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Hi,
I am thinking about Solitons modeling and would like to know which “out of the box” software products are readily available and how they compare with developing one’s own Linux/Unix based package e.g. in C/C++ or Java??
Replies from Experienced N Solitons modeling researchers with links will be much appreciated!
Open Source solutions are Always Preferred ones.
Best regards,
George Yury Matveev
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I'm searching for new collaborations. I'm focused mostly on numerics on obtaining solitons bifurcating from band edges and localized modes. I'm specialized in shooting methods with more than one parameter, NR optimized for CUDA obtaining solutions, parametric curve step to obtain branches that cannot be described by a function. Optimization of split-step FFT dynamics with coupled Nonlinear partial differential equations, Ex: SHG was the harder one. I worked previously with V. Konotop, F. Abdullaev and B. Malomed. I was first author on all papers. They provided the equations and researched on obtaining solutions.
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To get a soliton train pulses from a ring laser cavity host a Saturable Absorber , we need to make a balance between the dispersion of the cavity active and passive elements . What is the EDF length that have to use and SFM 28 for example ?
Thank you for your advises !
Regardes .
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It is well confirmed that the nonlinear evolution equation is widely used as a leading mathematical equation for describing the physical significance of many physics branches. But it is a challenging task to solve them. Recently, many researchers have proposed various types of analytic methods to construct the soliton solutions. Unfortunately, most of these methods lead to equivalent solutions. In my opinion, the methods with the equivalent solutions of NLEEs are not helpful for further verification in the laboratory. That is why my discussion topic is "Which are the most appropriate analytic methods to construct the soliton solutions for nonlinear evolution equations?".
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It's ok, sir.
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Hello scientific community,
In your point of view, is it possible to include the terms of variation of the geometry of a channel in the analytical expression of the Serre-Green-Naghdi soliton?
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The geomorphic relationships known as hydraulic geometry In general Hydraulic geometry deals with variation in channel characteristics in relation to variations in discharge. Two sets of variations take place: variations at a particular cross section (at-a-station) and variations along the length of the stream (downstream variations). Characteristics responsive to analysis by hydraulic geometry include width (water-surface width), depth (mean water depth), velocity (mean velocity through the cross section), sediment (usually concentration or transport, or both, of suspended sediment), downstream slope, and channel friction.
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I am looking to understand what is soliton in qft, not in optical devices because it seemed to me that solitons look very different in optical field than in qft, but I am trying to understand what this means and I am very confused about topological solitons , the solitons are like taking L = 1 2 ∂µφ∂µφ - U (φ) and after use a metric like mikosky metric, after this I could use euler lagrange for the field of motion equation and this would give me a non-relativistic lagragian , and the problem is that I cant get the static solution because I don't know how to do it, are the solitons a static solution for some potential? , Maybe solitons should exist only in the 1d + 1d field which has local symmetry u (1) or are there solitons in 3d + 1d? ,what means solitons in BRST quantazation bacuase i fell that brst dosent have any relatiship with solitons.
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Thank you, I have been studying qft for my pre-university research project for my application, thanks for the comments.
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Is it possible to reformulate general relativity in terms of elasticity theory assuming spacetime is some isotropic elastic continuum? This will also lead to concepts of spacetime strains, stresses and stretching forces, also expressed as tensors in elasticity theory. Curving of spacetime would have a meaning of displacement from undisturbed state. If this is the case, what would be an analog of Young Module E of space?
The question is related to the concept of Atomic Solitons and AString Spacetime Quanta composing spacetime like in image attached
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I do not think that the Millette theory about Elastodynamics of the spacetime continuum introduces any extension to GR, such that classical GR might be seen as a lower order approximation of such theory. Instead, Millette just propose a new viewpoint in which spacetime behaves as a matter field itself, so providing a stress-energy tensor for it to be used as a source in the GR equations. In my opinion this should not appear as that strange, being the same GR field equations self-consistent. However, this theory shows some very challenging points:
1. it is a continuum theory, that would be normal at large or cosmic scales, but the author projects the continuum condition also at microscales... the realm of QM and QFT.
2. as a consequence of the above point, QM schema looks completely concerned, the wave-particle duality is seen as the way in which strein waves of the spacetime "medium" propagates; the same mass of a particle is viewed as a localized dilatation of the spacetime at the wavefront of a propagating strein wave.
Apart from the fact that that theory seems to give the old Ether concept a new life, which is of course questionable due to SR, the whole basis of the idea has some appealing aspects that could worth some detailed discussion, before expressing any definitive judgement on it.
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All mountain ranges are distinctly wavy, with sequential peaks and valleys, which implies some kind of wavy process during their formations.
The question to geologists:
- what process forms those geological waves; why waves?
- what the conventional geological theory (orogeny?) tells about WAVY mountains;
- is there a specific wave pattern (like number of peaks) in mountain ranges?
In a book Soliton Nature (https://www.amazon.com/-/e/B082F3MQ4L), we published the hypothesis of 'solitonic geological wave packets'
and YouTube video
But, not being specialist in geology, unsure about validity of solitonic hypothesis.
Any help with references to solid researches would be appreciated.
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This is how a profile of some mountain ranges is described by sech function following from Nonlinear Shrodinger Equation describing envelope solitons in.many branches of physics.
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How to distinguish between travelling wave solutions and solitary wave solutions of nonlinear dispersive equation like KdV-Burgers' equation? While analysing the wave propagation of solution figue, how actually one identifies it as soliton?
Thanks in advance
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Dear Amitha,
Travelling waves arise naturally in many physically systems, usually described by partial differential equations. Solitary waves, also known as 'solitons', are are a particular class of travelling waves that have special properties.
Solitons can usually propagate over large distances without dissipation due to certain nonlinear effects cancelling out dispersive effects. They also have the additional property that they can interact with other solitons such that they emerge following a collision without changing shape, apart for a small phase change.
John Scott-Russel was the first person to report, in 1844, the sight of a solitary wave on the Edinburgh-Glasgow canal. It travelled approximately 2 miles at around 9 miles/hour.
There is a very comprehensive literature on the subject, and a famous equation that sustains solitons is the Korteweg-deVries equation - see Wikipedia entry. Many additional resources will be found on the Internet.
I hope this helps a little.
Kind regards,
Graham W Griffiths
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A soliton (such as, KdV soliton) has to satisfy an infinite number of conservation laws. What are the physical names and significances of such solitonic conserved properties? Please explain elaborately with examples, illustrations and references.
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Thanks a lot for nice reply. Please send some relevant reference books, papers, or links.
Best wishes and regards.
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I would like to know if soliton pulses are still employed in existing fiber optic communication systems. If yes, then please mention some of the existing commercial systems that use soliton pulses.
Considering the special conditions that are required for propagation of soliton pulses, I think it would be quite difficult to maintain soliton pulses in a WDM system.
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I'm sorry I'm not aware of any article mentioning this piece of info explicitly.
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About zero dispersion fiber and pulse shaping.
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what is the approximate speed of the soliotn in fiber optics? OR Does it depends on the fiber nonlinearities ??
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why do we normalized propagation distance and time , pulse ,in nonlinear Schrodinger equation ?
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Welcome
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what is advantage of dark solitons in comparison with bright solitons ?
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Bright and dark solitons are usually considered from the perspective of modulation instability of the carrier wave train: bright solitons are observed when the carrier wave is unstable with respect to long-wave modulations, while dark solitons are observed when the carrier wave is modulationally stable.
For details see the following paper
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Solitons is the common but we are changing the structures which are also based on the common photonic crystal. Is there possibility of same kind of soliton in all three structures.
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In general, a soliton wave is a nonlinear localized wave possessing a particle-like nature that maintaining its shape during propagation even after an elastic collision with another soliton wave. The possibility of soliton propagation in the anomalous dispersion regime of an optical material was predicted by analyzing theoretically the nonlinear Schrodinger equation (NLSE).
In optics, soliton wave can arise due to the balance between Kerr nonlinear effect and dispersion effect (GVD). Based on confinement in the time or space domain, one can have either temporal or spatial solitons. This induces to an intensity-dependent refractive index of the medium and then leads to temporal self-phase modulation (SPM) and spatial self-focusing. A temporal soliton is formed when the SPM effect compensates the dispersion-induced pulse broadening. In the same way, a spatial soliton is formed when the self-focusing effect counteracts the natural diffraction-induced pulse broadening.
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Paper offers simple formula for spacetime quanta as solitonic kink-like function AString(x) which satisfy the equation AString'(x) = AString(2x+1) - AString(2x-1) and capable to represent direction x in space as a superposition of quanta x = … AString(x-2s) + AString(x-s) + AString(x) + AString(x+s) + AString(x+2s)… as shown in pictures. AString is an integral and compounding block of well-known finite Atomic Function up(x) introduced in 1971 by V.L. Rvachev.
The question to Spacetime Quantisation (STQ) specialists : is there other alternatives to represent a spacetime continuum via other kind of functions?
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To use the Concrete Volume Calculator, simply enter the width, length, and thickness of your pour, then click on the Calculate button. The calculator will estimate the number of cubic yards of concrete that will be required.
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The light,optical pulse and solitons propagation in nonlinear medium? Specify the main physics concepts? What are the necessary conditions for propagation and to develop other effects .
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A classic containing your answer is enclosed as
J. R. Taylor (ed), “Optical Solitons – Theory and Experiment”, Cambridge University Press, Cambridge, UK (1992).
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Recently, we have derived a new equation from Geophysical Fluid mechanics with the following presentation in the attached file.
We wonder about two facts:
(1) When is it integrable?
{2} Could we find the solutions for the equation? such as the solitons, N-solitons etc..?
Have you ever encountered the equation, and I am looking forward to your help. Thank you for your suggestive references!
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It is in continuation with our earlier discussion on the topic. Since, derived in a realistic geo-fluidic configuration presumed to be closed by usual basic equations, such a derived PDE is indeed believed to exist in Nature without loss of any usual conservation law. The Painleve analysis may be executed to detect and confirm global integrability. An asymptotic analytic (approximate, traveling wave) solution may also be tried to check further reliability and validation of the derived PDE.
The following book may be useful for the purpose:
"Nonlinear PDE's in Condensed Matter and Reactive Flows",
edited by
Henri Berestycki et al.
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Their are number of solitons are possible in photonic crystals. Some of them are available in literature.
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I don't think any kind of solitons can happen in photonic crystals, because solitons require non-linear medium and photonic crystals are from linear physics.
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Non-Linear waves propagating in plasmas
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It implies the shape conservation property of the propagating KdV soliton. In other words, the plasma perturbations propagate in such a way that the strength of the nonlinear wave steepening is always proportional to the linear dispersive wave broadening on the spatiotemporal scales of our observation.
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which expands theory of solitons to new domains - similar to your project.
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Hi Dimitri, can you download the paper from the link above?
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Is this meant as solitons? I think that I read that EM field does not have solitonslike solutions in most cases. But if they are solitons how is their ability accounted for to 'feel' entire space in almost 0 time (as is evident from Feynman trajectories approach)?
In a box the exitations imerge immediately and comprise the whole length of the box. So there is a probability to detect a photon far from the source. How is this to be consistent with the constancy of speed of light c?
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Dear Ilian,
Transient processes are not considered for quantum transitions. Photons creation or phonon it does not matter.
Quantum theory is a theory of metamorphosis. Each quantum process is metamorphosis of initial state to final state. Even if initial and final states are the same, the process is considered as a sort of metamorphosis.
If one consider field in each point as oscillator, these oscillators have to be coupled, because one CAN NOT initiate independent oscillations of each oscillator. If one consider each standing wave as oscillator, these oscillators ARE independent.
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Many nonlinear equations are solved for soliton solutions using Hirota bilinearization method but those equations are proven to be integrable either in inverse scattering sense or Lax pair method. Is it correct to use the Hirota bilinearization for nonintegrable systems? 
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A short answer is: yes, it is, because the existence of HIrota bilinear representation per se does not guarantee integrability. It more or less guarantees existence of a solution of a special form that in many cases can be interpreted as a two-soliton solution. Integrability, however, requires existence of four-soliton solution, see e.g.
Hietarinta, J. Hirota's bilinear method and its connection with integrability. Integrability, 279–314, Lecture Notes in Phys., 767, Springer, Berlin, 2009.
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  Domain walls have different shape and functions in ferroics, so are they topological defects?
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I believe they are topological defects in one-dimensional system.
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pedestal energy of ultra short soliton
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You have to measure the AC trace of the soliton pulse at first. Then, you should fit the pulse with a Sceh^2 profile. Finally, you can calculate the pedestal energy according to the area.
Please see Fig 7(b) in J. Lightwave Technol. 35, 17, 3780, 2017.
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I am doing research on evanescent fields and I'm curious as to whether or not a decaying exponential field can have a complex shape if it passes through nanolithography scale structures. (Examples of papers that state this would be helpful) Perhaps someone with COMSOL (or a similar simulation software) could help me out?
From thinking about it I'm fairly certain it would, say if you had in a simple case an evanescent field generated at an interface where there was a nanolithographic step, you would probably get a transition at the step where the field would not have the same penetration depth as it does in either part of the step.
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Well, I don't necessarily want to simulate it as I am making a device that I think will be capable of measuring it. It's just that I'd like to make sure it's possible for that to occur so that making a device to measure it makes sense.
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I know how it works but I want to know the fundamental reason which relates eigan value analysis to the stability analysis of a linear system. Everywhere I searched, I came across examples about how eigan values have been used but I want to understand how eigan values came to play such a big role in the field of stability analysis. How is it helpful in uncovering the "underlying structure" of a system, which is what it does actually ? Can someone point me towards some good literature that would enlighten me in that direction ?
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For linear time invariant (LTI) systems, eigenvalues are the same as poles of the open-loop system. Let the system be described by
dx(t)/dt = A x(t) + B u(t) and y(t)=C x(t)
where x(t) is the state vector, u(t) the input vector and y(t) is output vector and the matrices A,B and C of appropriate sizes are constants. Without trying full transient solution using state transition matrices, for stability analysis of the LTI system, we simply assume that the initial conditions x(0)=0 and take laplace transform of both sides of the equations.
s x(s) = A x(s) + B u(s) and y(s)= C x(s) = C Inv(s I - A) B u(s)
G(s) = y(s)/u(s) = C Adj(s I - A) B/ det (s I - A)
where G(s) is the transfer function matrix, s is the complex variable and I is an identity matrix. The poles of G(s) are roots of det (sI - A) =0, i.e., a, b, .... and are easily related to roots of det (lamda * I -A) =0 or the eigenvalues.
The output y(t), when the input u(t)=delta(t) is an impulse or u(s)=1, is simply obtained from
y(t) = Inv Laplace G(s)
which can be obtained after partial fractions of the form G(s) = A/(s-a) + B/(s-b) + ..
The solution is y(t)=A*exp(a*t)+B*exp (b*t) + ... and it converges to zero if real parts of roots a, b, ... are negative. It is the same result that we expect from eigenvalue analysis.
Physically speaking, the eigenvalues identify the modal frequencies and the decay rates (effective damping) of decoupled/orthogonal modal oscillators. The whole unforced system response is a combination of all modal responses in a particular way (called modal participation) which depends on the initial conditions.
An interesting note about modal damping: It is not physical damping. Consider a unstable oscillating system needs to be stabilized and thus, it is mounted on a flexible damped foundation. If the foundation does not provide sufficient damping then the whole new system remains unstable, i.e., new mode has still negative modal damping. If the foundation provides good deal of damping then the foundation would not move (similar to being jammed) and the whole system would still remain unstable. The foundation damping needs to be tuned in such a way that it dissipates maximum energy exactly at the modal frequency of the unstable mode.
Such physical intuition comes from practice/experience. Similar problems related to damper optimization are found in many fields: floating mass/root damper in turbine blades where floating mass and its shape need optimization to improve friction damping, squeeze-film dampers in rotor systems where anchoring spring needs optimization to improve damping, etc. No book can explain these things. When you work on various industrial problems, you will learn by yourself.
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Waves in nonlinear media with dissipation are called shock waves. The waves in nonlinear media with a dispersion are called solitons. But what are the name of waves in nonlinear media with dispersion and dissipation? And how this type of waves can be doubtlessly identified in the experiment?
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Dear Dr. Karelin,
According to Maiden et al. (2016) (Phys. Rev. Let. DOI: 10.1103/PhysRevLett.116.174501), we can use the term DSW (Dispersive Shock Wave) in order to characterize the waves described in your question. Their general characteristics are summarized succinctly:
"A DSW is an expanding, oscillatory train of amplitude-ordered nonlinear waves composed of a large amplitude solitonic wave adjacent to a monotonically decreasing wave envelope that terminates with a packet of small amplitude dispersive waves"
Sincerely, Janusz
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In the coefficients of Mie scattering Riccati- Bessel function appears. What is the significance of these functions in the context of Mie -scattering?
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There are two independent Riccati Bessel solutions F which vanish at the center of the scattering and G which are singular there whenever scattering occurs the radial part of the scattered solutions far from the influence of the scattering center are mixing of F and G ie of the form \psi= a F+ b G. The incident radial wave can be considered F or G after scattering they mix.
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Anyone
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Thanks to all for response, may someone send me the book  Optical Solitons by Agrawal and Kivshar.
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It is well known that certain integrable equations such as the Camassa-Holm, Novikov and Qiao admit the so-called peakon solutions. However, all of them are not evolution equations. Then I would like to know if there is some evolution equation admitting such property.
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I have finally discovered an answer to my question and it is positive.  An example of an evolution equation admiting peakon solutions can be found in the link below.
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Is there an initial condition for collison of solitons as Gaussian initial condition and undular bore initial condition? Could somebody help me, please?
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You can try a_1*sech[ b_1*(x-x_1)] * exp(i*k_1*x)  + a_2*sech[b_2*(x-x_2)] * exp(i*k_2*x).  You can choose, for simplicity, a_1= a_2, b_1 = b_2,  x_1 = - x_2, and play with k_1 and k_2.
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Individual photons travel in "empty" space for very long distances maintaining (except for red shift and other subtle effects) their wavelength, energy, momentum, and do not disperse or diffuse.
If a pair of photons have paths that cross each other and they coincide in time (empty space understood) apparently they interact and then leave the interaction to run their errands further down their previous trajectories. The idea of a photon being some sort of classical three dimensional electromagnetic soliton is therefore a natural one. But Maxwell equations do not seem to handle these properties.
Is a good, consistent, understandable treatment of photons as electromagnetic solitons known to someone?
Cordially,
Daniel
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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.
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I have generated explicit soltion solutions for coupled nonlocal nonlinear schrodinger equation. Now, want to check the stability of the solution by NUMERICAL ANALYSIS. I am not familiar with any methods can anyone suggest me to do it.
Thanks in advance.
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Take your solution as input and run  numerical simulation programme ( may be based on split step Fourier method). In case of stable soliton, it will soon settle down to the solution, otherwise it will disintegrate or explode. Hope you get good result. Best regards and best luck.
SKonar
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Good morning, I am a graduate student, new in researching the transmission and stability of optical solitons. I've read some articles about it, and a method named 'Vakhitov–Kolokolov criterion' have been metioned many times. I am curious about it, also confused.
Could anyone tell me about Vakhitov–Kolokolov criterion in briefly, or introduce some articals deducing it detailedly to me? 
I will be  deeply grateful.
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Dear Prof. Gert Van der Zwan
You are right.
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This is less a question, more a request for confirmation of my understanding of some reading I've done. Is the following true?
For asymptotically flat solitons (generalisations of Bartnik and McKinnon's su(2) solutions), it appears that the asymptotic values of the gauge fields (i.e. for r tending to infinity) are fixed to certain integer values, which means that the tangential pressure P vanishes at infinity. This in turn means the solutions carry no global magnetic charge, for if they did, the Einstein equations would be singular at infinity.
If anyone could either confirm this, or else point out my error(s), I'd much appreciate it.
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In the static situation, using Bogomol'nyi type analysis, it can be derived a positive-definite energy functional which has a lower bound. Specializing to the gauge group SU(2) and the t'Hooft-Polyakov ansatz for the gauge and Higgs fields, we seek static, spherically symmetric solutions to the coupled system of equations in both the isotropic and standard coordinate systems. In both cases, in the spontaneously broken symmetry situation, it is found great simplications reducing the solutions of the coupled system to the solution of a single non-linear differential equation, different one in each case, but well-known in other contexts of physics. They found abelian and non-abelian monopole solutions with gravitational fields playing the role of Higgs fields in providing attraction that balances the repulsion due to the gauge fields. Numerical solutions indicate the possibility of blackhole horizons inside the monopoles enclosing the singularity at the origin. Such non-abelian monopoles are then the analogs of Reissner-Nordstr\"om blackholes with magnetic charge.
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If I just put up a thulium fiber laser system which is working in the analmalous region, however, I just want to use some optical devices to change the whole dispersion in the cavity from negative dispersion into positive dispersion. Has anyone known this?
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a pair of prisms, see the textbook
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Skyrmions, Magnetic Solitons - Domain walls are considered topological defects 
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Zhang, Haijun, et al. Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface." Nature Physics 5, 2009, 438; Chen, Y. L., et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3, Science, 325, 2009, 178. Hasan, M. Zahid, and Charles L. Kane. Colloquium: topological insulators, Reviews of Modern Physics, 82, 2010, 3045  
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If we consider a two component BEC, in such regime, the solitonic solutions leads to the formation of a bound state. This bound state formation occurs only when one component follows dark soliton and other one is bright soliton. My question is what is the importance of such bound state in the literature? By importance, I mean to say what are the applications of such bound states in BEC?
Thanks in advance for your kind comments.
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I mean acoustic or lattice solitary wave.
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That is the review paper on ''the solitons in materials Science ''  which contains more then 200 references up to 1983 covers almost the complete spectrum of MaterScience. 
The following text book  entitled as  'A first course in soliton theory' by Academician  Barat Nuriyev, 1996, which was published by Middle East Technical University, Department of Mathematics, METU-ANKARA,   may be very useful for any one interested with the subject.
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I'm trying to simulate the pulse propagation in HC-PCF using the professor Govind Agrawal's NLSE solver software [attached]. I think this software can't fulfill all the HC-PCF parameters requirements. Thats why I'm not getting the desired output even i'm using the same parameter values from the papers (attached).
Can anyone please help me in this regard ?
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It is possible by boundary elements method. 
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It is known that Self-Dual Yang Mills theory reduces to Korteweg-deVries equation or its modified form (mKdV), see the file included. Then it seems possible to consider hadrons as solitons. For a review of various models, see Stephan Hartmann's paper: http://core.kmi.open.ac.uk/download/pdf/11921144.pdf
But as far as I know, there is no convincing theory which is able to describe soliton model of quark masses. Or does anybody know a recent progress in this soliton approach of hadrons? Your comments are welcome.
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It is possible to consider baryons as solitons in the large N_c limit. This observation is due to Witten in 1979. There was proposal by Skyrme dating back to 1959 considering  baryons as topological solitons in the nonlinear sigma model for pseudoscalar mesons. These solitons are called skyrmions. Skyrme's  work fits in nicely with the large N_c expectation. See papers
‘Exotic Levels from Topology in the Quantum Chromodynamic Effective Lagrangian,’ A.P. Balachandran, V.P. Nair, S.G. Rajeev and A. Stern, Phys. Rev. Lett. 49, 1124 (1982).
(This shows that skyrmions carry nonzero baryon number.)
E. Witten, Current algebra,baryons and quark confinement, Nucl. Phys. B223, 433 (1983) (This shows that skyrmions in the sigma model are fermions; also demonstrates nonzero baryon number.)
G. Adkins, C. Nappi and E. Witten, Static proprties of nucleons in the Skyrme model,Nucl. Phys. B228, 552 (1983).
There are literally over a 1000 papers elaborating on these early works. Skyrme's work, which was way ahead of its time, can be traced from these papers.
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Using the Grammian Determinant one can write N-soliton solution for the Complex Field equations like NLS equation. But is it possible to write N-soliton solution in the case of real field integrable equations (KdV,mKdV,extended KdV equations)?
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Thank you...
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As I know, all of the studies have explored the spatial profile of light passing through a nonlinear nanosuspension via solution of spatial part of Nonlinear Schrodinger Equation (NLSE) or wave equation. (For example they have derived soliton stripes) However, no other paper can be found to describe time evolution of light inside such nonlinear medium. Why?
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Dear Morteza,
I know two groups of scientists who can help your with your question. Please, check names of (1) Dr. Johen Guck (Dresden: http://www.biotec.tu-dresden.de/research/guck.html) and (2) Dr. Ido Perlman (Michigan: http://www.rappaport.org.il/Rappaport/Templates/ShowPage.asp?DBID=1&TMID=610&FID=77&PID=0&IID=246) who are studying light propagation through different tissues and materials. If they cannot help, they can redirect you to physicists studying the subject. Cordially, Serguei.
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Solitons are the 1D topological defect with phase singularity of \pi, whereas vortices have \2pi phase jump around its core. Recently the obsevation of solitonic vortex in BEC has been confirmed. Can anybody elaborate solitonic vortex clearly?
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For solitary waves, there are two types. Bright solitons can exist in BECs with attractive interactions, while dark solitons are topological excitation in repulsive BECs.  Bright solitons are self-focusing, nondispersive, particlelike solitary waves (see PRA.85.013627 for detailed description of one dimensional bright solitons in condensates with and without a trap).  They have uniform phase distribution. A dark soliton is an envelope soliton that has the form of a density dip with a phase jump across its density minimum (see review in J. Phys. A: Math. Theor. 43 (2010) 213001).  Vortices are also topological excitations in condensates. The quantization of circulation of the fluid leads to quantized vortex states which are characterized by the phase circulation about the vortex core being $2s\pi$ (Phys. Rev. Lett. 81, 5477, Phys. Rev. A 64, 031601(R) ), where $s$, the winding number or vortex charge, is a nonzero integer. This phase singularity is independent of atomic and trap parameters. Vortices with opposite charge (opposite phase circulation, i.e. clockwise and anticlockwise) are called vortex and antivortex. If the winding number satisfies the relation $|s|>1$, vortices are not stable when perturbed. They will dissociate into singly charged vortics ($s=\pm 1$). 
Svortices are vortices confined to essentially one-dimensional dynamics, which obey a similar phase-offset–velocity relationship as solitons. Marking the transition between solitons and vortices, svortices are a distinct class of symmetry-breaking stationary and uniformly rotating excited solutions of the 2D and 3D Gross-Pitaevskii equation, , which have properties of both vortices and solitons (J. Phys. B: At. Mol. Opt. Phys. 34 L113, New J. Phys. 15 113028).
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I understand this is common in femtosecond fiber lasers. This is because of instability in the soliton formed due to the long nonlinear interaction lengths in fibers. But what if I observe similar sideband from the Ti:Sapphire laser oscillator. Is this detrimental to the laser performance ?
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The lowest-order Kelly sidebands actually appear in a Kerr-lens mode-locked Ti:sapphire laser, where the overall cavity dispersion crosses zero. In my experience, they are not detrimental to the laser performance unless they are too close to the wavelength range where the laser is above threshold. Then, slight fluctuations of the pulse energy might move them in and out of the spectrum, leading to leaps in the pulse energy (interestingly not in the average power). Of course, they drain power out of the pulses into the cw spike(s) but this is usually negligible. Also, according to Haus (IEEE Journal on Sel. Topics in QE, vol. 6 (2000) p.1173), they limit the pulse width that can be achieved by KLM from a certain cavity configuration.
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I have seen that in different problems, different types of transformation are used. Why can't we use the same form for all problems? What does it actually mean?
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Stretching the coordinates helps see in detail what happens on different scales of distance or time. For example, a boundary layer is usually thin, but a lot happens there. By stretching the coordinates in the vicinity of the boundary layer, one can see in detail what goes on there.
Different stretching "recipes" are required for different physical problems because of the different small domains, over which the interesting physics takes place.
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Does anyone know how to simulate soliton propagation in a ring? How to adjust the dispersive and nonlinearity coefficients? Shall it be solved in 2D or 3D?
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yes, i can ,i worked by COMSOL for my theeses such as laminar flow particle tracing and magnetic field and heat transfer
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The system of equations
u_t + (\partial_x^2 + \partial_y^2)u_x + u u_x/2 = -(p_x u_x + p_y u_y)/2 (Eq. 1)
p_xx + p_yy = u_x (Eq. 2)
reduces to the KdV equation if u and p are independent of y. The KdV equation can of course be solved using the inverse scattering transformation (IST). Can the system of equations (1) and (2) be solved by the IST when u and p depend on x, y and t? [These equations arise in the theory of electromigration in thin films --- see R. M. Bradley, Physica D 158, 216 (2001)].
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Note that if we set u=U(X,T), where X = (cos theta)x + (sin theta) y and T= (cos theta) t, then we get U_T + U_XXX + U U_X = 0, which is the KdV equation. However, the disturbance is still one-dimensional --- it just depends on X. I'm that hoping that the equations could be solved for an arbitrary initial two-dimensional condition.
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In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. Can they be considered as a viable optical energy carrier without attenuation issues?
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This question will shortly celebrate I believe 40 years old birthday :) There are heap of the paublications (peak was in about 1990). I have been working in telecom company in the USA and can tell you that from the point of view of power budget the solitons are not very attractive carrier inspite of their relative robustness to the noise and other perturbations. They could be interesting for ultra long, but not high capacity links. If you want more information, send me an e-mail, we can organize skype chat some time.
Best wishes and good luck!
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how can we identify Gap-Soliton in Bose-Einstein Condensates (BEC)?
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Dear Sudharsan, The measure of chemical potential can identify Gap-Soliton as chemical potential corresponding to Gap-Soliton in the presence of optical lattice (OL) lie in the gap of the band spectrum of the linear Bloch-wave spectrum (energy spectrum of OL ). Another indicator can be the non-dispersive behaviour in the time evolution of matter-wave.