Questions tagged [non-linear-systems]

The term non-linear or nonlinear has several definitions but is generally used to describe a system that cannot be approximated by a superposition principle or perturbative approach.

66 questions with no upvoted or accepted answers
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8
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0answers
187 views

Nonlinear stability question

I am looking for a simple example where a system is linearly unstable, but nonlinearly unstable or stable, depending on the sign of the initial perturbation. For instance, assume the linear normal ...
6
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1answer
391 views

Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch ...
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1answer
141 views

Pink noise in low-dimensional systems

Pink noise (1/f) is often cited as a signature of complex or critical systems. Is it possible for a low-dimensional time-independent first-order system to generate pink noise? Intuitively it seems ...
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360 views

Driven Pendulum

If the point of suspension of a pendulum is driven periodically in the vertical direction , we can derive the equation of motion for the suspended mass to be of the form, $\ddot{\theta}(t) + (a-b\cos{...
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64 views

Solutions of nonlinear systems invariant wrt. perturbations (looking for applications)

I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics. The problem in a nutshell: describe properties of solution sets of real ...
4
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89 views

Reference for the Landau-Lifshitz system

I'm interested in understanding the dynamics of the discrete Landau-Lifshitz system. It's solutions to equations like $$\frac{\partial X_n}{\partial t} = X_n\times (X_{n-1}+X_{n+1})$$ where the $X_n$ ...
3
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116 views

Beyond Schwinger field strength

What happens to a field theory when it surpasses the Schwinger limit? Can we use the same theory or should we modify it? That is, how is a field theory defined in the "supercritical" regime with ...
3
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679 views

What is the mechanism of subharmonic oscillations?

It's clear to me from linear systems theory that energy manifested within a fundamental mode of resonance can saturate with the excess energy spilling over into harmonic frequencies greater than the ...
3
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150 views

Non-linear Wave Equation - Numerical Methods

Motivation: I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where $'=\frac{...
2
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0answers
259 views

Exact solution for non-linear Fokker-Planck equation

I'm searching for exact (analytical) results for FP equation in 2 variables (such as $x$ and $p$ in 1D) with a steady state. Kramer's like (with force due to confining potential, such as harmonic ...
2
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42 views

How to use Belinsky-Zakharov transformation

I know it might be trivial. When using BZ transformation [1] to generate soliton solutions of Einstein’s field equations, one need a seed solution $g_{0}$ which gives $A_{0}$ and $B_{0}$. Taking them ...
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142 views

Differential equation with nonlinear nonlocal interaction

I am studying literature on the nonlinear Schroedinger equation. More generally, I am wondering how to tackle an equation of motion such as $\bigl(i\partial_t-\Delta\bigr)\psi(\vec{x},t)+V(\vec{x},t)=...
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29 views

Fermi-Pasta-Ulam for the beam equation

The Fermi-Pasta-Ulam numerical experiment is based upon the discrete wave equation, with a small non-linearity added to the forcing term. Does anybody know of similar research performed on the beam ...
2
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54 views

Why do vortices scatter at right-angles

I have been taking a course on non-perturbative physics and currently the teacher is away so I cannot ask him. In the lectures, he made the claim that a pair of vortices in the abelian-Higgs model ...
2
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1answer
176 views

Wave vector relation in nonlinear material

A light wave ($k_1,\omega_1$) travels in a medium of refractive index $n_1$ and then encounters a nonlinear medium ($n_2$) under the angle $\theta_1$. Snell's law tells us the wave's direction in the ...
2
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1answer
74 views

integrability and area-preservation property of dynamical systems

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
2
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236 views

What are the equations of motion that model near light speed orbits of a massive body about incredibly massive bodies?

In Kip Thorne's recently published book, The Science of Interstellar, he describes, by means of an illustration, the complex nature of a spacecraft orbiting a massive black hole with velocities ~ 0....
2
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91 views

Categorization of electromagnetic solitons?

I've seen over the years several mentions of electromagnetic solitons that appear in the high-intensity regime (where vacuum polarization becomes important). Some of these are coupled with plasmas, ...
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62 views

Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. ...
2
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153 views

KdV equation and classical linear wave equation

Like we know, the standard form of KdV equation is $$u_{t}-6uu_{x}+u_{xxx}=0,\tag{1}$$ where this equation describes a solitary wave propagation and $u=u(x,t)$. On the other hand, we know the ...
2
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208 views

value of $\omega$ in nonlinear equation

In this article the authors wrote a nonlinear equation as $$-\frac{\partial^2 \phi}{\partial t^2}+ \nabla \phi= \phi+ \sum_{k=2}^\infty g_k \phi^k$$ They have scaled the time as, $$\tau=\omega(\...
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736 views

Nonlinear refraction index of vacuum above Schwinger limit

This question is more about trying to feel the waters in our current abilities to compute (or roughly estimate) the refraction index of vacuum, specifically when high numbers of electromagnetic quanta ...
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0answers
114 views

Macroscopic chromodynamics

Lately I've been reading about gamma ray lasing phenomena, and I've been wondering about the applications of this. More concretely, the above fantastic question led me to wonder if we could somehow ...
2
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2answers
61 views

Finding dispersion relations

I was wondering if there is a general (theoretical, not experimental) method for finding the dispersion relation for waves in a medium, say given the equation governing purturbations in the medium? ...
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28 views

Is there any nonlinear equations depending on Fourier coefficients?

A nonlinear partial differential equation is an expression depending on derivatives of $u$ $$f(x,t,u,u_x,u_t,\cdots)=0,$$ where the derivatives of $u$ can be obtained from the Taylor series of $u$. ...
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2answers
36 views

Is the trajectory of a particle with constant velocity (though its direction can change by collisions) always non-chaotic?

Suppose we have a particle that travels with constant velocity, without heat losses by friction, and no forces acting on it except for occasionally collisions with much bigger wall-like masses than ...
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48 views

Nonlinearities arising from linear equations

I have sources that tell me that the Bloch Equations, which describe the magnetization vector in nuclear magnetic resonance, are nonlinear. In vector form, without relaxation, they are: $$ \partial_t \...
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177 views

Nonlinear acoustic processes in air - how to quantitatively estimate power in four-wave mixing?

I'd like to understand if non-linear mixing of human speech with a high-power ultrasonic "carrier" in air could produce sidebands with more power than the original speech. 30 kHz to 300 kHz has a ...
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0answers
335 views

Trial function in variational principle - why do we choose this ansatz?

I am reading this paper right now and I cannot understand the intuition behind their choice of ansatz in the variational principle. Specifically this passage on the second page of the paper puzzles ...
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104 views

Why do the Manley-Rowe relations use this Hamiltonian and how can I derive it?

The derivation of the Manley-Rowe relations begin with the following Hamiltonian for 3 classical, coupled harmonic oscillators: $H=(p_1^2+p_2^2+p_3^2)/2m+1/2(k_1q_1^2+k_2q_2^2+k_3q_3^2)+\lambda ...
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34 views

Supercontinoum lasers

In my work I've encountered the phrase "supercontinuoum (CW) laser" quite a bit. After reading the wikipedia page, I'm interested in a more theoretical introductory. I'm mostly interested in ...
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0answers
82 views

Nonlinear Saturated Schrodinger Equation in 1D- Physical Models

I'm studying the Nonlinear 1d Schrodinger equation $$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, ,$$ and specifically, its solitary ...
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38 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? Clearly my question looks at the same time fairly ...
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154 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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0answers
91 views

Good textbooks on nonlinear electrodynamics?

Looking for suggestions for a good textbook on nonlinear electrodynamics, not going into optics immediately as most textbooks tend to do but perhaps a rigorous mathematical exposition on the ...
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83 views

Non-Linear Behavior of Iterated Functional Maps

The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see http://...
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1answer
51 views

Attractors in Duffing equation

The Duffing equation in its full form is $$\ddot{x} + \delta \dot{x} -ax + \beta x^3 = \gamma \cos(\omega t)$$ Now for specific values of the parameters several attractors exist (or not). Let's ...
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0answers
236 views

Lagrangian of non-linear 3 mass, 2 spring system

Given 3 masses connected by 2 springs with the angle of intersection constant, but the springs themselves bending. Young's modulus, which is a variation of Hook's Law, applies to the flexing that ...
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0answers
93 views

Generalized long wave and KdV equation

I have read many papers about benjamin-bona-mahony (BBM) equation or Regularized Long Wave (RLW) equation and found that BBM equation can be derived from KdV equation. from other papers i got others ...
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30 views

Change of variables to apply Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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1answer
72 views

Tangent and Normal accelerations position estimation

How can I derive a particle position given it's last known positions (x,y), velocities in it's components (vx, vy), tangential and centripetal (normal) accelerations? (this is the only available data) ...
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0answers
267 views

Hysteresis in the Lorenz Equations

I was going through Strogatz's wonderful book on nonlinear dynamics and while reading through one problem he posed at the end of the chapter, I did not really understand what was going on. So I hope ...
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0answers
48 views

Data Collection of oscillatory motion

I'd like to study nonlinear oscillatory motion this semester. I plan to build several different mechanical systems (pendula, masses on springs, etc with and without driving forces, large/ small drag, ...
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77 views

Showing the Hamiltonian of the $\alpha$ FPU is real

I am studying the $\alpha$ FPU chain which is a model of coupled oscillators with small non-linearity. For these systems, I derived the following Hamiltonian $H$ which is given by $$ H=\sum_{j=1}^{N} \...
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0answers
183 views

Implicit Differentiation, A doubt

$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes ...
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1answer
100 views

How do nonlinear phenomena arise from linear theories

How is it possible that linear theories, for example maxwells equations or the schroedinger equation, produce nonlinear physics?
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21 views

Spontaneous synchronization references

Can someone suggest references for an introduction on spontaneous synchronization, theory/examples. I am trying to understand it so I can test it for some problems I am working on. I have no prior ...
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1answer
55 views

Vacuum birefringence

Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example: ...
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20 views

About FPU non linear problem, in reference to the original article

I'm reading the original article about the Fermi-Pasta-Ulam-Tsingou (FPUT) problem and I have some problems about the conclusion. Here the behavior of the system as was reported in the article: $$x_i=(...
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22 views

Distinguishing a LTI from not with unknown inputs

Linear time invariant (LTI) systems are a staple of physics. They appear in many situations. But how do you know a system is a LTI? In particular, if you are provided with a black box which ...