Some explicit expressions and interesting bifurcation phenomena for nonlinear waves in generalized Zakharov equations (Q370282)

Summary: Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations \(u_{tt} - c^2_su_{xx} = \beta(|E|^2)_{xx}\), \(iE_t + \alpha E_{xx} - \delta_1uE + \delta_2|E|^2E + \delta_3|E|^4E = 0\) where \(\alpha, \beta, \delta_1, \delta_2, \delta_3\), and \(c_s\) are real parameters, \(E = E(x, t)\) is a complex function, and \(u = u(x, t)\) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.