Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system

Solitary-wa v e solutions to a dual equation o f the Kaup-Boussinesq system Jiangb o Zhou ∗ , Lixin Tian , Xingh ua F an Nonline ar Scientific R ese ar ch Center, F a culty of Scienc e, Jiangsu University, Zhenjiang, Jiangsu 212013, China Abstract In this pap er, w e emplo y the bifurcation theory o f planar dynamical systems to inv estigate the trav elling-wa v e solutions to a d ual equ ation of the Kaup- Boussinesq system. The expressions for smo oth solitary-wa v e s olutions are obtained. Key wor ds: dual equation of the Kaup-Boussinesq system, solitary-wa v e solution, bif urcat io n metho d 2000 MSC: 35Q51 , 34C23, 3 7 G10, 35Q35 1. In tro duction Since the theory o f solito ns has very wide applications in fluid dynamics, nonlinear optics, bio che mistry , microbiolog y , ph ysics and many other fields, the study of soliton solutions to nonlinear part ia l differential equations has b ecome an increasingly imp o rtan t area of researc h [1]-[5]. It is kno wn that, solitons are the solitary w av es that retain their individualit y under in teraction and ev entually tra v el with their original shap es a nd sp eeds. Therefore, to ∗ Corresp o nding author. T el: + 86-5 11-88 969336, F ax : +8 6-51 1-889 69336 Email addr ess: zho ujian gbo@y ahoo.cn (Jiangb o Zhou) Novemb er 9, 2018 in v estigate the soliton solutions, one must firstly obtain the solitary- wa ve solutions. Man y efforts hav e b een denoted t o seeking solitary-w av e solutions to nonlinear partial differen tial equations (see, e.g., [6]-[13]). Recen tly , Guha [14] studied the dual counter part of the f ollo wing Kaup- Boussinesq system [15],    u t = v xxx + 2( uv ) x , v t = u x + 2 v v x , (1.1) where u ( x, t ) denotes the height of the w a ter surface a b ov e a horizon tal b ot- tom and v ( x, t ) is related to t he ho r izontal v elo city field. Using momen t of inertia op erators metho d and the frozen Lie-P oisson structure, Guha deriv ed the dual counte r part of system (1.1) , that is    m t + ( mu + 1 2 u 2 x + u 2 + 2 uv ) x = 0 , p t + ( pu ) x = 0 , (1.2) where m = u xx + µu + λv , p = λu + v . System (1 .2) is a t w o comp onent in tegrable system [14]. When µ = λ = 0 , it b ecomes    u xxt + ( uu xx + 1 2 u 2 x + u 2 + 2 uv ) x = 0 , v t + ( uv ) x = 0 . (1.3) V arious asp ects o f the Kaup-Boussinesq system (1.1) ha v e b een studied. F or instance , Smirno v [16] obta ined real finite gap r egula r solutions to sy stem (1.1), and Boriso v et al. [17] applied the proliferation sc heme to system (1.1). Also the closely related v arian t of system (1.1) hav e b een studied in tensiv ely (see [18]-[2 9]). How ev er, it seems that the dual equation o f system ( 1.1) has attracted little a t t en tion. 2 In this pap er, we use the bifurcation theory of planar dynamical systems (see [3 0]-[3 3 ]) to inv estigate the trav elling-w av e solutions to system (1.3) and obtain analytic expressions for its smo oth solitary-w av e solutions. T o the b est of our knowledge, the solitary-w av e solutions to system (1.3) ha v e not b een rep orted in the literature. The bifurcatio n metho d was first used by Li and Liu [34] to obtain smo oth and non-smo oth tr a v elling-w av e solutions to a nonlinearly disp ersiv e equation and w as later employ ed b y many authors to deriv e a v ariety of trav elling-w av e solutions to a la rge num b er of nonlinear partial differential equations [35]-[42]. The remainder of the pap er is org anized as follo ws. In Section 2, using the tra v elling-wa v e ansatz, w e transform system (1.3) into a plana r dynamical system and then discuss bifurcations of phase p o rtraits of this system. In Section 3, we obtain the expressions f o r smo oth solitary-w av e solutions to system (1.3) . A short conclusion is giv en in Section 4. 2. Bifurcation analysis Let ξ = x − ct , where c is the wa ve sp eed. By using the trav elling-w av e ansatz u ( x, t ) = ϕ ( x − ct ) = ϕ ( ξ ), v ( x, t ) = ψ ( x − ct ) = ψ ( ξ ), we reduce system (1.3) to the follo wing o r dina r y differential equations:    − cϕ ′′′ + ( ϕϕ ′′ + 1 2 ϕ ′ 2 + ϕ 2 + 2 ϕψ ) ′ = 0 , − cψ ′ + ( ϕψ ) ′ = 0 . (2.1) In tegrating (2.1) once with resp ect to ξ , we hav e    − cϕ ′′ + ϕϕ ′′ + 1 2 ϕ ′ 2 + ϕ 2 + 2 ϕψ = g 1 , − cψ + ϕψ = g 2 , (2.2) 3 where g 1 , g 2 are tw o inte gral constants. F rom the second equation in system (2.2), w e obtain ψ = g 2 ϕ − c . (2.3) Substituting (2.3) in to the first equation in system (2.2) yields ϕ ′′ = ( ϕ − c )( g 1 − 1 2 ϕ ′ 2 − ϕ 2 ) − 2 g 2 ϕ ( ϕ − c ) 2 . (2.4) Let ϕ ′ = √ 2 2 y , then w e get the followin g planar dynamical system:      dϕ dξ = √ 2 2 y , dy dξ = √ 2(( ϕ − c )( g 1 − 1 2 y 2 − ϕ 2 ) − 2 g 2 ϕ ) ( ϕ − c ) 2 . (2.5) This is a planar Hamiltonian system with Hamiltonian function H ( ϕ, y ) = ϕ 4 − 4 c 3 ϕ 3 + (4 g 2 − 2 g 1 ) ϕ 2 + 4 cg 1 ϕ + ( ϕ − c ) 2 y 2 = h, (2.6) where h is a constant. Note that (2.5) has a singular line ϕ = c . T o a v oid the line temp orarily w e mak e transformation dξ = √ 2 2 ( ϕ − c ) 2 dζ . Under this transformation, Eq.(2.5) b ecomes    dϕ dζ = 1 2 ( ϕ − c ) 2 y , dy dζ = ( ϕ − c )( g 1 − 1 2 y 2 − ϕ 2 ) − 2 g 2 ϕ. (2.7) System (2.5) and system (2 .7) hav e the same first in tegral as (2.6) . Con- sequen tly , system (2.7 ) has the same top ological phase p ortraits as system (2.5) except for the straig h t line ϕ = c . F or a fixed h , (2.6) determines a set of inv ariant curv es of syste m (2.7). As h is v aried, (2.6) determines differen t families of orbits of system (2.7) havin g 4 differen t dynamical b eha viors. Let M ( ϕ e , y e ) b e the co efficien t matrix of the linearized v ersion o f system (2.7) at t he equilibrium p o in t ( ϕ e , y e ), then M ( ϕ e , y e ) =   ( ϕ e − c ) y e 1 2 ( ϕ e − c ) 2 − 3 ϕ 2 e + 2 cϕ e + g 1 − 2 g 2 − 1 2 y 2 − ( ϕ e − c ) y e   (2.8) and at this equilibrium p oin t, w e hav e J ( ϕ e , y e ) = det M ( ϕ e , y e ) = − ( ϕ e − c ) 2 y 2 e + 1 2 ( ϕ e − c ) 2 (3 ϕ 2 e − 2 cϕ e − g 1 + 2 g 2 + 1 2 y 2 ) , (2.9) p ( ϕ e , y e ) = tra ce( M ( ϕ e , y e )) = 0 . (2.10) It is easy t o see that the equilibrium p oin t of system (2.7) is in the form of ( ϕ e , 0). At this equilibrium p oint, w e hav e J ( ϕ e , 0) = 1 2 ( ϕ e − c ) 2 (3 ϕ 2 e − 2 cϕ e − g 1 + 2 g 2 ). By using the bifurcation theory of planar dynamical system, w e kno w that if J ( ϕ e , 0) > 0 (or < 0), then the equilibrium ( ϕ e , 0) is a cen ter (or saddle) p o in t; if J ( ϕ e , 0) = 0, and the Poincar ´ e index of the equilibrium p oint is 0, then it is a cusp. Usually , a solitary-w a v e solution t o system (1.3) corresp o nds to a ho mo - clinic orbit of system (2.7). Therefore, to obtain solitary-wa v e solutions to system (1.3), w e need only to seek homo clinic orbits o f system (2.7) and so only the saddle p oints are of interes t. Firstly , w e need to lo ok for the po ssible zeros of the f unction f ( ϕ ) = − ϕ 3 + cϕ 2 + aϕ + b , (2.11) where a = g 1 − 2 g 2 , b = − cg 1 . In or der to find a ll p ossible zeros of f ( ϕ ), w e set f ′ ( ϕ ) = − 3 ϕ 2 + 2 cϕ + a = 0 . (2.12) 5 When ∆ = 4 c 2 + 12 a > 0, we find t w o real ro ots to Eq.(2.12) as follows : ϕ ∗ 1 = 1 3 ( c − √ c 2 + 3 a ) , (2.13) ϕ ∗ 2 = 1 3 ( c + √ c 2 + 3 a ) , (2.14) with ϕ ∗ 1 < ϕ ∗ 2 . When ∆ = 4 c 2 + 12 a ≤ 0, the inequalit y f ′ ( ϕ ) ≤ 0 holds. In this case, if ( ϕ e , 0) is an equilibrium p oin t of system (2.7), then it is a cen ter p oint (o r a cusp) b ecause J ( ϕ e , 0) = − 1 2 ( ϕ e − c ) 2 f ′ ( ϕ e ) ≥ 0. Therefore, in the fo llowing, w e alw a ys supp ose ∆ = 4 c 2 + 12 a > 0. Substitute (2.13) a nd (2.1 4) in to (2.11), resp ectiv ely , we get f 1 = f ( ϕ ∗ 1 ) = 2 c 3 27 − 2 c 2 27 √ c 2 + 3 a − 2 a 9 √ c 2 + 3 a + ac 3 + b, (2.15) f 2 = f ( ϕ ∗ 2 ) = 2 c 3 27 + 2 c 2 27 √ c 2 + 3 a + 2 a 9 √ c 2 + 3 a + ac 3 + b, (2.16) with f 1 − f 2 = − 2 c 2 27 √ c 2 + 3 a ( c 2 + 3 a ) < 0 . (2.17) The equilibrium p oin ts of system (2.7) ha v e the following prop erties. Theorem 2.1. (1 ) If f 2 < 0 , then s ystem (2.7) has only one e quilibrium p oint, den ote d by ( ϕ 1 , 0)( ϕ 1 < ϕ ∗ 1 < ϕ ∗ 2 ) , which is a c enter p oint; (2) I f f 1 > 0 , then system (2.7) has only one e quilib rium p oint, den ote d by ( ϕ 2 , 0)( ϕ ∗ 1 < ϕ ∗ 2 < ϕ 2 ) , wh ich is a c enter p oint; (3) If f 1 = 0 , then system (2.7) has two e quilibrium p oints, denote d b y ( ϕ 3 , 0) , ( ϕ 4 , 0)( ϕ 3 = ϕ ∗ 1 < ϕ ∗ 2 < ϕ 4 ) . ( ϕ 3 , 0) is a c usp, while ( ϕ 4 , 0) is a c enter p oint; 6 (4) If f 2 = 0 , then system (2.7) has two e quilibrium p oints, denote d b y ( ϕ 5 , 0) , ( ϕ 6 , 0)( ϕ 5 < ϕ ∗ 1 < ϕ ∗ 2 = ϕ 6 ) . ( ϕ 5 , 0) is a c enter p oint, while ( ϕ 6 , 0) is a cusp; (5) If f 1 < 0 , f 2 > 0 , then system (2.7) has t hr e e e quilibrium p oints, denote d by ( ϕ 7 , 0) , ( ϕ 8 , 0) and ( ϕ 9 , 0)( ϕ 7 < ϕ ∗ 1 < ϕ 8 < ϕ ∗ 2 < ϕ 9 ) . ( ϕ 7 , 0) and ( ϕ 9 , 0) ar e two c enter p o ints, wh ile ( ϕ 8 , 0) is a sadd le p oint. In this pap er, we only consider the case c > 0 b ecause in the case c < 0 w e will get analogous result. I n order to giv e the details of t he bif urcat io n, w e fix t he para meter a = − 1. Thus , we obtain the follo wing t w o bifurcation curv es of system (2.7). L 1 : b = − 2 c 3 27 + 2 c 2 27 √ c 2 − 3 − 2 9 √ c 2 − 3 + c 3 , L 2 : b = − 2 c 3 27 − 2 c 2 27 √ c 2 − 3 + 2 9 √ c 2 − 3 + c 3 . The ab ov e bifurcation curv es divide the parameter space in to three re- gions (see Fig. 1) in whic h differen t phase p ortraits exist. By theorem 2.1 , w e can see that o nly in regions (I I), can system (2.7) has saddle p oints. See Fig. 2 for an example of t he corresp onding phase p o r t raits. 3. Solitary-w a v e solutions to system (1.3) F rom the discussions in Section 2, we can see that, when the para meters a = − 1, ( b, c ) ∈ (I I), system (2.7) has infinite man y saddle points. So there a re infinite ma ny ho mo clinic orbits and system (1.3) has infinite man y solitary-w a v e solutions accordingly . 7 L 1 II I L 2 III c 2.0 2.5 3.0 3.5 4.0 4.5 b K 0.6 K 0.4 K 0.2 0.0 0.2 0.4 Figure 1: The bifurcation sets and bifurcation curv es of system (2.7) for the parameter c > √ 3. In order to obtain exact expressions for solitary-w a v e solutions, w e fix b = − 2 c 3 27 + c 3 . Case I : √ 3 < c < 3 In this case, there are tw o homo clinic orbits connecting with the saddle p oint ( c 3 , 0), see Fig. 2(a) for an example. The tw o homo clinic orbits of system (2.7) or (2.5) can b e expressed resp ectiv ely as y = ± ( ϕ − c 3 ) q − ϕ 2 + 2 c 3 ϕ + 5 c 2 9 − 2 ϕ − c for ϕ − ≤ ϕ ≤ c 3 , (3.1) y = ± ( ϕ − c 3 ) q − ϕ 2 + 2 c 3 ϕ + 5 c 2 9 − 2 ϕ − c for c 3 ≤ ϕ ≤ ϕ + , (3.2) where ϕ ± = 1 3 ( c ± √ 6 c 2 − 18). Substituting (3.1), (3 .2 ) in to the first equation of system (2.5), resp ec- tiv ely , and in tegrating alo ng the cor r esp onding homo clinic orbit, w e hav e Z ϕ ϕ − s − c ( s − c 3 ) q − s 2 + 2 c 3 s + 5 c 2 9 − 2 ds = √ 2 2 | ξ | , (3.3) 8 4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 y K 0.08 K 0.06 K 0.04 K 0.02 0.00 0.02 0.04 0.06 0.08 (a) 4 K 1 1 2 y K 0.8 K 0.2 0.2 0.8 (b) Figure 2: The phase p or t raits of system (2.7) when the para meters c > √ 3, a = − 1 and b = − 2 c 3 27 + c 3 . (a) c = 1 . 8; (b) c = 4 . Z ϕ + ϕ s − c ( s − c 3 ) q − s 2 + 2 c 3 s + 5 c 2 9 − 2 ds = − √ 2 2 | ξ | , (3.4) It follows from (3.3), (3.4 ) that π 2 + arctan( α ( ϕ )) + 2 c √ 6 c 2 − 18 ln( β ( ϕ )) = √ 2 2 | ξ | , ϕ − ≤ ϕ ≤ c 3 , (3.5) and π 2 − arctan( α ( ϕ )) − 2 c √ 6 c 2 − 18 ln( − β ( ϕ )) = √ 2 2 | ξ | , c 3 ≤ ϕ ≤ ϕ + , (3.6) where α ( ϕ ) = 3 ϕ − c p − 9 ϕ 2 + 6 cϕ + 5 c 2 − 18 , (3.7) β ( ϕ ) = √ 6 c 2 − 18 + p − 9 ϕ 2 + 6 cϕ + 5 c 2 − 18 c − 3 ϕ . (3.8) Therefore, w e obta in tw o solitary-w av e solutions to system (1.3) in the 9 follo wing parametric forms:    ξ = ± √ 2( π 2 + arctan( α ( ϕ )) + 2 c √ 6 c 2 − 18 ln( β ( ϕ )) , ϕ = ϕ, ( ϕ − ≤ ϕ ≤ c 3 ) , (3.9)    ξ = ± √ 2( π 2 + arctan( α ( ϕ )) + 2 c √ 6 c 2 − 18 ln( β ( ϕ )) , ψ = g 2 ϕ − c , ( ϕ − ≤ ϕ ≤ c 3 ) , (3 .1 0) and    ξ = ± √ 2( π 2 − arctan( α ( ϕ )) − 2 c √ 6 c 2 − 18 ln( − β ( ϕ )) , ϕ = ϕ, ( c 3 ≤ ϕ ≤ ϕ + ) , (3 .1 1)    ξ = ± √ 2( π 2 − arctan( α ( ϕ )) − 2 c √ 6 c 2 − 18 ln( − β ( ϕ )) , ψ = g 2 ϕ − c , ( c 3 ≤ ϕ ≤ ϕ + ) . (3 .1 2) No w we ta ke a set of data and employ Maple to display the graphs of the ab ov e obtained solitar y-w a v e solutions in Fig. 3 . Case I I: c ≥ 3 In this case, there is one homo clinic o r bit connecting with t he saddle p oint ( c 3 , 0), see Fig. 2 ( b) for an example. Similar to the Case I, we can obta in a solitary-w av e solution to system (1.3), g iven as (3 .9), (3.10). A t ypical suc h solution is shown in Fig. 4. 4. Conclusion In summary , b y using the bifurcatio n metho d, w e obta in ana lytic ex- pressions f or smo oth solita ry-w a v e solutions to a dual equation of the Kaup- Boussinesq system (1 .3). The results of this pap er suggest that, in addition to solving many single-comp o nen t par t ia l differen tial equations, the bifurcation metho d can b e used to obtain t r av elling-wa ve solutions of tw o- comp onen t systems . 10 x K 30 K 20 K 10 0 10 20 30 4 0.3 0.4 0.5 0.6 (a) x K 30 K 20 K 10 0 10 20 30 y K 0.80 K 0.75 K 0.70 K 0.65 (b) x K 20 K 10 0 10 20 4 0.7 0.8 0.9 1.0 (c) x K 20 K 10 0 10 20 y K 1.2 K 1.1 K 1.0 K 0.9 (d) Figure 3: So lit a ry-w a v e solutions t o system (1.3) when the par a meters c = 1 . 8, a = − 1, b = 0 . 168000 a nd g 2 = 1. References [1] R. Camassa, D. D . Ho lm, An integrable shallow w ater equation with p eak ed solitons, Ph ys. Rev. Lett. 7 1 (19 9 3) 16 6 1-1664 . [2] A. Constan tin, W. 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