Planar and radial kinks in nonlinear Klein-Gordon models: Existence, stability, and dynamics

We consider effectively one-dimensional planar and radial kinks in two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon model and the ${\varphi }^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schrödinger equations, and we study the transverse stability of these kinks. This enables us to characterize one-dimensional planar kinks as solitonic filaments, whose stationary states and corresponding spectral stability can be characterized not only in the homogeneous case, but also in the presence of external potentials. Beyond that, the full nonlinear (transverse) dynamics of such filaments are described using the reduced, one-dimensional, adiabatic invariant formulation. For radial kinks, this approach confirms their azimuthal stability. It also predicts the possibility of creating stationary and stable ringlike kinks. In all cases, we corroborate the results of our methodology with full numerics on the original sine-Gordon and ${\varphi }^4$ models.

Figure 1

Top view of a Josephson junction corresponding to the 2D sG Hamiltonian of Eq. (20). The region OCDEFO is the junction area where the oxide layer is thin, enabling Josephson coupling. In the two surrounding passive regions ABCOA and OFGHO, labeled P, the oxide layer is thicker so that no tunneling is present.

Figure 2

The top panels show examples of the sG kinks with a radial external potential of the form $V_{\mathrm{ext}}\left(r\right)=A\mathrm{sech}\left(r\right)$. The left panels are for $A=0$ (i.e., in the absence of the potential for a “standard” 2D sG model), while the right panels are for $A=-4$. The bottom panels show the eigenvalues associated with the kinks in the linearized stability analysis. In the bottom left panel, the numerically computed eigenvalues from the full system (33) (blue circles) are found in excellent agreement with the theory (red stars), see Eq. (18) with $A=0$, namely $\lambda =ik\pi /L_y$.

Figure 3

Exactly the same panels as with Fig. 2, and with exactly the same potential [and for the cases $A=0$ (left) and $A=-4$ (right)], but now for the ${\varphi }^4$ model, to illustrate the generic nature of our findings.

Figure 4

An example of the sG model with a longitudinal potential $V_{\mathrm{ext}}\left(x\right)=A{\mathrm{sech}}^2\left(x\right)$ that conforms to the kink symmetry. Importantly, this potential renders the kink immediately unstable. The case example of $A=0.25$ is shown in the top panel for the kink profile, while the bottom shows the corresponding eigenvalues as computed numerically (blue circles) and as calculated by the analytical theory (red stars); see Eq. (18).

Figure 5

Exactly the same panels as with Fig. 4, and with exactly the same potential but now for the ${\varphi }^4$ model, and specifically for $A=0.1$. Theoretical values (red stars) are estimated from Eq. (19).

Figure 6

Stability spectra for the sG model as a function of the amplitude of the longitudinal potential $V_{\mathrm{ext}}\left(x\right)=A{\mathrm{sech}}^2\left(x\right)$. The left, middle, and right panels depict, respectively, ${\lambda }^2, {\lambda }_i=\mathrm{Im}\left(\lambda \right)$, and ${\lambda }_r=\mathrm{Re}\left(\lambda \right)$. The thick colored lines correspond to the numerically computed spectra and the thin black lines correspond to our theoretical predictions given by Eq. (18). Note that for the unstable modes (see the right panel) the different $k$-modes have been identified.

Figure 7

Same as in Fig. 6 but for the ${\varphi }^4$ model. Theoretical predictions are given by Eq. (19).

Figure 8

Unstable eigenfunctions for the sG (left group of panels) and the ${\varphi }^4$ (right group of panels) models corresponding to the steady states shown in Figs. 4 and 5 in the presence of the longitudinal potential $V_{\mathrm{ext}}\left(x\right)=A{\mathrm{sech}}^2\left(x\right)$ for $A=0.25$ and 0.1, respectively. Note that for these parameter values, the steady state for the sG model possesses four unstable eigenvalues ($k=0$, 1, 2, and 3) while the steady state for the ${\varphi }^4$ model has three unstable eigenvalues ($k=0$, 1, and 2).

Figure 9

Dynamics of a sG kink under the longitudinal potential $V_{\mathrm{ext}}\left(x\right)=A{\mathrm{sech}}^2\left(x\right)$ for $A=0.1$ (top panels) and $A=0.25$ (bottom panels) at the indicated times. The background image depicts the field $u(x,y,t)$ while the magenta line depicts the corresponding dynamics using the reduced AI PDE (15) with $P^{\prime }\left(X\right)$ defined in Eq. (16). Both systems are initialized with the same initial condition corresponding to a stationary kink perturbed with the $k=2$ longitudinal mode with amplitude 0.1. Namely, the initial location of the kink is given by $X(y,t=0)=\varepsilon sin(\pi ky/L_y)$ with $\varepsilon =0.1$ and $k=2$. See movie-sG-1 and movie-sG-2 of the Supplemental Material for animations depicting the corresponding dynamics [44].

Figure 10

Same as in Fig. 9 but for $A=0$ (top panels) and $A=0.25$ (bottom panels) for an initial condition corresponding to a stationary kink perturbed with a (linear) combination of the modes $k=2,4,6,8,10$. Specifically, $X(y,t=0)={\sum }_{j=1}^5{\varepsilon }_{j}sin(2\pi jy/L_y+{\phi }_j)$ with ${\varepsilon }_j=0.3, {\phi }_j=(j-1)L_y\pi /10$, and $k=2j$. See movie-sG-3 and movie-sG-4 in the Supplemental Material for animations depicting the corresponding dynamics [44].

Figure 11

Same as in Fig. 9 but for the ${\varphi }^4$ model. See movie-phi4-1 and movie-phi4-2 in the Supplemental Material for animations depicting the corresponding dynamics [44].

Figure 12

Same as in Fig. 10 but for the ${\varphi }^4$ model. See movie-phi4-3 and movie-phi4-4 in the Supplemental Material for animations depicting the corresponding dynamics [44].

Figure 13

Stationary radial kink for the sG model under an external potential of the form $V_{\mathrm{ext}}\left(r\right)=A\mathrm{sech}\left(r\right)$. The top two sets of panels correspond to $A=-6$ (left) and $A=-16$ (right). The top panels show the two-dimensional radial kink solution, while the middle panels confirm that these solutions are dynamically stable by showing their spectrum (residing on the imaginary axis). The bottom panels compare the exact numerical solution (solid blue line) with the theoretical prediction $u_{\mathrm{th}}$ (red dashed line) for such an equilibrium given by a sG kink centered at $R_0$ given by solving Eq. (26). The corresponding theoretical values for the equilibrium radius are $R_0=1.2212$ for $A=-6$ and $R_0=2.4541$ for $A=-16$.

Figure 14

Exactly the same diagnostics as for the sG equation and for the same potential but now for the case of the radial potential in the ${\varphi }^4$ model. The corresponding theoretical values for the equilibrium radius from Eq. (26) are $R_0=1.5565$ for $A=-6$ and $R_0=2.2812$ for $A=-16$.

Figure 15

An example of the oscillatory radial evolution dynamics of the initial condition of Eq. (38) for the radial PDE of Eq. (10) and $V_{\mathrm{ext}}\left(r\right)=-4\mathrm{sech}\left(r\right)$. The left panel shows the $(r,t)$ space-time contour plot illustrating the stable vibrating dynamics of the radial kink. The right panel portrays individual cuts involving the state at $t=0$ (orange dashed) and the result at $t=13$ (yellow dotted; this time is denoted by a vertical line on the left panel). For comparison, the exact stationary, stable radial configuration is depicted by a blue solid line.