Bifurcations in the generalized Korteweg–de Vries equation

We consider the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with boundary condition by space variable. For different values of the parameters in a sufficiently small neighborhood of the zero equilibrium state we construct the asymptotic behavior of periodic solutions and invariant tori. Separately we consider the case of the characteristic equation has a countable number of roots in the range of stability of the zero solution. In this situation we build a special nonlinear boundary-value problem, which plays the role of a normal form and determines the dynamics of the original problem.