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| [[File:Soliton de Peregrine.png|thumb|upright=1.8|3D view of the spatio-temporal evolution of a Peregrine soliton]]
| | There are a number of essential tips that you might want to bear in mind when teaching your puppy the basics about good behavior. Exercising the right training techniques is what"ll make or break your training program along with your dog. Teaching your pup and follow these five essential guidelines is likely to be easier than ever before. <br><br>1 - Be Gentle - Your new pet is going to be acutely painful and sensitive initially, and as a result will not have the capacity to handle whatever is too demanding on both an emotional and a physical level. Now is the time where your puppy will react badly to stress o-r being educated too tough, although studying generally speaking easily takes place. If fears are found too easily during the training process, then it could inhibit the puppy"s power to understand, therefore be sure to be gentle but firm in your training. <br><br>2 - Keep Things Brief - Puppies have even shorter attention spans than children. Your puppy is only going to understand when his or her attention is on you, and you"ll not begin to see the benefits that you are seeking once your puppy is tired physically or mentally. Make sure to be temporary when getting your pup through education activities, and you then can move ahead. Should people want to discover more on [http://www.streetfire.net/profile/designerdognzn.htm designerdognzn - StreetFire Member in US], we recommend tons of online libraries people might think about pursuing. My family friend found out about [http://social.xfire.com/blog/yorknyjbd Xfire - Gaming Simplified] by browsing webpages. <br><br>3 - Exercise Patience - Expecting overnight effects is planning to irritate you and cause your training program to get rid of its focus. Curl up, and understand that puppies understand in bursts, and such things as this may take time. Puppies also do go through short memory falls so do not allow yourself to become overwhelmed if your puppy seems to forget some of its training from time to another. Exercise tolerance you"ll be just fine and in regards to education. <br><br>4 - Exercise Simplicity - Teaching your puppy should be done in a step by step process if you"d like to achieve the best results. This is actually the easiest way that your puppy will learn. Exercise a straightforward, step by step approach and your puppy will learn more easily and will take pleasure in the process more carefully than if you were to employ a more interval training routine. <br><br>5 - Build Confidence - confidence starts with building confidence in-a young dog, and Confidence is the core of every healthier adult dog. Building confidence in your puppy isn"t hard at all to do; all you should do is spend time with your puppy as usually as you possibly can. To read additional info, we recommend you check out: [http://jazztimes.com/community/profiles/400492-franciscocabnk home page]. This may help to develop self esteem in your pup. You shouldn"t often be in training mode when you first get your puppy, but instead sometimes you should step right back and play with your dog, having a good time with him or her along the way. Education is essential, but above everything else your puppy has to know that you are friends. <br><br>These five fundamental training fundamentals are important in planning your puppy for a successful training program and will travel better results when properly incorporated into your step-by-step puppy training process..<br><br>If you have any sort of inquiries regarding where and how you can use list of health insurance companies, [http://fallaciousaccor38.xtgem.com discover here],, you can call us at our website. |
| The '''Peregrine [[soliton]]''' (or '''Peregrine [[breather]]''') is an [[analytic solution]] of the [[nonlinear Schrödinger equation]].<ref>{{cite journal| title=Water waves, nonlinear Schrödinger equations and their solutions | first=D. H. | last=Peregrine | journal=J. Austral. Math. Soc. |series= B | volume=25 | year=1983 | pages=16–43 |doi= 10.1017/S0334270000003891 }}</ref> This solution has been proposed in 1983 by [[Howell Peregrine]], researcher at the mathematics department of the [[University of Bristol]].
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| == Main properties ==
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| Contrary to the usual fundamental [[soliton]] that can maintain its profile unchanged during propagation, the Peregrine soliton presents a double spatio-temporal localization. Therefore, starting from a weak oscillation on a continuous background, the Peregrine soliton develops undergoing a progressive increase of its amplitude and a narrowing of its temporal duration. At the point of maximum compression, the amplitude is three times the level of the continuous background (and if one considers the intensity as it is relevant in optics, there is a factor 9 between the peak intensity and the surrounding background). After this point of maximal compression, the wave's amplitude decreases and its width increases and it finally vanishes.
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| These features of the Peregrine soliton are fully consistent with the quantitative criteria usually used in order to qualify a wave as a [[rogue wave]]. Therefore, the Peregrine soliton is an attractive hypothesis to explain the formation of those waves which have a high amplitude and may appear from nowhere and disappear without a trace.<ref>{{cite journal |last1= Shrira|first1=V.I. |last2= Geogjaev |first2= V.V. | year= 2009 |title= What makes the Peregrine soliton so special as a prototype of freak waves ?|journal= J. Eng. Math.}}</ref>
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| == Mathematical expression ==
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| === In the spatio-temporal domain ===
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| [[File:Peregrine soliton profile.png|thumb|upright 1.4|Spatial and temporal profiles of a Peregrine soliton obtained at the point of maximum compression]]
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| The Peregrine soliton is a solution of the one-dimensional nonlinear Schrödinger equation that can be written in normalized units as follows :
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| : <math>i \frac{\partial \psi}{\partial \tau} + \frac{1}{2} \frac{\partial^2 \psi}{\partial \xi^2} + |\psi|^2 \psi = 0</math>
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| with <math>\xi</math> the spatial coordinate and <math>\tau</math> the temporal coordinate. <math>\psi (\xi, \tau)</math> being the [[Envelope (mathematics)|envelope]] of a surface wave in deep water. The [[Dispersion relation|dispersion]] is anomalous and the nonlinearity is [[self-focusing]] (note that similar results could be obtained for a normally dispersive medium combined with a defocusing nonlinearity).
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| The Peregrine analytical expression is :
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| : <math> \psi (\xi, \tau) = \left[ 1-\frac{4 (1 + 2 i \tau)}{1+4 \xi^2 + 4 \tau^2} \right] e^{i \tau} </math>
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| so that the temporal and spatial maxima are obtained for <math>\xi = 0</math> and <math>\tau = 0</math>.
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| === In the spectral domain ===
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| [[File:Evolution of the spectrum of a Peregrine soliton.png|thumb|upright=1.5|Evolution of the spectrum of the Peregrine soliton <ref name="PLA Akhmediev"/>]] | |
| It is also possible to mathematically express the Peregrine soliton according to the spatial frequency <math>\eta</math>:<ref name="PLA Akhmediev">{{cite journal | title=Universal triangular spectra in parametrically-driven systems | author=Akhmediev, N., Ankiewicz, A. , Soto-Crespo, J. M. and Dudley J. M. | journal=Phys. Lett. A | year=2011 | volume=375 | pages=775–779|bibcode = 2011PhLA..375..775A |doi = 10.1016/j.physleta.2010.11.044 }}</ref>
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| <math> \tilde{\psi} (\eta, \tau) = \frac{1}{\sqrt{2 \pi}} \int{\psi (\xi, \tau) e^{i \eta \xi} d\xi} = \sqrt{2 \pi} e^{i \tau} \left[ \frac{1+2 i \tau}{\sqrt{1+4 \tau^2}} \exp \left( -\frac{|\eta|}{2} \sqrt{1+4 \tau^2} \right) - \delta(\eta) \right] </math>
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| with <math>\delta</math> being the [[Dirac delta function]].
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| This corresponds to a [[Absolute value|modulus]] (with the constant continuous background here omitted) :
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| <math> |\tilde{\psi} (\eta, \tau)| = \sqrt{2 \pi} \exp \left( -\frac{|\eta|}{2} \sqrt{1+4 \tau^2} \right). </math>
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| One can notice that for any given time <math>\tau</math>, the modulus of the spectrum exhibits a typical triangular shape when plotted on a logarithmic scale. The broadest spectrum is obtained for <math>\tau = 0 </math>, which corresponds to the maximum of compression of the spatio-temporal nonlinear structure.
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| === Different interpretations of the Peregrine soliton===
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| [[File:Peregrine soliton and other nonlinear solutions.png|thumb|Peregrine soliton and other nonlinear solutions]]
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| ==== As a rational soliton ====
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| The Peregrine soliton is a first-order rational soliton.
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| ==== As an Akhmediev breather ====
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| The Peregrine soliton can also be seen as the limiting case of the space-periodic Akhmediev [[breather]] when the period tends to infinity.<ref name="NP Kibler">{{cite journal |last1= Kibler|first1=B. |last2= Fatome |first2= J. |last3= Finot |first3= C. | last4= Millot |first4= G. | last5= Dias |first5= F. | last6= Genty |first6= G. | last7= Akhmediev |first7= N. | last8= Dudley |first8= J.M. | year= 2010 |title= The Peregrine soliton in nonlinear fibre optics|journal= Nature Physics|doi= 10.1038/nphys1740 |volume= 6 |issue= 10 |pages= 790–795|bibcode = 2010NatPh...6..790K}}</ref>
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| ==== As a Kuznetsov-Ma soliton ====
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| The Peregrine soliton can also be seen as the limiting case of the time-periodic Kuznetsov-Ma breather when the period tends to infinity.
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| == Experimental demonstration ==
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| Mathematical predictions by H. Peregrine had initially been established in the domain of [[hydrodynamics]]. This is however very different from where the Peregrine soliton has been for the first time experimentally generated and characterized.
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| === Generation in optics ===
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| [[File:Peregrine soliton in optics.png|thumb|upright|Record of the temporal profile of a Peregrine soliton in optics <ref name="OL Hammani">{{cite journal|title=Peregrine soliton generation and breakup in standard telecommunications fiber | first1=K. | last1=Hammani | first2=B. | last2=Kibler | first3=C. | last3=Finot |first4=P. | last4=Morin | first5=J. | last5= Fatome | first6= J.M. | last6=Dudley | first7= G. | last7 = Millot |journal=Optics Letters | volume= 36 | year= 2011 |pages= 112–114 |doi= 10.1364/OL.36.000112| issue=2| pmid=21263470 |bibcode = 2011OptL...36..112H }}</ref>
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| ]]
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| In 2010, more than 25 years after the initial work of Peregrine, researchers took advantage of the analogy that can be drawn between hydrodynamics and optics in order to generate Peregrine solitons in [[optical fiber]]s.<ref name="NP Kibler"/><ref>{{cite web | title = Peregrine’s 'Soliton' observed at last | publisher= bris.ac.uk | url = http://www.bris.ac.uk/news/2010/7184.html | accessdate = 2010-08-24 }}</ref> In fact, the evolution of light in fiber optics and the evolution of surface waves in deep water are both modelled by the nonlinear Schrödinger equation (note however that spatial and temporal variables have to be switched). Such an analogy has been exploited in the past in order to generate [[Soliton (optics)|optical solitons]] in optical fibers.
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| More precisely, the nonlinear Schrödinger equation can be written in the context of optical fibers under the following dimensional form :
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| <math>i \frac{\partial \psi}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 \psi}{\partial t^2 } + \gamma |\psi|^2 \psi = 0</math>
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| with <math>\beta_2</math> being the second order dispersion (supposed to be anomalous, i.e. <math>\beta_2 < 0</math>) and <math>\gamma</math> being the nonlinear Kerr coefficient. <math>z</math> and <math>t</math> are the propagation distance and the temporal coordinate respestively.
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| In this context, the Peregrine soliton has the following dimensionnal expression:<ref name="OL Hammani"/>
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| : <math>\psi (z,t) = \sqrt{P_0} \left[ 1-\frac{4 \left( 1 + 2 i \dfrac{z}{L_{NL}} \right) }{1+4 \left( \dfrac{t}{T_0} \right) ^2 + 4 \left( \dfrac{z}{L_{NL}} \right) ^2} \right] e^{ \dfrac{i z}{L_{NL}}} </math>.
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| <math>L_{NL}</math> is a nonlinear length defined as <math>L_{NL} = \dfrac{1}{\gamma P_0}</math> and <math>T_0</math> is a duration defined as <math>T_0 = \dfrac{1}{\sqrt{\beta_2 L_{NL}}}</math>. <math>P_0</math> is the power of the continuous background.
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| By using exclusively standard [[optical communication]] components, it has been shown that even with an approximate initial condition (in the case of this work, an initial sinosoidal beating), a profile very close to the ideal Peregrine soliton can be generated.<ref name="OL Hammani"/><ref>{{cite journal |last1= Erkintalo|first1=M. |last2= Genty|first2= G. | last3= Wetzel |first3= B. | last4= Dudley |first4= J. M. | year= 2011 |title= Akhmediev breather evolution in optical fiber for realistic initial conditions|journal= Phys. Lett. A | volume= 375 | pages= 2029–2034 |doi= 10.1016/j.physleta.2011.04.002 |issue= 19 |bibcode = 2011PhLA..375.2029E }}</ref> However, the non-ideal input condition lead to substructures that appear after the point of maximum compression. Those substructures have also a profile close to a Peregrine soliton,<ref name="OL Hammani"/> which can be analytically explained using a [[Jean Gaston Darboux|Darboux]] transformation.<ref>{{cite journal |last1= Erkintalo|first1=M. |last2= Kibler|first2= B. | last3= Hammani|first3= K. | last4= Finot |first4= C. | last5= Akhmediev |first5= N. | last6= Dudley |first6= J.M. | last7= Genty |first7= G. | year= 2011|title= Higher-Order Modulation Instability in Nonlinear Fiber Optics |journal= Physical Review Letters | volume= 107 | pages= 253901 |doi= 10.1103/PhysRevLett.107.253901 |bibcode = 2011PhRvL.107y3901E }}</ref>
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| The typical triangular spectral shape has also been experimentally confirmed.<ref name="NP Kibler"/><ref name="OL Hammani"/><ref>{{cite journal | title=Spectral dynamics of modulation instability described using Akhmediev breather theory | author=Hammani K., Wetzel B. , Kibler B. , Fatome J., Finot C. , Millot G., Akhmediev N., and Dudley J. M. | journal=Opt. Lett. | year=2011 | volume=36 | issue=2140-2142 | doi=10.1364/OL.36.002140|bibcode = 2011OptL...36.2140H }}</ref>
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| === Generation in hydrodynamics ===
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| These results in optics have been confirmed in 2011 in hydrodynamics<ref>{{cite journal |last1= Chabchoub|first1=A. |last2= Hoffmann |first2= N.P. | last3= Akhmediev |first3= N. | year= 2011 |title= Rogue wave observation in a water wave tank|journal= Phys. Rev. Lett.|doi= 10.1103/PhysRevLett.106.204502 |volume= 106 |issue= 20 |bibcode=2011PhRvL.106t4502C}}</ref><ref>{{cite web | title = Rogue waves captured | publisher= www.sciencenews.org | url = http://www.sciencenews.org/view/generic/id/74610/title/Rogue_waves_captured | accessdate = 2011-06-03}}</ref> with experiments carried out in a 15-m long water [[wave flume|wave tank]]. In 2013, complementary experiments using scaled chemical tanker have discussed the potential devastating effects on the ship.<ref>{{cite journal |last1= Onorato|first1=M. |last2= Proment |first2= D. | last3= Clauss|first3= G. | last4= Clauss|first4= M. | year= 2013 |title= Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test|journal= Plos One|doi= 10.1371/journal.pone.0054629 |volume= 8 |bibcode = 2013PLoSO...854629O }}</ref>
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| === Generation in other fields of physics ===
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| Other experiments carried out in the [[Plasma (physics)|physics of plasmas]] have also highlighted the emergence of Peregrine solitons in other fields ruled by the nonlinear Schrödinger equation.<ref>{{cite journal |last1= Bailung|first1=H. |last2= Sharma |first2= S. K. | last3= Nakamura |first3= Y. | year= 2011 |title= Observation of Peregrine solitons in a multicomponent plasma with negative ions|journal= Phys. Rev. Lett.}}</ref>
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| ==See also==
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| * [[Nonlinear Schrödinger equation]]
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| * [[Breather]]
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| * [[Rogue wave]]
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| == Notes and references ==
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| {{reflist}}
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| {{Commons category|Peregrine soliton}}
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| {{DEFAULTSORT:Peregrine Soliton}}
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| [[Category:Solitons]]
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| [[Category:Fluid dynamics]]
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| [[Category:Waves]]
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| [[Category:Nonlinear optics]]
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| [[Category:Water waves]]
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There are a number of essential tips that you might want to bear in mind when teaching your puppy the basics about good behavior. Exercising the right training techniques is what"ll make or break your training program along with your dog. Teaching your pup and follow these five essential guidelines is likely to be easier than ever before.
1 - Be Gentle - Your new pet is going to be acutely painful and sensitive initially, and as a result will not have the capacity to handle whatever is too demanding on both an emotional and a physical level. Now is the time where your puppy will react badly to stress o-r being educated too tough, although studying generally speaking easily takes place. If fears are found too easily during the training process, then it could inhibit the puppy"s power to understand, therefore be sure to be gentle but firm in your training.
2 - Keep Things Brief - Puppies have even shorter attention spans than children. Your puppy is only going to understand when his or her attention is on you, and you"ll not begin to see the benefits that you are seeking once your puppy is tired physically or mentally. Make sure to be temporary when getting your pup through education activities, and you then can move ahead. Should people want to discover more on designerdognzn - StreetFire Member in US, we recommend tons of online libraries people might think about pursuing. My family friend found out about Xfire - Gaming Simplified by browsing webpages.
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4 - Exercise Simplicity - Teaching your puppy should be done in a step by step process if you"d like to achieve the best results. This is actually the easiest way that your puppy will learn. Exercise a straightforward, step by step approach and your puppy will learn more easily and will take pleasure in the process more carefully than if you were to employ a more interval training routine.
5 - Build Confidence - confidence starts with building confidence in-a young dog, and Confidence is the core of every healthier adult dog. Building confidence in your puppy isn"t hard at all to do; all you should do is spend time with your puppy as usually as you possibly can. To read additional info, we recommend you check out: home page. This may help to develop self esteem in your pup. You shouldn"t often be in training mode when you first get your puppy, but instead sometimes you should step right back and play with your dog, having a good time with him or her along the way. Education is essential, but above everything else your puppy has to know that you are friends.
These five fundamental training fundamentals are important in planning your puppy for a successful training program and will travel better results when properly incorporated into your step-by-step puppy training process..
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