^{1}

^{2}

^{1}

^{1}

^{2}

We study the existence and stability of fundamental bright discrete solitons in a parity-time- (

A system of equations is

The most basic configuration having

When nonlinearity is present in a

In the continuous limit, the coupled equations without gain-loss have been studied in [

The stability of bright discrete solitons in

In this work, we determine the eigenvalues of discrete solitons in

The manuscript is outlined as follows. In Section

The governing equations describing

The focusing system has static localised solutions that can be obtained from substituting

In the uncoupled limit, that is, when

Using perturbation expansion, solutions of the coupler (

It is well-known that there are two natural fundamental solutions representing bright discrete solitons that may exist for any

In the uncoupled limit, the mode structures

For the first-order correction due to the weak coupling, writing

Equations (

For the onsite soliton, that is, a one-excited-site discrete mode, one can perform the same computations to obtain the mode structure of the form

After we find discrete solitons, their linear stability is then determined by solving a corresponding linear eigenvalue problem. To do so, we introduce the linearisation ansatz

The spectrum of (

Following the weak-coupling analysis as in Section

At order

The intersite soliton I (i.e., the symmetric intersite soliton) has three pairs of eigenvalues for small

The intersite soliton II, that is, the intersite soliton that is antisymmetric between the arms, has three pairs of eigenvalues given by

The onsite soliton has only one eigenvalue for small

As for the second type of the onsite soliton, we have

We have solved the steady-state equation (

First, we consider the discrete intersite soliton I. We show in Figure

Eigenvalues of intersite soliton I with

In Figure

The same as in Figure

In both figures, we also plot the approximate eigenvalues in solid (blue) curves, where good agreement is obtained for small

From numerical computations, we conjecture that if in the limit

Next, we consider intersite solitons II (i.e., antisymmetric intersite solitons). Shown in Figure

The spectra of intersite soliton II with

We also study onsite solitons. Shown in Figures

(a, c) The spectrum of onsite soliton I in the complex plane for

The same as in Figure

Figure

Figure

Unlike intersite discrete solitons that are always unstable, onsite discrete solitons may be stable. In Figure

The stability region of the onsite soliton type I (a) and II (b) in the

Finally, we present in Figure

The typical dynamics of the instability of the discrete solitons in the previous figures. Here,

We have presented a systematic method to determine the stability of discrete solitons in a

As mentioned in Section

Performing the expansion in

At

The steps of finding the coefficients

Solve the eigenvalue problem (

Determine

Solve (

Determine

The procedure repeats if one would like to calculate the higher order terms.

The leading order eigenvalue

The authors declare that there is no conflict of interests regarding the publication of this paper.

Alhaji A. Bachtiar and Hadi Susanto are grateful to the University of Nottingham for the 2013 Visiting Fellowship Scheme and British Council for the 2015 Indonesia Second City Partnership Travel Grant.