コロキウム

In this talk, in the first part we briefly describe the solutions of local NLS equation. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. Employing a two-mode approximation, we systematically develop two cubic-quintic ordinary differential equations and assess the contributions of the long-range interactions in each of the relevant prefactors, gauging how to simplify the ensuing dynamical system. Finally, we obtain a reduced canonical description for the conjugate variables of relative population imbalance and relative phase between the two wells and proceed to a dynamical systems analysis of the resulting pair of ordinary differential equations. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. Next, we study the stability and internal modes of one-dimensional gap soliton employing the modified nonlinear Schrodinger equation with a sinusoidal potential together with the present of a weak nonlocality. Using an analytical theory, it is proved that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and one of these is always unstable. Also, we study the oscillatory instabilities and internal modes of the modified nonlinear Schrodinger equation. The analytical results are in excellent agreement with numerical results.