Nonlinear Definition and 558 Threads

Hi everybody, In Robert W. Boyd's book "Nonlinear Optics", the quantum treatment of the nonlinear optical susceptibility lead to the next expression, for the second order case: \chi^{(2)}_{ijk}(\omega_{\sigma},\omega_q,\omega_p)=\frac{N}{\hbar^2} P_F\sum_{mn}...

Homework Statement y'=(x^2 +xy-y)/((x^2(y)) -2x^2)[/B]Homework EquationsThe Attempt at a Solution I know that really the only way to solve this one is to use an integrating factor, and make it into an exact equation. My DE teacher said that to make it into a exact equation you need to take...

Hello, I'm a doctoral student in civil engineering. In my research I came across a differential equation for the net force acting on an object as it impacts a granular medium at low velocities. z'' + a[ z' ]^2 + b[ z ] = c Where a, b, and c are all constants I believe that this equation will...

I am solving a nonlinear ODE in the form of Newton's Second Law. In the equation, there is a Heaviside Theta Function of the function which I am solving (##\theta (x(t)##). Since it is quite troublesome to have both the left side of the ODE and the imput of the ODE to contain function of unknown...

I'm am almost certain that the answer to this question is "yes". If so, what in nature can be a nonlinear traveling wave?

Homework Statement Find whether this system is stable or unstable at the steady state (x1,x2)=(0,0) dx1/dt = -x1+2sin(x1)+x2 dx2/dt=2sin(x2) Homework Equations The Attempt at a Solution z1=x1-0 z2=x2-0 dz1/dt=-z1+z2+2z1 dz2/dt=2z2 Jacobian = [ 1 1 ] [ 0 2 ] so the system is unstable. This...

Dear forum colleagues, I'm looking for universities which have a good research lab in the field of nonlinear optics, located in Canada. Can you please give me some hints or contacts? PS: If anyone need information about Brazil and Portugal, please don't hesitate and just ask! ;) Luis

In my introductory ODE class we have focused mostly on linear differential equations. I know that nonlinear differential equations are much harder to solve, and I am wondering what exactly the "state of the art" methods are for dealing with them, or also what recent developments have been made...

Hi guys! I have a question on applying constraint on Linkage systems. Assumed that there is a two dimensional one-bar linkage, one end can only rotate and one end is free (Such as the figure above, please neglect the damper-spring system if you want). This link can rotate only 180 degrees, not...

I have some experimental circuit element that exhibits both resistance and capacitance, and results in nonlinear I-V curves. can i extract capacitance and resistance of this element just from the I-V curve? or do I need time-axis data? a suggestion on what equation to use to fit this data? thanks.

So I'm reading the Example on page 161 of Differential Equations, Dynamical Systems and an Introduction to Chaos by Hirsh, Smale, and Devaney. I'm not understanding everything. So given the system x' = x + y^2 y' = -y we see this is non-linear. I get it that near the origin, y^2 tends to...

I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems". the question is about the equation \partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3 Part A: with the ansatz u=\sum_{n=0}^\infty a_n cos(nx) show that the...

Can the nonlinear hydrodynamic be ignored in the process of a ship leaning slowly， and the ship capsized ultimately. Thanks！

Given a DE in the general form of either y'' = y^2 or y'' = (y-1)^2, is there a general method to solve these? I separated the equations to get y''(y^-2) = 1 and then integrated, which left me with (-y^-1)dy = (t + c)dt, and then integrated once more. Is this correct so far? I have essentially...

Homework Statement This is not the exact problem that I want to solve but I will use this as a guidance tool: ##y'' - (y')^2 + y^3 = 0## where y is the function of x 2. The attempt at a solution I tried doing a substitution ##u(x) = y'(x)## which leads to ##u' - u^2 + y^3 = 0## where both u...

While I am studying the wave propagation in fluids, the amplitude modulation seems to be governed by the Nonlinear Schrodinger (NLS) equation. In some of the journal papers the nonlinearity parameter, N seems to be of high value (N≈O(104)) and so on. I understand that weak nonlinearity...

Homework Statement By truncating the differential equation below at n=12, derive the form of the solution, obtaining expressions for all the ancoefficients in terms of the parameter \lambda .Homework Equations The ODE is: \frac{\mathrm{d^2}\phi }{\mathrm{d} x^2} = \frac{\phi^{3/2}}{x^{1/2}}...

Hi, everyone. I am having a hard time finding explicit values of non-linear susceptibility tensor values for any sort of crystals. Specifically, I'm looking for values of a BBO crystal, but I would like to know where to find others for my future research. I should say that I am looking for the...

Homework Statement I've determined the dispersion relation for a particular traveling wave and have found that it contains both a real and an imaginary part. So, I let k=\alpha+i\beta and solved for \alpha and \beta I found that there are \pm signs in the solutions for both \alpha and \beta...

This is part of a personal project... I've recently become addicted to modeling various physical systems from scratch, such that I find explicit solutions of position as a function of time, and graph em. But I've hit a glass ceiling trying to find an analytic solution to the 1-dimensional...

Is it possible to solve a nonlinear differential equation of the form below such that the dependent variable y can be expressed as a function of time t? (second time derivative of y) = y + y squared + y cube

Hi, I need help solving this ode, when I try to solve it I end up with a big crazy answer and I believe it should be simpler. (dy/dx)^2=((ay^4)/2)-(a+1)y^2+1 y(0)=0, y'(0)=1 and a is within [0,1]

Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] I am pretty much lost on how to go about solving this. It would be greatly appreciated if someone could point me in the right direction on this. Thanks!

Homework Statement Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] It would be greatly appreciated if someone could point me in the right direction on this. Thanks! Homework Equations The Attempt at a...

Homework Statement A nonlinear spring has a temperature dependent force law, F = -\frac{K}{T}(L-L_o)^3 At a temperature T = T_o and length L = L_o the specific heat at a constant length is C_L = C_o. What is the specific heat at T = T_o when the spring is stretched to length 2L_o...

Hi everyone, I need some help to solve this differential equation. The question states "Use the perturbation or multiple scale method to find the third-order approximate solution for the following system: diff(x(t), t, t)+w^2*x(t)*(1+epsilon*x(t)^2) = 0 " Currently, I am still...

I completed the book 'Nonlinear Dynamics and Chaos' by Strogatz. What will be next good book to learn about nonlinear dynamics?

Hi everyone I am trying to solve non linear heat conduction where thermal conductivity is function of temperature, I am solving it by finite difference method. this is my equation ∂2t/∂x2+∂2t/∂y2 *k(t)= -q (x,y) i have solved the equation taking k(t)= a-b*t,and when i further solved the...

Hi everyone, i am trying to solve and program a non linear differential equation in MATLAB where thermal conductivity depends on temparature.I am trying it to solve by explicit finite difference method. the given equation is ∂2t/∂x2+∂2t/∂y2 *k(t)= -q (x,y) i have solved the equation taking...

i just signed up here so i hope this is the right place. i need to solve a set of 2 non-linear ordinary differencial equations. i tryed using NDSolve but it doesn't really work so I am not sure what's wrong with my code. here is my code (copy paste): c = 0.1; Subscript[sys, B]...

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber \end{equation}...

[SIZE="4"]Definition/Summary Nonlinear optical processes that occur due to the presence of a second-order nonlinear susceptibility are termed 2nd order processes, or three-wave mixing processes. There are four second order processes, second harmonic generation, sum and difference frequency...

[SIZE="4"]Definition/Summary Light propagating through a vacuum will obey the principle of superposition, however this is not generally true for light propagating through gaseous or condensed media. As light propagates through transparent media, it induces a dipole moment on any atoms present...

Hey there, I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present. u′′=au3+bu2+cu the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand. so...

The three non-linear equations are given by \begin{equation} c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0 \end{equation} \begin{equation} s[2.001 *c + 835(1-q)]-2.001*c =0 \end{equation} \begin{equation} q[2.73 + (5.98*10^{10})c]-(5.98 *10^{10})c =0 \end{equation} Using...

A new variational principle is presented in this paper: http://arxiv.org/ftp/arxiv/papers/1112/1112.2286.pdf When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set up the functional as such: $$...

Hello everyone I really need to ask if someone could show me a paper for the maximum gain peak of SBS in HNLF or any parameters having to do with SBS for highly non linear fiber ?? I really tried to look at many papers but nothing concerning Stimulated Brillouin Scattering(SBS) Thank you

Can anyone help me solve this problem? It seemed straightforward at first, but I am not getting the correct answer of -12 Nm. Thank you! A nonlinear spring is modeled by a force law given by F(x) = -10x + 3x^2, where F is measured in Newtons and x in meters. How much work is done stretching the...

Homework Statement Consider a classical one-dimensional nonlinear oscillator whose energy is given by \epsilon=\frac{p^{2}}{2m}+ax^{4} where x,p, and m have their usual meanings; the paramater, a, is a constant a) If the oscillator is in equilibrium with a heat bath at temperature T...

This is a problem from Boyd Nonlinear Optics chptr 1 problem 2. Homework Statement Numerical estimate of nonlinear optical quantities. A laser beam of frequency ω carrying 1 W of power is focused to a spot size of 30μm diameter in a crystal having a refractive index of n =2 and a second order...

Hello. I have a set of ODE where 1) \frac{dv_x}{dt}=\frac{q(t)B}{m}v_y 2) \frac{dv_y}{dt}=\frac{q(t)B}{m}v_x 3) \frac{dv_z}{dt}=0 Following the strategy to solve a simple harmonic oscillator, I differentiate (1) to get 4) \frac{d^2v_x}{dt^2}=\frac{q(t)B}{m}\frac{dv_y}{dt}+q'(t)v_y...

Hello, I'm studying basic nonlinear optics and I would like to solve a couple doubts about (basic) photon interaction. Let a monocromatic (of frequency ω) electromagnetic field propagate through a nonlinear medium and let the third(and higher)-order terms in the relation between the...

I've been doing linear algebra for so long, that I've become quite a dunce at solving nonlinear systems of equations. tl;dr: Can anyone suggest a fruitful plan of attack for the following system of equations? \begin{align} \cos\phi_1 \cos\phi_2 \cos\phi_3 + \sigma \sin\phi_1...

Hi all. It's been a few years since I've posted here, but it's remained a great go-to resource for me. Any time I have dealt with mechanical vibrations, the fundamental frequency was based on a constant stiffness. However, I have never encountered the subject of finding the fundamental...

Hello, So I was hoping to get some help implementing a nonlinear least squares fitting algorithm. Technically this is an extension of my previous thread, however the problem I am having now is correctly computing the algorithm So the problem definition is this: Given two sets of n 3D points...

Hi guys, I need to simulate wave propagation for a nonlinear dispersive wave PDE and since I can't find proper resources for handling nonlinear PDEs numerically, I would appreciate any help and clues. the PDE is in the form of utt-(au+bu2+cu3+duxx)xx=0 Romik Ps: BC: Clamped at both ends IC...

dy/dx=2x+y^2 By the way, methods of solving linear differential equation are useless, such as integrating factor and Bernoulli method.

Hi there, I am currently a student at Heriot-Watt University and have been given a project of deriving a nonlinear equation of motion for an a380 wing engine. The engine is to be considered as a lumped mass attached to the cantilever beam and the main fuselage is considered having translational...

I have the following system of 3 nonlinear equations that I need to solve in python: 7 = -10zt + 4yzt - 5yt + 4tz^2 3 = 2yzt + 5yt 1 = - 10t + 2yt + 4zt Therefore I need to solve for y,z, and t. Attempt to solve the problem: def equations(p): y,z,t = p f1 = -10*z*t + 4*y*z*t - 5*y*t...

Hi, I'm an electrical engineering student starting research in nonlinear optics, and I'd like some good books to do with nonlinear optics. I'm looking for book similar in style to Nonlinear Optics by Robert Boyd as I really quite like that book. Other books I've gone through include Optical...