Perhaps this paper may help.

Conference Paper On the Size of a Photon

For this formulatio,s a look to the web site:

  http://www.lescientifique-upvd.com

A plane wave with an infinite extension and a non-zero amplitude owns infinite energy. That contradicts the energy quantum h*f.

Photon-like objects are real massless time-stable and spatially finite physical objects with an intrinsically compatible translational-rotational dynamical structure.

They carry energy- momentum and propagate as a whole in a translational-rotational periodic manner by the speed of light. The corresponding integral action for one period T is given by the Planck-like constant ”h = ET”, where ”E” is the full

energy of the photon-like object. They are composite objects, each one consists of

two time recognizable and energy-momentum exchanging  subsystems

carrying the SAME stress-energy-momentum and being in a state of dynamical equilibrium. The mutually exchanged energy for one period gives the elementary action ”h”. Photon-like objects follow the rule: no translation as a whole is possible without local rotation, and no local rotation is possible without translation as a whole.The adequate mathematics we came to was Extended Lie derivative and Frobenius integrability/nonintegrability theory of geometric distributions.

Maxwell equations can not describe such soliton-like 3d-objects because every component of the two fields (E,B) MUST satisfy the D'Alambert wave equation, which does not admit time-stable 3d-spatially finite solutions.

Appropriate nonlinearization of   of Maxwell charge-free equations (see arXiv:math-phys.1210.8323, pictures on page 288,325, authors: Stoil Donev, Maria Tashkova) gives such solutions. The pre-metric approach to this nonlinearization see in arXiv:math-ph 1603.09360, authors: Stoil Donev, Maria Tashkova.

Here is the corresponding "Motion picture":

The Maxwell equation perfectly well handles these properties. That's an exception, the wave is not dispersive, the phase velocity is always c in vacuum because the rest mass is zero. Though, it is dispersive in a material medium. There are interactions between photons only at very high energy, when particle anti-particle pairs can be created, since the wave equation is linear. Then it is no wonder they pass through each other. Solitons obey another type of superposition principle, but they are self-interacting, and thus can't have a vanishing rest mass, Cf. the kinks of the sine-Gordon equation.

Calling an elementary particle a "soliton",or, perhaps, more appropriately, an "instanton" or a "compacton" is of no value, unless one has identified a nonlinear evolution equation that governs the evolution in space and time of the density of something associated with the particle, e.g., a mass density, a charge density, or the probability density of a localized wave function.  In general, we do not have such equations for elementary particles.  Important steps in this direction have started with the seminal 1974 work of 't Hooft and Polyakov.  An exciting example in recent years has been Manton's work on Skyrmions.  There are many others. Until you identify a nonlinear evolution equation that governs the propagation in space and time of some density associated with a single photon (energy density, or probability density?) it helps nothing to call it a soliton.  Electromagnetic waves are plane wave solutions of the (linear) Maxwell equations.  As such, they represent an entity, the electromagnetic filed, which is completely not localized.  Pictorially, one might say that this is because it is made of an infinite number of photons, scattered all over space.  You do get solitons out of photons, when you form wave packets that are solutions of some nonlinear evolution equation, such as the Nonlinear Schrödinger equation, which is obtained as an approximation from Maxwell's equations in matter, as they become nonlinear for high amplitude electromagnetic fields.  But then, again, the NLS soliotn is a wave packet made up of many photons.

Hello Yair

What is in a name?

I checked definition, history and current usage of "soliton" and it seems that in many circumstances the term makes sense, even in absence of explicit references to equations. The Zabusky-Kruskal paper where the word was originally introduced did not put equational restrictions for its use. I even know that the title "Phenomenological Solitons" has been given to a book.

Since it makes no sense to assume intentions to establish prohibitions, perhaps you meant that a satisfactory non-linear equation describing photons is missing in Physics, in which case I fully agree. In fact my question is a request of information on such equation --not yet known to me-- and to discuss its virtues. Or exchange ideas on difficulties: why the equation has not yet been found. Or thoughts on impossibilities: why it cannot exist, if that were the case.

But one fact seems true: I did succeed in conveying certain meaning with "Why photons are not considered three dimensional electromagnetic solitons?".

Allow me to add some general comments on non-linear evolution equations.

When (ordinary) evolution equations appeared --linear and non-linear, one, two or more dependent variables, first, second and higher orders-- they were mostly what we now call "vector fields in Euclidean spaces" (the jet spaces of Euclidean spaces are themselves Euclidean). Thus the unknown functions they sought were functions of time, taking values in an Euclidean space. Later it was found that vector fields make sense in smooth, finite dimensional manifolds $M$. The unknown function is then a function of time taking values in $M$. Here the dependent variable is Euclidean only locally. When $M$ is not globally diffeomorphic to an Euclidean space then the vector fields --equivalently the equations-- are non-linear simply because linearity does not make sense in $M$.

The situation is similar with partial differential equations. Practitioners of PDEs, linear and non-linear, usually formulate their equations in infinite dimensional linear spaces. The equations are in $L^2$, or in $C^\infty$, etc. Or explicit reference to the actual space may be missing, particularly if practical results are the target. But what we usually have is that the unknown function, say $u$, belongs to an infinite dimensional linear space of functions. They can be added and multiplied by scalars. Partial differential equations of evolution type can often be considered as vector fields on these linear function spaces (first order) or on their jet spaces (higher order). But more generally $u$ may lie in a non-linear infinite dimensional manifold where addition of the the various $u$ make no sense. This non-linear framework is called Global Analysis. In my modest opinion it may be within Global Analysis that a non-linear equation for "photons are solitons" can be found. I am attempting the task, but unfortunately have not yet reached the goal. Such are my reasons, hopefully not vain, for requesting information.

Most cordial regards, Yair, and with great respect for your time and comments, which I am sure everyone else equally appreciates

Daniel Crespin

The definition of soliton makes explicit use of the features of solutions of an evolution equation, since they are functions of space and time, whatever their geometry or topology, and irrespective of the structure of the space of functions or the target space. Linear equations can't exhibit solitons, from their very definitions.

What about single photons, emmited by a two-level quantum system, they will travel in space with a constant shape of the decaying exponent. (reference : http://www.eg.bucknell.edu/physics/ligare/ajp_visualization.pdf). I think, they are very similar to solitons, but why they are not?

Article The calculated photon: Visualization of a quantum field

Elementary particles could be described by a soliton-like solution. For example, the electron itself could be a soliton-like state. But quantum mechanics describe its evolution with no information on its shape evolution, since the electron is discussed as a mass piont with certain mass, charge and spin. This is the case like photon. 

 Stable localized states are  usually  seen in nonlinear systems, such as optics in Kerr metarial, BEC, or water system. Therefore, we usually assume the stable localized state could be soliton solution of a nonlinear evolution equation. In fact, linear system could also admit stable localized states. But our knowledge on stable localized state in linear theory is less than nonlinear theory. In quantum mechanics, coherent state is also a stable localized state, which admits the smallest uncertainty relation. We have calculated the uncertainty relation of classical bright soliton of Nonlinear Schrodinger equation, which is very close to the one of the coherent state. What the reason of this? or are there any hints for describing photon or electron's fundamental property?

Coherent states are non-dispersive states in a harmonic oscillator.  The non-linear term in the non-linear Schrödinger equation in free space brings the analog of the harmonic potential.  With the Schrödinger equation, we can only describe the properties of the Schrödinger equation.

Not answering this question, do many soliton photons make a plane TEM wave? The number of photons per second is equivalent with the amplitudes of the TEM wave fields, and the energy content of the photon is equivalent with the TEM wave frequency. The emission of just 1 photon soliton (in stead of many) might represent the TEM wave with the lowest field amplitude possible for that particular frequency.

I do not understand why the energy of a photon depends on the TEM wave frequency. Perhaps there is a standard photon/soliton with fixed energy (like electron charge, it is always the same), then there is a fixed number of unit photons squeezed in 1 wavelenght of a TEM wave with particular frequency and amplitude. If the frequency is doubled, then twice as much unit photons will travel per second, which gives a false impression of the 'unit' photon energy dependence on wave frequency.

For a lowest possible TEM wave amplitude we might have 1 or many photon-solitons with unit energy, that constitute 1 wavelenght of the macroscopic TEM wave, and this 'number' does not depend on the TEM wave frequency (just free thinking here).

My view on the nature and formal representation of photon-like objects can be found in our book "Geometric view on photon-like objects" (ArXiV/1210-8323). The basic moment is to replace the linear vacuum Maxwell equations with corresponding physically motivated nonlinear equations, representing the time-stability of photons as result of appropriate time-stable energy-momentum exchange process between two recognizable subsystems, formally represented on Minkowski space-time by the couple of bivectors (F,G) and their flows across the standard volume 4-form, which flows are formally represented by two differential 2-forms.

A theory of bosons and fermions can be based on nonlinear electromagnetics. Applying Maxwell’s curl equations in a space region in which energy density follows the function sech2(r), it becomes obvious that both bosons and fermions are solutions of the nonlinear Helmholtz equation, and that fermions have quantized structure.

As Yair Zarmi notices, “Calling an elementary particle a "soliton", … is of no value, unless one has identified a nonlinear evolution equation that governs the evolution in space and time of the density of something associated with the particle...”

I may add that the process of emission of photons is completely missing from modern physics. This process can be modeled with the Kadomtsev–Petviashvili (KP) equation.

These things are a part of a larger theory. A link to it can be found on my profile page, in the project log.

In my opinion, it should first be clarified how a point source emits electromagnetic energy. It makes little sense to discuss or measure this process at very small wavelengths. Whenever the wavelength and the distance between possible sources of emissions are much smaller than the smallest sensors, everything somehow becomes punctiform and eludes the analysis.

NMR provides the best conditions: The wavelength of the generated photons (2 kHz