Nonlinear Dynamics of Kink Configurations: From Small to Large Kink Collisions

Abstract

This study explores the scattering dynamics of kinks within a nonlinear system governed by a parameterized potential $U_λ(χ)$, examining the distinct behaviors of small and large kinks across a range of $λ$ values and initial velocities. For small kinks, we investigate the critical velocity for separation, the influence of vibrational modes, resonance phenomena, and the conditions under which large kinks emerge from collisions. Our findings reveal that the critical velocity exhibits a non-monotonic dependence on the parameter $λ$, reflecting the evolving stability of small kinks, while the decreasing frequency of vibrational modes with increasing $λ$ diminishes resonance effects, leading to simpler scattering dynamics at higher $λ$. The formation of large kinks from small kink collisions is favored at lower $λ$, where the mass difference between small and large kinks is reduced. Conversely, large kink scattering consistently results in the production of small kinks, with the number of small kink pairs growing as both $λ$ and initial velocity increase, a process driven by energy transfer from the translational modes of large kinks to the potential energy required for small kink creation. The absence of vibrational modes in large kinks contrasts with their presence in small kinks, where such modes give rise to complex phenomena like bion formation and resonance. These results underscore the pivotal role of $λ$ in shaping kink interactions and offer valuable insights into the dynamics of topological defects in nonlinear systems, with potential implications for understanding similar phenomena in condensed matter physics and related fields.

Figures (6)

• Figure 1: (a) The deformed potential $U_\lambda(\chi)$ as defined in Eq. \ref{['pot1']} for different values of $\lambda$. (b) Spatial profiles of large kinks, $\chi_{L,\lambda}(x)$, from Eq. \ref{['largek']} for different values of $\lambda$. (c) Spatial profiles of small kinks, $\chi_{S,\lambda}(x)$, from Eq. \ref{['smallk']} for different values of $\lambda$. (d) Masses of small ($m$) and large ($M$) kinks as functions of $\lambda$, calculated using Eqs. \ref{['smallkinkmass']} and \ref{['largekinkmass']}, respectively.
• Figure 2: (a) The stability potential $v_{S,\lambda}(x)$ for small kinks, as defined in Eq. \ref{['eq:qmp_small']}, for $\lambda = 0, 0.25, 0.50, 0.75, 0.90$. One sees that increasing $\lambda$, the potential well which has the Poeschl-Teller shape, becomes shallower and wider. (b) The stability potential $v_{L,\lambda}(x)$ for large kinks, as defined in Eq. \ref{['eq:qmp_large']}, for the same $\lambda$ values. For $\lambda$ from 0 to 0.54, the wells deepen and narrow, while for $\lambda > 0.54$, they become shallower and wider. We also note a change in the shape of the potential, from the reflectionless modified Poeschl-Teller to the volcano type as $\lambda$ increases towards unit.
• Figure 3: (a) The critical velocity $v_c$ as a function of $\lambda$, showing a peak at $\lambda = 0.54$. (b) The first vibrational mode $\omega_1^2$ of small kinks as a function of $\lambda$, decreasing from 3.938 at $\lambda = 0.09$ to $0.067$ at $\lambda = 0.99$. (c) Final velocity $v_f$ versus initial velocity $v_i$ for $\lambda = 0.07$, showing bion formation ($v_i < 0.15$), small kink separation ($0.15 < v_i < 0.53$), and large kink creation ($v_i > 0.53$).
• Figure 4: Illustration of the small kinks scattering for $\lambda=0.07$ and different values of initial velocities, with initial positions $X_{0k}=-X_{0ak}=-10$ for all cases. (a) At $v_i = 0.1$, the kink and antikink form a bion. (b) At $v_i = 0.2$, the kink and antikink separate after collision. (c) At $v_i = 0.5$, separation occurs with a higher final velocity. (d) At $v_i = 0.6$, large kinks are created.
• Figure 5: Small kink-antikink collision, (a) and (b) correspond to simulations with parameter $\lambda=0.25$, (c) and (d) correspond to $\lambda=0.5$, while (e) and (f) correspond to $\lambda=0.75$. (a), (c), and (e) depict the number of bounces as a function of initial velocity and (b), (d), and (f) present the final velocity for cases with one and two bounces as a function of initial velocity. We are using $X_{0k}=-10$.