Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

Abstract

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations $u_{tt}-c_s^{\mathrm{2}}{u}_{xx}=\beta (|E{|}^{\mathrm{2}}{)}_{xx},\mathrm{}i{E}_t+\alpha E_{xx}-{\delta }_{\mathrm{1}}uE+{\delta }_{\mathrm{2}}|E{|}^{\mathrm{2}}E+{\delta }_{\mathrm{3}}|E{|}^{\mathrm{4}}E=\mathrm{0},$ where $\alpha ,\beta ,{\delta }_{\mathrm{1}},{\delta }_{\mathrm{2}},{\delta }_{\mathrm{3}}$, 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.