Coupled Ablowitz-Ladik equations with branched dispersion

TL;DR

This paper investigates a multicomponent Ablowitz-Ladik system with branched dispersion, demonstrating its complete integrability and multisoliton solutions, and shows how to derive these from a simpler diagonal system via periodic reduction.

Contribution

It introduces a multicomponent Ablowitz-Ladik system with branched dispersion and establishes its integrability and multisoliton solutions, extending previous models.

Findings

Proves complete integrability of the multicomponent system

Constructs multisoliton solutions for the system

Shows derivation from a diagonal Ablowitz-Ladik equation via periodic reduction

Abstract

Complete integrability and multisoliton solutions are discussed for a multicomponent Ablowitz-Ladik system with branched dispersion relation. It is also shown that starting from a "diagonal" (in two-dimensions) completely integrable Ablowitz-Ladik equation, one can obtain all the results using a periodic reduction.

Equations

$$i\frac{d}{dt}{Q}_n=(1+Q_n{Q}_n^{\ast })(E_{{\sigma }_1}{Q}_{n+1}{E}_{{\sigma }_2}+E_{{\sigma }_2}{Q}_{n-1}{E}_{{\sigma }_1}),$$

$$Q_{n}=\left(\begin{array}[]{ccccccc}q_{1}(n,t)&0&0&......&0\\ 0&q_{2}(n,t)&0&......&0\\ 0&0&q_{3}(n,t)&......&0\\ ....&....&....&.....&......&\\ 0&0&0&......&q_{N}(n,t)\\ \end{array}\right)$$

$$\sigma_{1}=\left(\begin{array}[]{ccccccc}1&2&.&.&.&.&N\\ 2&3&.&.&.&.&1\\ \end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{ccccccc}1&2&.&.&.&.&N\\ N&1&2&.&.&.&N-1\\ \end{array}\right).$$

$$i{\dot{q}}_1$$

$$i{\dot{q}}_2$$

$$i{\dot{q}}_{N-1}$$

$$i{\dot{q}}_N$$

$$i{\dot{q}}_1$$

$$i{\dot{q}}_2$$

$$i{\mathbf{D}}_{\mathbf{t}}{G}_1\cdot F_1$$

$$i{\mathbf{D}}_{\mathbf{t}}{G}_2\cdot F_2$$

$$F_1^2+|G_1{|}^2$$

$$F_2^2+|G_2{|}^2$$

$$G_1={\alpha }_1{e}^{{\eta }_1},G_2={\alpha }_2{e}^{{\eta }_1},F_1=1+e^{{\eta }_1+{\eta }_1^{\ast }+{\varphi }_1},F_2=1+e^{{\eta }_1+{\eta }_1^{\ast }+{\varphi }_2}$$

$${\omega }_1=-2iϵ\cosh (k_1),ϵ\in \{\pm 1\}.$$

$$G_1=e^{{\eta }_1},G_2=ϵe^{{\eta }_1},F_1=F_2=1+e^{{\eta }_1+{\eta }_1^{\ast }+{\varphi }_{12}}$$

$$e^{{\varphi }_{12}}=\frac{1}{2[\cosh (k_1+k_1^{\ast })-1]}$$

$${\eta }_1=k_{1}n+{\omega }_{1}t+{\eta }_1^{(0)}.$$

$$q_1=\frac{e^{k_{1}n-2iϵ\cosh (k_1)t+{\eta }_1^{(0)}}}{1+\frac{1}{2(\cosh (k_1+k_1^{\ast })-1)}{e}^{(k_1+k_1^{\ast })n-2iϵ[\cosh (k_1)-\cosh (k_1^{\ast })]t+{\eta }_1^{(0)}+{\eta }_1^{(0)\ast }}}$$

$$q_2=\frac{ϵe^{k_{1}n-2iϵ\cosh (k_1)t+{\eta }_1^{(0)}}}{1+\frac{1}{2(\cosh (k_1+k_1^{\ast })-1)}{e}^{(k_1+k_1^{\ast })n-2iϵ[\cosh (k_1)-\cosh (k_1^{\ast })]t+{\eta }_1^{(0)}+{\eta }_1^{(0)\ast }}}$$

$$G_1$$

$$G_2$$

$$F_1$$

$$F_2$$

$$e^{{\varphi }_{ij}}=\{\begin{array}{ll}\frac{1}{2}{\left[{ϵ}_i{ϵ}_j\cosh (k_i+k_j^{\ast })-1\right]}^{-1},& \text{mbox}ifi=1,2\text{mbox}andj=3,4;\text{vspace}0.3cm\\ 2\left[{ϵ}_i{ϵ}_j\cosh (k_i-k_j)-1\right],& \text{mbox}ifi=1,2\text{mbox}andj=1,2\\ & \text{mbox}ori=3,4\text{mbox}andj=3,4;\end{array}$$

$${\eta }_j=k_{j}n+{\omega }_{j}t+{\eta }_j^{(0)},k_{j+2}=k_j^{\ast }$$

$${\omega }_j=-2i{ϵ}_j\cosh (k_j),{ϵ}_j\in \{-1,1\},j=1,2.$$

$$G_1$$

$$G_2$$

$$F_1$$

$$F_2$$

$$e^{{\varphi }_{ij}}=\{\begin{array}{ll}\frac{1}{2}{\left[{ϵ}_i{ϵ}_j\cosh (k_i+k_j^{\ast })-1\right]}^{-1},& \text{mbox}ifi=1,2,3\text{mbox}andj=4,5,6;\text{vspace}0.3cm\\ 2\left[{ϵ}_i{ϵ}_j\cosh (k_i-k_j)-1\right],& \text{mbox}ifi=1,2,3\text{mbox}andj=1,2,3\\ & \text{mbox}ori=4,5,6\text{mbox}andj=4,5,6;\end{array}$$

$${\eta }_j=k_{j}n+{\omega }_{j}t+{\eta }_j^{(0)},k_{j+3}=k_j^{\ast }$$

$${\omega }_j=-2i{ϵ}_j\cosh (k_j),{ϵ}_j\in \{-1,1\},j=1,2,3.$$

$$\displaystyle L_{n}=\left(\begin{array}[]{cccc}0&z&0&q_{2}\\ z&0&q_{1}&0\\ 0&-q_{2}^{*}&0&z^{-1}\\ -q_{1}^{*}&0&z^{-1}&0\\ \end{array}\right),$$

Full text

1 Introduction

The study of discrete nonlinear Schrödinger equation is extremely important since the seventies. The equation appeared for the first time in the biophysical context [1] and since then the topic developed rapidly in nonlinear optics, Bose-Einstein condensates, etc. Initially, it was used for dynamics of wave guide arrays [2] and then in the study of discrete diffraction, Peierls barriers, dispersion management, etc. [3]. It was shown afterwards that nonlinear localized waves in optically-induced lattices and photo refractive media are accurately described by discrete nonlinear Schrödinger equation (NLS). However, these phenomena are more or less described by the non-integrable variant of discrete NLS. But quite recently it was shown that motion of discrete curves (modelling polymer’s motions) can be modelled by the completely integrable variant of discrete NLS, namely Ablowitz-Ladik [4] and Hirota-Tsujimoto [5] equations. Integrable coupled semi-discrete and discrete equations represent a topic which is still not well developed. They have a very interesting phenomenology because of the extra-degrees of freedom coming from the matrix structure. Coupled soliton equations have been proposed by several authors [6, 7, 8, 9]. Integrable discretization of such systems is in general quite complicated. The Hirota bilinear formalism [10] turns out to be very effective inasmuch as the bilinear form can be discretized straightforwardly and multisoliton solution or Bäcklund transformations can be obtained directly.

In this paper we are discussing a generalisation of the Ablowitz-Ladik model [11] to the multicomponent (matrix) case, but with many branches of dispersion relation (being different from the well known discrete Manakov system [12]). We will show complete integrability of this system using Hirota bilinear formalism and displaying the Lax pair. The main feature of such system is the structure of dispersion relation (having multiple branches) and phases of the components parametrised by the order $N$ roots of unity. The existence of many branches of the dispersion relation allows more freedom in soliton interaction. We do the analysis directly on the system using Hirota bilinear formalism and we provide also the Lax pairs proving thus the complete integrability. Furthermore we will show how starting from a diagonal Ablowitz-Ladik (having a diagonal one-dimensional evolution in two discrete independent variables) one can obtain all the results, in a very simple way, using a periodic reduction on the second discrete independent variable. The paper is organised as follows: in the first and second chapter we discuss the soliton solutions and Lax pairs for two and three component system and in the third one we discuss the general case and the periodic reduction.

2 Coupled Ablowitz-Ladik equations

The general differential-difference Ablowitz-Ladik system with branched dispersion is:

[TABLE]

where $Q(n,t)$ is a diagonal matrix of complex functions $q_{i}(n,t)$ given by:

[TABLE]

and $E_{\sigma_{i}}$ are permutation matrices corresponding to the following permutations:

[TABLE]

On the components, system (1) has the following expression:

[TABLE]

where $q_{\mu}=q_{\mu}(n,t)$ and $\overline{q_{\mu}}=q_{\mu}(n+1,t),\underline{q_{\mu}}=q_{\mu}(n-1,t)$ for any $\mu=1,...,N$.

For $N=1$, system (2) reduces to the classical Ablowitz-Ladik (AL) equation:

[TABLE]

for which the Lax pair and the multisoliton solutions are known [11].

For $N=2$, we get the following system:

[TABLE]

which, to our knowledge, was very little investigated. The main goal of this paper is to investigate the integrability and the multisoliton solutions for the general case. There are many methods for investigating the integrability of a dynamical system. One of them consists in a direct computation of the conserved quantities [13]. Another one is based on the computation of Lie symmetries of the system [14], [15]. Also very interesting results appeared about symmetries and conservation laws on (semi)discrete nonlinear Schrödinger equation in [20], [21]

In this paper we are going to use the Hirota bilinear formalism and we will provide the Lax pairs as well.

2.1 Soliton solutions and Lax pairs for $N=2$

We consider the coupled Ablowitz-Ladik with two equations (2), for which we apply the Hirota bilinear formalism. Using the nonlinear substitutions $q_{1}=G_{1}/F_{1}$ and $q_{2}=G_{2}/F_{2}$, we cast (2) into the bilinear form:

[TABLE]

where $F_{1},F_{2}$ are real functions, while $G_{1},G_{2}$ and complex valued functions. In order to solve this bilinear system we take for the one soliton solution the following ansatz:

[TABLE]

where amplitudes are given by the components of the “polarization vector” $(\alpha_{1},\alpha_{2})$. From the first two bilinear equations we get an homogeneous algebraic system with the unknowns $\alpha_{1},\alpha_{2}$. Its compatibility condition gives the dispersion relation:

[TABLE]

Next, two bilinear equations give the following two possible relations $e^{\phi_{1}}=e^{\phi_{2}}$ and $e^{\phi_{1}}=-e^{\phi_{2}}$. The last one gives $\alpha_{1}=\alpha_{2}=0$ so remains only $\phi_{1}=\phi_{2}\equiv\phi_{12}$. Also, from the first bilinear equations we get $\epsilon\alpha_{1}=\alpha_{2}$. Now fixing $\alpha_{1}=1$ we can write down the one soliton solution in the simplest form:

[TABLE]

where:

[TABLE]

and

[TABLE]

Explicitly, the 1-soliton solution of (2) is:

[TABLE]

Remark: Because physically the interest is in the square modulus amplitudes, here they are fixed (unlike the case of Manakov system where the polarization vector is free) and only the propagation direction can be different.

In constructing the two-soliton solution we have to take into account that all types of solitons must be considered (in this case both directions of propagation). Straightforward calculation gives 2-soliton solution that describes interaction of two solitons for any propagation direction:

[TABLE]

where:

[TABLE]

[TABLE]

[TABLE]

Long but straightforward calculations show the three-soliton solution in the form:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where:

[TABLE]

[TABLE]

[TABLE]

This solution describes interaction of three solitons for any propagation direction. Although the existence of three-soliton solution in Hirota form for NLS-type equations is a strong indicator for the complete integrability [16], we also give here the Lax pair:

[TABLE]

[TABLE]

where $z$ is the spectral parameter and $Z=-\frac{1}{2}\left(z^{2}+z^{-2}\right)$. The compatibility condition $\partial_{t}L+LB-\overline{B}L=0$ gives precisely system (2).

Remark: The above spectral Lax operator can be put in a block Ablowitz-Ladik spectral operator as in [22]

Because the system is completely integrable we can write easily the $N$-soliton solution:

[TABLE]

where:

[TABLE]

[TABLE]

and where:

[TABLE]

[TABLE]

2.2 Soliton solutions and Lax pairs for $N=3$

We consider the coupled Ablowitz-Ladik with three equations:

[TABLE]

The system is again a completely integrable one. The Lax pair is the following

[TABLE]

[TABLE]

where $Z_{+}=1-z^{2},Z_{-}=-1-z^{-2}$.

The compatibility condition $\partial_{t}L+LB-\overline{B}L=0$ gives precisely our system (2.2). In the last section we will show how these Lax pairs were found.

In order to see the solutions we build the Hirota bilinear form, considering the nonlinear substitutions: $q_{1}=G_{1}/F_{1}$, $q_{2}=G_{2}/F_{2}$ and $q_{3}=G_{3}/F_{3}$:

[TABLE]

where $F_{1},F_{2},F_{3}$ are real functions, while $G_{1},G_{2},G_{3}$ and complex functions. Doing the same machinery as in the case of $N=2$, we can start with the following ansatz $G_{i}=\alpha_{i}e^{\eta},\quad F_{i}=1+e^{\eta_{i}+\eta_{i}^{*}+\phi_{i}}$ for $i=1,2,3$. Plugging into the bilinear equations, we will find that $\phi_{1}=\phi_{2}=\phi_{3}\equiv\phi_{12}$ and $\alpha_{1},\alpha_{2},\alpha_{3}$ are ordered as the cubic roots of unity. Accordingly, the 1-soliton solution is:

[TABLE]

where:

[TABLE]

and

[TABLE]

with three branches of dispersion:

[TABLE]

The 2-soliton solution has the following form:

[TABLE]

where:

[TABLE]

and:

[TABLE]

with the three branches of dispersion ($j=1,2$):

[TABLE]

$3$-soliton solution can be constructed easily and this validates the Lax integrability.

[TABLE]

where:

[TABLE]

and:

[TABLE]

[TABLE]

Since $\epsilon_{j}\in\left\{1,-\frac{1}{2}\pm i\frac{\sqrt{3}}{2}\right\}$, each of the three solitons can have any of the following three branches of dispersion:

[TABLE]

Extension to $N$-soliton solution is straightforward. Formulas are the same except that the sums in (2.2) are taken from 1 to $2N$ and all the definitions with $i+3$ and $j+3$ will turn into $i+N$ and $j+N$

3 General case and periodic reduction

Starting from the general system of coupled Ablowitz-Ladik with $N$ coupled equations, given in (2):

[TABLE]

and using the nonlinear substitutions: $q_{\nu}=G_{\nu}/F_{\nu}$, ${\nu}=1,...,N$, we obtain the Hirota bilinear form:

[TABLE]

The 3-soliton solution for (2) has the expressions for $G_{\nu}$, $F_{\nu}$, $(\nu=1,...,N)$:

[TABLE]

where $D_{1}(\underline{\mu})$, $D_{2}(\underline{\mu})$, $\eta_{i}$, $\phi_{ij}$ are given in (2.2) with the difference that now we have $N$ branches of dispersion for each soliton ($k_{j}$ is the wave number of the $j$-soliton where $j=1,2,3$):

[TABLE]

So the branches of dispersion are labelled by the index $l$. So the parameter $\epsilon_{j}$ which characterizes the $j$-soliton ($j=1,2,3$) can have $N$ values, the order $N$ roots of unity.

3.1 The periodic reduction

One could obtain the above results starting from a general “diagonal” equation in two dimensions and performing periodic reduction. This idea has been used for the first time in [17] where coupled semidiscrete KdV equations were obtained modelling a modular genetic network. The idea is to consider that the independent discrete variable of ordinary Ablowitz-Ladik equation is in fact a diagonal in a two-dimensional (or $d$-dimensional) lattice. Imposing periodic reduction on the one such coordinate in that 2D-lattice, then we will obtain coupled systems of Ablowitz-Ladik equations. In order to see it clearly let us start with the following equation:

[TABLE]

This equation is of course completely integrable. It has the following Lax pair:

[TABLE]

[TABLE]

where $Z=-\frac{1}{2}\left(z^{2}+z^{-2}\right)$, the compatibility condition $\partial_{t}L_{n,m}+L_{n,m}B_{n,m}-B_{n+1,m+1}L_{n,m}=0$ gives exactly (24).

Now lets consider the periodic 2-reduction on the $m$ direction (meaning that $Q(n,m)$ is a periodic function only with respect to $m$ and the period is 2). This means that $Q(n,m)\equiv q_{1}(n),\>Q(n,m+1)\equiv q_{2}(n),\>Q(n,m+2)\equiv q_{1}(n),\>Q(n,m-1)\equiv q_{2}(n),\>etc.$ Introducing this reduction in (24) we get precisely:

[TABLE]

In the same way, if we impose periodic-3 reduction, we get the system with three equations (2.2) and so on. Accordingly, the general system can be obtained from this diagonal Ablowitz-Ladik equation (24), but choosing a periodic $N$-reduction on $m$.

The Hirota bilinear form can be obtained by $Q_{n,m}(t)=G^{m}_{n}/F^{m}_{n}$ (in this notation $m$ is not exponent). Then we cast (24) into:

[TABLE]

where $F_{n}^{m}$ is real function, while $G_{n}^{m}$ is complex valued function.

The 1-soliton solution is:

[TABLE]

where:

[TABLE]

For 2-soliton solution we obtain:

[TABLE]

where:

[TABLE]

[TABLE]

[TABLE]

The three-soliton solution is given by:

[TABLE]

where:

[TABLE]

[TABLE]

[TABLE]

The $N$-soliton solution has the following form for $G^{m}_{n}$ and $F^{m}_{n}$ (being essentially the multisoliton solution of Ablowitz-Ladik):

[TABLE]

where:

[TABLE]

[TABLE]

[TABLE]

and where:

[TABLE]

[TABLE]

Now, all the results are coming straightforward from the diagonal Ablowitz-Ladik (24). This can be seen immediately looking at the two bilinear systems (3.1) and (3). The systems are the same (considering the bar to be the up-shift/down-shift in $m$-direction). In the case $N=2$, the $m$-dependence is dropped, $p$ from the definition of $\eta$ or $\omega(k,p)$ will be $-i\pi$ or $+i\pi$, making the dispersion relation to have two branches (allowing solitons to move either in the same direction or in the opposite one). In the case $N=3$, we have similarly $p$, its exponentials are the cubic roots of the unity. The Lax pairs can be obtained in the same way but imposing periodicity on the spectral functions namely:

[TABLE]

[TABLE]

Imposing, let us say $N=3$ periodicity, we get $\psi_{i}(n,m)\equiv u^{i}_{1}(n),\>\psi_{i}(n,m+1)=u^{i}_{2}(n),\>\psi_{i}(n,m+2)=u^{i}_{3}(n),\>\psi_{i}(n,m+3)=u^{i}_{1}(n),\>etc.$ for any $i=1,2$. We will obtain 6 equations and $6\times 6$ Lax pair matrices that we have already found above (20).

4 Time discretization

We are going to perform fully integrable discretzation of our system ($t$ discretization) using again the Hirota bilinear formalism [18]. It is easy to discretize the bilinear equation. Just replace the continuous bilinear operator with a discrete one and impose the gauge-invariance (i.e. invariance with respect to the multiplication with exponential of linears) [18]:

[TABLE]

where $\delta$ is the discrete step.

[TABLE]

where $\tilde{a}=a((m+1)\delta,n)$ and $t$ becomes $m\delta$.

So, our general system (2), as mentioned in (3), has the following bilinear form:

[TABLE]

where $\mu=1,...,N,\>F_{0}=F_{N},\>F_{N+1}=F_{1},\>G_{0}=G_{N},\>G_{N+1}=G_{1}$.

Discretizing the first bilinear equation we get:

[TABLE]

where $F_{\mu}=F_{\mu}(m\delta,n)$, $G_{\mu}=G_{\mu}(m\delta,n)$, $\widetilde{\overline{G_{\mu}}}=G_{\mu}((m+1)\delta,n+1)$ and $\widetilde{\overline{F_{\mu}}}=F_{\mu}((m+1)\delta,n+1)$.

Using Hirota-Tsujimoto approach [18], [19] we are not changing the second bilinear equation (otherwise the bilinear system will not have two-soliton solution). Our fully discrete bilinear system will be:

[TABLE]

Integrability of the bilinear system (33), (32) can be seen from the existence of 3-soliton solutions, which has the same form as (4), namely:

[TABLE]

where $D_{1}(\underline{\nu})$, $D_{2}(\underline{\nu})$, $\eta_{i}$, $\phi_{ij}$ are given in (2.2) with the difference that now we have $N$ branches of dispersion for each soliton with the following fully discrete form:

[TABLE]

where $k_{j}$ is the wave number and $j$ is the index of the soliton. The Lax pair of this systems is unknown to us. However because we have three soliton solution we expect the system to be completely integrable.

Dividing (32) by $F_{\mu}\widetilde{F}_{\mu}$ and putting $q_{\mu}=G_{\mu}/F_{\mu}$ and $\Gamma_{\mu}={\widetilde{\overline{F_{\mu+1}}}\underline{F_{\mu-1}}}/{F_{\mu}\widetilde{F}_{\mu}}$, we get:

[TABLE]

But, using (36) we get:

[TABLE]

Finally, our fully discrete Ablowitz-Ladik is the following system:

[TABLE]

Now, eliminating $\Gamma_{\mu}$, we will obtain the following (higher order) fully discrete Ablowitz-Ladik system:

[TABLE]

5 Conclusions

In this paper we studied coupled Ablowitz-Ladik equations with branched dispersion relations. The main motivation was to see that the integrability survives and how to study collisions of solitons in different directions. It was shown by Hirota bilinear formalism and Lax pairs that the system is integrable and moreover it was shown that with a very simple periodic reduction of an 2D integrable Ablowitz-Ladik equation all the results are recovered. This simple method was applied initially in [17] but we think that it can be widely extended to a lot of semidiscrete and discrete equations to obtain integrable matrix systems with branched dispersions. In fact, for any completely integrable semidiscrete equation $F(\dot{u}_{n},u_{n},u_{n+1},u_{n-1})=0$, its 2D variant $F(\dot{u}_{n,m}u_{n,m}u_{n+1,m+1},u_{n-1,m-1})=0$, if it is integrable, then by various periodic reduction in $m$, one can obtain integrable matrix systems.

From the point of view of applications we do not know any physical application. However, because the semidiscrete NLS can model motion of discrete curves [4] we expect that the systems described in this paper can describe motion of modular polymers (containing different types of segments), exactly as in the case of modular gene networks which can be modelled [17] by semidiscrete KdV systems with branched dispersion. We intend to do this in a future publication.

Acknowledgements: The work is supported by the project PN-II-ID-PCE-2011-3-0083, Romanian Ministery of Education and Research. ASC is supported by the project PN-II-ID-PCE-2011-3-0137, Romanian Ministery of Education and Research. Also we are very grateful to the referees for all the obervations.