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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Nonlinear Pharmacokinetics: Overview01:19

Nonlinear Pharmacokinetics: Overview

Nonlinear or dose-dependent pharmacokinetics is a phenomenon that occurs when the pharmacokinetic parameters of certain drugs deviate from linear pharmacokinetics at higher doses. These drugs do not follow the expected first-order kinetics, where the rate of drug elimination is directly proportional to the drug concentration. Instead, they exhibit a nonlinear relationship, which can be attributed to several factors.
Nonlinearity can arise due to the saturation of plasma protein-binding or...
Nonlinear Pharmacokinetics: Causes of Nonlinearity01:22

Nonlinear Pharmacokinetics: Causes of Nonlinearity

Nonlinearity in drug pharmacokinetics is caused by various factors influencing how a drug is absorbed, distributed, metabolized, and excreted. Understanding these nonlinear processes is crucial for predicting drug behavior in the body and optimizing drug dosing regimens.
Nonlinear drug absorption can occur when the process is rate-limited by solubility, carrier-mediated transport systems, or saturation of the presystemic gut wall or hepatic metabolism. For instance, high doses of riboflavin...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...
Nonlinear Pharmacokinetics: Bioavailability and Protein-Drug Binding01:22

Nonlinear Pharmacokinetics: Bioavailability and Protein-Drug Binding

When a drug follows nonlinear pharmacokinetics, its bioavailability, the amount of the drug that reaches the systemic circulation, can change with different doses. This is due to the presence of a saturable pathway. The pathway becomes saturated as the drug concentration increases, decreasing the absorption rate. Consequently, the drug's bioavailability may be lower than expected at higher doses.
To quantify the extent of bioavailability, pharmacologists often use a parameter called .

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Related Experiment Video

Updated: May 21, 2026

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle
15:06

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle

Published on: January 3, 2016

Forced nonlinear Schrödinger equation with arbitrary nonlinearity.

Fred Cooper1, Avinash Khare, Niurka R Quintero

  • 1Santa Fe Institute, Santa Fe, New Mexico 87501, USA. cooper@santafe.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 12, 2012
PubMed
Summary
This summary is machine-generated.

We found new exact solutions for the nonlinear Schrödinger equation (NLSE) with external forces, showing conserved solitary wave momentum. Variational approximations and stability criteria were validated by numerical simulations for NLSE dynamics.

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

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Last Updated: May 21, 2026

Measurement of Scattering Nonlinearities from a Single Plasmonic Nanoparticle
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Published on: January 3, 2016

Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics
  • Quantum Mechanics

Background:

  • The nonlinear Schrödinger equation (NLSE) describes various physical phenomena, including wave propagation in nonlinear media.
  • External forcing terms can significantly alter the behavior of solutions, such as solitary waves.
  • Understanding the stability and dynamics of solitary waves under external influences is crucial.

Purpose of the Study:

  • To find new exact solutions for the NLSE with specific self-interaction and external forcing terms.
  • To investigate the behavior and stability of solitary wave solutions in the presence of these forces.
  • To develop and validate a variational approximation and stability criteria for NLSE dynamics.

Main Methods:

  • Derivation of new exact solutions for the NLSE.
  • Application of a variational approximation allowing for time-dependent collective coordinates (position, momentum, width, phase).
  • Analysis of stability using a dynamical condition and phase portraits.
  • Numerical simulations to validate the variational approximation and stability criteria.

Main Results:

  • New exact solutions for the forced NLSE were found, with conserved solitary wave momentum in a specific moving frame.
  • The variational approximation yielded stationary solutions close to the exact ones.
  • Stability criteria were proposed and found to be accurate for small forcing terms.
  • Numerical simulations confirmed the effectiveness of the variational approach in capturing solitary wave behavior.

Conclusions:

  • The study provides new exact solutions and a robust framework for analyzing forced NLSE dynamics.
  • The developed variational approximation and stability criteria offer valuable tools for understanding solitary wave behavior.
  • The findings are particularly relevant for systems where external forces are weak compared to intrinsic nonlinearities.