Questions tagged [solitons]
Solitons are self-stabilizing solitary wave packets maintaining their shape propagating at a constant velocity. They are caused by a balance of nonlinear and dispersive (where the speed of the waves varies with frequency) effects in the medium.
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What are the two different $\mathbb{S}^n$ in the construction of the homotopy group $\pi_n(\mathbb{S}^n)$ that classifies topological defects?
According to Mukhanov's Physical Foundations of Cosmology,
Homotopy groups give us a useful unifying description of topological defects. Maps of the $n$-dimensional sphere $\mathbb{S}^n$ into a ...
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Peak splitting in one-component reaction–diffusion equations
I am studying a one-component reaction–diffusion equation:
$$ \partial_t u(x,t) = D \partial^2_x u(x,t) + R\left(u(x,t)\right)$$
Looking at systems that exhibit a peak solution (solitary localized ...
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Spontaneous discrete symmetry breaking always implies domain walls
I've read several times that if a discrete symmetry is spontaneously broken, then there exist domain walls that interpolate between the different vacua. However, Weinberg says that if the former ...
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Sum of topological charges is the Euler characteristic
I have seen many places claiming that the given a collection of topological defects on a 2-dimensional surface, the sum of the topological charges is $2\pi\chi$ (where $\chi$ is the Euler ...
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Stability and topological charge of kink (anti-kink) solutions (soliton)
I am reading the book << Gauge theory of elementary particle physics >>. In chapter 15, it presents a model having finite-energy solution.
First, we have a $1+1D$ spacetime model
\begin{...
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Must a field approach one of its vacua to have finite energy?
I'm reading these Cornell lectures on solitons (link doesn't work right now, but it just worked yesterday), and I can't seem to prove what I thought would be a simple analysis exercise.
Namely, ...
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Soliton solution of the NLS equation
My understanding of soliton - it is a moving pulse in a medium which does not change its structure with time. It has other properties like no interaction with other solitons (this could certainly be ...
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Completely Integrable Frustrated Lattice Systems
The Toda lattice is a prime example of a lattice system that is completely integrable, in the sense that it admits a Lax pair,
https://doi.org/10.1143/PTP.51.703,
making it easy to find soliton ...
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Set of zeroes as coset space
I am currently studying Chapter 6 of Coleman S. - Aspects of Symmetry.
We study a spontaneously broken gauge theory in two spatial dimensions where the Lagrangian reads:
$$
\mathcal{L} = -\frac{1}{4}...
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Intuition about ADHM construction
I'm trying to understand reasons, why self-dual Yang-Mills equation can be reduced to algebraic equations. It's seem like a miracle.
In article Construction of Instanton and Monopole Solutions and ...
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Doubt on Lax formulation of Korteweg–de Vries equation
The Korteweg–de Vries equation is given by:
$$\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$$
This equation can be formulated using ...
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Spin of skyrmion
Baryons can be considered as solitions in Skyrme model(See also this post.):
Such Lagrangian haven't any information about number of colors. Bosonic or fermionic nature of baryons depends on number ...
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Periodic traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ for a $\phi^4$ model
Consider
\begin{equation}\label{1}
\partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1)
\end{equation}
the $\phi^4$ model.
I know that
$$H(x)=\...
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Theory on domain walls
In Baryons in Quantum Chromodynamics, Zohar Komargodski have slide:
I wanna understand:
Why domein wall can have nontrivial worldvolume theory?
When such solitonic objects have interior degrees of ...
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$\phi^4$ theory kinks as fermions?
In 1+1 dimensions there is duality between models of fermions and bosons called bosonization (or fermionization). For instance the sine-Gordon theory $$\mathcal{L}= \frac{1}{2}\partial_\mu \phi \...
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