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| In the theory of [[integrable system]]s, a '''peakon''' ("peaked soliton") is a [[soliton]] with [[discontinuous]] first [[derivative]]; the wave profile is shaped like the graph of the function <math>e^{-|x|}</math>. Some examples of [[non-linear partial differential equation]]s with (multi-)peakon solutions are the [[Camassa–Holm equation|Camassa–Holm shallow water wave equation]], the [[Degasperis–Procesi equation]] and the [[Fornberg–Whitham equation]].
| | If you present photography effectively, it helps you look much more properly at the globe around you. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. * A community forum for debate of the product together with some other customers in the comments spot. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. We also help to integrate various plug-ins to expand the functionalities of the web application. It primarily lays emphasis on improving the search engine results of your website whenever a related query is typed in the search box. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. But the Joomla was created as the CMS over years of hard work. Websites that do rank highly, do so becaue they use keyword-heavy post titles. So, we have to add our social media sharing buttons in website. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. An ease of use which pertains to both internet site back-end and front-end users alike. When you have virtually any questions about in which as well as the way to employ [http://dinky.in/?WordpressBackupPlugin344213 wordpress backup], you are able to e-mail us at our own web-page. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Thus, Word - Press is a good alternative if you are looking for free blogging software. Get started today so that people searching for your type of business will be directed to you. |
| Since peakon solutions are only piecewise differentiable, they must be interpreted in a suitable [[weak solution|weak sense]].
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| The concept was introduced in 1993 by Camassa and Holm in the short but much cited paper where they derived their shallow water equation.<ref>Camassa & Holm 1993</ref>
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| == A family of equations with peakon solutions ==
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| The primary example of a PDE which supports peakon solutions is
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| :<math>
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| u_t - u_{xxt} + (b+1) u u_x = b u_x u_{xx} + u u_{xxx}, \,
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| </math> | |
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| where <math>u(x,t)</math> is the unknown function, and ''b'' is a parameter.<ref>Degasperis, Holm & Hone 2002</ref>
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| In terms of the auxiliary function <math>m(x,t)</math> defined by the relation <math>m = u-u_{xx}</math>, the equation takes the simpler form
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| :<math>
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| m_t + m_x u + b m u_x = 0. \,
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| </math>
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| This equation is [[integrable system|integrable]] for exactly two values of ''b'', namely ''b'' = 2 (the [[Camassa–Holm equation]]) and ''b'' = 3 (the [[Degasperis–Procesi equation]]).
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| == The single peakon solution ==
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| The PDE above admits the travelling wave solution <math>u(x,t) = c \, e^{-|x-ct|}</math>,
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| which is a peaked solitary wave with amplitude ''c'' and speed ''c''.
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| This solution is called a (single) peakon solution,
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| or simply a '''peakon'''.
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| If ''c'' is negative, the wave moves to the left with the peak pointing downwards,
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| and then it is sometimes called an '''antipeakon'''.
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| It is not immediately obvious in what sense the peakon solution satisfies the PDE. | |
| Since the derivative ''u''<sub>''x''</sub> has a jump discontinuity at the peak,
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| the second derivative ''u''<sub>''xx''</sub> must be taken in the sense of [[distribution (mathematics)|distributions]] and will contain a [[Dirac delta function]];
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| in fact, <math>m = u - u_{xx} = c \, \delta(x-ct)</math>.
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| Now the product <math>m u_x</math> occurring in the PDE seems to be undefined, since the distribution ''m'' is supported at the very point where the derivative ''u''<sub>''x''</sub> is undefined. An [[ad hoc]] interpretation is to take the value of ''u''<sub>''x''</sub> at that point to equal the average of its left and right limits (zero, in this case). A more satisfactory way to make sense of the solution is to invert the relationship between ''u'' and ''m'' by writing <math>m = (G/2) * u</math>, where <math>G(x) = \exp(-|x|)</math>, and use this to rewrite the PDE as a (nonlocal) [[Hyperbolic partial differential equation#Hyperbolic system and conservation laws|hyperbolic conservation law]]: | |
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| :<math>
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| \partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \left(\frac{b u^2}{2} + \frac{(3-b) u_x^2}{2} \right) \right] = 0.
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| </math>
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| (The star denotes [[convolution]] with respect to ''x''.)
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| In this formulation the function ''u'' can simply be interpreted as a [[weak solution]] in the usual sense.<ref>Constantin & McKean 1999 (who treat the Camassa–Holm case ''b'' = 2; the general case is very similar)</ref>
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| == Multipeakon solutions ==
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| [[Image:Two-peakon.svg|thumb|400px|Two-peakon wave profile (solid curve) formed by adding two peakons (dashed curves): <math>u = m_1 \, e^{-|x-x_1|} + m_2 \, e^{-|x-x_2|}</math>]]
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| Multipeakon solutions are formed by taking a linear combination of several peakons, each with its own time-dependent amplitude and position. (This is a very simple structure compared to the multisoliton solutions of most other integrable PDEs, like the [[Korteweg–de Vries equation]] for instance.)
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| The ''n''-peakon solution thus takes the form
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| :<math>
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| u(x,t) = \sum_{i=1}^n m_i(t) \, e^{-|x-x_i(t)|},
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| </math> | |
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| where the 2''n'' functions <math>x_i(t)</math> and <math>m_i(t)</math>
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| must be chosen suitably in order for ''u'' to satisfy the PDE.
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| For the "''b''-family" above it turns out that this ansatz indeed gives a solution, provided that the system of [[ordinary differential equation|ODEs]]
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| :<math>
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| \dot{x}_k = \sum_{i=1}^n m_i e^{-|x_k-x_i|},
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| \qquad
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| \dot{m}_k = (b-1) \sum_{i=1}^n m_k m_i \sgn(x_k-x_i) e^{-|x_k-x_i|}
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| \qquad
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| (k = 1,\dots,n)
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| </math>
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| is satisfied. (Here sgn denotes the [[sign function]].)
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| Note that the right-hand side of the equation for <math>x_k</math> is obtained by substituting <math>x=x_k</math> in the formula for ''u''.
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| Similarly, the equation for <math>m_k</math> can be expressed in terms of <math>u_x</math>, if one interprets the derivative of <math>\exp(-|x|)</math> at ''x'' = 0 as being zero.
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| This gives the following convenient shorthand notation for the system:
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| :<math>
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| \dot{x}_k = u(x_k),
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| \qquad
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| \dot{m}_k = -(b-1) m_k u_x(x_k)
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| \qquad
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| (k = 1,\dots,n).
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| </math>
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| The first equation provides some useful intuition about peakon dynamics: the velocity of each peakon equals the elevation of the wave at that point.
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| == Explicit solution formulas ==
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| In the integrable cases ''b'' = 2 and ''b'' = 3, the system of ODEs describing the peakon dynamics can be solved explicitly for arbitrary ''n'' in terms of elementary functions, using inverse spectral techniques. For example, the solution for ''n'' = 3 in the Camassa–Holm case ''b'' = 2 is given by<ref>Beals, Sattinger & Szmigielski 2000 (where a different normalization and sign convention is used)</ref>
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| :<math>
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| \begin{align}
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| x_1(t) &= \log\frac{(\lambda_1-\lambda_2)^2 (\lambda_1-\lambda_3)^2 (\lambda_2-\lambda_3)^2 a_1 a_2 a_3}{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}
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| \\
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| x_2(t) &= \log\frac{\sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3}
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| \\
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| x_3(t) &= \log(a_1+a_2+a_3)
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| \\
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| m_1(t) &= \frac{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1 \lambda_2 \lambda_3 \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
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| \\
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| m_2(t) &= \frac{ \left( \lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3 \right) \sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{ \left( \lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3 \right) \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
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| \\
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| m_3(t) &= \frac{a_1+a_2+a_3}{\lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3}
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| \end{align}
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| </math>
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| where <math>a_k(t) = a_k(0) e^{t/\lambda_k}</math>, and where the 2''n'' constants <math>a_k(0)</math> and <math>\lambda_k</math> are determined from initial conditions. The general solution for arbitrary ''n'' can be expressed in terms of [[symmetric function]]s of <math>a_k</math> and <math>\lambda_k</math>. The general ''n''-peakon solution in the Degasperis–Procesi case ''b'' = 3 is similar in flavour, although the detailed structure is more complicated.<ref>Lundmark & Szmigielski 2005</ref>
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{Citation
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| | last = Beals | |
| | first = Richard
| |
| | author-link =
| |
| | last2 = Sattinger
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| | first2 = David H.
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| | last3 = Szmigielski
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| | first3 = Jacek
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| | year = 2000
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| | title = Multipeakons and the classical moment problem
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| | periodical = Adv. Math.
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| | volume = 154
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| | issue = 2
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| | pages = 229–257
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| | doi = 10.1006/aima.1999.1883
| |
| }}
| |
| | |
| *{{Citation
| |
| | last = Camassa
| |
| | first = Roberto
| |
| | author-link =
| |
| | last2 = Holm
| |
| | first2 = Darryl D.
| |
| | year = 1993
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| | title = An integrable shallow water equation with peaked solitons
| |
| | periodical = Phys. Rev. Lett.
| |
| | volume = 71
| |
| | issue = 11
| |
| | pages = 1661–1664
| |
| | doi = 10.1103/PhysRevLett.71.1661
| |
| | bibcode=1993PhRvL..71.1661C
| |
| }}
| |
| | |
| *{{Citation
| |
| | last = Constantin
| |
| | first = Adrian
| |
| | author-link =
| |
| | last2 = McKean
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| | first2 = Henry P.
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| | year = 1999
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| | title = A shallow water equation on the circle
| |
| | periodical = Commun. Pure Appl. Math.
| |
| | volume = 52
| |
| | issue = 8
| |
| | pages = 949–982
| |
| | doi = 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
| |
| }}
| |
| | |
| *{{Citation
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| | last = Degasperis
| |
| | first = Antonio
| |
| | author-link =
| |
| | last2 = Holm
| |
| | first2 = Darryl D.
| |
| | last3 = Hone
| |
| | first3 = Andrew N. W.
| |
| | year = 2002
| |
| | title = A new integrable equation with peakon solutions
| |
| | periodical = Theoretical and Mathematical Physics
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| | volume = 133
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| | issue = 2
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| | pages = 1463–1474
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| | arxiv = nlin.SI/0205023
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| | doi = 10.1023/A:1021186408422
| |
| }}
| |
| | |
| *{{Citation
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| | last = Lundmark
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| | first = Hans
| |
| | author-link =
| |
| | last2 = Szmigielski
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| | first2 = Jacek
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| | year = 2005
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| | title = Degasperis–Procesi peakons and the discrete cubic string
| |
| | periodical = International Mathematics Research Papers
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| | volume = 2005
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| | issue = 2
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| | pages = 53–116
| |
| | arxiv = nlin.SI/0503036
| |
| | doi = 10.1155/IMRP.2005.53
| |
| }}
| |
| | |
| [[Category:Solitons]]
| |
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