nonlinear optics, BEC, superfluidity, superconductivity, ferromagnetism, etc. [book]{}.

The concept of the SSB in nonlinear systems of the NLS type was, plausibly, introduced for the first time in 1979 by E. B. Davies [@Davies], who addressed a nonlinear extension of the Schrödinger equation for a pair of quantum particles with an isotropic interaction potential. In this context, the SSB was predicted as the breaking of the rotational symmetry in the GS. Another early work, which predicted the SSB in a relatively simple form, addressed the *self-trapping model*, based on a system of linearly coupled ordinary differential equations including the self-attractive cubic terms [@Scott].

In the effectively one-dimensional geometry, the SSB can be studied in the framework of the scaled NLSE/GPE with potential $H(x)$ of the DWP type, for the amplitude of the electromagnetic wave, or the single-particle wave function, $\psi \left( x,z\right) $: $$i\frac{\partial \psi }{\partial z}=-\frac{1}{2}\frac{\partial ^{2}\psi }{%
\partial x^{2}}-\left\vert \psi \right\vert ^{2}\psi +H(x)\psi ,
\label{NLSE}$$where $z$ is the propagation distance in optics, or time in the GPE. This equation can be reduced to a system of coupled ordinary differential equations for two amplitudes, $u_{1,2}(z)$, by means of the *tight-binding approximation* [@tight], which replaces $\psi (x,z)$ by a linear 