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Our exposition will be structured as follows: in Sec. II we provide an overview of the theoretical properties of both the discrete and the continuous model. In Sec. III, we numerically explore the existence, stability and dynamical features of the models, while in Sec. IV we summarize our findings and present our conclusions as well as some interesting directions for future work.

II. THEORETICAL SETUP

Following the setup of [18, 19], the two-dimensional (continuum) Dirac model of relevance to the binary waveguide problem is of the form:

2 i∂tψ1 = (i∂x + ∂y)ψ2 + (m g ψ1 )ψ1, − − | |2 i∂tψ = (i∂x ∂y)ψ (m + g ψ )ψ , (1) 2 − − 1 − | 2| 2 where (ψ1, ψ2) denotes the spinor field (mode amplitude), while m represents the mass associated with the propagationmismatch between the two different types of waveguides. The cubic nonlinearity stems from the Kerr effect and breaks the Lorentz symmetry (contrary, e.g., to what is the case with the Soler model; cf. [30, 31]). We consider this model both at the discrete and at the continuum level.

A. Continuous model

The analysis of the continuum model of Eq. (1) can be performed in radial coordinates, following a procedure similar to