All Questions
Tagged with solitons non-linear-systems
16
questions
2
votes
1
answer
643
views
Dimension analysis in Derrick theorem
The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe:
What I don't understand from the above statement:
why $e(\mu)$ has minimum for ...
- 207k
1
vote
0
answers
46
views
Wave propagation speed in non-linear differential equations
Could it happen than a solitary travelling wave (soliton) had a different propagation speed when seen from the usual wave equations from that in a non-linear equation. I mean, suppose a solution $F=f(...
- 207k
1
vote
1
answer
124
views
Intuition behind focusing vs defocusing in integrable systems like NLS, KdV, mKdV
The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description)
Non-Linear Schrodinger equation
Korteweg-de Vries equation
...
- 11
0
votes
0
answers
39
views
Do you know about any book which discusses solitons in Benjamin-Ono Equation?
Benjamin-Ono equation is an integrable equation with soliton solutions. There are many books on solitons. The ones I know about mainly discuss solitons in Korteweg de-Vries and related equations. Do ...
- 2,646
0
votes
1
answer
66
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Doubt on Lax formulation of Korteweg–de Vries equation
The Korteweg–de Vries equation is given by:
$$\frac{\partial u(x,t)}{\partial t}-6u\frac{\partial u(x,t)}{\partial x}+\frac{\partial^3 u(x,t)}{\partial x^3}=0$$
This equation can be formulated using ...
- 605
0
votes
1
answer
126
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Periodic traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ for a $\phi^4$ model
Consider
\begin{equation}\label{1}
\partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1)
\end{equation}
the $\phi^4$ model.
I know that
$$H(x)=\...
- 103
6
votes
3
answers
525
views
Link between integrability and soliton solutions
I have been doing some research on the properties and dynamics of solitons (in particular, solitons in superfluids) and several works and papers mention the link between solitonic solutions and ...
- 348
3
votes
0
answers
118
views
How to use Belinsky-Zakharov transformation
I know it might be trivial. When using BZ transformation [1] to generate soliton solutions of Einstein’s field equations, one need a seed solution $g_{0}$ which gives $A_{0}$ and $B_{0}$. Taking them ...
- 341
2
votes
0
answers
261
views
Solving Higher-Order Kinetic Energy Term (Gross-Pitaevskii equation) [closed]
Consider now propagation of non-linear waves in one-dimensional chain of dimers governed by the non-linear Schrödinger equation for the normalized wave envelope $\Psi(x,t)$,
$$
i \frac{\partial \Psi}{...
- 139
1
vote
0
answers
107
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 ...
- 153
12
votes
2
answers
7k
views
Explanation of the waves on the water planet in the movie Interstellar?
We will ignore some of the more obvious issues with the movie and assume all other things are consistent to have fun with some of these questions.
Simple [hopefully] Pre-questions:
1) If the water ...
- 15.6k
2
votes
0
answers
62
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 ...
- 207k
2
votes
1
answer
89
views
Solitons and its infinite extension
A soliton, for example the KdV equation solution, has the profile proportional to a hyperbolic secant squared ${\text{sech}}^{2}(x-ct)$. And since it is hyperbolic it has an exponential dependence, so ...
- 5,097
2
votes
0
answers
96
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, ...
- 14.5k
2
votes
0
answers
197
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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 ...
- 11.5k