A solution showing the existence of both bright and dark spatial solitons by a simple modification of a previous approach is obtained for a model of reflection grating with modulated Kerr media without imposing specific restrictions on the physical parameters. The resulted relations of the system parameters and the pertinent physical quantities reveal an interesting algebraic structure and symmetry properties containing the previous results as special cases, with higher degrees of symmetry corresponding to more specific restrictions of the parameters. Detailed analysis of the associated parameter space exhibits clearly delineated regions for the different species of spatial solitons with wide ranging variations of characteristics as functions of the parameters. A practical working scheme is then formulated for tailoring the soliton characteristics according to a relatively simple rule. This is exemplified by a specific choice of soliton characteristics supported by a realistic set of system parameters in connection with the study of the dynamical process of soliton propagation in the system. A further study conducted on the dynamics of linear superpositions of two exact bright soliton pairs shows similar relative-phase dependent, attractive, and repulsive interactions as observed in a homogeneous medium while exhibiting visible envelope broadening due to the diffraction effect. The diffraction effect is shown to result in a wobbling phenomenon for the case of two bright solitons with asymmetric amplitudes. In the case of wide-stripe dark solitons, the propagation features a rapid split-off into several narrow stripes, and the number of new stripes depends proportionally on the width of the initial stripes as well as the ratio between its forward and backward amplitudes. Finally, an enhanced breathing phenomenon was demonstrated as a result of increased nonlinear modulation of the system.
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