This paper studies advanced mathematical methods like sin-cos and sinh-cosh approachesto find precise solutions by considering inter-modal dispersion and spatio-temporal dynamics with kerrlaw NONLINEARity in the Resonant shrödinger EQUATION. These methods are useful for solving NONLINEARpartial differential EQUATIONs. To obtain the ordinary differential EQUATION for the traveling wave solution,we initially deal with the general partial differential EQUATION (PDE). Then, a series of optical solitonsolutions, including cusp and dark solitons, are derived for the Resonant shrödinger EQUATION using thiseffective approach.

The NONLINEAR SCHRODINGER EQUATION (NLS) is one of the most important EQUATIONs in quantum physics. It is frequently useful in describing the waveguide movement of minute particles, for example an electron in the atom. NLS can be divided into three different cases: critical, supercritical, and subcritical. In this paper we try to show numerical methods that solve critical case of NLS, for short CNLS, in two dimensions. We also study the effects of the discretization of the EQUATION in the solutions. There are some initial conditions for which the solutions of CNLS become singular in the finite time interval, but by using the finite difference method for the discretization of the Laplacian term in the EQUATION, it is shown that the resulting discrete NLS represents a more accurate discretized version of the modified CNLS EQUATION which can have local solution as well. So, as such modified CNLS EQUATIONs are simply obtained by inserting small perturbations in the original EQUATION, evidently by using this method it is almost possible to avoid blowup in the numerical solutions of these EQUATIONs.

The unification between the theory of general relativity and the standard model of particle physics predicts the existence of a minimal measurable length on the order of the Planck length. Nowadays phenomenological studies of field theory in the presence of a minimal observable distance are extensively performed. The existence of a minimal measurable length leads to the generalized uncertainty principle (GUP). In this work, we obtain at first the angular wave frequency in the presence of a minimal observable length by considering the generalized uncertainty principle. Then, by expanding the generalized angular wave frequency, the NONLINEAR SCHRODINGER EQUATION is found. Also, the soliton solution of the generalized NONLINEAR SCHRODINGER EQUATION is obtained. In the limit𝛽 → 0, the soliton solution in generalized space becomes the same as usual soliton solution. The value of minimal observable length is considered about10 𝑚 .

In this paper, we have studied some NONLINEAR SCHRODINGER EQUATIONs appeared inmany body systems such as a nanowire with a soft polar layer, a quantum well in the presence of the electron-electron interactions, Grass-Pitaevskii EQUATION for Bose-Einstein condensate in the presence on the two and three body interactions. Calculation of the ground state energy of these systems by analytical methods is very difficult. Solution of these EQUATIONs through numerical methods like self-consistent one also needs complicated computer programming. In this paper, Ground state energy have been obtained by means of the Euler-Lagrange Variational method as well as simple and appropriate trial wave-functions.Comparison of the result we have obtained with the Varational technique and the available methods in the literatures shows the high accuracy of this method.