Swinging Waves in the Ablowitz-Ladik Equation

Abstract

We construct a novel family of exact cnoidal wave and soliton solutions of the focusing and defocusing Ablowitz-Ladik equations. Unlike cnoidal waves that were obtained by earlier authors, the phase variable of the new solutions exhibits a nonlinear dependence on time and site number; the wave ``swings". Our approach hinges on the existence of a two-point map governing the absolute value of the complex field; this map gives rise to standing waves centred arbitrarily relative to the lattice sites. Having derived stationary solutions, we use these as a basis for constructing waves with nonzero velocity. The localised members of the new family comprise dark solitons with the nontrivial asymptotic behaviour. We identify periodic and quasiperiodic patterns and establish an explicit quantisation rule for the velocity of the wave circulating in a closed loop of $N$ sites.

Figures (4)

• Figure 1: The modulus (a,c) and nonlinear phase (b,d) of periodic solutions to the stationary Ablowitz-Ladik equation \ref{['A2']} with $\sigma=1$ (top) and $\sigma=-1$ (bottom). Parameter values: $N=60,\;m=0.9,\;\mu=6K(m)/N$. The background amplitude $A$ is chosen to satisfy the periodicity condition $\theta_N-\theta_0=2\pi$ for $\sigma=1$ and $\theta_N-\theta_0=6 \pi$ for $\sigma=-1$. (Equation \ref{['Y53']} of section \ref{['stanqua']} gives, approximately, $A=A_1= 2.003\times 10^{-3}$ for $\sigma=1$ and $A=A_3=1.080\times 10^{-1}$ for $\sigma=-1$. These amplitude values occur as the horizontal coordinates of the blue blobs in Fig \ref{['Aw']} (a) and (b), respectively.)
• Figure 2: Parameter domain of the standing wave solutions on the $(m,A)$ plane, with the fixed $\mu$ and $x_0$, in the focusing (a) and defocusing (b) case. In both cases we chose $\mu> \pi$ and let $x_0=0$. (The diagrams with $\mu< \pi$ look similar, the only difference being that $m_0=0$.) In the focusing situation (a), the upper boundary is given by $A=(1-m)\frac{\text{sn}^2(\mu,m)}{\text{cn}^2(\mu,m)}$. In the defocusing case (b), the domain is bounded from below by $A=m\text{sn}^2(\mu,m)$. Solutions indicated along the boundaries of the parameter domain are either real standing waves documented in section \ref{['sec:constant_phase']} (equations \ref{['F10']}, \ref{['F9']}, \ref{['F8']}) or linear-phase solutions with a constant modulus. The family of stationary dark solitons found along the $m=1$ boundary of the tinted domain in (b) is obtained in section \ref{['solitons']}. The notation $\mathrm{tn} (\mu,m)$ stands for $\text{sn} (\mu,m) / \text{cn} (\mu,m)$ and $\mathrm{sd} (\mu,m)$ denotes $\text{sn} (\mu,m) / \text{dn} (\mu,m)$.
• Figure 3: Nonlinear-phase wavetrains of the focusing (a,b) and defocusing (c,d) Ablowitz-Ladik equation \ref{['A1']}. In both cases we have chosen $I_1>0$ and $\omega>0$, with $N=60,\;m=0.9,\;\mu=6K(m)/N$, and $k=3\pi/10$. The same amplitude values as in Fig \ref{['fig:stat_foc_defoc']} are used: $A=2.003\times 10^{-3}$ for $\sigma=1$ and $A=1.080\times 10^{-1}$ for $\sigma=-1$. (Note that the resulting winding numbers are not the same as the winding numbers in Fig \ref{['fig:stat_foc_defoc']}. The wavetrains in (a,b) satisfy $\Theta_N-\Theta_0=20 \pi$; those in (c,d) have $\Theta_N-\Theta_0=24 \pi$.) The time interval between the phase rolls (panels (b) and (d)) exhibits periodic shortening; the wave "swings".
• Figure 4: The right-hand side of equation \ref{['Y53']} for a fixed $m$ and varied $A$. The left panel pertains to $\sigma=1$ and the right one to $\sigma=-1$; in either case $N=60$, $m=0.9$ and $\ell=3$. Intersections of the $w(A)$ curve with the horizontal gridlines (corresponding to the integer values of $w$) give the quantised standing wave amplitudes, $A_w$. Thus, the horizontal coordinate of the blue blob in (a) is $A_1$ and the coordinate of a similar blob in (b) is $A_3$. The notation $m^\prime \mathrm{tn}^2 \mu$ in the left panel stands for $(1-m) \text{sn}^2 (\mu_\ell, m) / \text{cn}^2 (\mu_\ell, m)$. In the right panel, $m \text{sn}^2 \mu= m \text{sn}^2(\mu_\ell, m)$.