Semiclassical dynamics and coherent soliton ensembles in the derivative nonlinear Schrödinger equation with periodic initial conditions

Abstract

The semiclassical limit of the derivative nonlinear Schrodinger 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.

Figures (7)

• Figure 1: Left column: Density plots of the amplitude $|q(x,t)|$ of the solution of the DNLS equation in the semi-classical limit with different initial conditions and $\epsilon = 0.05$. Right column: Corresponding snapshots of the solution at $t=2.5$. Top row: the "raised cosine" from equation \ref{['raisedcos']}. Middle row: the "exp sine" potential from equation \ref{['expsin']}. Bottom row: the "dn" potential from equation \ref{['dn']} with elliptic parameter $m=0.8$.
• Figure 2: Plots of the numerically computed Floquet spectrum for the three initial conditions in equations \ref{['ics']}, plus a plot showing the convergence of the spectrum to the real axis as $\epsilon\downarrow0$. Top Left: the raised cosine initial condition with $\epsilon=0.07$. Top Right: the exp sine initial condition with $\epsilon=0.07.$. Bottom left: the dn initial condition with $m=0.8$ and $\epsilon=0.07$. Bottom Right: convergence to the real axis as $\epsilon\downarrow0$. The way deviation from real axis is measured is the magnitude of Im$(\zeta^2)$. Circles: numerically calculated maximum of Im$(\zeta^2)$ over all points in the Lax spectrum as a function of $\epsilon$, for a range of values of $\epsilon$ from $\epsilon=10^{-4}$ to $\epsilon=10^{-7}$. Solid lines: linear regressions of these points in a loglog space.
• Figure 3: Plots of $S_1(\lambda)$ (left) and $S_2(\lambda)$ (right). One of the dashed lines in the $S_1(\lambda)$ plot is at $-e^{-2}$, the upper $\lambda$ boundary of range (iii) for the $\exp(-\sin^2(x))$ potential. The other dashed line corresponds to the upper boundary of range (iii) for the dn potential with $m=0.8.$
• Figure 4: Comparison between the numerically computed Floquet discriminant $\Delta = \mathop{\rm tr}\nolimits M/2$ and the WKB approximation. Top left: raised cosine potential with $\epsilon=0.1$. Top right: exp sine potential with $\epsilon=0.05$. Bottom left: dn potential with $m=0.2$ and $\epsilon=0.037$. Bottom right: dn potential with $m=0.8$ and $\epsilon=0.025$. Red dashed lines: WKB approximation of $\mathop{\rm tr}\nolimits M$ as a function $\lambda$. Blue solid lines: results of numerically integrating the first half of the Lax pair \ref{['e:firstLax']}. Vertical dashed lines: the four ranges of $\lambda$, ordered from left to right as (i), (iii), (iv) and (ii). Note that the raised cosine does not have a range (iv). Horizontal lines: the values $\mathop{\rm tr}\nolimits M=\pm2$, corresponding to the edges of the spectrum. Since the amplitude of the oscillations grows exponentially, following OsborneBergamascoBO2020 the function $f(\Delta)$ was plotted instead of $\Delta$, in order to capture the whole behavior in a single plot, with $f(y)$ defined as $f(y)=y$ for $|y|\leq1$ and $f(y)=\text{sgn}(y)(1+\log_{10}|y|)$ for $|y|>1$.
• Figure 5: Left: the number of bands in the range $(-q^2_\mathrm{max},-q^2_\mathrm{min})$ as predicted by equation \ref{['e:bandcount']} (solid lines) as a function of $\epsilon$ and the numerically computed number of bands from the monodromy matrix (circles). Right: the relative band width of the second spectral band $W_2$ as predicted by equation \ref{['e:relativewidth']} as solid lines for the three examined potentials. The solid dots are numerically computed values for relative width $W_2$ calculated using band edges generated with Floquet-Hill's method.