On the Existence of Bright Solitons in Cubic-Quintic Nonlinear Schrödinger Equation with Inhomogeneous Nonlinearity

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameter λ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method

Theory and applications to brain tumor dynamics

Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios

Existence of dark solitons in a class of stationary nonlinear Schrödinger equations with periodically modulated nonlinearity and periodic asymptotics

In this paper, we give a proof of the existence of stationary dark solitonsolutions or heteroclinic orbits of nonlinear equations of Schrödinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of dark solitons for cubic and photorefractive nonlinearities.MICINN project FIS2008-0484

On the Existence of Bright Solitons in Cubic-Quintic Nonlinear Schrödinger Equation with Inhomogeneous Nonlinearity

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameter λ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubicquintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method

Solitons for the cubic-quintic nonlinear Schrödinger equation with time and space modulated coefficients

In this paper, we construct, by means of similarity transformations, explicit solutions to the cubic–quintic nonlinear Schrödinger equation with potentials and nonlinearities depending on both time and spatial coordinates. We present the general approach and use it to calculate bright and dark soliton solutions for nonlinearities and potentials of physical interest in applications to Bose–Einstein condensates and nonlinear optics.project FIS2008-02873 Ministerio de Ciencia e Innovació

Solitary waves in coupled nonlinear Schrodinger equations with spatially inhomogeneous nonlinearities

Using Lie group theory we construct explicit solitary wave solutions of coupled nonlinear Schrodinger systems with spatially inhomogeneous nonlinearities. We present the general theory, use it to construct different families of explicit solutions and study their linear and dynamical stability

Ecuaciones de Schrödinger no lineales con no linealidad espacialmente inhomogenea

En esta tesis, se estudia una variante de la ecuación de Schrodinger no lineal: la ecuación de Schrodinger no lineal con no linealidad espacialmente inhomogenea. La tesis se divide en 4 partes: después de un capitulo introductorio sobre la ecuación de Schrodinger no lineal, se divide el estudio de la ecuación de Schrodinger no lineal inhomogenea en 3 bloques: En el primero se dan resultados sobre la existencia y estabilidad de soluciones de esta ecuación. Para ello se utilizan diferentes técnicas, tales como la aproximación variacional, técnicas de sistemas dinámicos, problema de autovalores... En el segundo bloque, una vez probada la existencia de soluciones, se procede a calcular soluciones analíticas de esta ecuación utilizando diferentes métodos analíticos, tales como el método de las simetrías de lie, transformación de similaridad, etc. En el tercer y último bloque, se estudian algunas aplicaciones físicas de esta ecuación a los condensados de bose Einstein y a la óptica no lineal

Localized and periodic exact solutions to the nonlinear Schrodinger equation with spatially modulated parameters: Linear and nonlinear lattices

Using similarity transformations we construct explicit solutions of the nonlinear Schrodinger equation with linear and nonlinear periodic potentials. We present explicit forms of spatially localized and periodic solutions, and study their properties. We put our results in the framework of the exploited perturbation techniques and discuss their implications on the properties of associated linear periodic potentials and on the possibilities of stabilization of gap solitons using polychromatic lattices.Comment: 17 pages, 5 figure