Inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

An inverse scattering transform for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions is derived by introducing an affine parameter to avoid constructing Riemann sheets. A one-soliton solution simpler than that in the literature is obtained, which is a breather and degenerates to a bright or dark soliton as the discrete eigenvalue becomes purely imaginary. The solution is mapped to that of the modified nonlinear Schrödinger equation by a gaugelike transformation, predicting some sub-picosecond solitons in optical fibers.

Figure 1

Integration path for Eq. (55). The radius of the dashed circle is $\rho$. The radius of the large solid circle approaches $\infty$.

Figure 2

Time evolution of the breather, Eq. (82), in three periods, where $\rho =2$, ${\gamma }_1=0.08$, ${\beta }_1=\pi ∕10$, $x_0=0$, and ${\phi }_0=3\pi ∕2$.