Narrow Results

In this study, the integrability conditions on mixed derivative nonlinear Schrödinger equations are the focus of this work. A. Rangwala mathematically defined these effects and dubbed this form the Rangwala–Rao equation (${ℛ}{ℛ}$) in 1990. Using innovative soliton wave solutions and their interactions, we hope to better understand how dispersion affects the electric field and pulse propagation in optical fibers. Generalized Khater (GKhat.) provides unique solitary wave solutions to the ${ℛ}{ℛ}$ problem. The pulses’ dynamical behavior through optical fibers is seen in these numerical simulations. The originality of the paper’s conclusions may be seen by contrasting our findings with those of other researchers.

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation u_t + a(1 + bu²)u²u_x + u_xxx = 0.

(i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves.

(ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively.

(iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves.

We also show that the common expressions include many results given by pioneers.