Localized mode interactions in $0\text{−}\pi$ Josephson junctions

A long Josephson junction containing regions with a phase shift of $\pi$ is considered. By exploiting the defect modes due to the discontinuities present in the system, it is shown that Josephson junctions with phase shift can be an ideal setting for studying localized mode interactions. A phase-shift configuration acting as a double-well potential is considered and shown to admit mode tunnelings between the wells. When the phase-shift configuration is periodic, it is shown that localized excitations forming bright and dark solitons can be created. Multimode approximations are derived confirming the numerical results.

Figure 1

(Color online) The eigenvalues of the ground state as a function of $a$ for $L=2$. The dashed vertical line indicates the bifurcation point of the nonuniform ground state, where on the left and on the right of the vertical line $u=0$ is stable and unstable, respectively. The dashed lines show the eigenvalues of the uniform solution in its instability region. The inset presents a ground state when $L=2$ and $a=1.65$.

Figure 2

(Color online) Panel (a) shows the time dynamics of an initially localized excitation in the left well with $L=2$ and $a=1$. In panel (b), the oscillation amplitudes of the dynamics presented in panel (a) are plotted against time. Panel (c) depicts the oscillation amplitude on top of a nonzero background with a small initial amplitude for a double well with $L=2$ and $a=1.65$. Coupling coefficients appearing in Eq. (8) as functions of $a$ for $L=2$ are shown in panel (d). In calculating the coefficients, the eigenfunctions have been normalized to ${\Vert {\Phi }_{\pm }\Vert }_{\infty }=1$. Approximations obtained from the two-mode Eq. (8) are shown as (red) dashed curves in panels (b) and (c).

Figure 3

(Color online) The left panels show the profile of numerically exact bright and dark lattice solitons obtained from the lattice Eq. (16). The insets depict the Floquet multipliers of the solution. The right panels present the corresponding time dynamics of Eq. (1) using the initial condition (15) with $A_n\left(0\right)$ shown in the adjacent left panel. All panels have $L=10$, the upper two have $a=0.5$ and the lower $a=1.6$.