Persistence of travelling fronts of KdV–Burgers–Kuramoto equation

Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers–KdV equation

“…It is significant to find new solutions, since either new exact solutions or the asymptotic behavior of traveling wave solutions may provide more information for understanding the physical phenomena. The solutions of Burgers-KdV equation (6) possess their actual physical applications [25].…”

“…Since then, the method has evolved and found that way toward applications, which has received a great deal of interest and has been used by many researchers to obtain the existence of traveling waves for generalized KdV equations [20][21][22][23][24], KdV-Burgers-Kuramoto equation [25], nonlinear dispersive-dissipative equation [26][27][28], reaction-diffusion equations [29,30], stochastic differential equations [31], slow-fast dynamic systems [32][33][34][35], Liénard equations [36][37][38] and biological models [18,[39][40][41][42][43], etc.…”

Multi-form solitary wave solutions of the KdV-Burgers-Kuramoto equation

“…Thus, the human being in general and the physicist in particular model the dynamics of nonlinear phenomena by mathematical equations of all outputs, among which are the nonlinear differential equations. They vary most often according to the physical system studied [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. If one thing is to get these equations, in order to analyze and understand the dynamics of these physical systems, another thing is to solve them and get solutions that are closer to reality.…”

Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation

“…It is just the relation that offers an effective way to study traveling waves of a PDE from the point of view of differential dynamical system. So, the dynamical system methods paly important roles in our paper, which is the most remarkable difference from [7]. It will be seen that this method allows detailed analysis on phase space geometry of traveling wave system of the KBK equation so that all possible bounded traveling waves and corresponding existence conditions can be identified clearly.…”

“…Our strategy is to transform it to a three-dimensional differential dynamical system with singular perturbation. This idea is firstly introduced by Fu and Liu [7] in 2010. From geometric singular perturbation point of view, they give the existence condition of kink waves by proving that a strictly increasing traveling front of the KdV-Burgers equation can persist in the KBK equation for sufficiently small dissipation parameterγ.…”