# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 2 Setup and Prior Background  
### Model and Theoretical Considerations  
The system of interest is a two-component BEC strongly elongated along the \(x\)-direction, subject to a tight transverse harmonic trap of frequency \(\omega_{\perp}\). Such a mixture can e.g., be composed of two different hyperfine states of the same alkali isotope, like \({}^{87}\)Rb. Within mean field theory and after integrating out the frozen transverse degrees of freedom, this mixture is described by the following two coupled (1+1)-dimensional Gross-Pitaevskii equations (GPEs) [51; 52]:  
\[i\hbar\partial_{t}\psi_{j}=\left(-\frac{\hbar^{2}}{2m}\partial_{x}^{2}-\mu_{j} +\sum_{k=1}^{2}g_{jk}|\psi_{k}|^{2}\right)\psi_{j}.\qquad(j=1,2). \tag{1}\]  
In the above equation, \(\psi_{j}(x,t)\) denote the mean-field wave functions of the two components normalized to the numbers of atoms \(N_{j}=\int_{-\infty}^{+\infty}|\psi_{j}|^{2}dx\), while \(m\) and \(\mu_{j}\) are the atomic mass (identical for both components) and chemical potentials, respectively. The effective one-dimensional coupling constants are given by \(g_{jk}=2\hbar\omega_{\perp}a_{jk}\), where \(a_{jk}\) denote the three \(s\)-wave scattering lengths that account for collisions between atoms belonging to the same (\(a_{jj}\)) or different (\(a_{12}=a_{21}\)) species. We restrict our considerations to purely repulsive interactions, i.e., all \(g_{jk}>0\), and consider an idealized homogeneous setting with no longitudinal trapping potential along the \(x\)-axis.  
By measuring densities \(|\psi_{j}|^{2}\), length, time and energy in units of \(2a_{11}\), \(a_{\perp}=\sqrt{\hbar/\left(m\omega_{\perp}\right)}\), \(\omega_{\perp}^{-1}\), \(\hbar\omega_{\perp}\), respectively, and in a second step rescaling space-time coordinates as \(t\to\mu_{1}t\), \(x\to\sqrt{\mu_{1}}x\), and the densities \(|\psi_{1,2}|^{2}\to\mu_{1}^{-1}|\psi_{1,2}|^{2}\), the system of Equation (1) can be written in the following dimensionless form:  
\[i\partial_{t}\psi_{d}+\frac{1}{2}\partial_{x}^{2}\psi_{d}-(|\psi _{d}|^{2}+g_{12}|\psi_{b}|^{2}-1)\psi_{d} = 0, \tag{2}\] \[i\partial_{t}\psi_{b}+\frac{1}{2}\partial_{x}^{2}\psi_{b}-(g_{12 }|\psi_{d}|^{2}+g_{22}|\psi_{b}|^{2}-\mu)\psi_{b} = 0. \tag{3}\]  
In the above system of equations, we slightly changed the notation, using \(\psi_{1}\equiv\psi_{d}\) and \(\psi_{2}\equiv\psi_{b}\), indicating this way that the component 1 (2) will be supporting dark (bright) solitons. Furthermore, \(\mu\equiv\mu_{2}/\mu_{1}\) is the rescaled chemical potential, while the interaction coefficients are normalized to the scattering length \(a_{11}\), i.e., \(g_{12}\equiv g_{12}/g_{11}\), and \(g_{22}\equiv g_{22}/g_{11}\). The system of Equations (2) and (3) conserves the total energy, \(E\), the rescaled total number of atoms in each component (\(N_{d}\) and \(N_{b}\), respectively) and the rescaled total number of atoms \(N\), where:\[E = \frac{1}{2}\int_{-\infty}^{+\infty}\Big{[}|\partial_{x}\psi_{d}|^{2} +|\partial_{x}\psi_{b}|^{2}+(|\psi_{d}|^{2}-1)^{2}+g_{22}|\psi_{b}|^{4}-2\mu| \psi_{b}|^{2}+2g_{12}|\psi_{d}|^{2}|\psi_{b}|^{2}\Big{]}dx, \tag{4}\] \[N = N_{d}+N_{b}=\sum_{i=d,b}\int_{-\infty}^{\infty}dx|\psi_{i}|^{2}. \tag{5}\]