We study solitons (strictly speaking, solitary waves) in a model of two linearly coupled waveguides with the Kerr nonlinearity and resonant gratings, neglecting the material dispersion. An effective dispersion is induced by the linear couplings between the forward and backward Bragg-scattered waves and between the two cores. First, we consider a transition from the obvious symmetric solitons to nontrivial asymmetric ones for quiescent (standing) solitons. The solutions are found in an approximate analytical form by means of the variational approximation and, independently, by direct finite-difference numerical simulations. Results produced by the two methods are in a fairly good agreement. We further establish the stability of the asymmetric solitons by direct simulations, while showing that the symmetric solitons coexisting with the asymmetric ones are always unstable. Next, we consider traveling solitary waves. We fix the frequency detuning, while the strength of the coupling between the two cores and the velocity of the moving soliton are varied. In this case the solutions are found only by direct numerical methods, revealing that moving asymmetric solitons exist and are stable. Similar to the case of the quiescent solitary waves, the symmetric solitons coexisting with the asymmetric ones prove to be always unstable.
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