. 306 (2005) 72. [4] J.P. Boy, Appl. Math. Comput. 81 (1997) 173. [5] J.B. Li an Z.R. Liu, Appl. Math. Moel. 25 (2000) 41. [6] Z.R. Liu an R. Wang, Chaos, Solitons an Fractals 19 (2004) 77. [7] J.B. Zhou an L.X. Tian, Nonlinear Anal-Real 11 (2010) 356. [8] B. Jiang, Y. Lu, J.H. Zhang, an Q.S. Bi, Appl. Math. Comput. 228 (2014) 220. [9] L.J. Qiao, S.Q. Tang, an H.X. Zhao, Commun. Theor. Phys. 63 (2015) 731. [10] L.J.F. Broer, Appl. Sci. Res. 31 (1975) 377. [11] D.J. Kaup, Prog. Theor. Phys. 54 (1975) 396. [12] S.Y. Lou an X.B. Hu, J. Math. Phys. 38 (1997) [13] B.A. Kupershmit, Cormrmn. Math. Phys. 99 (1985) 51. [14] G. Chavchanize, Mem. Dierential Equations Math. Phys. 36 (2005) 81. [15] Z.J. Zhou an Z.B. Li, Acta Phys. Sin. 52 (2003) 262. [16] J. Satuma, K. Kajiwara, J. Matsukiaira, an J. Hietarinta, J. Phys. Soc. Jpn. 61 (1992) [17] A.K. Svinin, Inverse Prolems 17 (2001) [18] C.L. Bai an S.Y. Lou, Chin. Phys. Lett. 30 (2013) [19] Q. Meng, W. Li, an B. He, Commun. Theor. Phys. 62 (2014) 308. [20] P.F. Byr an M.D. Frieman, Hanook o Elliptic Integrals or Engineers an Scientists, Springer, New York (1971).