One-dimensional anyons with competing $\delta$-function and derivative $\delta$-function potentials

We propose an exactly solvable model of one-dimensional anyons with competing $\delta$-function and derivative $\delta$-function interaction potentials. The Bethe ansatz equations are derived in terms of the $N$-particle sector for the quantum anyonic field model of the generalized derivative nonlinear Schr\"{o}dinger equation. This more general anyon model exhibits richer physics than that of the recently studied one-dimensional model of $\delta$-function interacting anyons. We show that the anyonic signature is inextricably related to the velocities of the colliding particles and the pairwise dynamical interaction between particles.