Stable Nonlinear Modes Sustained by Gauge Fields

We reveal the universal effect of gauge fields on the existence, evolution, and stability of solitons in the spinor multidimensional nonlinear Schrödinger equation. Focusing on the two-dimensional case, we show that when gauge field can be split in a pure gauge and a nonpure gauge generating effective potential, the roles of these components in soliton dynamics are different: the localization characteristics of emerging states are determined by the curvature, while pure gauge affects the stability of the modes. Respectively the solutions can be exactly represented as the envelopes which may depend on the pure gauge implicitly through the effective potential, and modulating stationary carrier-mode states, which are independent of the curvature. Our central finding is that nonzero curvature can lead to the existence of unusual modes, in particular, enabling stable localized self-trapped fundamental and vortex-carrying states in media with constant repulsive interactions without additional external confining potentials and even in the expulsive external traps.

Figure 1

Absolute value $|{\psi }_1|$ (upper row) and phase distributions (middle row) for the fundamental soliton [(a),(b)] in the attractive medium, $g=-1$, and self-trapped fundamental [(c),(d)] and vortex [(e),(f)] pseudosolitons in the repulsive medium, $g=1$ (corresponding $\mu$ values are indicated in the plots and by red dots in Fig. 2 below). These are stable and have identical $|{\psi }_1|$ and $|{\psi }_2|$ distributions but different phases. (g)–(i) Decay of the unstable vortex soliton with $\mu =1.05$ in the attractive medium. Numerical results shown here and below were obtained for the original system (1) and for $B=1$.

Figure 2

Norm $N$ (a), maximal amplitude ${\psi }_{1,2}^{\max }$ (b), and integral width $W$ (c) of the fundamental solitons ($\mu <1/2$, $g=-1$) and pseudosolitons ($\mu >1/2$, $g=+1$), and norm of vortex solitons and vortex pseudosolitons (d) vs $\mu$. The $N(\mu )$ curves for families bifurcating from ${\mathrm{\Psi }}_{1,2}^{\mathrm{lin}}$ coincide. The curves in (a) also exactly coincide with those for solitons bifurcating from the linear superposition ${\mathrm{\Psi }}_1^{\mathrm{lin}}+{\mathrm{\Psi }}_2^{\mathrm{lin}}$. They also coincide with families obtained for non-SU(2)-symmetric nonlinearities $g=-1.4$, $\tilde{g}=-0.6$ (at $\mu <1/2$) and $g=1.4$, $\tilde{g}=0.6$ (at $\mu >1/2$). The family bifurcating from ${\mathrm{\Psi }}_1^{\mathrm{lin}}+{\mathrm{\Psi }}_2^{\mathrm{lin}}$, however, is unstable at $\mu \gtrsim 1.05$ [indicated by red curve in (a) coinciding with stable branch for the family bifurcating from ${\mathrm{\Psi }}_1^{\mathrm{lin}}$]. Stable (unstable) families are shown black (red).

Figure 3

Absolute values $|{\psi }_{1,2}|$ for solitons and pseudosolitons bifurcating from the state ${\mathrm{\Psi }}_1^{\mathrm{lin}}+{\mathrm{\Psi }}_2^{\mathrm{lin}}$ in the attractive (a),(b), $\mu =0$, and repulsive (c),(d), $\mu =2$, media, respectively. (e),(f) Decay of the unstable pseudosoliton of this type at $\mu =1.4$ in the repulsive medium.

Figure 4

Norm $N$ (a) and integral width $W$ (c) vs $\mu$ for fundamental soliton in the expulsive parabolic trap with ${\omega }_0=8^{-1/2}$.

Figure 5

A stable pseudosoliton with $\mu =0.1$ in the repulsive medium with expulsive potential. $|{\psi }_1|$ and $|{\psi }_2|$ distributions are identical.