From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

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1 PRAMANA c Indian Academ of Sciences Vol. 74, No. journal of Januar 00 phsics pp. 9 6 From bell-shaped solitar wave to W/M-shaped solitar wave solutions in an integrable nonlinear wave equation AIYONG CHEN,,, JIBIN LI, CHUNHAI LI and YUANDUO ZHANG 3 Center of Nonlinear Science Studies, Kunming Universit of Science and Technolog, Kunming, Yunnan, , People s Republic of China School of Mathematics and Computing Science, Guilin Universit of Electronic Technolog, Guilin, Guangxi, 54004, People s Republic of China 3 Foundation Department, Southwest Forestr Universit, Kunming, Yunnan, 6504, People s Republic of China Corresponding author. aiongchen@63.com MS received 7 October 008; revised Jul 009; accepted 9 September 009 Abstract. The bifurcation theor of dnamical sstems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitar waves of this equation evolve from bell-shaped solitar waves to W/M-shaped solitar waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitar wave solutions are obtained. Kewords. Bifurcation method; periodic wave solution; solitar wave solution; W/Mshaped solitar wave solutions. PACS Nos Yv; 0.30.Jr; 0.30.Oz; Kf; 47.0.Kf; 5.35.Sb. Introduction The KdV equation u t = 6uu x + u xxx (.) is a well-known nonlinear partial differential equation originall formulated to model unidirectional propagation of shallow water gravit waves in one dimension []. It describes the long time evolution of weakl nonlinear dispersive waves of small but finite amplitude. The original experimental observations of Russell [] in 844 and the pioneering studies b Korteweg and de Vries [] in 895 showed the balance between the weak nonlinear term 6uu x and the dispersion term u xxx which gave rise to unidirectional solitar wave. 9

2 Aiong Chen et al Because of its role as a model equation in describing a variet of phsical sstems, the KdV equation has been widel investigated in recent decades. Some similar models similar to the KdV equations were proposed in [3,4]. In 997, Rosenau [4] studied the nonanaltic solitar waves of the following integrable nonlinear wave equation: (u u xx ) t = au x + [(u u x)(u u xx )] x, (.) where a( 0) is a constant. The equation is obtained through a reshuffling procedure of the Hamiltonian operators underling the bi-hamiltonian structure of mkdv equations u t = u xxx + 3 u u x. (.3) Equation (.) supports peakons. Among the nonanaltic entities, the peakon, a soliton with a finite discontinuit in gradient at its crest, is perhaps the weakest nonanalticit observable b the ee. In [4], the author has studied the peakons of eq. (.) and pointed out that the interaction of nonlinear dispersion with nonlinear convection generates exactl compact structures. Unfortunatel, as the author has pointed out in [4], a lack of proper mathematical tools makes this goal at the present time prett much beond our reach. In this paper, we shall point out that the existence of singular curves in the phase plane of the travelling wave sstem is the original reason for the appearance of nonsmooth travelling wave solutions in our travelling wave models b using the theor of dnamical sstems. Recentl, Qiao [5] proposed the following completel integrable wave equation: (u u xx ) t + (u u xx ) x (u u x) + (u u xx ) u x = 0, (.4) where u is the fluid velocit and subscripts denote the partial derivatives. This equation can also be derived from the two-dimensional Euler equation using the approximation procedure. The author obtained the so-called W/M -shaped-peak solitons. More recentl, Li and Zhang [6] used the method of dnamical sstems to eq. (.4), and explained wh the so-called W/M -shaped-peak solitons can be created. In this paper, we stud the solitar wave solutions of eq. (.) using the bifurcation theor of dnamical sstems, which was developed b Li et al in [6 9]. We show that there exist smooth solitar wave solutions of eq. (.) when some parameter conditions are satisfied. In addition, we point out that the solitar waves of eq. (.) evolve from bell-shaped solitar waves to W/M-shaped solitar waves when wave speed passes certain critical wave speed. To investigate the travelling wave solutions of eq. (.), substituting u = u(x ct) = u(ξ) into eq. (.), we obtain c(φ φ ) = aφ + [(φ φ )(φ φ )], (.5) where φ is the derivative with respect to ξ. Integrating eq. (.5) once and neglecting the constant of integration we find 0 Pramana J. Phs., Vol. 74, No., Januar 00

3 From bell-shaped solitar wave to W/M-shaped solitar wave solutions (φ φ + c)φ = φ 3 + (a + c)φ φφ. (.6) Clearl, eq. (.6) is equivalent to the two-dimensional sstem dφ dξ =, d dξ = φ(φ + (a + c)) φ, + c (.7) which has the first integral H(φ, ) = ( φ c) + 4aφ = h. (.8) On the singular curve φ +c = 0, sstem (.7) is discontinuous. Such a sstem is called a singular travelling wave sstem b Li and Zhang [6].. Bifurcations of phase portraits of sstem (.7) In this section, we discuss bifurcations of phase portraits of sstem (.7) and the existence of critical wave speed. It is known that sstem (.7) has the same phase portraits as the sstem dφ dζ = (φ + c), d dζ = φ(φ + (a + c)), (.) where dξ = (φ + c)dζ, for φ + c 0. The distribution of the equilibrium points of sstem (.) is as follows. () For c < 0, when a+c 0, sstem (.) has onl one equilibrium point O(0, 0); when a + c < 0, sstem (.) has three equilibrium points O(0, 0) and P ± (±φ, 0), where φ = (a + c). () For c > 0, when a + c 0, sstem (.) has three equilibrium points O(0, 0) and S ± (0, ± c); when a + c < 0, sstem (.) has five equilibrium points O(0, 0), P ± (±φ, 0) and S ± (0, ± c). In addition, from eq. (.8), we have h 0 = H(0, 0) = 4c, h = H(±φ, 0) = 8a(a c), h = H(0, ± c) = 0. (.) It is known that a solitar wave solution of eq. (.) corresponds to a homoclinic orbit of sstem (.). Obviousl, for c < 0, a > 0, a+c < 0, there exists a homoclinic orbit of sstem (.) homoclinic to O(0, 0) defined b H(φ, ) = h 0. Hence, we are onl interested in the case c < 0, a > 0, a + c < 0. Pramana J. Phs., Vol. 74, No., Januar 00

4 Aiong Chen et al (-) (-) (-3) (-4) Figure. The change of phase portraits of sstem (.) for c < 0, a > 0. (-) a c < 0, (-) a < c < a, (-3) c = a, (-4) c < a. 4 PROPOSITION. Suppose that c < 0, a > 0, a+c < 0, we have the following conclusions about critical wave speed c = a: () For c < c < a, the homoclinic orbit defined b H(φ, ) = h 0 does not intersect the singular curve φ + c = 0. Equation (.) has two smooth bell-shaped solitar wave solutions. () For c < c, the homoclinic orbit defined b H(φ, ) = h 0 intersects the singular curve φ + c = 0. Equation (.) has a W-shaped solitar wave and an M-shaped solitar wave solutions. The phase portraits of sstem (.) can be shown in figure for c < 0, a > Solitar wave solutions of eq. (.) In this section, we will give some exact parametric representations of solitar wave solutions of eq. (.). We alwas assume that c < 0, a > 0, a + c < 0. Pramana J. Phs., Vol. 74, No., Januar 00

5 From bell-shaped solitar wave to W/M-shaped solitar wave solutions 3. The case a < c < a In this case, the homoclinic orbit defined b H(φ, ) = h 0 has no intersection point with the hperbola φ + c = 0. Thus, eq. (.) has a smooth bell-shaped solitar wave solution of valle-tpe and a smooth solitar wave solution of peaktpe. To find the exact explicit parametric representations of solitar wave solutions, we have the algebraic equation of homoclinic orbit = φ + c ± c aφ. (3.) The signs before the term c aφ are dependent on the interval of φ. Under the condition a < c < a, for φ ( (a + c), (a + c)), we need to take + before the term c aφ. Setting ψ = c aφ, we have = a (ψ ψ )(ψ ψ), where ψ = c, ψ = a + c. B the first equation of sstem (.7), we obtain the following exact parametric representations of smooth bell-shaped solitar wave solutions of eq. (.) c ψ φ(η) = ± (η), a ψ(η) = (a + c)( + cosh( c(a + c)η)), ξ(η) = x ct = ( ) c(a + c) η 4(a + c) ln(χ), χ = 4c(a + c) (a + c)(ψ + c) c(a + c)(ψ c)(ψ (a + c)). a(ψ + c) (3.) The homoclinic orbit and profiles of bell-shaped solitar waves are shown in figure. 3. The case c = a In this case, we have algebraic equation of homoclinic orbit = φ 4a + 4a aφ. (3.3) Setting ψ = 4a aφ, we have = a ψ(ψ ψ), where ψ = a. B the first equation of sstem (.7), we obtain the following exact parametric representations of smooth solitar wave solutions of eq. (.) 4a ψ φ(η) = ± (η), a ψ(η) = a( + cosh( aη)), ξ(η) = x ct = ( ( a η + 4a ln a + 3ψ )) ψ(ψ + a). (3.4) a ψ Pramana J. Phs., Vol. 74, No., Januar 00 3

6 Aiong Chen et al xi 0.5 (-) (-) Figure. The homoclinic orbit (-) and the profile of a bell-shaped solitar wave (-) for a < c < a. 3.3 The case c < a The hperbola φ +c = 0 intersects the homoclinic orbit defined b H(φ, ) = h 0 at four points Q ± ( φ, ± ) and Q ± (φ, ± ), where φ = c a/a, = c(a + c)/a. We have = φ +c+ c aφ in the interval between negative and positive half branches of the hperbola φ +c = 0. While in the left-hand side of the negative half branch and right-hand side of the positive half branch of the hperbola φ + c = 0, we have = φ + c c aφ. Therefore, we can respectivel write that = a ( ψ + aψ + c + ac) = a (ψ ψ )(ψ ψ), = a ( ψ aψ + c + ac) = a (ψ + ψ )(ψ + ψ), (3.5) where ψ = c, ψ = a + c. We next define a value η b satisfing φ(η c ψ ) = (η) = c a a a = φ. (3.6) It is eas to see that for η (, η ) and η (η, + ), we have the same parametric representations of solitar wave solution as eq. (3.). For η ( η, η ), we have ψ dψ = dξ. (3.7) (ψ + ψ ) (ψ ψ )(ψ + ψ ) Integrating (3.7), we obtain the following exact parametric representations of solitar wave solutions of eq. (.) 4 Pramana J. Phs., Vol. 74, No., Januar 00

7 From bell-shaped solitar wave to W/M-shaped solitar wave solutions xi 4 (3-) (3-) Figure 3. The homoclinic orbit (3-) and the profile of W/M-shaped solitar wave (3-) for c < a. 4 c ψ φ(η) = ± (η), a ψ(η) = (a + c) a cosh( c(a + c)η), ξ(η) = x ct = ( ) c(a + c) η + 4(a + c) ln(χ), χ = 4c(a + c) + (a + c)(ψ c) c(a + c)(ψ + c)(ψ + a + c). (3.8) a(ψ c) The homoclinic orbit and profiles of W/M-shaped solitar waves are shown in figure Conclusion In this paper,we have studied the dnamical behaviour and solitar wave solutions of an integrable nonlinear wave equation using bifurcation theor of dnamical sstems. It is pointed out that the solitar waves of this equation evolve from bell-shaped solitar waves to W/M-shaped solitar waves when wave speed passes certain critical wave speed. Equation (.) naturall has a phsical meaning since it is derived from the two-dimensional Euler equation. It can be cast into the Newton equation, u = P (u) P (A), of a particle with a new potential P (u) = u ± (A + c) + 4a(A u ), where A = lim ξ ± u. In this paper, we successfull solve this Newton equation with W/M-shaped solitar wave solutions. These solitar wave solutions ma be applied to neuroscience for providing a mathematical model and explaining electrophsiological responses of visceral nociceptive neurons and sensitization of dorsal root reflexes [0]. The mathematical results we have obtained about the singular travelling wave equation provide a deep insight into the nonlinear wave model and will be useful for phsicists to comprehend the dnamical behaviour of nonlinear wave models. Pramana J. Phs., Vol. 74, No., Januar 00 5

8 Aiong Chen et al Acknowledgements The authors wish to thank the anonmous reviewer for his/her helpful comments and suggestions. This work is supported b the National Natural Science Foundation of China (Nos 0960 and ). References [] D J Korteweg and G de Vries, Philos. Mag. 39, 4 (895) [] J S Russell, On waves, in: Report of 4th Meeting of the British Association for the Advancement of Science, York (844) [3] R Camassa and D Holm, Phs. Rev. Lett. 7, 66 (993) [4] P Rosenau, Phs. Lett. A30, 305 (997) [5] Z Qiao, J. Math. Phs. 47, 70 (006) [6] J B Li and Y Zhang, Nonlin. Anal. 0, 797 (009) [7] J B Li and Z R Liu, Appl. Math. Modell. 5, 4 (000) [8] D H Feng and J B Li, Pramana J. Phs. 68, 863 (007) [9] A Y Chen and Z J Ma, Pramana J. Phs. 7, 57 (008) [0] J H Chen, H R Weng and P M Doughert, Neuroscience 6, 743 (004) 6 Pramana J. Phs., Vol. 74, No., Januar 00