In is paper, we have studied the propagation of SPATIAL SOLITONS in the medium with a SPATIALly modulated nonlinearity. Wave equation includes the terms of diffraction and periodic self- focusing. To solve the wave equation, we have employed numerical method including Monte Carlo based Euler-Lagrange variational schema. The effect of the nonlinearity strength related to periodic Kerr self-focusing, on the physical properties of the systems such as maximum intensity and soliton width oscillations are considered.

In this paper, we study pointwise projectively related Finsler gradient Ricci SOLITONS. We obtain an equation that characterizes the relationship between two pointwise projectively related Finsler gradient Ricci SOLITONS. Further, if two Finsler gradient Ricci SOLITONS (M, F, dV ˜ F˜) and (M, F, dVF ) satisfy F˜,k = µ ∂[F˜2 ] ∂yk and some extra conditions, where “, ” denotes the horizontal covariant derivative with respect to F, we characterize their relationships along the geodesics. In particular, if two Finsler gradient Ricci SOLITONS are both complete, then (M, F, dVF ) is expanding or shrinking and (M, F, dV ˜ F˜) is shrinking.

CSs have been theoretically predicted and recently experimentally demonstrated in broad area, vertical cavity driven semiconductor lasers (VCSELs) slightly below the lasing threshold. Above threshold, the simple adiabatic elimination of the polarization variable is not correct, leading to oscillatory instabilities with a spuriously high critical wave-number. To achieve real insight on the complete dynamical problem, we study here the complete system of equations and find regimes where a Hopf instability, typical of lasers above threshold, affects the lower intensity branch of the homogeneous steady state, while the higher intensity branch is unstable due to a Turing instability. Numerical results obtained by direct integration of the dynamical equations show that writable/erasable CSs are possible in this regimes, sitting on unstable background.

This paper recovers optical soliton solutions with $\chi^{(2)}$--nonlinear susceptibility. Bright, dark, singular, bright--dark combo SOLITONS are recovered. A variety of algorithms are implemented. These include Riccati equation approach, exp--function expansion, modified simple equation method, sine-Gordon equation scheme, $F$--expansion, trial function and functional variable approaches.%This paper recovers optical soliton solutions with $\chi^{(2)}$--nonlinear susceptibility. Bright, dark, singular, bright--dark combo SOLITONS are recovered. A variety of algorithms are implemented. These include Riccati equation approach, exp--function expansion, modified simple equation method, sine-Gordon equation scheme, $F$--expansion, trial function and functional variable approaches.%This paper recovers optical soliton solutions with $\chi^{(2)}$--nonlinear susceptibility. Bright, dark, singular, bright--dark combo SOLITONS are recovered. A variety of algorithms are implemented. These include Riccati equation approach, exp--function expansion, modified simple equation method, sine-Gordon equation scheme, $F$--expansion, trial function and functional variable approaches.

The class of homogeneous Siklos space-times is considered from algebraic point of view and the generalized Ricci SOLITONS are completely classified.