Bright Fractional Single and Multi-Solitons in a Prototypical Nonlinear Schr{ö}dinger Paradigm: Existence, Stability and Dynamics

Abstract

In the present work we explore features of single and pairs of solitary waves in a fractional variant of the nonlinear Schr{ö}dinger equation. Motivated by the recent experimental realization of arbitrary fractional exponents, upon quantifying the tail properties of such coherent structures, we detail their destabilization when the fractional exponent $α$ acquires values $α<1$ and showcase how the relevant destabilization is associated with collapse type phenomena. We then turn to in- and out-of-phase pairs of such waveforms and illustrate how they generically exist for arbitrary $α$ when we cross the harmonic limit, i.e., for $α>2$. Importantly, we use the parameter $α$ as a ``bifurcation parameter'' in order to connect the harmonic ($α=2$) and biharmonic ($α=4$) limits. Remarkably, not only do we retrieve the instability of all solitonic pairs in the biharmonic case, but showcase a stabilization feature of particular branches of such multipulses that is {\it unique} to the fractional case and does not arise -- to our knowledge -- for integer multi-pulse settings. We explain systematically this stabilization via spectral analysis and expand upon the implications of our results for the potential observability of fractional multipulse solitary waves.

Figures (16)

• Figure 1: Right tails of the solitary wave solutions for various values of $\alpha$ in a linear scale (top) and a semilog scale (bottom).
• Figure 2: Tails of a stationary solution, and fitted curves for $\alpha=3.99995$. The stationary solution is a black dash-dot line, the oscillating part of the tail (the "near tail") is approximated by $\phi(x)=1.618\,{\mathrm{e}}^{-0.7085\,x} \,\sin \left(0.7066\,x+0.4376\right)$ (magenta curve) and the non-oscillating (power law) part (the "far tail") is approximated by $\phi(x)=\frac{0.001886}{x^{4.998} }$ (green curve). The left panel is presented in a loglog plot and the right one in a semilog plot.
• Figure 3: Single soliton plots and spectral plots: $\alpha=0.9$ for panels (a) and (b) and $\alpha=1.3$ for panels (c) and (d). Eigenvalue plots include only the positive imaginary axis, as the negative part is symmetric about the real axis. Isolated non-zero real valued eigenvalues in (b) are given by $\lambda=\pm 0.9593$ and isolated non-zero imaginary eigenvalues in (d) are $\lambda=\pm 0.8977i$. Arrows indicate the motion of the eigenvalues as $\alpha$ increases. The eigenvalues that are on the $x$-axis for $\alpha=0.9$ pass through the origin and switch to the imaginary axis at $\alpha=1.3$; the switch occurs at $\alpha=1$ (as indicated in Figure \ref{['alphaSquared']}).
• Figure 4: $\lambda^2$ as a function of $\alpha$ for the isolated (i.e., point spectrum) eigenvalue pair $\pm\lambda$ that depends on $\alpha$, for the single soliton state (as shown in Figure \ref{['fig_spectralOneSoliton']}). We see that the pair of real eigenvalues in Figure \ref{['fig_spectralOneSoliton']} first moves toward the origin, then moves away from the origin on the imaginary axis. Thus, $\lambda^2$ decreases towards zero for real values ($\alpha \le 1$), then smoothly transition to negative (and decreasing) for imaginary values ($\alpha>1$). This is in line with the expected fundamental solution instability within this model for $\alpha < d$ (where $d$ is the dimension of the operator).
• Figure 5: Eigenvectors and dynamical evolution of a single soliton for $\alpha=0.9$ and real eigenvalues $\lambda=\pm0.956$; top row is $\lambda=0.956$, bottom row $\lambda=-0.956$. Panels (a), (b), (f), (g) are real part of eigenvector (left) and imaginary part of eigenvector (right). Panels (c)-(e) and (h)-(j) are the results of a dynamic simulation using as initial condition "steady state + $\varepsilon \times$eigenvector" for the presented eigenvector. The surface plots of $|u(x,t)|$ for $\varepsilon=10^{-6}$ (panels (c) and (h)) and for $\varepsilon=-10^{-6}$ (panels (d) and (i)) are shown, as well as the projection of $u(x,t)-u(x,0)\exp(it)$ onto the given eigenvector (panels (d) and (j)) for both $\varepsilon=10^{-6}$ (in blue) and $\varepsilon=-10^{-6}$ (in red).

Theorems & Definitions (1)

• Definition 1