|
|
| Line 1: |
Line 1: |
| {{Refimprove|date=October 2008}}
| | The author's name is Andera and she believes it sounds fairly good. My wife and I live in Mississippi but now I'm considering other options. He works as a bookkeeper. To climb is some thing I truly enjoy performing.<br><br>Review my homepage; online reader - [https://www.machlitim.org.il/subdomain/megila/end/node/12300 www.machlitim.org.il], |
| [[File:Ile de ré.JPG|thumb|right|300px|Crossing [[swell (ocean)|swell]]s, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of [[Île de Ré]] (Isle of Rhé), France, in the [[Atlantic Ocean]]. The interaction of such near-[[soliton]]s in shallow water may be modeled through the Kadomtsev–Petviashvili equation.]]
| |
| In [[mathematics]] and [[physics]], the '''Kadomtsev–Petviashvili equation''' – or '''KP equation''', named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a [[partial differential equation]] to describe [[nonlinear]] [[wave motion]]. The KP equation is usually written as:
| |
| :<math>\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0</math>
| |
| where <math>\lambda=\pm 1</math>. The above form shows that the KP equation is a generalization to two [[spatial dimension]]s, ''x'' and ''y'', of the one-dimensional [[Korteweg–de Vries equation|Korteweg–de Vries (KdV) equation]]. To be physically meaningful, the wave propagation direction has to be not-too-far from the ''x'' direction, i.e. with only slow variations of solutions in the ''y'' direction.
| |
| | |
| Like the KdV equation, the KP equation is completely integrable. It can also be solved using the [[inverse scattering transform]] much like the [[nonlinear Schrödinger equation]].
| |
| | |
| ==History==
| |
| | |
| [[File:RIAN archive 151311 Russian physicist Boris Kadomtsev.jpg|thumb|right|180px|Boris Kadomtsev.]]
| |
| The KP equation was first written in 1970 by Soviet physicists Boris B. Kadomtsev (1928–1998) and Vladimir I. Petviashvili (1936–1993); it came as a natural generalization of the KdV equation (derived by Korteweg and De Vries in 1895). Whereas in the KdV equation waves are strictly one-dimensional, in the KP equation this restriction is relaxed. Still, both in the KdV and the KP equation, waves have to travel in the positive ''x''-direction.
| |
| | |
| ==Connections to physics==
| |
| | |
| The KP equation can be used to model [[water wave]]s of long [[wavelength]] with weakly non-linear restoring forces and [[dispersion (water waves)|frequency dispersion]]. If [[surface tension]] is weak compared to [[Earth's gravity|gravitational forces]], <math>\lambda=+1</math> is used; if surface tension is strong, then <math>\lambda=-1</math>. Because of the asymmetry in the way ''x''- and ''y''-terms enter the equation, the waves described by the KP equation behave differently in the direction of propagation (''x''-direction) and transverse (''y'') direction; oscillations in the ''y''-direction tend to be smoother (be of small-deviation).
| |
| | |
| The KP equation can also be used to model waves in [[Ferromagnetism|ferromagnetic]] media, as well as two-dimensional matter–wave pulses in [[Bose–Einstein condensate]]s.
| |
| | |
| ==Limiting behavior==
| |
| | |
| For <math>\epsilon\ll 1</math>, typical ''x''-dependent oscillations have a wavelength of <math>O(1/\epsilon)</math> giving a singular limiting regime as <math>\epsilon\rightarrow 0</math>. The limit <math>\epsilon\rightarrow 0</math> is called the [[Dispersionless equation|dispersion]]less limit.
| |
| | |
| If we also assume that the solutions are independent of ''y'' as <math>\epsilon\rightarrow 0</math>, then they also satisfy [[Burgers' equation]]:
| |
| :<math>\displaystyle \partial_t u+u\partial_x u=0.</math>
| |
| | |
| Suppose the amplitude of oscillations of a solution is asymptotically small — <math>O(\epsilon)</math> — in the dispersionless limit. Then the amplitude satisfies a mean-field equation of [[Davey–Stewartson equations|Davey–Stewartson]] type.
| |
| | |
| ==See also==
| |
| * [[Novikov–Veselov equation]]
| |
| | |
| ==References==
| |
| * {{Cite journal|first1=B. B.|last1=Kadomtsev|first2=V. I.|last2=Petviashvili|title=On the stability of solitary waves in weakly dispersive media|journal=Sov. Phys. Dokl.|volume=15|year=1970|pages=539–541|bibcode = 1970SPhD...15..539K }}. Translation of {{cite journal| title=Об устойчивости уединенных волн в слабо диспергирующих средах | journal=[[Doklady Akademii Nauk SSSR]] | volume=192 | issue= | pages=753–756}}
| |
| * {{Springer|id=K/k120110|first=Emma|last=Previato}}
| |
| | |
| ==External links==
| |
| * {{mathworld|urlname=Kadomtsev-PetviashviliEquation|title=Kadomtsev–Petviashvili equation}}
| |
| * {{scholarpedia | urlname=Kadomtsev-Petviashvili_equation | title=Kadomtsev–Petviashvili equation | curator=Gioni Biondini and Dmitri Pelinovsky }}
| |
| * {{cite web | url=http://www.amath.washington.edu/~bernard/kp.html | title=The KP page | author=Bernard Deconinck | publisher=[[University of Washington]], Department of Applied Mathematics }}
| |
| | |
| {{DEFAULTSORT:Kadomtsev-Petviashvili equation}}
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Exactly solvable models]]
| |
| [[Category:Solitons]]
| |
| [[Category:Equations of fluid dynamics]]
| |
The author's name is Andera and she believes it sounds fairly good. My wife and I live in Mississippi but now I'm considering other options. He works as a bookkeeper. To climb is some thing I truly enjoy performing.
Review my homepage; online reader - www.machlitim.org.il,