| In the first part of this thesis, an equation of Boussinesq-type of the form{dollar}{dollar}(BQ)qquad usb{lcub}tt{rcub} - usb{lcub}xx{rcub} + usb{lcub}xxxx{rcub} + f(u)sb{lcub}xx{rcub}=0,qquad {lcub}rm for{rcub} t > 0,xin{lcub}bf R{rcub}{dollar}{dollar}is considered. It is shown that solitary wave may be stable or unstable, depending on the range of the wave's speed of propagation and on the nonlinearity. Sharp conditions to that effect are given.; In the second part, we investigate conditions of finite-time blowing up of solutions for (BQ). The conditions are expressed in terms of the energy of the ground state. In particular, there exists the initial data arbitrarily close to the ground state, whose solutions of (BQ) blow up in a finite time.; In the last part, we consider a generalized KdV equation{dollar}{dollar}(gKdV)qquad usb{lcub}t{rcub} + usb{lcub}xxx{rcub} + usp{lcub}p-1{rcub}usb{lcub}x{rcub}=0,qquad {lcub}rm for{rcub} t > 0,xin{lcub}bf R{rcub}{dollar}{dollar}For {dollar}2le p < 5,{dollar} with some initial conditions, we obtain the asymptotic stability of solitary wave for (gKdV). |
