On the discrete Kuznetsov–Ma solutions for the defocusing Ablowitz-Ladik equation with large background amplitude

The Ablowitz-Ladik (AL) model:

was introduced in [2, 3] as an integrable spatial discretization of the focusing ($\sigma=-1$) and defocusing ($\sigma=1$) nonlinear Schrödinger (NLS) equation:

to which they reduce as the lattice spacing $h\to 0$ with $nh\to x$.

Besides being used as numerical schemes for their continuous counterparts, the AL equations describe the quantum correlation function of the $XY$-model of spins [21, 24]. When $\sigma=-1$, the equation plays a significant role in the study of anharmonic lattices [35] and lossless nonlinear electric lattices [27], and it is gauge equivalent to an integrable discretization of the Heisenberg spin chain [20]. Additionally, the focusing AL hierarchy describes the integrable motions of a discrete curve on the sphere [17]. Importantly, in addition to its intrinsic value, the AL model provides a springboard towards exploring solutions in the non-integrable, yet more widespread discrete variant of the nonlinear Schrödinger model, namely the so called discrete NLS equation [22]. Indeed, this has been leveraged for solitary wave dynamics in early works, such as, e.g., [11], but also for more complex structures including periodic and rogue waves, e.g., in the more recent work of [33]. This justifies a multi-faceted interest in the periodic (and rogue) wave solutions of the Ablowitz-Ladik model [5], from a theoretical point of view in their own right, but also from a computational and applications point of view, as well.

In this work we are interested in characterizing special solutions of the defocusing AL equation on a nontrivial background. Specifically, we will consider the equation:

with boundary conditions:

Modulo a rescaling of dependent and independent variables ($Q_{n}=hq_{n}$ and $\tau=t/h^{-2}$), the above equation is gauge-equivalent to Eq. (1) (the gauge transformation $Q_{n}\to Q_{n}=e^{-2iQ_{o}^{2}\tau}$ has simply the effect of allowing for constant boundary conditions as $n\to\pm\infty$).

As mentioned above, the AL equations are integrable, and their initial-value problem can be solved by means of the Inverse Scattering Transform (IST). The IST for Eq. (2) was initially developed under the assumption that the background amplitude $Q_{o}$ satisfies a “small norm” condition, $0<Q_{o}<1$ [36, 14, 1]. This condition is actually not just a technical assumption: in [28] it was shown that the defocusing AL equation, which is modulationally stable for $0\leq Q_{o}<1$, becomes unstable if $Q_{o}>1$, and exhibits discrete rogue wave solutions, some of which are regular for all times. This finding sparked renewed interest in the defocusing AL equation on a background. In [29] the IST was generalized to the case of arbitrarily large background $Q_{o}>1$. As a by-product of the IST, explicit solutions were also obtained in [29], which are the discrete analog of the celebrated Kuznetsov–Ma [23, 26], Akhmediev [7, 8], Tajiri-Watanabe [34] and Peregrine solutions [30, 6, 5], and which mimic the corresponding solutions for the focusing AL equation [9, 10, 32, 31].

The recent work [15] further investigated the stability of background solutions of the AL equations in the periodic setting, confirming in particular that, unlike its continuous counterpart, in the defocusing case ($AL_{-}$, in the terminology of [15]) the background solution is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1. A discrete Kuznetsov-Ma (KM) breather solution was presented in [29], and it was observed that the solution could be regular for certain choices of the soliton parameters, but its regularity was not analyzed in detail.

The goal of the present work is to provide a systematic investigation of the conditions on the background and on the spectral parameters (discrete eigenvalue and associated norming constant) that guarantee the KM solution to be non-singular for all $n\in\mathbb{Z}$ and all $\tau\in\mathbb{R}$. Furthermore, a novel KM-type breather solution will be presented which is also regular on the lattice under the same conditions. We also employ the Darboux transformations derived in [15] to analyze Akhmediev breathers, and, by generalizing the construction to the case of KM breathers, we obtain multi-KM breather solutions. In particular, we show that parameters choices exist for which a double KM breather is regular on the lattice. Finally, we corroborate the analytical findings by numerical computations where for the obtained exact solutions, a spectral stability analysis is performed and dynamical evolution of the lattice model with such waveforms as initial data is explored. The latter will often lead to unbounded growth of the solution due to the manifestation of the modulational instability of its background.

Our presentation is structured as follows. First, we provide some relevant background of the model, including its Lax pair and associated discrete spectrum. We then move to the construction of the KM breathers, followed by an analysis of the conditions under which the relevant waveforms are regular. We then proceed to provide a construction of the solutions via Darboux transformations, before exploring the stability and direct numerical dynamics of the associated waveforms. We close with a discussion of the conclusions of our findings and some directions for future research.

The Lax pair for the AL system (2) is given by [2]:

where $v_{n}$ is a $2$-component vector, $z\in\mathbb{C}$ is the scattering parameter, and $Q_{n}(\tau)$ is the AL potential.

If $Q_{n}(\tau)$ satisfies (3) as $n\to\pm\infty$, the asymptotic behavior of the simultaneous solutions of the Lax pair is given by $\lambda^{n}(ir)^{n}e^{i\Omega\tau}$ or $\lambda^{-n}(ir)^{-n}e^{-i\Omega\tau}$, where $\lambda$ as a function of $z$ is defined by

and the dispersion relation is given by:

Clearly, $\lambda=\lambda(z)$ has branching, with 4 branch points on the imaginary axis located at:

Note that $-r-Q_{o}<-1<r-Q_{o}<0<Q_{o}-r<1<Q_{o}+r$, which specifies the order of the branch points.

It is convenient to introduce the variable $\xi=(z+1/z)/2ir$, and define a 2-sheeted Riemann surface cut across the segment $(-1,1)$ on the real $\xi$-axis, with local coordinates on each sheet given as in the left panel of Fig. 1:

such that on Sheet I/II $\lambda(z)=\xi\pm\sqrt{\xi^{2}-1}$. The IST developed in [29] is formulated in terms of a uniformization variable

for which one has

showing that all these quantities, as well as all quantities which are even functions of $\lambda$ and $z$, are meromorphic functions of $\zeta$. It is worth noticing that $|\lambda^{2}(\zeta)|\lessgtr 1$ for $\zeta\in D_{\pm}$, where $D_{\pm}=\left\{\zeta\in\mathbb{C}:\,|\zeta-ir|\lessgtr Q_{o}\right\}$, and $|\lambda(\zeta)|^{2}=1$ for $\zeta\in\Sigma$, where $\Sigma=\left\{\zeta\in\mathbb{C}:\,|\zeta-ir|=Q_{o}\right\}$. We refer the reader to [29] for the details about the IST, while noting that in Appendix B we provide a short errata to correct some typos and small inaccuracies in [29]. For the purposes of this work, it suffices to recall that the solitons/breathers of the AL equation are characterized by special values of the spectral parameter ($z$, or $\zeta$) for which bounded eigenfunctions of the Lax pair exist. In terms of the uniformization variable $\zeta$ such discrete eigenvalues appear in symmetric quartets: $\left\{\zeta_{k},\,\frac{\zeta_{k}^{*}+ir}{ir\zeta_{k}^{*}+1},\,-1/\zeta_{k}^{*},\,\frac{ir\zeta_{k}-1}{\zeta_{k}-ir}\right\}$, with the first two in $D_{-}$ and the last two in $D_{+}$. It is convenient to then group the eigenvalues as follows:

and $\bar{\zeta}_{k}=-1/\zeta_{k}^{*}$ for $k=1,\dots,2J$. As is typical, the discrete eigenvalue specifies the velocity and amplitude of the corresponding soliton/breather. To each discrete eigenvalue $\bar{\zeta}_{k}\in D_{+}$ is associated a norming constant $\bar{C}_{k}$:

which in turn characterizes the soliton/breather position and phase.

The general discrete breather solution of Eq. (2) on the nontrivial background (3) is given in Eq. (89) in [29], and it is an analog of the discrete Tajiri-Watanabe (TW) breather of the focusing AL equation obtained in [31]. The solution is given in terms of a discrete eigenvalue $\bar{\zeta}_{1}\in D_{+}$ and its associated norming constant $\bar{C}_{1}$:

where

According to (11), $\lambda^{2}(\zeta)=\zeta(\zeta-ir)/(ir\zeta-1)$ and

Also, $\bar{\zeta}_{2}=(\bar{\zeta}_{1}^{*}+ir)/(ir\bar{\zeta}_{1}^{*}+1)$ and $\zeta_{1}=-1/\bar{\zeta}_{1}^{*}$, and we recall that $|\lambda^{2}(\zeta)|<1$ for any $\zeta\in D_{+}$ (hence, $\rho>1$). It is also important to point out that $\alpha>1$ for any choice of the soliton parameters $\bar{\zeta}_{1}$ and $\bar{C}_{1}$, which means that in general the TW solution is singular. This singularity, as well as ways to avoid it, is crucial for our considerations that follow next. We refer the reader to [29] for the details of the derivation of Eq. (14), while noting that we corrected a typo in the expressions for $\xi,\eta$ (see also Appendix B).

The discrete analog of the Kuznetsov-Ma (KM) solutions is obtained when the discrete eigenvalue $\bar{\zeta}_{1}\in D_{+}\setminus\left\{0,r\right\}$ is purely imaginary, i.e., $\bar{\zeta}_{1}=i\kappa$ with

where we recall $r=\sqrt{Q_{o}^{2}-1}>0$. In this case, the parameter $\bar{\lambda}_{1}^{2}=\lambda^{2}(\bar{\zeta}_{1})\in\mathbb{R}$, but note that it is not necessarily positive. Specifically, one has

and one can check that $-1/r<-\sqrt{1+r^{2}}+r$ for any $r$, therefore

In [29] the implicit assumption was made that $\bar{\lambda}_{1}^{2}>0$, but in the parametrization $\bar{\lambda}_{1}^{2}=e^{i\phi}/\rho$ one needs to take $\phi=\pi$ if $\bar{\lambda}_{1}^{2}<0$, while indeed $\phi=0$ when $\bar{\lambda}_{1}^{2}>0$. This will not affect the regularity of the solution (see below), but it will give rise to a different solution if one chooses values of $r,\kappa$ for which $\bar{\lambda}_{1}^{2}<0$. Based on the above discussion, we have:

and

Now, considering (15) when $\bar{\zeta}_{1}=i\kappa$, one can immediately see that $\delta_{3}=0$ and

The remaining coefficients (15) in the TW solution reduce to

and

where we have written the norming constant $\bar{C}_{1}=c\,e^{i\chi}$, $c>0$, and the background as $Q_{+}=Q_{o}\,e^{i\theta}\equiv e^{i\theta}\sqrt{1+r^{2}}$, and to simplify the expressions of $\cos\eta_{o},\sin\eta_{o}$ we have taken into account that $\kappa^{2}-2r\kappa-1<0$ for $\kappa$ in the allowed range, and that the remaining factor in $\tilde{A}_{1}+\gamma$ is simply $\bar{\lambda}_{1}^{2}$. Note the additional term $n\phi$ in the expression (23a) of $\eta$ compared to [29], which, we reiterate, is only present if $0<\kappa<r$.

For simplicity, we can assume below that $\theta_{+}=0$, i.e., $Q_{+}=Q_{o}$, and also take the norming constant to be real and positive, i.e., $\chi=0$ and $\bar{C}_{1}=c>0$. Then, if $r,\kappa$ are chosen so that $\bar{\lambda}_{1}^{2}>0$, $\eta_{o}=\pi$ and from (14) we obtain

Conversely, still assuming that the phases of $Q_{+}$ and $\bar{C}_{1}$ are zero, if $r,\kappa$ are chosen so that $\bar{\lambda}_{1}^{2}<0$, then $\eta_{o}=0$, and (14) yields

Eq. (24) was already obtained in [29], and we only corrected a typo. In the remainder of this work, we will refer to this solution as KM1. On the other hand, to the best of our knowledge Eq. (25) is a novel KM breather solution of the defocusing AL equation with background $Q_{o}>1$, which we will refer to as KM2. Some prototypical case examples of both families of solutions are shown in Fig. 2. There, it can be seen that the examples of KM2 breathers, carrying the staggered spatial structure evident in Eq. (25), feature peaks that bear an alternating structure in time, contrary to what is observed for the KM1 breathers.

Note that $\alpha>1$ for any $\kappa\neq r$ [see Eq. (23b)], which means that the denominators in Eq. (24) and Eq. (25) will generically have zeros, and hence the solution will be singular unless, for a given $r$ (specified by $Q_{o}$), one can choose the soliton parameters $\kappa$ and $\bar{C}_{1}$ so that no integer falls in the interval where $\cosh(n\log\rho-\xi_{o})\leq\alpha$.

Indeed, the minimum value for the $\cosh$ term in the denominator is 1, and it is achieved when $x=\xi_{o}/\log\rho$, where $\xi_{o}=\log\sqrt{\delta_{4}}$. The interval where $\cosh$ is less than or equal to $\alpha$ is given by

and the width of this interval is $2\mathop{\rm arccosh}\nolimits\alpha/\log\rho$, which is independent of the norming constant $\bar{C}_{1}$, while the center of the interval, which is the center of the soliton, $\xi_{o}/\log\rho$, is determined by $|\bar{C}_{1}|$ via $\delta_{4}$.

A necessary condition in order for the above interval (26) not to contain any integer is that the width of the interval be strictly less than 1, and hence a necessary condition for regularity of the solution is that

The latter is an inequality for the two parameters $r,\kappa$, and one can find pairs of values $r_{o},\kappa_{o}$ with $r_{o}>0$ and $-\sqrt{r_{o}^{2}+1}+r<\kappa_{o}<\sqrt{r_{o}^{2}+1}+r_{o}$, $\kappa_{o}\neq 0,r_{o}$ for which this necessary condition for regularity is satisfied; see, in particular, the representation of the left panel of Fig. 3. Note that the set of loci of $(r,\kappa)$ for which (27) is satisfied is certainly not empty, since $\log\rho$ in the second term is positive (recall $\rho>1$), and it becomes arbitrarily large as $\kappa\to r$ from either side. The contour plot of $f(r,\kappa)$ is shown on the left panel in Fig. 3. A stronger regularity condition, i.e., requiring the width of the interval where $\cosh(n\log\rho-\xi_{o})-\alpha\leq 0$ to be smaller than 1/2, can be imposed to ensure a larger lower bound on the denominator of the solution. The corresponding contour plot is shown on the right panel in Fig. 3.

Once $r_{o},\kappa_{o}$ are chosen to satisfy (27) so that the interval has width less than 1, we need to choose the center of the interval so that it does not contain any integer. We can fix $\bar{n}\in\mathbb{Z}$ arbitrarily, and place the center the soliton in the interval $(\bar{n},\bar{n}+1)$ by requiring:

where the half width of the interval, $\Delta$, is

[Note the width of the above interval is exactly $1-2\Delta>0$.] The smaller the value of $\Delta$, the further away from the singularities one can choose the closest integers, and therefore the smaller the value of $Q_{n}(\tau)$.

Recalling that $\xi_{o}=\log\sqrt{\delta_{4}}$, the inequality (28) can be written as

where we have used the definition of $\sqrt{\delta_{4}}$ in terms of $\bar{C}_{1}$ and

Any choice of $c=|\bar{C}_{1}|$ in the above intervals guarantees that for the corresponding values of $r,\kappa$ the denominator of $Q_{n}(\tau)$ will not vanish at the integers. For instance, if one sets $\bar{n}=0$, then $n=0$ and $n=1$ are the integers where the denominator will be the smallest, and one would want to choose $c$ in each range so that $n=0$ and $n=1$ are equidistant from the soliton center. Recall that the soliton center is $\xi_{o}/\log\rho$, and the choice of $c$ in the above intervals guarantees that $0<\Delta<\xi_{o}/\log\rho<1-\Delta<1$. The plots in Fig. 4 show $\cosh{(n\log\rho-\xi_{o})}-\alpha$ at the integer values $n=0,1$. It is clear that the optimal choice for $c$ in each case is the value at the intersection of the two curves.

Note that for any fixed $r$ the frequency, $\bar{\Omega}_{1}$, and $\log\rho$, which controls the width of the breather in each period, become singular at $\kappa=0,r$ (cf Fig 5). As it is clear from the contour plots in Fig 3, solutions are regular in left and right neighborhoods of $\kappa=0,r$, but in order to obtain solutions whose frequency is not too large and width not too small, one wants to keep sufficiently far from those singular values. On the other hand, the further away one moves from $\kappa=0,r$ in the intervals where the solution remains bounded for all $n\in\mathbb{Z}$, the closer $\Delta$ is to its limiting value of 1/2, and correspondingly the larger the solution is going to be. So a careful balance is required in the choice of parameters in order to obtain more meaningful KM solutions which are regular for all $n\in\mathbb{Z}$, $\tau\in\mathbb{R}$. In Table 1 we provide some such examples, for various values of $r>0$. For the parameters in the Table, we have chosen $\bar{n}=0$ for simplicity.

A Darboux transformation is meant to take a known solution of an integrable PDE or ODE into a new solution. In [15], the Darboux transformation for the focusing and defocusing AL equations (AL_±, respectively) on a nontrivial background were obtained to derive and study the stability of periodic Akhmediev breather solutions. The construction of the Darboux matrix (DM) for the Lax pair (4a) follows the one proposed in [18, 25] for the case of rapidly decaying solutions (see also [13], where the Darboux transformation is used to obtain rogue wave solutions of the focusing AL equation superimposed to standing periodic waves). The DM $D_{n}(z,\tau)$ takes a fundamental solution $\Psi_{n}^{[0]}(z,\tau)$ of the Lax pair (4a) with potential $Q_{n}^{[0]}(\tau)$ into a new fundamental solution $\Psi_{n}(z,\tau)=D_{n}(z,\tau)\Psi_{n}^{[0]}$ corresponding to a potential $Q_{n}(\tau)$. The requirement that both $\Psi_{n}^{[0]}$ and $\Psi_{n}$ satisfy the equations in the Lax pair implies that the DM is

Furthermore, the symmetries of the Lax pair and of the boundary conditions require that

The details of the construction of the DM are given in [15]; in Appendix B we report, in our notations, what is needed below in order to reproduce the KM solutions presented in Sec 3, as well as some examples of double KM breathers.

Below, we will apply the DM to the case of KM breathers. In order to compare the solutions in Sec 3 with the ones one can derive using the DM, we need to take into account that the latter is given in terms of the original spectral parameter $z$ (and $\lambda(z)$, as defined in (5)), while the former is expressed in terms of the uniform variable $\zeta$ (cf. Fig 6). For this reason, we start here by converting the purely imaginary discrete eigenvalues $\bar{\zeta}_{j}=i\kappa_{j}\in D_{+}$ into the corresponding values for $z_{j}$ and $\lambda_{j}$.

Consider purely imaginary values of $\zeta=i\kappa$, for all $\kappa$ in the interval: $(-Q_{o}+r,Q_{o}+r)\setminus\left\{0,r\right\}$. It follows from (11) that

showing that $z^{2}>0$ in the interval $0<\kappa<r$ where $\lambda^{2}<0$, whereas $z^{2}<0$ for $-Q_{o}+r<\kappa<0$ and $r<\kappa<r+\sqrt{1+r^{2}}$ where $\lambda^{2}>0$. This implies that when $z\in\mathbb{R}$ then $\lambda\in i\mathbb{R}$ and vice versa. Note also that the third equation above allows one to determine the relative signs of $z$ and $\lambda$.

Specifically, we obtain the following.

For $0<\kappa<r$:

$$z=\pm\mu,\qquad\lambda=\pm i\kappa\mu\qquad\text{with }\mu=\sqrt{{\frac{r-\kappa}{\kappa(1+r\kappa)}}}\in\mathbb{R}^{+}\,.$$

(33)

Since $\mu\to 0^{+}$ as $\kappa\to r^{-}$ and $\mu\to+\infty$ as $\kappa\to 0^{+}$, considering both signs in $z=\pm\mu$ we see that the entire real $z$-axis corresponds to the segment $i[0,r]$ in the $\zeta$-plane.

For $-Q_{o}+r<\kappa<0$:

$$z=\pm i\mu\,,\qquad\lambda=\pm|\kappa|\mu\qquad\text{with }\mu=\sqrt{\frac{r-\kappa}{|\kappa|(1+r\kappa)}}\in\mathbb{R}^{+}.$$

(34)

As $\kappa\to(-Q_{o}+r)^{+}$, one has $\mu\to Q_{o}+r$, and $\mu\to+\infty$ as $\kappa\to 0^{-}$, so in this case, considering both signs for $z=\pm i\mu$, the segment $i[-Q_{o}+r,0]$ in the $\zeta$-plane corresponds to the semilines on the imaginary $z$-axis: $z\in i(-\infty,-(Q_{o}+r)]\cup i[Q_{o}+r,+\infty)$.

For $r<\kappa<r+Q_{o}$:

$$z=\pm i\mu\,,\qquad\lambda=\mp\kappa\mu\qquad\text{with }\mu=\sqrt{\frac{|r-\kappa|}{\kappa(1+r\kappa)}}\in\mathbb{R}^{+}.$$

(35)

And since $\mu\to Q_{o}-r$ as $\kappa\to(r+Q_{o})^{-}$ and $\mu\to 0$ as $\kappa\to r^{+}$, it follows, again considering both signs in $z=\pm i\mu$ it follows that the segment $i[r,r+Q_{o}]$ in the $\zeta$-plane corresponds to segment $i[-Q_{o}+r,Q_{o}-r]$ along the imaginary $z$-axis.

For the purpose of the DM, one wants to parametrize both $z$ and $\lambda$ above using one single parameter, and we can use the same parametrization for $\lambda$ which was used for our KM breathers, namely:

and further simplify the latter to:

based on the fact that for KM breathers $\rho>1$, $\phi/2=0,\pi,\pm\pi/2$ depending on which of the above cases for $\kappa$ we are considering, and also that $\mathop{\rm Im}\nolimits\Omega=0$. As far as $z$ is concerned, that can be obtained from (5), and the appropriate sign chosen according to the cases indicated before. We need to establish the range of values of $\rho>1$. This can be done by looking at each interval of values for $\kappa$ below, determine what the corresponding interval of values for $\rho$ should be by solving (18) for $\rho$ in terms of $\kappa$. For any fixed $r$, computing $\partial_{\kappa}\rho$ when $0<\kappa<r$ yields two stationary points, located at $\kappa=(\pm Q_{o}-1)/r$. These correspond to $\zeta=i\kappa$ coinciding with the points $\lambda_{o}^{\pm}$ (see Fig. 6); and $\kappa=(Q_{o}-1)/r\in(0,r)$. In conclusion, for each given $r$, one has:

On the other hand, when $-Q_{o}+r<\kappa<0$ or $r<\kappa<Q_{o}+r$, one finds that $\partial_{\kappa}\rho=(-r^{2}\kappa^{2}-2\kappa+r)/\kappa^{2}(\kappa-r)^{2}$, with the same two stationary points $\kappa=-i\lambda_{o}^{\pm}$. In these cases, however, the stationary points are both outside the ranges of values for $\kappa$, so $\rho$ is a monotonic function of $\kappa$. Furthermore, clearly $\rho\to+\infty$ as either $\kappa\to 0^{-}$ or as $\kappa\to r^{-}$, while $\rho\to 1$ as $\kappa\to r\pm Q_{o}$, confirming that in both cases $\rho$ can take any value greater than 1.