Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Perturbation method for a phase coexistence system

+ 2 like - 0 dislike
954 views

The differential equation with $k$ dimensions has the form $\frac{d\mathbf{F}}{d\mathbf{x}} = \mathbf{F}(\mathbf{x},\epsilon)$, where $\mathbf{x}$ is the variable vector and $\epsilon$ is a parameter.  I focus on the steady state, which yields the solution of $\mathbf{F}(\mathbf{x},\epsilon)=0$. Assume that the equation has a perturbation solution $\mathbf{x} = \mathbf{a}\epsilon +\mathcal{O}(\epsilon^2)$ for the parameter $\epsilon\rightarrow 0$ and another perturbation solution $\mathbf{x} = \mathbf{b}_0-\mathbf{b}_1\epsilon^{-1} +\mathcal{O}(\epsilon^{-2})$ for the parameter $\epsilon\rightarrow \infty$.

Now I already know (from numerical simulations) that the above two kinds of solutions i.e., $x_i\rightarrow 0$ for $i\in I$ and $x_j\rightarrow b_0$ for $j\in J$, coexists for the parameter $\epsilon$ of a specific value interval. ($K=\{1,2,..., k\}$ and the sets $I,J\subseteq K$, $I\cap J=\emptyset$).
My question is how to identify the solution sets $I$ and $J$ by perturbation method? or I should consider its Hamiltonian?

asked Jul 19, 2019 in Computational Physics by anonymous [ no revision ]
edited Jul 19, 2019

1 Answer

+ 2 like - 0 dislike

Make an Laurent series ansatz $x=\sum_{k\in Z} z_k \varepsilon^{k}$, insert it into the differential equation, expand into a Laurent series and compare coeffiicients to get differential equations for the $z_k$. Then you can analyze a truncated version of this system to find out which coefficients can be set to zero without impairing solvability.

answered Jul 19, 2019 by Arnold Neumaier (15,787 points) [ revision history ]
edited Jul 21, 2019 by Dilaton

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsO$\varnothing$erflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...