We investigate solitons and nonlinear Bloch waves in Bose-Einsteincondensates trapped in optical lattices. By introducing specially designedlocalized profiles of the spatial modulation of the attractive nonlinearity, weconstruct an infinite number of exact soliton solutions in terms of the Mathieuand elliptic functions, with the chemical potential belonging to thesemi-infinite bandgap of the optical-lattice-induced spectrum. Starting fromthe exact solutions, we employ the relaxation method to construct genericfamilies of soliton solutions in a numerical form. The stability of thesolitons is investigated through the computation of the eigenvalues for smallperturbations, and also by direct simulations. Finally, we demonstrate avirtually exact (in the numerical sense) composition relation between nonlinearBloch waves and solitons.
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