Floquet solitons in square lattices: Existence, stability, and dynamics

In the present work, we revisit a recently proposed and experimentally realized topological two-dimensional lattice with periodically time-dependent interactions. We identify the fundamental solitons, previously observed in experiments and direct numerical simulations, as exact, exponentially localized, periodic in time solutions. This is done for a variety of phase-shift angles of the central nodes upon an oscillation period of the coupling strength. Subsequently, we perform a systematic Floquet stability analysis of the relevant structures. We analyze both their point and their continuous spectrum and find that the solutions are generically stable, aside from the possible emergence of complex quartets due to the collision of bands of continuous spectrum. The relevant instabilities become weaker as the lattice size gets larger. Finally, we also consider multisoliton analogs of these Floquet states, inspired by the corresponding discrete nonlinear Schrödinger (DNLS) lattice. When exciting initially multiple sites in phase, we find that the solutions reflect the instability of their DNLS multi-soliton counterparts, while for configurations with multiple excited sites in alternating phases, the Floquet states are spectrally stable, again analogously to their DNLS counterparts.

Figure 1

(a) Drawing of $z$-dependent nearest-neighbor couplings. (b) Coupling functions $J_j\left(z\right)$ over one period of the coupling for the choice of the period $T=2\pi$ with steepness parameter $k=20$.

Figure 2

(a) Color map of the ${log}_{10}$ max intensity at each site of the fundamental breather over one period. (b) Intensity of the solution at the four central sites of the fundamental breather over one period $T$; the four sites are labeled in (a). (c) Floquet spectrum of the fundamental breather, with top inset showing the four isolated modes associated with the point spectrum of the solution, and bottom inset detailing the spectrum near (1,0). (d) Color map of the ${log}_{10}$ intensity of the isolated Floquet mode at $z=0$ of the fundamental breather solution corresponding to $\lambda =0.5953+0.8035i$; intensity maps of the other isolated Floquet modes are similar. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 3

(a) Floquet spectrum of the fundamental breather solution with inset showing the isolated modes. (b) Floquet spectrum of background state. $20\times 20$ lattice, coupling constant $C=1.05$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 4

(a) Norm of solution at $z=0$ vs. $C$, representing the continuation of the fundamental solutions as a function of the coupling strength, for phase shifts $\theta =\pi ,\pi /2$. (b) Continuous spectrum angle $\alpha$ vs $C$, for phase shifts $\theta =\pi ,\pi /2,\pi /3,\pi /4$. This is periodic in $C$, with period $\omega =2\pi /T$. $20\times 20$ lattice, coupling period $T=2\pi$, steepness factor $k=10$.

Figure 5

(a)–(b) Floquet spectrum of the fundamental breather for (a) $C=1.0625$ and (b) $C=1.5$. (c) Maximum absolute value of Floquet multiplier $\lambda$ vs $C$ for increasing grid size. (d) Log of the maximum absolute difference between the perturbed solution $u\left(z\right)$ and the fundamental breather $\varphi \left(z\right)$; initial condition for perturbation is $\varphi \left(0\right)+\epsilon v$, where $\epsilon ={10}^{-10}$ and $v$ is the eigenfunction corresponding to the largest Floquet multiplier ($\left|\lambda \right|=1.033$ for $C=1.25$). The line represents the least-squares regression line of the early growth stage of the dynamics. $20\times 20$ lattice, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 6

(a)–(d) Solution at the four central sites for evolution in $z$ of the perturbed fundamental breather (a) $C=1$, (b) $C=1.0625$, (c) $C=1.125$, and (d) $C=1.2083$. Initial condition is a single excited lattice site with the same intensity as the fundamental breather. Fourth-order Runge-Kutta scheme, step size $\pi /50$. Lattice size $20\times 20$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 7

(a) Maximum absolute value of Floquet multiplier $\lambda$ vs $C$. Inset shows Floquet spectrum of fundamental breather for $C=1.17$. (b) Unstable Floquet eigenmodes for $C=1.17,1.18$, and 1.19. $20\times 20$ lattice, period $T=2\pi$, phase shift $\theta =\pi /2$, steepness factor $k=10$.

Figure 8

(a) Color map of the ${log}_{10}$ intensity of the unstable Floquet mode ($\lambda =3.6748+0.1238i, |\lambda =3.6769|$). (b) Log of the maximum absolute difference between the perturbed solution $u\left(z\right)$ and the fundamental breather $\varphi \left(z\right)$; initial condition for perturbation is $\varphi \left(0\right)+\epsilon v$, where $\epsilon ={10}^{-5}$ and $v$ is the eigenfunction corresponding to the most unstable Floquet eigenmode. The line represents the least squares regression fit of the growth portion of the curve. $20\times 20$ lattice, $C=1.19$, period $T=2\pi$, phase shift $\theta =\pi /2$, steepness factor $k=10$.

Figure 9

(a)–(b) Evolution in $z$ (a) and log power spectrum (b) of central site 1 [see Fig. 2] for the unperturbed fundamental breather on $z$ interval $[0,10\pi ]$. (c)–(d) Evolution in $z$ (c) and log power spectrum (d) of central site 1 for the perturbed fundamental breather from Fig. 8 on the $z$ interval $[250\pi ,500\pi ]$; this corresponds to the flat part of Fig. 8, after the perturbation growth has sufficiently saturated. Only the end of this interval is shown in (c). $20\times 20$ lattice, $C=1.19$, period $T=2\pi$, phase shift $\theta =\pi /2$, steepness factor $k=10$.

Figure 10

(a) Color map of the ${log}_{10}$ max intensity at each site of the diagonal breather over one period. Note that the intensity is the same for the in-phase and out-of-phase breathers. (b) Floquet spectrum of diagonal breather for opposite sites initialized out-of-phase (top) and in-phase (bottom). For the unstable in-phase breather, the largest Floquet multiplier is $\lambda =1.1770$. (c)–(d) Intensity of the solution at the primary sites where the breather is localized. Note that the intensity at site 3 partially overlaps with site 1 in (c) and with site 6 in (d). The intensity in (a), (c), and (d) is the same for in-phase and out-of-phase breathers, although their stability is not. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 11

(a) Color map of the ${log}_{10}$ max intensity at each site of the adjacent breather over one period. (b) Floquet spectrum of the adjacent breather for adjacent sites initialized out-of-phase (top) and in-phase (bottom). For the unstable in-phase breather, the largest Floquet multiplier is $\lambda =32.8933$. (c) Intensity of the solution at the primary sites where the breather is localized. The intensity in (a) and (c) is the same for in-phase and out-of-phase breathers, although their stability is not. (d) Log of maximum absolute difference between the perturbed solution $u\left(z\right)$ and the in-phase adjacent breather $\varphi \left(z\right)$; initial condition for perturbation is $\varphi \left(0\right)+\epsilon v$, where $\epsilon ={10}^{-5}$ and $v$ is the eigenfunction corresponding to the largest Floquet multiplier. The least-squares linear regression line is also shown with slope 0.2783. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 12

(a)–(b) Color map of the ${log}_{10}$ max intensity at each site for the Floquet eigenfunction corresponding to the largest Floquet multiplier for the unstable diagonal breather (a) and the unstable adjacent breather (b). One can note the different spatial structure of the corresponding eigenfunctions. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 13

(a)–(b) Solution for four of the seven central sites at the end of the $z$ interval for the evolution of the perturbed opposite breather with out-of-phase initialization (a) and in-phase initialization (b). Initial condition is two adjacent sites initialized with the same intensity. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.

Figure 14

(a)–(b) Solution for four of the six central sites at the end of the $z$ interval for the evolution of the perturbed adjacent breather with out-of-phase initialization on $[0,{10}^5\pi ]$ (a) and the unperturbed adjacent breather with in-phase initialization on $[0,2000\pi ]$ (b). Initial condition is two adjacent sites initialized with the same intensity. $20\times 20$ lattice, coupling constant $C=1$, period $T=2\pi$, phase shift $\theta =\pi$, steepness factor $k=10$.