Exact solutions and dynamics of generalized AKNS equations associated with the nonisospectral depending on exponential function

No matter constructing or solving nonlinear evolution equations (NLEEs), it is important and interesting in the field of nonlinear science. In this paper, generalized Ablowitz–Kaup–Newell–Segur (AKNS) equations are constructed and solved exactly. To be specific, the famous AKNS spectral problem is first generalized by embedding a nonisospectral parameter whose varying with time obeys the exponential function of spectral parameter. Based on the generalized AKNS spectral problem and its corresponding time evolution equation, we then derive a generalized AKNS equation with infinite number of terms. Furthermore, exact solutions of the generalized AKNS equations are formulated through the inverse scattering transform method. Finally, in the case of reflectionless potentials, the obtained exact solutions are reduced to explicit n-soliton solutions. It is shown that the dynamical evolutions of such soliton solutions possess not only time-varying speeds and amplitudes but also singular points in the process of propagations.