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In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with $Maple$. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods.

The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

The existence of breaking wave solutions of the second class of singular nonlinear wave equations is proved by methods from the dynamical systems theory. For the second class of singular nonlinear traveling wave equations, dynamical behaviors of the traveling wave solutions are completely classified and thoroughly discussed. Corresponding to some bounded orbits of the traveling systems, exact parametric representations of traveling wave solutions are derived within different parameter regions of the parameter space.

In this paper, we consider the degenerate coupled multi-KdV equations. Depending on the coupled multiplicity $l$, the study of the traveling wave solutions for this model derives a series of planar dynamical systems. We consider the cases of $l=2,3,4.$ On the basis of the investigation on the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) under different parameter conditions.

Using the dynamical systems analysis and singular traveling wave theory developed by Li and Chen [2007] to the classical and modified Serre shallow water wave equations, it is shown that, in different regions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) can be obtained. More than 28 explicit and exact parametric representations are precisely derived. It is demonstrated that, more interestingly, the modified Serre equation has uncountably infinitely many smooth solitary wave solutions and uncountably infinitely many pseudo-peakon solutions. Moreover, it is found that, differing from the well-known peakon solution of the Camassa–Holm equation, the modified Serre equation has four new forms of peakon solutions.

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation $u(x,t)=\varphi (\xi ),$$\xi =kx+v{t}^{\alpha }$, we reduce the PDE to an ODE which depends on the fractional order $\alpha$, then the analysis depends on the order $\alpha$. Moreover, as $\alpha =1$, the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order $\alpha$. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

For the generalized Serre–Green–Naghdi equations with surface tension, using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] for their traveling wave systems, in different parameter conditions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) are obtained. More than 26 explicit exact parametric representations are given. It is interesting to find that this fully nonlinear water waves equation coexists with uncountably infinitely many smooth solitary wave solutions or infinitely many pseudo-peakon solutions with periodic solutions or compacton solutions. Differing from the well-known peakon solution of the Camassa–Holm equation, the generalized Serre–Green–Naghdi equations have four new forms of peakon solutions.

The breather, rogue and interaction waves of the (3+1)-dimensional integrable fourth-order nonlinear equation are reviewed. The breather and rouge waves are attained with the aid of the extended homoclinic test. Thereafter, the interaction solutions between a lump wave and a 1-kink or 2-kink soliton are researched. Additionally, four kinds of interaction solutions between lump, kink, and periodic waves through a “rational-cosh-cos” test function are established. Furthermore, the dynamic attributions of the attained solutions are presented utilizing the graphical analysis.