Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation

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1 Computational and Applied Mathematics Journal 2017; 3(6): ISSN: (Print); ISSN: (Online) Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation Minzhi Wei, Junning Cai * Department of Information and Statistics, Guangxi University of Finance and Economics, Nanning, P. R. China address xiaoyanxiong123@163.com (Junning Cai) * Corresponding author Citation Minzhi Wei, Junning Cai. Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation. Computational and Applied Mathematics Journal. Vol. 3, No. 6, 2017, pp Received: January 12, 2018; Accepted: January 29, 2018; Published: February 12, 2018 Abstract: The dynamical behavior of traveling wave solutions in the generalized Camassa-Holm equation is analyzed by using the bifurcation theory and the method of phase portraits analysis. The condition under which smooth solitary waves periodic waves appear are also given. What more interesting is it gives rise to M-shape and W-sharp type solutions. Keywords: Bifurcation Theory, Smooth Solitary Wave Solutions, Periodic Wave Solutions, M/W-Shape Type Solutions, Generalized Camassa-Holm Equation 1. Introduction There is increasing attention has been paid to certain nonlinear evolution equations that holds solitary wave solutions in the past decades. A famous representative of such equations is the Camassa-Holm (CH) equation [1], it is a nonlinear dispersive wave equation that takes the form u u + 3uu = 2 uu + uu, (1) t xxt x x xxx where k is a constant related to the critical shallow water wave speed. This equation arises from the theory of shallow water waves [1, 2] by an asymptotic expansion of Eulers equations for inviscid fluid flow. It provides a model of wave breaking for a large class of solutions in which the wave slope blows up in a finite time while the wave amplitude remains bounded [3]. Especially, (1) has peakons of the form x ct u(, t x) = ce, where c is thewave speed. A peakon is a weak solution in some Sobolev space with corner at its crest. Qian and Tang studied the peakons and the periodic cusp wave solution of the generalized Camassa-Holm equation [12] u u auu + 2ku = 2 uu + uu, (2) t xxt x x x xxx This work is supported by Guangxi College Enhancing Youths Capacity Project (KY2016LX315) and Guangxi University of Finance and Economics Youth Progresss Project (2017QNB09). Where k is a constant related to the critical shallow water wave speed. They constructed some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions. Constantin gave a mathematical description of the existence of interacting solitary waves [13]. Boyd and Liu not only discovered the periodic cusp wave of (1), but also revealed the relation between the periodic cusp wave solutions and the peaked wave solutions [14, 15]. Liu and Qian [16] investigated the peakons of the following generalized Camassa-Holm equation m t xxt x x x xxx u u + au u + 2ku = 2 uu + uu, (3) with a > 0, k R, m Z. They obtained the integral representations of compactons and the implicit or integral representations of the generalized kink waves from the bifurcations of phase portrait and the bounded integral curves. More recently, Li and Qiao [10] considered the following generalized Camassa-Holm equation with both quadratic and cubic nonlinearties: mt = bux + k1[ m( u ux)] x + k2(2 mux + mxu), m = u u xx. (4) 2 2

2 Computational and Applied Mathematics Journal 2017; 3(6): For k1 = 2, k2 = 0, (4) is exactly the cubic nonlinear FORQ equation [17, 18, 19, 20]: 2 2 t x m u ux x Recio and Anco [4] derived the generalized Camassa-Holm equation: m bu + [ ( )] = 0, m = u u xx. (5) 2 2 p p 1 t x x x x m + ( u u ) u m + ( u( u u ) m) = 0, m = u u xx, (6) by possessing the Hamiltonian structures of Camassa-Hlom equation. If p = 1, (6) reduces to the camassa-holm equation (1). By substituting the traveling wave solution u( x, t) = ϕ( x ct) = ϕ( ξ), where c is the wave speed. Integrating once and then the partial differential equation (6) can be transferred to the following ordinary differential equation which is equivalent to p 2 2 p 1 c( ϕ ϕ ) + ( ϕ ϕ ) + ( ϕ ϕ ) ϕ( ϕ ϕ ) = g, (7) 2p where g is the integrable constant. (7) is equivalent to the two-dimensional systems as follows y, dξ = 1 g + cϕ ( ϕ y ) ϕ ( ϕ y ) dy 2p = d ξ 2 2 p 1 c ϕ( ϕ y ) 2 2 p p 1. (8) In this paper, by quoting bifurcation method, the generalized Camassa-Holm equation (6) is investigated when p = 2 and p = 3. For p = 2, (8) can be cast into y, dξ = g cϕ ϕ ϕ y + y dy = dξ c + ϕ ϕy (9) with the first integral c H1( ϕ, y) = gϕ + ϕ ϕ + ( ϕ ϕy c) y = h. (10) For p = 3, (8) can be cast into y, dξ = 1 g + cϕ ( ϕ y ) ϕ ( ϕ y ) dy 2p = d ξ c ϕ( ϕ y ) , (11) with the first integral y c H2( ϕ, y) = gϕ ϕ ϕ + ( c ϕ + ϕ y ϕy ) = hɶ. (12) Supposed that u( ξ ) is a continuous solution of (8) for ξ (, + ) and lim u( ξ) = α and lim u( ξ) = β. ξ ξ Recall that (i) u( x, t ) is called a solitary wave solution if α = β, (ii) u( x, t ) is called a kink or anti-kink solution α β. Usually, a solitary wave solution of (6) corresponds to a homoclinic orbit of (8), a kink (or anti-kink) wave solution (6) corresponds to a heteroclinic orbit (or the so-called connecting orbit) of (8). Similarly, a periodic orbit of (6) corresponds to a periodic traveling wave solution of (8). Thus, in order to investigate all possible bifurcations of solitary waves and periodic waves of (6), all periodic annuli and homoclinic, heteroclinic orbits of (8) should be found, which depend on the system parameters. The bifurcation theory of dynamical systems (see [5, 6]) plays an important role in this study. This paper is organized as follows. In Sec. 2, the bifurcations of phase portraits of system (6) will be discussed under different parameter conditions. In Sec.3, we prove the existence of solitary wave solutions, periodic solutions, M-shape type and W-shape type breaking wave solutions of. It is a brief conclusion in Sec.4.

3 54 Minzhi Wei and Junning Cai: Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation 2. Bifurcations of Phase Portraits of (9) and (11) 3 2 In this section, Eq. (9) will be considered the phase portraits. First, letd τ = ( ϕ ϕy c)dξ,eq. (9) becomes d dy dτ = (13) 3 2 ( ϕ ϕy c) y, τ = g cϕ ϕ ϕ y y 5 4 Noted that f1( ϕ) = g cϕ + ϕ and 4 f 3 1 ( ϕ) = c + 5 ϕ. When which implies the relations in the ( c, g) -parameter plane 1 c 3 0, ϕ = ϕ = 5 it takes c 5 c f1( ϕ) = g c, L a : g = c 5, g < 0. 4 Similarly, letd τ = ( c ϕ + 2 ϕ y ϕy )dξ,eq. (11) becomes ( c ϕ 2 ϕ y ϕy ) y, dτ = + dy g cϕ ϕ y (5ϕ 3 ϕ y y ). dξ = (14) Noted that 7 6 f1( ϕ) = g + cϕ ϕ and 6 f 5 2 ( ϕ) = c 7 ϕ. When which implies the relations in the ( c, g) -parameter plane 1 c 5 0, ϕ = ϕ = 7 it takes c 7 c f1( ϕ) = g + c, L b : g = c 7, g < 0. 6 In the ( c, g) parameter plane and the curves, the line g = 0 partition it into 6 regions (see figure 1). Figure 1. The bifurcation set of (8) in (c, g)-parameter plane. Obviously, (10) has the same topological phase portraits as 3 2 (9) except for the singular curve ϕ ϕy c = 0, and (13) and (9) are integrable, which have the same first integral as (10). For a fixed h, (10) determines a set of invariant

4 Computational and Applied Mathematics Journal 2017; 3(6): curves of (13), which contains different branches of curves. As h is varied, (10) defines different families of orbits of (13) with different dynamical behaviors (see the discussion below). It is similarly to (11) and (13). On the following, the bifurcations of phase portraits of (13) will be investigated. Let M1 ( ϕe, ye ) and M2 ( ϕe, ye ) be the coefficient matrixes of the linearized system of (13) and (14) at an equilibrium point ( ϕ, y ), and J1,2( ϕe, ye) = det( M1,2( ϕe, ye)), thus, e e M M =, (3 ϕe ye) ye ϕ e 3ϕeye c 1( ϕe, ye) c + 5ϕe 3 ϕeye (3 ϕe ye) ye = ( 5ϕe + 6 ϕe ye ye) ye ϕ e 3ϕeye c 2( ϕe, ye) c 7ϕe + 10ϕe ye 3 ϕeye ( 5ϕe + 6 ϕe ye ye) ye By the theory of planar dynamical systems, the equilibrium point is a saddle point if J < 0, the equilibrium point is a center point if J > 0 and Trace( M( ϕ,0)) = 0, if J > 0 and 2 ( Trace( M( ϕe,0))) 4 J( ϕe,0) > 0, then it is a node; the equilibrium point is a cusp, if J = 0and the Poincare index of e the equilibrium point is 0. From the above qualitative analysis, it can be known that the properties of equilibrium point and the description of orbits connecting the equilibrium points. By using the mathematical soft (Maple), we obtain the phase portraits as figure 1 and figure 2. Figure 2. Phase portraits of (8) on parameter conditions: (2-1) ( c, g) ( A1), (2-2) ( c, g) ( A2), (2-3) ( c, g) ( A3) and g = 0, c < 0, (2-4) ( c, g) ( A4), (2-5) ( c, g) ( A5),(2-6)( c, g) ( A6) and g = 0, c > 0.

5 56 Minzhi Wei and Junning Cai: Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation Figure 3. Phase portraits of (8) on parameter conditions: (3-1) ( c, g) ( B1), (3-2) ( c, g) ( B2), (3-3) ( c, g) ( B3), (3-4) ( c, g) ( B4), (3-5) ( c, g) ( B5), (3-6) ( c, g) ( B6), (3-7) g = 0, c > 0, (3-8) g = 0, c < 0. In figure 2 and figure 3, the red orbits are the singular 3 2 curves ϕ ϕy c = 0 and c ϕ + 2ϕ y ϕy = 0, the green orbits are defined by H( ϕ, y) = h, are the special orbits, others are the black orbits. 3. Dynamical Behaviors and Existence of Smooth and Non-smooth Traveling Wave Solutions of (9) and (11) In this section, the existence of smooth and non-smooth traveling wave solutions of (9) and (11) will be considered, and describe that periodic waves converge to solitary wave as h changes and smooth waves converge to non-smooth waves as parameter ( c, g ) changes. On the following, we claim that the traveling wave solutions is smooth when the wave is analytic, vice versa The Smooth Solitary Wave and Periodic Solutions of (9) (1) For ( c, g) ( A1), see figure (2-1). (i) Corresponding to the two heteroclinic orbit (the blue orbits) defined by h = H( ϕ, y) = h s 1, hs1 in (10) connecting two equilibrium points Es1( ϕ s1, ± Ys1) which is on the right-hand side of y-axis, it gives rise to a smooth solitary wave solution of (9). (ii) Corresponding to the periodic orbit defined by H( ϕ, y) = h ( h s 1, + ) inside the two heteroclinic orbit defined by h = H( ϕ, y) = h s 1, which enclosing the center point ( ϕ1,0), it implies there are a family of smooth periodic wave solutions of (9). However, when h h1, s the periodic orbits approaching to the two heteroclinic orbit, that is the period of the smooth periodic wave solutions tends to. (iii) Corresponding to the two heteroclinic orbit (the blue orbits) defined by h = H( ϕ, y) = h s 2, in (10) connecting two equilibrium points Es2( ϕ s2, ± Ys2) which is on the left-hand side of y-axis, it gives rise to another smooth solitary wave solution different from (i). (iv) Corresponding to the periodic orbit defined by H( ϕ, y) = h (, h s 2) inside the two heteroclinic orbit defined by h = H( ϕ, y) = h s 2, which enclosing the center point ( 2,0), it implies there are a family of smooth periodic wave solutions of (9). However, when h h 2, s the periodic orbits approaching to the two heteroclinic orbit, that is the period of the smooth periodic wave solutions tends to different from (ii). (2) For ( c, g) ( A2), see figure (2-2), it is symmetry to figure (2-1), thus, it has the same profiles of the

6 Computational and Applied Mathematics Journal 2017; 3(6): solutions to figure (2-1). (3) For ( c, g) ( A3), see figure (2-3). (i) Corresponding to the homoclinic orbit (the blue orbits) defined by h = H( ϕ, y) = h2, in (10) intersecting the equilibrium points ( 2, 0), it gives rise to a smooth quasi-solitary wave solution of (9)(see figure (4-1)). (ii) Corresponding to the periodic orbit defined by H( ϕ, y) = h ( h,0) inside the homoclinic orbit 2 defined byh = H( ϕ, y) = h2, which enclosing the center point ( 1,0), it implies that there are a family of smooth periodic wave solutions of (9) (see figure (4-3)). (4) For ( c, g) ( A4), see figure (2-2) and ( c, g) ( A5), see figure (2-5), we know that there are not any traveling wave solutions of (9). (5) For ( c, g) ( A6), see figure (2-6), it is symmetry to figure (2-1), thus, it has the same profiles of the solutions to figure (2-1)(see figure (4-2) and (4-4)). Figure 4. The smooth solitary wave and periodic solutions of (9). (4-1) solitary wave solution with peak type, (4-2) solitary wave solution with valley type, (4-3) and (4-4) smooth periodic wave solutions.

7 58 Minzhi Wei and Junning Cai: Bifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation 3.2. The M/W-Shape Type and Quasi-Periodic Wave Solutions of (11) As in [9], system (14) is called the associated regular system of (11). Unlike the first class of singular traveling system (see [7, 8]), for the second class of singular traveling wave systems determined by Eq. (14), even its associated regular system (14) has a family of smooth periodic solutions and homoclinic or heteroclinic orbits, the existence of singular curves (the red orbits in figure 2 and figure 3) defined by c ϕ + 2ϕ y ϕy = 0, implies the existence of breaking wave solutions ϕ( ξ ) of Eq. (11), when the phase orbits of these solutions intersect with a branch of the singular curve at two points. Noted that the curve c ϕ + 2ϕ y ϕy = 0, are three singular curves of the vector field defined by system (12). For example, consider the case of ϕ < 0 in figure (3-3), when τ is varied along the homoclinic orbit defined by H2( ϕ, y) = h2 and passes through the singular curve, on the singular curve the vector field defined by system (12) has opposite directions. This implies that the honoclinic orbit of system (12), defined by H2( ϕ, y) = h2, consists of three breaking wave solutions of Eq. (7). Similarly, for a closed orbit defined by H( ϕ, y) = hɶ with hɶ hɶ << 0, if it intersects the singular curves 2 c ϕ + 2ϕ y ϕy = 0, at two points then this closed orbit consists of two breaking wave solutions of Eq. (7). Generally, for the first class of singular traveling wave systems, near a singular straight curves c ϕ + 2ϕ y ϕy = 0, ξ is a slow time scale variable while τ is a fast time scale variable. But, for the second class of singular traveling wave systems, we know that both ξ and τ have the same time scale. (1) For ( c, g) ( B1), see figure (3-1) and for ( c, g) ( B2), see figure (3-2), we know that there are not any traveling wave solutions of (11). (2) For( c, g) ( B3), see figure (3-3). Corresponding to the homoclinic orbit (the blue orbits) defined by h = H( ϕ, y) = hɶ 1 in (10), connecting two equilibrium points E1 ( 1,0) which is on the left-hand side of y -axis, it gives rise to a W-shape type wave solution of (9)(see figure (5-1)). (3) For ( c, g) ( B6), see figure (3-6), it is symmetry to figure (3-3), thus, it has the same profiles of the solutions to figure (3-3)(see figure (5-2)). (4) For ( c, g) ( B3),( B4),( B7),( B8), see figure (3-3)- (3-4) and figure (3-7)-(3-8), it is known that there are not any traveling wave solutions of (11). Figure 5. (5-1) W-shape type solution, (5-1) M-shape type solution. As for the formulae of explicit exact traveling wave solution of (5), it is not easy to solve, so we ignore it, and it will be investigated it in the future. 4. Conclusion In summary, it obtains the bifurcation about the parameters and proves the existence of some traveling wave solutions of the generalized Camassa-Holm equation by using methods from dynamical systems theory and phase portraits in this paper. It is shown that generalized Camassa-Holm equation has smooth solitary wave solutions, periodic wave solutions and M/Wshape type solutions under some parameter conditions. Acknowledgements This work is supported by Guangxi College Enhancing Youth s Capacity Project (KY2016LX315) and Guangxi

8 Computational and Applied Mathematics Journal 2017; 3(6): University of Finance and Economics Youth Progress s Project (2017QNB09). References [1] R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), [2] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), [3] A. Constantin, J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 33 (2000), [4] E. Recio, S. C. Anco, A general family of multi-peakon equations, 2016, pdf. [5] S. N. Chow, J. K. Hale. Method of bifurcation theory. New York: Springer-Verlag, [6] J. Guckenheimer, P. J. Holmes. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York: Springer-Verlag, [7] J. B. Li, H. H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach, Science Press, Beijing, [8] J. B. Li, G. R. Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifurcat. and Chaos, 17 (2007), [9] J. B. Li, Zhao, X. G. R. Chen, On the breaking property for the second class of singular nonlinear traveling wave equations, Int. J. Bifurcat. and Chaos, 19, (2010), [10] J. B. Li, Z. J. Qiao, BIfurcatons and exact traveling wave solutipns for a genernalized Camassa-Holm equation, Int. Bifurcat. and Chaos., 3, (2013), [11] A. M. Wazwaz, The Camassa-Holm-KP equations with compact and noncompact travelling wave solutions, Appl. Math. Comput., 170 (2005), [12] T. F. Qian, M. Y. Tang, Peakons and periodic cusp waves in a generalized Camassa-Holm equation, Chaos, Solitons & Fractals, 12, (2001), [13] A. Constantin, Soliton interactions for the Camassa-Holm equation, Exposition Math, 15, (1997), [14] J. P. Boyd, Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation, App. Math. Comput., 81, (1997), [15] Z. R. Liu, Q. X. Li, and Q. M. Lin, New bounded traveling waves of Camassa-Holm equation, Int. Bifurcat. and Chaos, 14, (2014), [16] Z. R. Liu, T. F., Qian, Peakons and their bifurcation in a generalized Camassa-Holm equation. Int. J. Bifurcat. and Chaos, 11, (2001), [17] A. S. Fokas, On a class of physically important integrable equations, Phys. D, 87, (1995), [18] P. J. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev., 53, (1996), [19] B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), [20] Z. J. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006)