Bright soliton interactions in the variable coefficient Fokas-Lenells equation, Conservation laws, Modulation instability and Soliton tunneling

TL;DR

The paper analyzes bright soliton dynamics in a fiber model described by a variable-coefficient Fokas-Lenells equation with time-dependent dispersion, nonlinearity, and gain/loss. It develops a gauge-transformed vcFLE, establishes its Lax pair and conservation laws, and analyzes modulation instability to characterize stability on a generalized plane wave. Using a nonstandard Hirota bilinearization with an auxiliary function, it derives explicit one- and two-soliton solutions and provides a scheme for arbitrary N-soliton solutions, then studies soliton interactions, acceleration/retardation under time-varying coefficients, and nonlinear tunneling through dispersion or nonlinearity barriers. The results reveal elastic soliton collisions with phase shifts dependent only on soliton parameters, and demonstrate barrier-induced tunneling where solitons preserve shape, offering analytical insights for dispersion-managed systems and potential experimental guidance in ultrafast photonics.

Abstract

We present here a study of the bright soliton dynamics in an inhomogeneous fibre by means of variable coefficient Fokas-Lenells equation with time varying dispersion, nonlinearity and gain/loss parameter. At first, we propose our system that governs the propagation of ultrashort pulses in an inhomogeneous fibre. Secondly, under a suitable gauge transformation, we transform the system into a simplified form of variable coefficient Fokas-Lenells equation. The Lax integrability and conservation laws are exhibited. We also study the stability of the generalised plane wave against small amplitude perturbations. Thereafter, by using a nonstandard Hirota bilinearization method with the help of a suitable auxiliary function, we obtain the bright one soliton, two soliton and provide a scheme for obtaining N-bright soliton solutions. The elastic collision dynamics of the two solitons is studied using asymptotic analysis. We also investigate the soliton acceleration/retardation under a suitable choice of dispersion and nonlinearity coefficients. Finally, the dramatic effect of the nonlinear tunnelling of the bright one and two-soliton is also studied under some Gaussian dispersion or nonlinearity.

Figures (5)

• Figure 1: (a) is the 3D plot of 1SS with $p_1=1+i$, $\alpha_1=5+0.5i$. (b) is the 3D plot of 2SS with $p_1=1+i$, $p_2=2+2i$, $\alpha_1=5+0.5i$, and $\alpha_2=-5-0.7i$.
• Figure 2: (a) shows the periodically accelerated/retarded 1SS when $D(t)=e^{\sigma t}\cos(\phi t)$, $R(t)=e^{\sigma t}\cos(\phi t)$, $\Gamma(t)=0$, where $\sigma=0.1$, $\phi=\pi/6$, with $p_1=1+0.1i$, $\alpha_1=1+0.1i$. (b) shows the elastic interaction between 2SS when $D(t)=e^{\sigma t}\cos(\phi t)$, $R(t)=e^{\sigma t}\cos(\phi t)$, $\Gamma(t)=0$, where $\sigma=0.1$, $\phi=\pi/6$, with $p_1=1+i$, $p_2=3+2i$, $\alpha_1=5+0.5i$, $\alpha_2=-5-0.5i$. The corresponding density plots are shown in (c), (d) respectively under the same parametric values.
• Figure 3: The 3D plot of (a) a 1SS tunneling through the dispersion barrier $D(t)=1+e^{-0.1t^2}$, $R(t)=1$, and $\Gamma(t)=-\frac{0.1 t e^{-0.1t^2}}{1+ e^{-0.1 t^2}}$, with $p_1=1+i$, $\alpha_1=1+2i$. In (b) we show the tunneling through nonlinear barrier where $R(t)=1+e^{-0.1t^2}$, $D(t)=1$, and $\Gamma(t)=\frac{0.1 t e^{-0.1t^2}}{1+ e^{-0.1 t^2}}$, $p_1=1+i$, $\alpha_1=1+2i$. Plots (c), (d) show the density distributions of the corresponding 3D plots (a), (b) respectively under the same parametric values.
• Figure 4: The 3D plot of (a) 2SS tunneling through the dispersion barrier $D(t)=1+e^{-0.1t^2}$, $R(t)=1$, and $\Gamma(t)=-\frac{0.1 t e^{-0.1 t^2}}{e^{-0.1 t^2}+1}$, with $p_1=1.5+1.5i$, $p_2=2+2i$, $\alpha_1=1+0.5i$, $\alpha_2=-1-0.5i$. In (b) we show the tunneling through the nonlinear barrier $R(t)=1+e^{-0.1t^2}$, $D(t)=1$, and $\Gamma(t)=\frac{0.1 t e^{-0.1 t^2}}{e^{-0.1 t^2}+1}$, with $p_1=1.5+1.5i$, $p_2=2+2i$, $\alpha_1=1+0.5i$, $\alpha_2=-1-0.5i$. Plots (c), (d) show the density distribution of the corresponding 3D plots (a), (b) respectively under the same parametric values.
• Figure 5: 3D plot of (a) a 1SS tunneling through double dispersion barrier $D(t)=0.5+e^{-0.5(t-6)^2}+e^{-0.5(t+6)^2}$, $R(t)=1$, with $p_1=1+i$, $\alpha_1=5$, and (b) shows the tunneling of 1SS through double nonlinear barrier $R(t)=0.5+e^{-0.5(t-6)^2}+ e^{-0.5(t+6)^2}$, $D(t)=1$, $p_1=1+i$, $\alpha_1=5$. In (c) and (d) we show the 2SS tunneling through dispersion $D(t)=0.5+0.5e^{-0.5(t-6)^2}+0.5e^{-0.5(t+6)^2}$, $R(t)=1$ and nonlinearity barrier $R(t)=0.5+0.5e^{-0.5(t-6)^2}+0.5 e^{-0.5(t+6)^2}$, $D(t)=1$ respectively with the same parametric values as $p_1=1+i$, $p_2=2+2i$, $\alpha_1=5+0.5i$, $\alpha_2=-5-0.7i$. The loss and gain coefficients for (a) and (b) are $\Gamma(t)=\frac{- e^{-0.5 (t-6)^2} (t-6)- e^{-0.5 (t+6)^2} (t+6)}{2 \left(0.5\, +e^{-0.5 (t-6)^2}+e^{-0.5 (t+6)^2}\right)}$ and $\Gamma(t)=\frac{ e^{-0.5 (t-6)^2} (t-6)+ e^{-0.5 (t+6)^2} (t+6)}{2 \left(0.5\, +e^{-0.5 (t-6)^2}+e^{-0.5 (t+6)^2}\right)}$, respectively. The loss and gain coefficients for (c) and (d) are $\Gamma(t)=\frac{-0.5 e^{-0.5 (t-6)^2} (t-6)-0.5 e^{-0.5 (t+6)^2} (t+6)}{2 \left(0.5\, +0.5 e^{-0.5 (t-6)^2}+0.5e^{-0.5 (t+6)^2}\right)}$ and $\Gamma(t)=\frac{0.5 e^{-0.5 (t-6)^2} (t-6)+0.5 e^{-0.5 (t+6)^2} (t+6)}{2 \left(0.5\, +0.5e^{-0.5 (t-6)^2}+0.5e^{-0.5 (t+6)^2}\right)}$, respectively.