One- and two-dimensional solitons under the action of the inverted cubic-quintic nonlinearity

Abstract

The usual cubic-quintic (CQ) nonlinearity is proved to sustain one- and two-dimensional (1D and 2D) broad (flat-top) solitons. In this work, we demonstrate that 1D and 2D soliton families can be supported, in the semi-infinite bandgap (SIBG), by the interplay of a lattice potential and the nonlinearity including self-defocusing cubic and self-focusing quintic terms, with the sign combination inverted with respect to the usual CQ nonlinearity. The families include fundamental and dipole solitons in 1D, and fundamental, quadrupole, and vortex solitons in 2D. The power, shapes, and stability of the solitons are reported. The results are strongly affected by the positions of the solitons in SIBG, the families being unstable very close to or very far from the SIBG's edge. The inverted CQ nonlinearity, considered in this work, sustains sharp 1D and 2D stable solitons, which can be naturally used as bit pixels in photonic data-processing applications.

Figures (11)

• Figure 1: Bandgap spectra, power and amplitude of 1D soliton families. (a,b): The bandgap spectrum produced by the 1D version of the linearized equation (\ref{['NLSES']}) with potential (\ref{['V1D']}), for $V_{0}=1$ (a) and $V_{0}=6$ (b). The spectrum is plotted in the plane of the quasi-momentum $k$ of Bloch modes and propagation constant $b$. The red, blue, and green strips represent the first, second, and third Bloch bands, respectively. Acronyms SIBG and 1stBG stand for the semi-infinite and first bandgaps, respectively. (c,d): The soliton's power $P$ vs. propagation constant $b$ in the SIBG for families of 1D fundamental (c) and dipole (d) solitons in the deep lattice potential, with $V_{0}=6$. Blue and red segments of the curves represent stable and unstable solitons, respectively. (e,f): The amplitude (maximum value of $|U(x)|$) of the families of 1D fundamental (e) and dipole (f) solitons at $V_{0}=6$. The vertical grey areas in panels (c)--(f) stand for the first Bloch band.
• Figure 2: Profiles and eigenvalues of linear stability analysis for 1D soliton families. Profiles of the 1D stable fundamental (a) and dipole (b) soliton found at $b=-0.9$, which correspond, respectively, to labels A1 and B1 in Figs. \ref{['fig1']}(c,d). (c): Eigenvalues produced by the numerical solution of Eq. (\ref{['LSA']}) for the soliton in panel (b). Unstable fundamental and dipole solitons, found at $b=-1.55$, which correspond to labels A2 and B2 in Figs. \ref{['fig1']}(c,d), are plotted in panels (d) and (e), respectively. (f): Eigenvalues $\lambda$ produced by Eq. (\ref{['LSA']}) for the soliton in panel (e). The corresponding solutions of Eq. (\ref{['NLSES']}) are obtained for potential (\ref{['V1D']}) with $V_{0}=6$.
• Figure 3: Perturbed propagations of 1D soliton families. The top row displays the stable perturbed propagation of 1D solitons with $b=-0.9$: (a) the fundamental soliton (which corresponds to point A1 in Fig. \ref{['fig1']}(c)); (b) the dipole soliton (corresponding to point B1 in Fig. \ref{['fig1']}(d)); (c) the tripole soliton; (d) the quadrupole one. The middle row displays the unstable propagation of the solitons with $b=-1.55$: (e) the fundamental soliton (corresponding to point A2 in Fig. \ref{['fig1']}(c); (f) the dipole soliton (corresponding to point B2 in Fig. \ref{['fig1']}(d)); (g) the tripole soliton; (h) the quadrupole one. The bottom row displays the unstable propagation of the solitons with $b=-0.02$: (i) the fundamental soliton (corresponding to point A3 in Fig. \ref{['fig1']}(c); (j) the dipole soliton (corresponding to point B3 in Fig. \ref{['fig1']}(d)); (k) the tripole soliton; (l) the quadrupole one.
• Figure 4: The 2D Bandgap spectra. (a,b): The bandgap spectrum produced by the linear version of the 2D equation (\ref{['NLSES']}) with potential (\ref{['V2D']}), for $V_{0}=1$ (a) and $V_{0}=6$ (b). The spectrum shows the propagation constant $b$ of Bloch modes vs. the components $k_{x}$ and $k_{y}$ of their quasi-momentum. As in Fig. \ref{['fig1']}, acronyms SIBG and 1stBG stand for the semi-infinite and the first bandgaps, respectively. In panel (a), surfaces denote, from top tp bottom, the first, second, third, fourth, and fifth Bloch bands.
• Figure 5: Power and amplitude of 2D soliton families. The curves of the soliton power $P$ versus the propagation constant $b$ in the semi-infinite band gap of the 2D soliton families at $V_{0}=6$: (a) for the fundamental solitons; (b) for the quadrupoles. The amplitude (maximum value of $|U(x,y)|$) of the 2D soliton families at $V_{0}=6$: (c) for the fundamental solitons; (d) for the quadrupoles. The vertical grey areas in panels (a)--(d) stand for the first Bloch band.