Kink Waves and Their Evolution of the RLW-Burgers Equation

and Applied Analysis 3 Strauss 10 constructed periodic travelling waves with vorticity for the classical inviscid water wave problem under the influence of gravity, described by the Euler equation with a free surface over a flat bottom, and used global bifurcation theory to construct a connected set of such solutions, containing flat waves as well as waves that approach flows with stagnation points. In 2003 Huang et al. 11 employed the Hopf bifurcation theorem to established the existence of travelling front solutions and small amplitude travelling wave train solutions for a reaction-diffusion system based on a predator-prey model, which are equivalent to heteroclinic orbits and small amplitude periodic orbits in R4, respectively. Besides, many results on bifurcations of travelling waves for Camassa-Holm equation, modified dispersive water wave equation, and KdV equation can be found from 12–15 . Motivated by the reasons above, we try to seek all bounded travelling waves of the RLW-Burgers equation and investigate their dynamical behaviors. By some techniques including analyzing the ω-limit set of unstable manifold, investigating the degenerate equilibria at infinity to give global phase portrait, and so forth, we obtain existence and uniqueness of bounded travelling waves of the RLW-Burgers equation. We prove that there only exist two types of bounded travelling waves for the RLW-Burgers equation, a type of monotone kink waves and a type of oscillatory ones. For the oscillatory kink wave, the regularity of its maximum oscillation amplitude changing with parameters is discussed. In addition, exact and approximate expressions for the monotone kink waves and the oscillatory ones are obtained, respectively, by tanh function method in some special cases. By these results, all bounded travelling waves of the RLW-Burgers equation are identified under different parameter conditions. Furthermore, evolution of the two types of bounded travelling waves is discussed to explain the corresponding physical phenomena. It shows that the ratio μ/δ and the travelling wave velocity c are critical factors to affect the evolution of them. 2. Preliminaries It is well known that the travelling wave solution has the form u x, t u x − ct , where c / 0 is the wave velocity. So, we can make the transformation ξ x − ct to change 1.1 into its corresponding travelling wave system δcu′′′ − μu′′ α − c u′ βuu′ 0, 2.1 where ’ denotes d/dξ. Integrating 2.1 once, we get u′′ − gu′ − eu − fu2 0, 2.2 which has the equivalent form u′ v P u, v , v′ eu gv fu2 Q u, v , 2.3 where e c − a /δc, g μ/δc, and f −β/2δc. 4 Abstract and Applied Analysis In the following discussion, without loss of generality, we only need to consider the case e > 0, g < 0, and f < 0. In fact, if e < 0, we can make the transformation u U − e/f , v v which converts 2.3 into U′ v, v′ EU gv fU2, 2.4 where E −e > 0, that is, the case e > 0 for system 2.3 . If g > 0, we can make transformation v −V , ξ −τ which converts 2.3 into u′ V, V ′ eu GV fu2, 2.5 where G −g < 0, that is, the case g < 0 for system 2.3 . Similarly, if f > 0, we can make transformation u −U, v −V which converts 2.3 into U′ V, V ′ eU gV FU2, 2.6 where F −f < 0, that is, the case f < 0 for system 2.3 . System 2.3 has two equilibria E1 0, 0 and E2 −e/f, 0 with the Jacobian matrices, respectively, J E1 : ( 0 1 e g ) , J E2 : ( 0 1 −e g ) . 2.7 Obviously, E1 is a saddle and E2 is a stable node resp., focus for g2 − 4e ≥ 0 resp., g2 − 4e < 0 . As a special case, when g 0, E1 is a saddle and E2 is a center. In fact, in this case system 2.3 is a Hamiltonian system with the first integral H u, v : 1 2 v2 − e 2 u2 − f 3 u3. 2.8 By the properties of planar Hamiltonian system, we know there is a unique homoclinic orbit Υ0 connecting the saddle E1 0, 0 . Taking e 1, f −1, we can give the global phase portrait of system 2.3 in Figure 1 a . The homoclinic orbit Υ0 corresponds to the bell-shape solitary wave of system 1.1 as shown in Figure 1 b . The homoclinic orbit Υ0 corresponds to the level curve 1/2 v2 − e/2 u2 − f/3 u3 0 which intersects u-axis at the point u0, 0 , where u0 −3e/2f . Letting u 0 u0, from Abstract and Applied Analysis 5and Applied Analysis 5