# The Nonlinear Schr\"odinger Equation and Conservation Laws

Research paper by Elias Rios

Indexed on: 19 Mar '14Published on: 19 Mar '14Published in: Mathematics - Analysis of PDEs

#### Abstract

The purpose of this short note is to show some results using the equation solution to the nonlinear equation of Schr\"odinger, (NLS), is used the \textit{Conservation Lews}, that through them is obtained inequality that are of utmost importance, also show a variation of the theorem discussed in [8], \textit{classical local smoothing estimate in one dimension}, and the \textit{Principle of Least Action}, in Lagrangian mechanics. The nonlinear Schr\"odinger equation is a nonlinear partial differential equation, and where the solution is a complex-valued function of $d$-dimensional. We focus our attention on working in one dimension, that is to say, $d=1$. We consider the existence of a solution to (NLS) and it meets the condision to be a smooth function of a complex variable.