Semiclassical Dynamics and Coherent Soliton Ensembles in the Derivative Nonlinear Schrödinger Equation With Periodic Initial Conditions

The semiclassical limit of the derivative nonlinear Schrödinger equation with periodic initial conditions is studied analytically and numerically. The spectrum of the associated scattering problem for a certain class of initial conditions, referred to as periodic single-lobe potentials, is numerically computed, and it is shown that the spectrum becomes confined to the real and imaginary axes or the spectral parameter in the semiclassical limit. A formal Wentzel–Kramers–Brillouin expansion is computed for the scattering eigenfunctions, which allows one to obtain asymptotic expressions for the number, location, and size of the spectral bands and gaps. The results of these calculations suggest that, in the semiclassical limit, all excitations in the spectrum become effective solitons. Finally, the analytical predictions are compared with direct numerical simulations as well as with numerical calculations of the Lax spectrum, and the results are shown to be in excellent agreement.