Comparisons with numerical calculations

We have solved the steady-state equation (2.2.1) numerically using a Newton–Raphson method and analysed the stability of the numerical solution by solving the eigenvalue problem (2.2.3). In this section, we compare these numerical results with the analytical

calculations of the previous section. For the sake of simplicity, we set Λ = 1 in all the illustrative examples. This setting, however, does not lose generality as Λ > _{0 can be} scaled out to 1 by the transformation

un→ ^un

√Λ, ε→^{εΛ and γ}→^{γΛ. (2.2.40)}

2.2.2.1 Onsite bright solitons

Comparisons between numerical calculations and analytical approximations for the case of onsite bright solitons have been fully presented and discussed by Susanto et al. [91]. For the sake of completeness, we reproduce the results of [91] for the (in)stability domain of onsite bright solitons in the(ε, γ)-plane in Fig. 2.3 by introducing the colour representation for the maximum value of|^Im(ω)|. Approximations (2.2.29) and (2.2.30) are also shown there from which we can see that the former gives better prediction for the occurrence of the instability point than the latter. This is understandable as the onset of the instability approximated by Eq. (2.2.29) is indeed caused by the collision of the discrete eigenvalue with the upper band of the continuous spectrum and also typically occurs for small ε. This is not the case in approximation (2.2.30) where the ac-tual collision is with an eigenvalue bifurcating from the inner edge of the phonon band and occurs for relatively large ε. Moreover, for γ < 0, as shown in Fig. 2.3, the onsite bright soliton is always unstable for all ε. This is still in accordance with our analytical prediction, i.e., if we set γ to be negative (provided Λ > −γ) in approximate eigen-value (2.2.28), the eigeneigen-value ΩE becomes negative for any value of coupling constant ε.

2.2.2.2 Intersite bright solitons

For the stability of intersite bright solitons, we start by examining the validity of our analytical prediction for the critical eigenvalues as given by Eqs. (2.2.35) and (2.2.36).

In Fig. 2.4, we present a comparison between the analytical approximation and the nu-merics for some values of γ, specifically γ=0.1, 0.18, 0.5, to represent the three possible cases explained in the previous section. One should notice that the appearance of the branching curves for each value of γ in the figure manifests the fact (from numerics and analytics) that the double eigenvalue of an intersite bright soliton splits into two distinct eigenvalues once the coupling is turned on. The figure reveals the relative ac-curacy of the small-ε approximations, and we conclude that their range of validity is wider for the lower branches of each branching curve.

Next we turn to a description of the eigenvalue structure of this intersite configuration

ε

γ

0 1 2 3 4

−0.2 0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8 1 Λ=1 1.2

Figure 2.3:The (in)stability region of onsite bright solitons in(ε, γ)-space. For each value of ε and γ, the corresponding colour indicates the maximum value of

|^Im(ω)|(over all eigenvalues ω) for the steady-state solution at that point.

Stability is therefore indicated by the region in which Im(ω) = 0, namely the black region (recall that ω and ω are eigenvalues as the stability matrix of the EVP (2.2.3) is real-valued). White dashed and dash-dotted lines give the analytical approximations (2.2.29) and (2.2.30), respectively.

for the three values of γ given above; this is shown in Fig. 2.5, where the left and right panels respectively present the structure just before and just after the first collision that results in the mode instability. We now describe the results in more detail for the three values of γ above.

For γ= 0.1, when ε=0 the critical eigenvalues ω lie in the gap between the two parts of the continuous spectrum, and the instability is caused by a collision between one of the critical eigenvalues and its twin at the origin (see the top panels of Fig. 2.5). For γ = 0.18, the critical eigenvalues ω also lie in the gap between the two parts of the continuous spectrum, but the instability in this case is due to a collision between one of the critical eigenvalues and the inner edge of the continuous spectrum at ω = ±√_Ω

L

(see the middle panels of Fig. 2.5). In contrast to the two cases above, for γ = 0.5 the critical eigenvalues lie beyond the continuous spectrum, and the instability is caused by a collision between one of the critical eigenvalues and the outer boundary at ω =

±√_Ω

U (see the bottom panels of Fig. 2.5). All the numerical results presented here are in accordance with the sketch shown in Fig. 2.2. Back to Fig. 2.4, the critical eigenvalues which are most responsible for the instability as illustrated above are shown by the upper branch for γ=0.18 (the middle branching curve) and the lower ones for γ=0.5 and γ=0.1 (the other branching curves).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.4

0.6 0.8 1 1.2 1.4 1.6 1.8 2

ε

|ω|

Λ=1

Figure 2.4:Comparison between the critical eigenvalues of intersite bright solitons ob-tained numerically (solid lines) and their analytical approximation (dashed lines). The appearance of the branching curves confirms the fact of the splitting of the double eigenvalues as ε increases (see text). The upper and lower branching curves correspond, respectively, to γ = 0.5 and γ=0.1, whereas the middle one corresponds to γ=0.18. The upper and lower branches of each branching curve is approximated, respectively, by Eq. (2.2.35) and Eq. (2.2.36); note that these approximations for γ=0.1 (the lower branching curve) are indistinguishable from the numerical results.

Numerical solution of the EVP (2.2.3) for intersite bright solitons, for a relatively large range of ε and γ, give us the (in)stability domain of the bright solitons in the two-parameter(ε, γ)-plane, which is presented in Fig. 2.6. Again, we use colours to rep-resent the maximum of |^Im(ω)| as a function of ε and γ; thus solitons are stable in the black region. Our analytical predictions for the occurrence of instability, given by Eqs. (2.2.37)–(2.2.39), are also shown, respectively, by dash-dotted, dotted and dashed lines. Particularly for γ < 0, Fig. 2.6 reveals the unstable region of the intersite bright solitons for all ε. This fact also agrees with our analytical prediction shown by both ΩE,1

in Eq. (2.2.35) and ΩE,2in Eq. (2.2.36) being negative at any value of ε when driving γ is negative (provided Λ>−^γ).

We now move on the examination of the dynamics of the governing equation (2.1.1) for an intersite bright soliton. By using a Runge-Kutta time integrator with the perturbed intersite soliton solution as the initial condition, we show in Fig. 2.7 the numerical evo-lution of the soliton for the same parameter values as those used in Fig. 2.5, which corresponds to each of the instability scenarios. We can see from the figure that the dynamics of the unstable solitons (right panels) manifests itself into different mecha-nisms. In the top right panel, the soliton’s centre shifts to another site forming an onsite configuration. In contrast, the typical instability of the unstable soliton in the middle right panel is in the form of soliton decay, while in the bottom right panel the soliton collapses into different sites.