Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 893279, 12 pages http://dx.doi.org/10.1155/2014/893279 Research Article Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models Jianping Shi1 and Jibin Li2 1 2 Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China Correspondence should be addressed to Jianping Shi; Received 7 November 2013; Revised 22 January 2014; Accepted 30 January 2014; Published 18 March 2014 Academic Editor: Wen-Xiu Ma Copyright © 2014 J. Shi and J. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The paper considers the nonlocal hydrodynamic-type systems which are two-dimensional travelling wave systems with a fiveparameter group. We apply the method of dynamical systems to investigate the bifurcations of phase portraits depending on the parameters of systems and analyze the dynamical behavior of the travelling wave solutions. The existence of peakons, compactons, and periodic cusp wave solutions is discussed. When the parameter 𝑛 equals 2, namely, let the isochoric Gruneisen coefficient equal 1, some exact analytical solutions such as smooth bright solitary wave solution, smooth and nonsmooth dark solitary wave solution, and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions, are obtained. 1. Introduction 𝜎𝑅]+2 𝑌̇ A hydrodynamic system of balance equations for mass and momentum is considered by Vladimirov et al. [1]: = 𝐺𝑅 − [𝐷2 + 1 − ]𝐷 𝐺̇ = 𝜖 [𝑓 (𝑅) − 𝜉 (𝐶1 + )] , 1 + ]𝑅 I 𝑢𝑡 + 𝑝𝑥 = , 𝜌 𝜌𝑡 + 𝜌2 𝑢𝑥 = 0, 𝜏̂𝑝𝑡 + 𝑝 = 𝛽 ]+3 𝐷3 𝑅 + 𝜎 (] + 1) 𝑅]+1 𝑌2 ] − 𝜏𝜖 𝑌, ]+2 𝑅 (2) (1) 𝛽 ]+2 2 𝜌 + 𝜎 [𝜌]+1 𝜌𝑥𝑥 + (] + 1) 𝜌] (𝜌𝑥 ) ] , ]+2 where ] = 𝑛 − 2, 𝑛 = 1 + Γ𝑉∞, Γ𝑉∞ is the isochoric Gruneisen coefficient [2], and 𝜏̂, 𝛽, and 𝜎 are parameters. This system is closed by the dynamic equation of state, taking into account the effects of spatiotemporal nonlocalities. Using group theory reduction, the authors of [1] obtained the following system of the ordinary differential equations: which describes a set of approximate travelling wave solutions to the source system (1) (see the initial system (1) of PDEs in [1]), where 𝜏, 𝜉 and 𝐶1 , 𝐷 are constant parameters; 𝜖 ≪ 0. When 𝜖 = 0, it immediately obtains that 𝐺 = 𝐺1 = const, and the system (2) reduces to the following two-dimensional system: 𝑑𝑅 = 𝑌, 𝑑𝜁 2 ]+3 𝑑𝑌 𝐺1 𝑅 − [𝐷 + (𝛽/ (] + 2)) 𝑅 = 𝑑𝜁 𝜎𝑅]+2 + 𝜎 (] + 1) 𝑅]+1 𝑌2 ] , (3) 𝑅̇ = 𝑌, where 𝜁 = 𝑥 − 𝐷𝑡. 2 Abstract and Applied Analysis Assume that 𝐴 𝑗 (𝑅𝑗 , 0), 𝑗 = 1, 2, are two equilibrium points of system (3). Vladimirov et al. obtained the following conclusion (see [1, 3]). Theorem A. If ] > −2 and 𝐷2 > 𝛽𝑅1]+3 , then system (3) possesses a one-parameter family of periodic solutions, localized around the critical point 𝐴 2 (𝑅2 , 0) in a bounded set 𝑀. The boundary of this set is formed by the homoclinic intersection of separatrices of the saddle point 𝐴 1 (𝑅1 , 0). We notice that for a fixed ], system (3) is a four-parameter system depending on the parameter group (𝛽, 𝜎, 𝐷, 𝐺1 ). The bifurcations and dynamical behavior of solutions of system (3) have not be studied by [1, 3]. The conclusion of Theorem A is incomplete (see Theorems 1–3 of Section 3 below). In fact, system (3) is the first class of singular travelling wave systems defined by [4, 5], which has the singular straight line 𝑅 = 0. Depending on the changes of parameters, there are very interesting bifurcations and dynamical behaviors of the travelling wave solutions in this kind of singular systems, for example, it gives rise to so-called peakons, compactons, loop solutions, and others. As Fokas stated that peakons are peaked solitons [6], that is, solitons with discontinuous first derivative, compactons are solitons with compact support. In this paper, we will give complete description for the dynamics of solutions of system (3). Instead of ], we use 𝑛 = ] + 2 to rewrite (3) as follows: 𝑑𝑅 = 𝑌, 𝑑𝜁 2 𝑛+1 𝑛−1 𝑑𝑌 𝐺1 𝑅 − [𝐷 + (𝛽/𝑛) 𝑅 + 𝜎 (𝑛 − 1) 𝑅 = 𝑑𝜁 𝜎𝑅𝑛 𝑌2 ] (4) . 2. Bifurcations of the Phase Portraits of System (4) In this section, we study the phase portraits of system (4) in the (𝑅, 𝑌)-phase plane. Consider the associated regular system of (4): 𝑑𝑅 = 𝜎𝑅𝑛 𝑌, 𝑑𝜂 𝛽 𝑑𝑌 = 𝐺1 𝑅 − [𝐷2 + 𝑅𝑛+1 + 𝜎 (𝑛 − 1) 𝑅𝑛−1 𝑌2 ] , 𝑑𝜂 𝑛 (6) where 𝑑𝜁 = 𝜎𝑅𝑛 𝑑𝜂. Clearly, now the straight line 𝑅 = 0 is a solution of system (6). On the straight line 𝑅 = 0, system (6) has no equilibrium point. Consider the following formulas: 𝑓 (𝑅) = 𝛽 𝑛+1 𝑅 − 𝐺1 𝑅 + 𝐷2 , 𝑛 (𝑛 + 1) 𝛽 𝑛 𝑓 (𝑅) = 𝑅 − 𝐺1 . 𝑛 (7) 󸀠 Obviously, for 𝑛 = 2𝑚, 𝑚 = 1, 2, . . ., when 𝛽𝐺1 > 0, and 𝑅 = 𝑅𝑀± ≡ ±(𝑛𝐺1 /(𝑛 + 1)𝛽)1/𝑛 , 𝑓󸀠 (𝑅𝑀± ) = 0, while for 𝑛 = 2𝑚 + 1, 𝑚 = 1, 2, . . ., when 𝑅 = 𝑅𝑀 ≡ (𝑛𝐺1 /(𝑛 + 1)𝛽)1/𝑛 , 𝑓󸀠 (𝑅𝑀) = 0; we have 𝑓(𝑅𝑀) = 𝑓(𝑅𝑀+ ) = 𝑓(𝑅𝑀− ) = 𝐷2 − (𝑛𝐺1 /(𝑛 + 2 1))𝑅𝑀. When 𝐷2 = 𝐷𝑐𝑟 ≡ (𝑛𝐺1 /(𝑛 + 1))(𝑛𝐺1 /(𝑛 + 1)𝛽)1/𝑛 , 𝑓(𝑅𝑀) = 0. Since every real root 𝑅𝑗 of the function 𝑓(𝑅) gives rise to an equilibrium point 𝐴 𝑗 (𝑅𝑗 , 0) of system (6), by using (7), we can analyse the equilibrium points 𝐴 𝑗 (𝑅𝑗 , 0) of system (6) in different parameter conditions and have the following conclusions. 2.1. In the Case of 𝑛 = 2𝑚. Consider the following. This system has the first integral: 𝐻 (𝑅, 𝑌) = 𝑌2 𝑅2𝑛−2 + 𝛽 2𝑅𝑛−1 𝑛𝐷2 [ − 𝐺1 𝑅 + 𝑅𝑛+1 ] = ℎ. 𝑛𝜎 𝑛−1 2𝑛 (5) Clearly, in order to make 𝐻(𝑅, 𝑌) well defined, we assume that 𝑛 > 1; that is, ] > −1. This paper is organized as follows. In Section 2, we analyse the bifurcations of phase portraits of system (4) under different parameter conditions. Section 3 discusses the existence of periodic solutions in different parameter conditions. In particular, it discusses the existence of solitary cusp wave solutions (peakons) and periodic cusp wave solutions. In Section 4, for the case 𝑛 = ] + 2 = 2, namely, let the isochoric Gruneisen coefficient equal 1, we figure out explicit parametric expressions for the solitary wave solutions, periodic wave solutions, and uncountably infinitely many breaking wave solutions (compactons). Finally, we give a conclusion of this paper. (1) For 𝛽 > 0 and 𝐺1 > 0, when 𝐷2 < (𝑛𝐺1 /(𝑛 + 1))𝑅𝑀, and𝑓(𝑅𝑀− ) > 0, 𝑓(𝑅𝑀+ ) < 0. There exist three equilibrium points 𝐴 𝑗 (𝑅𝑗 , 0), 𝑗 = 1, 2, 3, of system (6), satisfying −∞ < 𝑅1 < 𝑅𝑀− < 0 < 𝑅2 < 𝑅𝑀+ < 𝑅3 < ∞. When 𝐷2 = (𝑛𝐺1 /(𝑛 + 1))𝑅𝑀, 𝑓(𝑅𝑀− ) > 0 and 𝑓(𝑅𝑀+ ) = 0. There exist two equilibrium points 𝐴 𝑗 (𝑅𝑗 , 0), 𝑗 = 1, 2, of system (6), satisfying −∞ < 𝑅1 < 𝑅𝑀− < 0 < 𝑅2 = 𝑅𝑀+ < ∞. When 𝐷2 > (𝑛𝐺1 /(𝑛+1))𝑅𝑀, 𝑓(𝑅𝑀− ) > 0, and 𝑓(𝑅𝑀+ ) > 0. There exists only one equilibrium point 𝐴 1 (𝑅1 , 0) of system (6), satisfying −∞ < 𝑅1 < 𝑅𝑀− < 0. (2) For 𝛽 < 0, and 𝐺1 < 0, when 𝐷2 < (𝑛|𝐺1 |/(𝑛 + 1)) 𝑅𝑀, 𝑓(𝑅𝑀− ) < 0, and 𝑓(𝑅𝑀+ ) > 0. There exist thre (...truncated)