Alfvén solitons and generalized Darboux transformation for a variable-coefficient derivative nonlinear Schrödinger equation in an inhomogeneous plasma

Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a variable-coefficient derivative nonlinear Schrödinger equation describing the propagation of the nonlinear Alfvén waves in an inhomogeneous plasma. Based on the existing Lax pair, with respect to the left polarized Alfvén wave, the $N$-fold generalized Darboux transformation, rational one/two Alfvén soliton solutions and mixed two Alfvén soliton solutions are derived, where $N=1,2,3\dots$. For those Alfvén solitons, we find that (1) the solitonic width cannot be affected by the dispersion coefficient $h\left(\tau \right)$ and the loss/gain coefficient $f\left(\tau \right)$, where $\tau$ is the stretched space variable; the solitonic amplitude grows (or decays) in the exponential rate $\exp [-\int f(\tau \left)d\tau \right]$; the solitonic amplitude cannot be affected by $h\left(\tau \right)$; the solitonic velocity and trajectory are related to $h\left(\tau \right)$, while they cannot be affected by $f\left(\tau \right)$; (2) the rational two Alfvén solitons experience no phase shift after the interaction; (3) the modulus square of mixed two Alfvén solitons can be decomposed into two single Alfvén solitons with the $\int h\left(\tau \right)d\tau$-dependent phase shifts; (4) the collapse of the rational one Alfvén soliton is displayed.