In this paper, the existence of solitary waves and periodic waves to a perturbed generalized KdV equation is established by applying the geometric singular perturbation theory and the regular perturbation analysis for a Hamiltonian system. Moreover, upper and lower bounds of the limit wave speed are obtained. Some previous results are extended.

We use the bifurcation method of dynamical systems to study the (2+1)-dimensional Broer-Kau-Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow-up wave solutions, periodic smooth wave solutions, periodic blow-up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow-up wave solutions converge to the solitary wave solutions.

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation ut + a(1 + bu2)u2ux + uxxx = 0. (i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves. (ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves. We also show that the common expressions include many results given by pioneers.

In this paper, we consider a modified disease model with distributed delay. The existence of traveling wave fronts connecting the zero equilibrium and the positive equilibrium is established by using an iterative technique and a nonstandard ordering for the set of profiles of the corresponding wave system. We also study the critical wave speed and give a detailed analysis on its location and asymptotic behavior with respect to the time delay. Our work extends some previous results.

No abstract