Starting from the Davey–Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion (CRE) solvability is studied with the help of a Riccati equation. It is significant that the soliton–cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given.
As a governing equation of wave propagation in a nonlinear, dispersive and dissipative media, KdV–Burgers type equation has received great attention. In this paper, the dynamical behavior of the generalized KdV–Burgers equation under a periodic perturbation is investigated numerically in detail. It is shown that dynamical chaos can occur when we choose appropriately systematic parameters and initial conditions. Abundant bifurcation structures and different routes to chaos such as period-doubling and inverse period-doubling cascades, intermittent bifurcation and crisis, are found by applying bifurcation diagrams, the Poincaré maps and phase portraits. To characterize chaotic behavior of this system, the spectrum of Lyapunov exponent and Lyapunov dimension of the attractor are also employed.
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