Characterization of soliton solutions in 2D nonlinear Schrِdinger lattices by using the spatial disorder

In this paper, the pattern of the soliton solutions to the discrete nonlinear Schrödinger (DNLS) equations in a 2D lattice is studied by the construction of horseshoes in l ∞ -spaces. The spatial disorder of the DNLS equations is the result of the strong amplitudes and stiffness of the nonlinearities. The complexity of this disorder is log ( N + 1 ) where N is the number of turning points of the nonlinearities. For the case N = 1 , there exist disjoint intervals I 0 and I 1 , for which the state u m , n at site ( m , n ) can be either dark ( u m , n ∈ I 0 ) or bright ( u m , n ∈ I 1 ) that depends on the configuration k m , n = 0 or 1, respectively. Bright soliton solutions of the DNLS equations with a cubic nonlinearity are also discussed.