**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Soliton

Summary

In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.
The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". The name was coined by Zabusky and Kruskal to describe solutions to the Korteweg–de Vries equation which models waves of the type seen by Russell, with the name meant to refer to the solitary nature of the waves and the 'on' suffix mirroring the usage for particles such as

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (24)

Related publications (93)

Loading

Loading

Loading

Related units (10)

Related courses (7)

PHYS-607: Nonlinear fibre optics

Presentation of the different sources of optical nonlinearities in an optical fibre

PHYS-449: Optics III

Réaliser un système d'optique complexe. Comprendre l'interaction de la lumière avec la matière en optique nonlinéaire et optique pulsé. Comprendre les méthodes de génération, d'amplification e de compression des impulsions courtes. Connaitre les techniques de spectroscopies résolues en temps.

PHYS-640: Neutron and X-ray Scattering of Quantum Materials

NNeutron and X-ray scattering are some of the most powerful and versatile experimental methods to study the structure and dynamics of materials on the atomic scale. This course covers basic theory, instrumentation and scientific applications of these experimental methods.

Related concepts (36)

Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applicatio

Korteweg–De Vries equation

In mathematics, the Korteweg–De Vries (KdV) equation is a partial differential equation (PDE) which serves as a mathematical model of waves on shallow water surfaces. It is particularly notable as

Integrable system

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with su

Solitons are stable, non-singular solutions to the classical equations of motion of non-linear field theory. Their energy is localized and finite and their shape remains unaltered during propagation. For this reason they represent particle-like states in field theory. Their mass and their size can be very large compared to those of the elementary particles in the theory. Therefore, a soliton can be viewed as a single particle-like object containing a large number of individual particles. The chiral Abelian Higgs model contains an interesting class of non-topological solitons, that carry a non-zero fermion number NF or Chern-Simons number NCS, which is the same because of the chiral anomaly. They consist of a bosonic configuration of gauge and Higgs fields characterized by NCS and are stable for sufficiently large NCS. In the first part of this thesis we study the properties of these anomalous solitons. We find that their energy-versus-fermion-number ratio is given by E ∼ NCS3/4 or E ∼ NCS2/3 depending on the structure of the scalar potential. For the former case we prove, using some inequalities from functional analysis, that there is a lower bound on the soliton energy, which reads E ≥ c NCS3/4 , where c is some parameter expressed through the masses and coupling constants of the theory. We construct the anomalous solitons numerically for two different choices for the potential accounting both for Higgs and gauge dynamics. Solutions are obtained as a function of NCS and the Higgs mass mH and we find that they are not spherically symmetric. In addition, we outline a relation between the structure of anomalous Abelian solitons and the intermediate state observed in type-I superconductors in external magnetic fields. In the limit of large NCS anomalous solitons can be described in the thin wall approximation, which allows us to remove the Higgs field from consideration. For absolute stability of anomalous solitons, it is essential that the gauge group is Abelian. If the gauge group is non-Abelian, fermions can always be converted to a gauge vacuum configuration with an arbitrary integer NCS. Therefore, if anomalous non-Abelian solitons exist, they could only be metastable. Interestingly, anomalous solitons can potentially exist in the electroweak theory, because this theory contains all necessary ingredients, namely chiral fermions and an Abelian gauge symmetry. In the second part of this thesis we investigate this possibility. Using the numerical solutions for anomalous Abelian solitons as a starting point, we construct the corresponding numerical solutions in electroweak theory. These solutions have a similar structure as the Abelian solitons with the Abelian gauge field replaced by the Z boson field. The charged boson fields W± vanish identically. However, for weak mixing angle θω > 0, the solutions have an associated magnetic field as well, that can be characterized by a magnetic dipole moment mem. Furthermore, the shape of the solutions and the structure of the gauge fields depend on θω. In the last part of this work we analyze the classical stability of the numerical solutions in the electroweak case. It is clear that the solutions are stable in the semilocal limit sin θω → 1, where the Abelian case is reproduced exactly. For arbitrary θω, we consider perturbations in the Higgs field and in the gauge fields Z and A and show that the solutions are stable with respect to these perturbations. For small θω however, the solutions are unstable with respect to the formation of a condensate of charged boson fields W± in the centre of the solution. This W-condensation instability is essentially the same, which also destabilizes the Z-string solution of electroweak theory.

The chiral Abelian Higgs model contains an interesting class of solitons found by Rubakov and Tavkhelidze. These objects carry non-zero fermion number NF (or Chern-Simons number NCS, what is the same because of the chiral anomaly) and are stable for sufficiently large NF. In this paper we study the properties of these anomalous solitons. We find that their energy-versus-fermion-number ratio is given by E ∼ NCS 3 / 4 or E ∼ NCS 2 / 3 depending on the structure of the scalar potential. For the former case we demonstrate that there is a lower bound on the soliton energy, which reads E ≥ c NCS 3 / 4, where c is some parameter expressed through the masses and coupling constants of the theory. We construct the anomalous solitons numerically accounting both for Higgs and gauge dynamics and show that they are not spherically symmetric. The thin wall approximation valid for macroscopic solutions with NCS ≫ 1 is discussed as well. © 2007 Elsevier B.V. All rights reserved.

2007Mikhail Churaev, Xinru Ji, Tobias Kippenberg, Junqiu Liu, Alexey Tikan, Aleksandr Tusnin, Rui Ning Wang

A photonic dimer composed of two evanescently coupled high-Q microresonators is a fundamental element of multimode soliton lattices. It has demonstrated a variety of emergent nonlinear phenomena, including supermode soliton generation and soliton hopping. Here, we present another aspect of dissipative soliton generation in coupled resonators, revealing the advantages of this system over conventional single-resonator platforms. Namely, we show that the accessibility of solitons markedly varies for symmetric and antisymmetric supermode families. Linear measurements reveal that the coupling between transverse modes, giving rise to avoided mode crossings, can be substantially suppressed. We explain the origin of this phenomenon and show its influence on the dissipative Kerr soliton generation in lattices of coupled resonators of any type. Choosing an example of the topological Su-Schrieffer-Heeger model, we demonstrate how the edge state can be protected from the interaction with higher--order modes, allowing for the formation of topological Kerr solitons.

Related lectures (16)