# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 2 Setup and Prior Background
### Interactions of Symbiotic Matter Waves beyond the Integrable Limit  
In the special case where the nonlinear coefficients are all equal to each other, i.e., \(g_{12}=g_{22}=1\), the system of Equations (2) and (3) admits exact single-DB-soliton solutions of the form [34]:  
\[\psi_{d}(x,t) = \cos\phi\tanh\left[D(x-x_{0}(t))\right]+i\sin\phi, \tag{6}\] \[\psi_{b}(x,t) = \eta\mathrm{sech}\left[D(x-x_{0}(t))\right]\times\exp\left[ikx+i \theta(t)+i(\mu-1)t\right], \tag{7}\]  
subject to the boundary conditions \(|\psi_{d}|^{2}\to 1\), and \(|\psi_{b}|^{2}\to 0\) for \(|x|\to\infty\). In the aforementioned solutions, \(\phi\) is the so-called soliton's phase angle, which fixes the \"grayness\" of the dark soliton, while \(\eta\) denotes the amplitude of the bright component. Furthermore, \(x_{0}(t)\) and \(D\) correspond to the soliton's center and inverse width, respectively, while \(k=D\tan\phi\) is the wave-number of the bright soliton, associated with the speed \(\dot{x}_{0}\) of the DB soliton, and \(\theta(t)\) is its phase. While such a general exact expression can only be obtained in the integrable limit, one can utilize an approximate ansatz based on the expressions given in Equations (6) and (7) and depart from the integrable limit. Such a method was employed in our very recent work of Ref. [50] in order to analyze effective interactions between symbiotic matter waves in the general case of different inter-atomic repulsion strengths within each and between the species. Our aim in what follows is to briefly comment on the key results obtained there, in order to be connected with the numerical findings that will be presented below. In particular, a pair of two equal-amplitude dark-bright solitons travelling in opposite directions was considered having the approximate form:  
\[\psi_{d}(x,t) = (\cos\phi\tanh X_{-}+i\sin\phi)\times(\cos\phi\tanh X_{+}-i\sin \phi)\, \tag{8}\] \[\psi_{b}(x,t) = \eta\mathrm{sech}X_{-}\,\mathrm{e}^{i[kx+\theta(t)+(\mu-1)t]}+ \eta\mathrm{sech}X_{+}\,\mathrm{e}^{i[-kx+\theta(t)+(\mu-1)t]}\,\mathrm{e}^{ i\Delta\theta}. \tag{9}\]  
In these expressions \(X_{\pm}=D\left(x\pm x_{0}(t)\right)\) where \(2x_{0}(t)\) is the distance between the two DB solitons, while \(\Delta\theta\) is the relative phase between the bright solitons. If \(\Delta\theta=0\), the bright solitons are in-phase (IP), while if \(\Delta\theta=\pi\), the bright solitons are out-of-phase (OP) or anti-symmetric. Within a Hamiltonian variational approach [3; 7; 37; 49], the approximate ansatz of Equations (8) and (9) is substituted into the total energy of the system given by Equation (4), and the relevant integrations are performed under the assumption that the soliton velocity is sufficiently small (\(\phi\approx 0\), \(k\approx 0\)). Full analytical results for all the integrals contributing to \(E\) were obtained in [50]. At large distances between the solitons, \(Dx_{0}\gg 1\), simplified asymptotic expressions can be derived by expanding with respect to \(\exp\left[-2 Dx_{0}\right]\) (see also the earlier findings for the integrable case [37]). The total energy can be decomposed as \(E=2E_{1}+E_{dd}+E_{bb}+E_{db}\), where \(E_{1}\) corresponds to the energy of a single DB soliton, contributing twice to the total energy, while the remaining three terms, namely, \(E_{dd}\), \(E_{bb}\) and \(E_{db}\), account for the interaction between: the two dark solitons, the two bright solitons, and the cross-interaction of a dark soliton in the first component with the bright soliton of the second component, respectively. Figure 1 illustrates our key findings regarding the interaction energy of two out-of-phase DB solitons as a function of their distance. Here and throughout this work, the chemical potential and the intra-species interaction coefficients are fixed to \(\mu=2/3\) and \(g_{22}=0.95\)[53; 54], respectively, while the interatomic interaction coefficient \(g_{12}\) is left to widely vary.