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Linear Approximation in Frequency Domain

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.

Classification of Systems-I

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

First Law: Particles in One-dimensional Equilibrium

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Newton's first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a net external force. It also states that there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This cause is a net external force. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt, due to the net force of friction. If we...

Linear Approximation in Time Domain

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First Law: Particles in Two-dimensional Equilibrium

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Linear time-invariant Systems

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Related Experiment Video

Updated: Jun 21, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Universal correlations in a nonlinear periodic 1D system.

Yaron Silberberg¹, Yoav Lahini, Yaron Bromberg

¹Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. yaron.silberberg@weizmann.ac.il

Physical Review Letters

|August 8, 2009

Summary

This summary is machine-generated.

Nonlinear systems with random phases spontaneously form spatial correlations, unlike linear systems. Strong nonlinearities lead to Gaussian intensity distributions and predictable phase correlations between neighboring sites.

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Area of Science:

Physics
Quantum Mechanics
Nonlinear Dynamics

Background:

Periodic 1D systems initialized with random phases typically evolve into uncorrelated thermal distributions.
Linear tight-binding models lack spatial correlations under such initial conditions.

Purpose of the Study:

To investigate the emergence of spatial correlations in nonlinear periodic 1D systems.
To characterize the behavior of intensity distributions and phase correlations in the presence of nonlinearity.

Main Methods:

Analysis of a nonlinear tight-binding model with uniform initial amplitudes and random phases.
Examination of intensity histograms and phase correlations between neighboring sites.

Main Results:

Strong nonlinearities induce spontaneous formation of spatial correlations.
Intensity histograms approach a narrow Gaussian distribution.
Neighboring sites exhibit phase correlations (out of phase for positive nonlinearity, in phase for negative).
Field correlation exhibits a universal, parameter-independent shape.

Conclusions:

Nonlinearity is crucial for developing spatial correlations in these systems.
The observed correlations and universal field shape are robust features.
Findings are applicable to 1D optical lattices and nonlinear optical waveguide arrays.