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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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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Moduli normalisation

Consider the action for a massless scalar $$ S = \int d^4x\big[\frac{1}{2}(\partial \phi)^2-V(\phi)\big], $$ $V(\phi) = \frac{g^2}{4}(\phi^2-v^2)^2$, which admits domain wall solutions centered at $...
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Can two plasma Double Layers interact with each other?

I have been recently reading about solitons propagating in plasma and found that two solitons interact elastically in a KdV (weakly non-linear) framework. I was also able to find research papers on ...
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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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Is it possible to find the linearized operator through the conserved quantities?

Let $$u_{tt}-u_{xx}= u-u^3 ,\: (t,x)\in \mathbb{R}\times \mathbb{R}.$$ I know that the linearized operator around a solution $u$ is given by $$\mathcal{L}=\frac{\partial^2}{\partial t^2}-\frac{\...
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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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Quantum solitons: derivation of $ \int {\phi^\prime}^2 dx = M$ using Lorentz invariance

I was reading through page 10 of this document (Chua, 2017) on quantum solitons, and came across the following statement relating to the equation for kinetic energy $$T = \left(\frac{da}{dt}\right)^2\...
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An interesting observation: Ordered, up and down movement of vortex rings in water

I was watching a video on David Tong's research work when I stumbled upon a peculiar movement of vortex rings in water. Around the 1:20 time mark, Baths and Quarks: Solitons explained, David Tong uses ...
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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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In which representation are monopoles of grand unifying theories classified?

In the context of grand unification theories A.Zee's book states that $SU(5)$ (or $SO(10)$ if $SU(5)$ is considered as outdated as GUT candidate) as GUT and as spontaneously broken non-abelian gauge ...
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Energy of a field configuration in General Relativity

I'm currently studying the article of Coleman and de Luccia about the bouncing configurations in an empty universe (DOI: https://doi.org/10.1103/PhysRevD.21.3305). In particular, they consider in the ...
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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 ...
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What does topologically stable mean?

I am working on an article about skyrmion manipulation and it is written that those particles are "topologically stable particle-like spin configurations that carry a characteristic topological charge ...
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How does the $U(1)$ global symmetry break in the gauged $XY$ model?

I'm studying the particle vortex duality, and I'm confused how we're able to say that in the Coulomb phase, the "hidden" $U(1)$ global magnetic symmetry spontaneously breaks. gauged XY model: $\...
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Will tsunami waves travel forever if there was no land?

If there was no land for tsunami waves to collide with, can the waves travel around the globe for forever?
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How solitons are related to particle physics?

Recently, I read a paper about introduction to solitons. Author said that the solutions of sine-Gordon equation can be candidate for modeling elementary particles and there are some applications in ...
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Different between droplet and a soliton

I am working on the droplet state in a Bose-Bose mixture. I have a question about the difference between the droplet liquid state and the soliton state: How we can treat a droplet state? And how do we ...
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Why do we consider solitons as a composite object?

Can someone explain why do we consider solitons as a composite object? I know that there are dual theories which the role of fundamental and solitonic objects can be mapped to each other, but I can't ...
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Can inhomogeneity in the medium accelerate particles

Suppose I have a charge which is moving in through a medium with constant velocity. Now, what will happen to the charge as it encounters an inhomogeneity in density? whether it will accelerate or ...
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What does the motion of water in tsunamis look like?

This is what normal wave motion looks like. Do tsunamis that travel at 60mph look any different?
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Photon number in optics

What does the term, photon number, mean in optics? I came across the term in research papers on squeezed light. One such line in a research paper reads: Quantum solitons consist of linear ...
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Topological solitons in general dimension

Let's begin with a simple model of a field theory: $$ \mathcal{H} = \int ( \nabla \phi ) ^2 $$ where $\phi$ is an angle valued field defined on some space. We suppose for the moment to freeze out ...
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Topological properties of dark solitons in superfluid systems

In the study of superfluid systems, vortices are often referred to as "topological excitations", because the winding of the phase of the superfluid order parameter around a vortex is a topological ...
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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 ...
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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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Asymptotic behaviour of soliton-antisoliton solution for the Sine Gordon equation

The question isn't about any actual homework, it's rather a (probably simple) intermediate step I've encountered on Rajaraman's Solitons and instantons : an introduction to solitons and instantons in ...
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Violation of Derrick's theorem for finite energy, time independent solutions?

How are vortices the finite energy time independent solutions for 2+1 dimensions abelian higgs model? Doesn't it violate derricks theorem that there are no finite energy time independent solutions in ...
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Non-chiral skyrmion v.s. Left/Right chiral skyrmion

A skyrmion in a 3-dimensional space (or a 3-dimensional spacetime) is detected by a topological index $$n= {\tfrac{1}{4\pi}}\int\mathbf{M}\cdot\left(\frac{\partial \mathbf{M}}{\partial x}\times\frac{...
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Are static black holes solitons?

If we start with the Einstein-Hilbert action with no matter, and consider time independent finite energy field configurations, then any static solution (e.g Schwarzchild metric) seems to be a soliton-...
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What is the intuition for topological currents?

The reason for topological stability of a kink solution in scalar field theory in $1+1$ dimensions is the fact that the finite energy scalar field cannot be continuously deformed into a vacuum. How ...
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KdV suggests a connection between waves in shallow water and the potential in the Schrödinger equation. What is the intuitive explanation?

The KdV equation $$v_t+\frac{1}{4}v_{xxx}-\frac{3}{2}vv_x=0$$ was originally invented to model waves in shallow water. However, it is well known that it also has applications in quantum mechanics. ...
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M branes/D branes are solitons?

I'm really confused. In M theory/String theory, the fundamental objects are M/D branes. However, branes by defintion are just solitons. Solitons are just waves that maintain there shape. So if a ...
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Is transition between field configurations a tunneling process?

I'm considering D=1+1 kink solution here. Given a D=2 theory with $\mathbb{Z}_2$ symmetry, there are 4 different mappings (or 2 distinct sectors---trivial and kink) from spacetime manifold (or just a ...
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Applications of Optical Solitons

It is well known for the past 50-60 years that intense laser beams can form into soliton/solitary waves. Those exist either spatially in CW beams or temporally in ultra-short pulses, and their ...
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Do solitons in QFT really exist? [closed]

In general, solitons are single-crest waves which travel at constant speed and don't loose their shape (due to their non-dispersivity), and there are many examples of them in the real world. Now in ...
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Why does this condition guarantees there exists only a finite number of discrete energy levels?

I'm reading section 2.2.1 of the book Solitons, Instantons and Twistors by Maciej Dunajski. The section is on the subject of direct scattering. It is claimed that, considering Schrodinger's equation ...
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Gauge potential over $\mathbb{S}^4$ vs. $\mathbb{R}^4$

NICHOLAS MANTON in "Topological Solitons" says "One may also regard the gauge potential as a connection on an $SU(2)$ bundle over $\mathbb{S}^4$, with field strength $F$. The fact that we can ...
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478 views

Finding the energy of a solution to the Sine-Gordon equation

I am delving into Quantum-Field Theory, and am stuck trying to work out how to compute the energy of a soliton solution to the Sine-Gordon equation in 1-1 spacetime. I start with the Lagrangian ...
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Are there general Soliton-Instanton correspondence?

In the symmetric double well potential, the solutions in $1+1$ static and real $\varphi^4$ theory, are solitons. However, we know that such theories are "dual" to one dimensional real $\varphi^4$ ...
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Systems with 'many' conserved quantities

The classical justification for the microcanonical ensemble relies on the fact that most many-body systems have just a 'small' (typically finite) number of conserved quantities (i.e. they violate ...
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Can massive particles be seen as soliton solutions?

I wonder if the common relativistic wave equations contain a sort of soliton solutions, which might be considered as particle localisations.
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Toda lattice solution for different algebras

It is well-known that Toda systems (Toda field theory) can possess different algebraic structure based on Cartan Matrix in the Hamiltonian's potential. But all solutions I have seen were written only ...
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Boundary condition for solitons in 1+1 dimensions to have finite energy

Suppose a classical field configuration of a real scalar field $\phi(x,t)$, in $1+1$ dimensions, has the energy $$E[\phi]=\int\limits_{-\infty}^{+\infty} dx\, \left[\frac{1}{2}\left(\frac{\partial\phi}...