Non-linear Schrödinger equation

Question

I have read about the existence of a non-linear scrhödinger equation. What is its utility and application? And how can it be derived? Is it in a relativistic or non-relativistic context?

Answer 1

Broadly speaking nonlinear Schrödinger equations are used to describe wave propagation in nonlinear media. There are many examples, like deep-water waves (where the linear shallow water approximation to the Navier-Stokes equations is not valid), or nonlinear optics.

There is for example an effect called "Kerr effect" in optical fibers where the refractive index depends on the intensity of the optical pulse. You can find a detailed derivation of a nonlinear Schrödinger equation describing electromagnetic waves in such a medium in the book

• M. J. Ablowitz, B. Prinari, A. D. Trubatch: "Discrete and Continuous Nonlinear Schrödinger Systems", Cambridge University Press, 2004

Edit: Fluids/Hydrodynamics: It is not easy to find a reference for the use of Schrödinger equations in Hydrodynamics, maybe because most authors write for a target audience that is not fluent in quantum mechanics, I don't know. But here is one:

• Andrew Majda and Andrea Bertozzi: "Vorticity and Incompressible Flow"

chapter 7.1, "Simplified Asymptotic Equations for Slender Vortex Filaments: The Self-Induced Approximation, Hasimoto's Transform and the Nonlinear Schrödinger Equation".

Answer 2

You can get some more information about the derivation from the related Wikipedia article about the Gross-Pitaevskii equation which is used in the context of Bose-Einstein condensation. There, one starts with a quantum field theory whose Hamiltonian is nonlinear in the fields, and replaces the field operator by a classical function that describes the condensate. So the reason why the same equation appears in so many different systems is that they are all described by an effective field theory with a nonlinear (fourth-order) interaction term. (The form of this effective theory is dictated by symmetry and the relevant degrees of freedom at low energy.) In the semiclassical approximation, this effective field theory reduces to the nonlinear "Schroedinger" equation.

Answer 3

The nonlinear Schrödinger equation (NLSE) in one dimension is $$ i\frac{\partial\psi}{\partial t}~+~\frac{\partial^2\psi}{\partial x^2}~+~f(|\psi|)\psi~=~0. $$ For $f(|\psi|)~=~|psi|^2$ this is the cubic Schrödinger equation. This is related to a couple of relativistic equations, such as the quartic equation for the Higgs field. The Dirac equation $i\gamma^\mu\partial_\mu\psi~+~({\bar\psi}\gamma^\mu \psi)\gamma_\mu\psi$ is the Thirring model of fermion condensates, related to BCS theory. This is a related nonlinear wave equation. In general the NLSE describes the motion of quantum waves through nonlinear media or in some process where the locality of standard Lagrangians (quadratic etc) is replaced with some nonlocal field.

A separable solution to the NLSE $\psi(x,t)~=~u(x)v(t)$ may be found with $v(t)~=~exp(-i\omega t)$ and the spatial function is given by a DE $u_{xx}~+~\omega u$ $f(|u|)u~=~0$defined as $$ \int\frac{du}{\sqrt{Au^2~-~2F(u)~+~B}}~=~C~\pm~x,~F(u)~=~\int uf(|u|)du $$ which is a traveling wave solution.

Answer 4

I've seen this used in the context of quantum simulations of bosonic systems. If you have a system of many identical bosons all in the same state, you can treat the squared magnitude of the single particle wave function as a density for all of the particles. Then a single particle interacting with this density becomes a good model for a system of interacting Bosons. See for example Yepez, Phys. Rev. Lett. 103, 084501 (2009).