A vectorial Darboux transformation for integrable matrix versions of the Fokas-Lenells equation

TL;DR

The paper develops a vectorial binary Darboux transformation for matrix versions of the first negative Kaup–Newell flow and, via a reduction, for the matrix Fokas–Lenells (FL) equation, enabling exact solution generation from a plane-wave or zero seed. Central to the approach is bidifferential calculus, which yields a Miura-type system linking the KN–1 and FL equations, and a Sylvester/Lyapunov framework that determines the Darboux potential via the matrix $\Omega$. The authors classify a broad spectrum of solutions for the two-component vector FL equation, including multi-solitons, dark/bright solitons, breathers, beating solitons, and rogue waves, obtained in a single non-iterative Darboux step and organized by seed data and spectral configurations. The method provides a nonlinear superposition principle at the level of data, and is extendable to larger matrix sizes and higher-component systems, offering a systematic route to rich integrable dynamics on plane-wave backgrounds with potential applications in optics and nonlinear wave theory.

Abstract

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup-Newell hierarchy. A reduction from the latter system to an integrable matrix version of the Fokas-Lenells equation is then shown to inherit a corresponding vectorial Darboux transformation. Matrix soliton solutions are derived from the trivial seed solution. Furthermore, the Darboux transformation is exploited to determine in a systematic way exact solutions of the two-component vector Fokas-Lenells equation on a plane wave background. This comprises breathers, dark solitons, rogue waves and "beating solitons".

Figures (19)

• Figure 1: Plots of the absolute value of solutions of the scalar FL equation (with $\sigma_1=\sigma_2=1$) obtained with an anti-conjugate pair of spectral parameters from the family in Example \ref{['ex:zero_seed_n=2_anti-conjugate_gammas']}. We set $\gamma = 1+\mathrm{i}$, $a_1 = (1,1)^T$, $a_2 = (\mathrm{i} \, \gamma,1)^T$ and $c_{12}=1$ (left plot), respectively $c_{12}=50$ (right plot).
• Figure 2: Plots of the absolute value of solutions of the scalar FL equation (with $\sigma_1=\sigma_2=1$) obtained with a $2\times 2$ Jordan block $\Gamma$ with eigenvalue $\gamma = -\mathrm{i}$, and $a_1 = (1, -\frac{1}{2} \mathrm{i})^T$, $a_2 = (1,0)^T$, $\mathrm{Im}(c_{12})=0$, and $\mathrm{Re}(c_{22})=0$, respectively $=50$. Also see Fig. \ref{['fig:scalarFL_2x2Jordan_imag_gamma_2']} for details on the first example.
• Figure 3: Plots of the absolute value of the first solution of the scalar FL equation displayed in Fig. \ref{['fig:scalarFL_2x2Jordan_imag_gamma']}, at $t=-2, -1,0, 1, 3$. Below is a plot of $\mathrm{Re}(u')$ and $\mathrm{Im}(u')$ at $t=0$. The apparent cusp of $\mathrm{Im}(u')$ at $t=0$ is just a sharp maximum.
• Figure 4: Plots of the absolute values of the first, and the real part (the imaginary part is similar) of the second component for an $n=1$ solution from the class in Example \ref{['ex:special_pw_breather']}. The parameters are $\alpha = \|A\| = 2$, $\gamma = 1 + \mathrm{i}$, $c_0=c_1=1$, $c_2=0$, but we replaced $w$ by $-w$. The second plot shows the "beating". The velocity is $v \approx 0.3$, the beating period $T' \approx 1$.
• Figure 5: Plots of the absolute value of the components of an $n=1$ beating soliton solution from the class in Example \ref{['ex:special_pw_beating_solitons']} at $t=-19.2$ (left), $t=0$ (middle) and $t=16.9$ (right). Here we set $A_2=A_1$, $\alpha = 1/(4 A_1^2)$ (so that $\beta=0$ and $w = \sqrt{1-\gamma^2/(16 A_1^4)}$), $A_1 = c_0 = c_1 = 1$, $c_2 =0$ and $\gamma=10$. Comparing the first and the third plot, we observe an exchange of the profiles of the absolute values of the components $u_1'$ and $u_2'$.

Theorems & Definitions (58)

• Proposition 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1
• Theorem 3.2
• proof